text:
\section{Introduction}

\emph{Algebras of chiral differential operators},
or \emph{algebras of CDOs},
are certain vertex algebras closely related to
the geometry of loop spaces and
the quantum theory of $2$-dimensional $\sigma$-models.
Locally, the algebra of CDOs on e.g.~an affine space
$\mathbb{A}^d$ is an elementary conformal vertex algebra
(a $\beta\gamma$-system) and it has some appropriate
holomorphic and smooth versions
(see \S\ref{algCDO}--\S\ref{sec.CDO.Rd}).
The global construction of CDOs,
first given by Gorbounov, Malikov and Schechtman,
shares various features with string geometry and $\sigma$-models.
In particular, the obstructions for a complex manifold $M$
to admit a sheaf of holomorphic CDOs $\mc{D}^{\mathrm{ch}}_M$ are certain
refinements of $c_1(M)$ and $c_2(M)$;
and if $M$ is closed, then the genus-$1$ partition function
of the conformal vertex superalgebra $H^*(M;\mc{D}^{\mathrm{ch}}_M)$ is
essentially the Witten genus of $M$.
\cite{GMS1,myCDO}
In the realm of algebraic geometry, Kapranov and Vasserot
have provided an interpretation of CDOs in terms of the notion of
\emph{formal loop spaces} they introduced.
\cite{KV1,KV}
Intuitively, the ``formal loop space of $X$'' is
the formal neighborhood of the subspace of constant loops
(i.e.~$X$ itself) inside the space of all loops,
but this has yet to be made precise for, say, smooth manifolds.
Meanwhile, physicists have explained how holomorphic CDOs
describe the so-called large volume limit of the half-twisted
$\sigma$-model.
\cite{Kapustin,Witten.CDO,Tan}
More recently, using his own mathematical formulation of quantum
field theory, Costello has given a new construction
of the Witten genus for complex manifolds, referring to
it as an ``analytic avatar'' of holomorphic CDOs.
\cite{Costello1,Costello2}

This paper investigates two related topics about
CDOs on \emph{smooth} manifolds:
(i) their interactions with smooth Lie group actions and
(ii) construction of geometrically meaningful modules.
First, anticipating an interpretation of CDOs in terms of
``formal loops'' (\`a la Kapranov-Vasserot), we study how
a Lie group action on a manifold can be lifted to
a ``formal loop group action'' on an algebra of CDOs.
This turns out to be a condition on the equivariant
first Pontrjagin class.
The case of a principal bundle receives particular attention
and gives rise to a vertex algebra that plays an important
role in the rest of the paper
(perhaps in the overall theory as well).
Using the abovementioned ``formal loop group actions'' and
semi-infinite cohomology, we then introduce a construction
of modules over CDOs.
Intuitively, each of these modules should be
the space of sections of some vector bundle over
a ``formal loop space'', as a module over
``differential operators''.
The first example we study leads to a new and more conceptual
construction of an arbitrary algebra of CDOs.
The other example, called the spinor module, may be useful
for formalizing the physical meaning of the Witten genus for
smooth string manifolds.

The following is a more detailed overview of the paper.

\begin{subsec}
{\bf Overview of the paper.}
\S\ref{sec.review} recalls the definition of
an algebra of CDOs on a smooth manifold
(Definition \ref{defn.CDO})
as well as the general construction and classification of
these vertex algebras
(Theorems \ref{thm.globalCDO} and \ref{thm.globalCDO.iso}).
The explicit generators-and-relations construction described
here should be compared to the more conceptual one
in \S\ref{CDOP.CDOM}.

Let $G$ be a compact connected Lie group,
$\mathfrak{g}$ its Lie algebra and
$\lambda$ an invariant symmetric bilinear form on $\mathfrak{g}$.
Notice that $\lambda$ determines a centrally extended
loop algebra $\hat{\mathfrak{g}}_\lambda$ (\S\ref{sec.Gmfld})
and also a vertex algebra $V_\lambda(\mathfrak{g})$
(Example \ref{VAoid.Lie}).
If $\mathfrak{g}$ is simple, we will use the more common notation
$V_k(\mathfrak{g})$ in place of $V_{k\lambda_0}(\mathfrak{g})$,
where $k\in\mathbb{C}$ and $\lambda_0$ is the normalized Killing
form.
Let us introduce the following terminology:
a \emph{formal loop group action} of level $\lambda$ on
a vector space $W$ is a $\hat{\mathfrak{g}}_\lambda$-action on $W$
whose restriction to $\mathfrak{g}\subset\hat{\mathfrak{g}}_\lambda$
integrates into a $G$-action;
and if $W$ is a vertex algebra then we also require that
$\mathfrak{g}$ acts by inner derivations.
(In the main text this is called
an \emph{inner $(\hat{\mathfrak{g}}_\lambda,G)$-action};
see Definition \ref{formalLG.action}.)
Notice that in the latter case the action is
induced by a map of vertex algebras
$V_\lambda(\mathfrak{g})\rightarrow W$.

Suppose $P$ is a smooth manifold with a smooth $G$-action
and $\mc{D}^{\mathrm{ch}}(P)$ is an algebra of CDOs on $P$.
Conjecturally, there should be a description of $\mc{D}^{\mathrm{ch}}(P)$
in terms of the ``formal loop space of $P$''.
This motivates the main result in \S\ref{sec.LGaction}:

\vspace{0.05in}
\noindent
{\bf Theorem \ref{thm.formalLG.action}.}
{\it
The $G$-action on $P$ lifts to a formal loop group action on
$\mc{D}^{\mathrm{ch}}(P)$ of level $\lambda$ if and only if
\begin{align*}
8\pi^2p_1(P)_G=\lambda(P)
\end{align*}
where $p_1(P)_G$ is the equivariant first Pontrjagin class
and $\lambda(P)$ is the image of $\lambda$ under the
characteristic map
$(\mathrm{Sym}^2\mathfrak{g}^\vee)^G\cong H^4(BG)\rightarrow H^4_G(M)$.
Moreover, the said action is primary with respect to
a suitably chosen conformal vector of $\mc{D}^{\mathrm{ch}}(P)$.
}

\vspace{0.05in}
\noindent
The key ideas behind this result are:
the use of the Cartan model for equivariant de Rham cohomology
(recalled in \S\ref{sec.Cartan});
and the observation that the $G$-action on $P$ and the vertex
algebra structure of $\mc{D}^{\mathrm{ch}}(P)$ together determine a Cartan
cocycle for $p_1(P)_G$ (Lemma \ref{lemma.Cartan.closed}).
In fact, there is a more refined statement detailing
a bijection between the formal loop group actions in question
and certain Cartan cochains.

From now, $P$ is the total space of a principal $G$-bundle
$\pi:P\rightarrow M$ and the algebra of CDOs $\mc{D}^{\mathrm{ch}}(P)$ is
equipped with a formal loop group action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(P)$.
\S\ref{sec.CDOP} is a more detailed study of the
vertex algebra $\mc{D}^{\mathrm{ch}}(P)$ in this situation.
First we consider the centralizer subalgebra
\begin{align*}
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}=C\big(\mc{D}^{\mathrm{ch}}(P),V_\lambda(\mathfrak{g})\big)
\end{align*}
i.e.~the subalgebra invariant under the formal loop group action.
While the structure of $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ is not fully
understood, a notable part of it is equivalent to
the Atiyah algebroid $(C^\infty(M),\mc{T}(P)^G)$ (\S\ref{sec.invCDO}).
For this reason the vertex algebra $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$
has more interesting modules compared to an algebra of CDOs
on $M$.
This will become more apparent later.

In the example $P=G$ with $G$ simple, the two $G$-actions on itself
by left and right multiplications can be lifted to two
\emph{commuting} formal loop group actions
$V_k(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(G)$ and
$V_{-k-2h^\vee}(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(G)$, where $k\in\mathbb{C}$
is arbitrary and $h^\vee$ is the dual Coxeter number
(\S\ref{CDOG}).
Hence there is an inclusion of, say, $V_k(\mathfrak{g})$ into
$C\big(\mc{D}^{\mathrm{ch}}(G),V_{-k-2h^\vee}(\mathfrak{g})\big)$.
If $k\neq-h^\vee$, then $V_k(\mathfrak{g})$ has a Sugawara vector
and it turns out to be also conformal for the above centralizer
(\S\ref{invCDOG}).
This seems to suggest that the said inclusion may in fact be
surjective, but the author has yet to find either a proof or
a counterexample.

In the case where $\pi$ is a principal frame bundle of $TM$,
we find a better description of $\mc{D}^{\mathrm{ch}}(P)$ such that
(among other things) the formal loop group action becomes
manifest.
This occupies the second half of \S\ref{sec.CDOP} and
leads to the following result, which is central to
the rest of the paper:

\vspace{0.05in}
\noindent
{\bf Theorem \ref{thm.CDOP}.}
{\it
Suppose we have
a principal $G$-bundle $\pi:P\rightarrow M$,
a representation $\rho:G\rightarrow SO(\mathbb{R}^d)$
and an isomorphism $P\times_\rho\mathbb{R}^d\cong TM$;
a connection $\Theta$ on $\pi$ that induces
the Levi-Civita connection on $TM$;
an invariant symmetric bilinear form $\lambda$ on
$\mathfrak{g}$;
and a basic $3$-form $H$ on $P$ with
$dH=(\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho)(\Omega\wedge\Omega)$,
where $\Omega$ is the curvature of $\Theta$
(see also \S\ref{conventions}).
These data determine a conformal vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$, which is generated by primary fields of
weights $0$ and $1$, and equipped with a primary formal loop
group action of level $\lambda$.
For its detailed definition see the main text.
}

\vspace{0.05in}
\noindent
Even though this vertex algebra arises as a particular
algebra of CDOs, it should really be regarded as a more
fundamental object for both conceptual and aesthetic reasons.
In fact, from vertex algebras of the form $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$,
there is a natural way to recover \emph{all} algebras of CDOs
and also construct many other interesting objects (see below).
Moreover, the definition of $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ is
arguably more appealing than that of an arbitrary algebra
of CDOs.

\S\ref{sec.ass} introduces the following construction:
given a vector space $W$ with a formal loop group action
of level $-\lambda-\lambda_{\t{\sf ad}}$, we can define a module over
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ as follows
(Definition \ref{defn.ass} \& Lemma \ref{lemma.ass})
\begin{align} \label{ass.module}
\Gamma^{\mathrm{ch}}(\pi,W)
:=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},\,
  \mc{D}^{\mathrm{ch}}(P)\otimes W\big).
\end{align}
The notation $H^{\frac{\infty}{2}+*}$ refers to
\emph{semi-infinite (a.k.a.~BRST) cohomology},
which can be defined using the Feigin complex
(recalled in \S\ref{sec.Feigin}--\S\ref{sec.semiinf}).
This is analogous to the construction of associated vector
bundles of $\pi:P\rightarrow M$.
In fact, $\Gamma^{\mathrm{ch}}(\pi,W)$ should be the space of sections of
a vector bundle over the ``formal loops of $M$'', as a module
over ``differential operators''.
Notice that if $W$ is a vertex algebra, then so is $\Gamma^{\mathrm{ch}}(\pi,W)$
and there is a map of vertex algebras
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}\rightarrow\Gamma^{\mathrm{ch}}(\pi,W)$.

In our first example of (\ref{ass.module}), $\pi$ is a principal
frame bundle of $TM$ and $W=\mathbb{C}$ (hence $\lambda=-\lambda_{\t{\sf ad}}$).
By Theorem \ref{thm.CDOP}, the definition of $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
depends on the choice of some $H\in\pi^*\Omega^3(M)$ that satisfies
$dH=\mathrm{Tr}\,\rho(\Omega)\wedge\rho(\Omega)$.
The main result in \S\ref{CDOP.CDOM} is a new description
of CDOs as alluded to earlier:

\vspace{0.05in}
\noindent
{\bf Theorem \ref{thm.CDO.semiinf}.}
{\it
The vertex algebra
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})=H^{\frac{\infty}{2}+0}(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P))$ is an algebra of CDOs on $M$.
Up to isomorphism every algebra of CDOs on $M$ is of this form.
}

\vspace{0.05in}
\noindent
The proof consists of two parts:
first we identify the two lowest weights of $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
(\S\ref{sec.ass.C.0} \& \S\ref{sec.ass.C.1})
and work out their structure
(Proposition \ref{prop.ass.C.VAoid});
then we use a certain property of its conformal vector
to deduce that $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is completely determined
by its two lowest weights
(Proposition \ref{prop.ass.C.conformal} \&
Corollary \ref{cor.ass.C.freeVA}).
There is also an extension of Theorem \ref{thm.CDO.semiinf}
for supermanifolds, including a special case that recovers
the chiral de Rham complex (\S\ref{CDO.cs}).

In our second and last example of (\ref{ass.module}),
$G$ is a spin group $\textrm{Spin}_{2d'}$,
$\pi$ is a spin structure on $M$ and
$W=S$ is the spinor representation of $\widehat{\mathfrak{so}}_{2d'}$.
It follows from Theorem \ref{thm.CDOP} that the definition of
the \emph{spinor module} $\Gamma^{\mathrm{ch}}(\pi,S)$ depends on the choice of
some $H\in\pi^*\Omega^3(M)$ that satisfies
$dH=\frac{1}{2}\mathrm{Tr}\,(\Omega\wedge\Omega)$ (\S\ref{sec.ass.S}).
Geometrically, $\Gamma^{\mathrm{ch}}(\pi,S)$ should be the space of sections
of the ``spinor bundle on the formal loops of $M$''.
In \S\ref{sec.FL.spinor} we carry out an analysis of $\Gamma^{\mathrm{ch}}(\pi,S)$
that parallels the one in \S\ref{CDOP.CDOM} and culminates
in a more explicit description of it in terms of
generating data (i.e.~a subspace and three types of fields)
and relations.
This is summarized in Theorem \ref{thm.ass.S}.
Notice that our work in \S\ref{sec.FL.spinor} should be viewed
mainly as preparation for further study.
The author \emph{speculates} that the spinor module $\Gamma^{\mathrm{ch}}(\pi,S)$
always admits an action of the $(0,1)$ superconformal algebra and
thereby provides at least a partial mathematical account
of the physical interpretation of the Witten genus
(\S\ref{ass.S.sconformal}).

The appendix reviews the notion of vertex algebroids
and their relations with vertex algebras.
Although vertex algebroids have a rather complicated definition,
they serve as a convenient tool for dealing with the vertex
algebras in this paper.
\end{subsec}

\begin{subsec} \label{conventions}
{\bf Notations and conventions.}
In this paper, every vertex algebra $V$ is graded by
nonnegative integers which we call \emph{weights};
its component of weight $k$ is denoted by $V_k$ and
its weight operator by $L_0$, i.e. $L_0|_{V_k}=k$.
For any $u\in V$, we always write the Fourier modes of
its vertex operator as $u_k$, $k\in\mathbb{Z}$, such that
$u_k$ has weight $-k$.
For any conformal vector $\nu\in V$ that we consider,
$\nu_0=L_0$.

Given a smooth manifold $M$, we write
$C^\infty(M)$, $\mc{T}(M)$, $\Omega^n(M)$ for its spaces of
smooth $\mathbb{C}$-valued functions, vector fields and $n$-forms.
Also, all ordinary cohomology groups and their equivariant
versions have complex coefficients.
Given a Lie algebra $\mathfrak{g}$ and a finite-dimensional
representation $\rho:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$, we write
$\lambda_\rho$ for the invariant symmetric bilinear form on
$\mathfrak{g}$ given by $\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
In particular, $\lambda_{\t{\sf ad}}$ is the Killing form.
Square brackets $[\;]$ are used for supercommutators between
operators of any parities, while curly brackets $\{\;\}$
are reserved for a different use (see \S\ref{VAoid}).
Repeated indices are always implicitly summed over all possible
values, unless a specific range is indicated.
\end{subsec}

\begin{subsec}
{\bf Acknowledgements.}
The author is currently supported by the ESPRC grant
EP/H040692/1.
He would like to thank Matthew Ando for a suggestion
that largely motivated Theorem \ref{thm.formalLG.action},
and Dennis Gaitsgory for answering some questions about
semi-infinite cohomology.
\end{subsec}

\newpage

\section{CDOs on Smooth Manifolds}
\label{sec.review}

A sheaf of CDOs is a sheaf of vertex algebras locally
modelled on a very basic vertex algebra (known as the
$\beta\gamma$-system), and provides a mathematical
approximation to a quantum field theory of much
interest.
\cite{MSV,GMS1,GMS2,KV,Kapustin,Witten.CDO,Tan}
This section reviews the construction and
classification of sheaves of CDOs on a smooth
manifold, following the formulation in \cite{myCDO}.

\begin{subsec} \label{algCDO}
{\bf The algebra of CDOs on $\mathbb{A}^d$.}
Let $d$ be a positive integer.
Define a unital associative algebra $\mathcal{U}$ with
the following generators and relations
\begin{eqnarray} \label{Weyl}
b^i_n,\;a_{i,n},\;n\in\mathbb{Z},\;i=1,\ldots,d,\qquad
[a_{i,n},b^j_m]=\delta^j_i\delta_{n,-m},\qquad
[b^i_n,b^j_m]=0=[a_{i,n},a_{j,m}]\,.
\end{eqnarray}
(This is an example of a Weyl algebra.)
The commutative subalgebra $\mathcal{U}_+$ generated
by $\{b^i_n\}_{n>0}$ and $\{a_{i,n}\}_{n\geq 0}$
has a trivial representation $\mathbb{C}$.
The induced $\mathcal{U}$-module
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}(\mathbb{A}^d):=\mathcal{U}\otimes_{\mathcal{U}_+}\mathbb{C}
\end{eqnarray*}
has the structure of a vertex algebra.
The vacuum is $\mathbf 1=1\otimes 1$.
The infinitesimal translation operator $T$ and weight
operator $L_0$ are determined by
\begin{eqnarray*}
\begin{array}{lll}
T\mathbf 1=0, &
[T,b^i_n]=(1-n)b^i_{n-1}, \phantom{aa} &
[T,a_{i,n}]=-na_{i,n-1} \vspace{0.03in} \\
L_0\mathbf 1=0, \phantom{aa} &
[L_0,b^i_n]=-nb^i_n, &
[L_0,a_{i,n}]=-na_{i,n}
\end{array}
\end{eqnarray*}
The fields (or vertex operators) of $b^i_0\mathbf 1\in\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_0$
and $a_{i,-1}\mathbf 1\in\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1$ are respectively
\begin{eqnarray*}
\begin{array}{l}
\sum_n b^i_n z^{-n},\qquad \sum_n a_{i,n}z^{-n-1}
\end{array}
\end{eqnarray*}
which determine the fields of other elements by
the Reconstruction Theorem \cite{FB-Z}.
This vertex algebra has a family of conformal elements
of central charge $2d$, namely
\footnote{
This vertex algebra is the tensor product of $d$ copies
of the $\beta\gamma$-system.
It admits other conformal elements that define different
conformal weights.
\cite{Kac}
}
\begin{eqnarray*}
\qquad\qquad\qquad
a_{i,-1}b^i_{-1}\mathbf 1+T^2 f,\qquad
f\in\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_0=\mathbb{C}[b^1_0,\ldots,b^d_0]\cdot\mathbf{1}.
\end{eqnarray*}

The vertex algebra $\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)$ is freely generated by
its associated vertex algebroid (see \S\ref{VA.VAoid} and
\S\ref{VAoid.VA}).
To describe the latter, consider the affine space
$\mathbb{A}^d=\textrm{Spec}\,\mathbb{C}[b^1,\cdots,b^d]$ and identify
the functions, $1$-forms and vector fields on $\mathbb{A}^d$
with the following subquotients of $\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)$: \vspace{0.03in} \\
\indent $\cdot$\;
$\mathcal{O}(\mathbb{A}^d)=\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_0$ via $b^i=b^i_0\boldsymbol{1}$,
$b^i b^j=b^i_0 b^j_0\boldsymbol{1}$, etc. \vspace{0.03in} \\
\indent $\cdot$\;
$\Omega^1(\mathbb{A}^d)\subset\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1$ via
$db^i=b^i_{-1}\boldsymbol{1}$ \vspace{0.03in} \\
\indent $\cdot$\;
$\mc{T}(\mathbb{A}^d)=\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1/\Omega^1(\mathbb{A}^d)$ via
$\partial_i=\partial/\partial b^i=$ coset of $a_{i,-1}\boldsymbol{1}$ \vspace{0.03in} \\
Under these identifications, the vertex algebroid associated to
$\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)$ is of the form
\begin{eqnarray*}
\big(\mathcal{O}(\mathbb{A}^d),\Omega^1(\mathbb{A}^d),\mc{T}(\mathbb{A}^d),
  \bullet,\{\;\},\{\;\}_{\Omega}\big).
\end{eqnarray*}
The extended Lie algebroid structure consists of
the usual differential on functions,
Lie bracket on vector fields,
Lie derivations by vector fields on functions and $1$-forms,
and pairing between $1$-forms and vector fields.
Using the splitting
\begin{eqnarray} \label{algCDO.splitting}
s:\mc{T}(\mathbb{A}^d)\rightarrow\mc{D}^{\mathrm{ch}}(\mathbb{A}^d)_1,\qquad
X=X^i\partial_i\;\mapsto\;a_{i,-1}X^i
\end{eqnarray}
the rest of the vertex algebroid structure, according to
(\ref{VA.VAoid.456}), reads as follows
\begin{align} \label{CDO.VAoid456}
X\bullet f=(\partial_j X^i)(\partial_i f)db^j,\quad
\{X,Y\}=-(\partial_j X^i)(\partial_i Y^j),\quad
\{X,Y\}_\Omega=-(\partial_k\partial_j X^i)(\partial_i Y^j)db^k
\end{align}
These expressions do not seem to have any obvious
global meaning;
however, see Theorem \ref{thm.globalCDO}.
\end{subsec}

\begin{subsec} \label{sec.CDO.Rd}
{\bf The sheaf of CDOs on $\mathbb{R}^d$.}
Now regard $b^1,\ldots,b^d$ as the coordinates of
$\mathbb{R}^d$.
The ($\mathbb{C}$-valued) smooth functions, $1$-forms
and vector fields on an open set
$W\subset\mathbb{R}^d$ form an extended Lie algebroid
just as in \S\ref{algCDO}, and the expressions in
(\ref{CDO.VAoid456}) again define a vertex
algebroid
\begin{eqnarray*}
\big(C^\infty(W),\Omega^1(W),\mc{T}(W),
  \bullet,\{\;\},\{\;\}_\Omega\big).
\end{eqnarray*}
The vertex algebra it freely generates
(see \S\ref{VAoid.VA}) will be denoted by $\mc{D}^{\mathrm{ch}}(W)$.
This vertex algebra also has a family of conformal
elements of central charge $2d$, namely
\begin{eqnarray} \label{localCDO.conformal}
\partial_{i,-1}db^i+\frac{1}{2}\,T\omega,\qquad
\omega\in\Omega^1(W),\;d\omega=0.
\end{eqnarray}
When $W$ varies, we obtain a sheaf
of conformal vertex algebras $\mc{D}^{\mathrm{ch}}$ on $\mathbb{R}^d$.
\end{subsec}

\begin{defn} \label{defn.CDO}
A {\bf sheaf of CDOs} on a smooth manifold $M$ of dimension $d$
is a sheaf of vertex algebras $\mathcal{V}$ with the following
properties:
\vspace{-0.05in}
\begin{itemize}
\item[$\cdot$]
its weight-zero component is $\mathcal{V}_0=C^\infty_M$, and
\vspace{-0.08in}
\item[$\cdot$]
each point of $M$ has a neighborhood $U$ such that $(U,\mathcal{V}|_U)$
is isomorphic to $(W,\mc{D}^{\mathrm{ch}}|_W)$ for some open set $W\subset\mathbb{R}^d$.
\vspace{-0.05in}
\end{itemize}
A {\bf conformal structure} on $\mathcal{V}$ is an element
$\nu\in\mathcal{V}(M)$ such that, under each isomorphism
as postulated above, $\nu|_U\in\mathcal{V}(U)$ corresponds
to one of the conformal elements
(\ref{localCDO.conformal}) of $\mc{D}^{\mathrm{ch}}(W)$.
\end{defn}

In order to state the results on the construction
and classification of sheaves of CDOs, let us
introduce a notation that will also appear
often in the sequel.

\begin{defn} \label{tnabla}
Let $M$ be a smooth manifold.
Given a connection $\nabla$ on $TM$ and $X\in\mc{T}(M)$, define
an operator $\nabla^t X\in \Gamma(\textrm{End}\,TM)$ by
\begin{eqnarray*}
(\nabla^t X)(Y):=\nabla_X Y-[X,Y],\qquad Y\in\mc{T}(M).
\end{eqnarray*}
Notice that if $\nabla$ is torsion-free, then
$\nabla^t X=\nabla X$.
\end{defn}

\begin{theorem} \label{thm.globalCDO}
\cite{myCDO}
Let $M$ be a smooth manifold of dimension $d$.

(a) Given a connection $\nabla$ on $TM$ with curvature $R$ and
$H\in\Omega^3(M)$ satisfying $dH=\mathrm{Tr}\,(R\wedge R)$, there is
a sheaf of vertex algebroids
$\big(C^\infty_M,\Omega^1_M,\mc{T}_M,\bullet,\{\;\},\{\;\}_\Omega\big)$
on $M$ defined by the following expressions
\begin{align}
\nonumber
X\bullet f\;\;\; &:= (\nabla X)f \\
\label{CDO.VAoid}
\{X,Y\}\;\; &:= -\mathrm{Tr}\,(\nabla^t X\cdot\nabla^t Y) \\
\nonumber
\{X,Y\}_\Omega &:=
  \mathrm{Tr}\,\bigg(-\nabla(\nabla^t X)\cdot\nabla^t Y
    +\nabla^t X\cdot\iota_Y R
    -\iota_X R\cdot\nabla^t Y\bigg)
  +\frac{1}{2}\iota_X\iota_Y H
\end{align}
and the sheaf of vertex algebras it freely generates
(see \S\ref{VAoid.VA}) is a sheaf of CDOs on $M$,
which we denote by $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$.
Up to isomorphism, every sheaf of CDOs on $M$ is
of this form.

(b) There is a one-to-one correspondence between
conformal structures
on $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$ and $\omega\in\Omega^1(M)$
satisfying $d\omega=\mathrm{Tr}\, R$.
Given such $\omega$, the corresponding conformal
structure, which we denote by $\nu^\omega$, has the
local expression
\begin{eqnarray} \label{CDO.conformal}
\nu^\omega|_U
=\sigma_{i,-1}\sigma^i
+\frac{1}{2}\mathrm{Tr}\,\big(\Gamma^\sigma_{-1}\Gamma^\sigma
  -\Gamma^\sigma_{-2}\mathbf{1}\big)
+\sigma^i([\sigma_j,\sigma_k])
  \sigma^k_{-1}(\Gamma^\sigma)^j_{\phantom{i}i}
+\frac{1}{2}\omega_{-2}\mathbf{1}
\end{eqnarray}
where $U\subset M$ is an open subset,
$\sigma=(\sigma_1,\ldots,\sigma_d)$ any $C^\infty(U)$-basis of
$\mc{T}(U)$, $(\sigma^1,\ldots,\sigma^d)$ the dual basis of
$\Omega^1(U)$, and
$\Gamma^\sigma\in\Omega^1(U)\otimes\mathfrak{gl}_d$ the connection
$1$-form of $\nabla$ with respect to $\sigma$
(i.e.~$\nabla\sigma_i=(\Gamma^\sigma)^j_{\phantom{i}i}\otimes\sigma_j$).
This conformal structure has central charge $2d=2\dim M$ and
the property that
\begin{eqnarray} \label{CDO.L1}
\nu^\omega_1\alpha=0\;\textrm{ for }\alpha\in\Omega^1(M),\qquad
\nu^\omega_1 X=\mathrm{Tr}\,\nabla^t X-\omega(X)\;\textrm{ for }X\in\mc{T}(M).
\quad\qedsymbol
\end{eqnarray}
\end{theorem}

{\it Remarks.}
(i) By Theorem \ref{thm.globalCDO}, a smooth manifold
$M$ admits sheaves of CDOs if and only if $p_1(M)$ is
trivial in de Rham cohomology, while conformal structures
always exist.
For example, if $\nabla$ is orthogonal with respect to
a Riemannian metric, then $\mathrm{Tr}\, R=0$ and a conformal
structure can be defined using, say, $\omega=0$.
However, this result generalizes to cs-manifolds
(supermanifolds with $\mathbb{C}$-valued structure sheaf)
in which case the obstruction to conformal structures
may well be nontrivial.~\cite{myCDO}

(ii) In \cite{myCDO}, we only obtained a local expression of
$\nu^\omega$ in terms of local coordinate vector fields,
but that implies the more general expression
(\ref{CDO.conformal}) by a straightforward calculation.

(iii) In the original work \cite{GMS2} as well as in
\cite{myCDO}, sheaves of CDOs and conformal structures
were constructed by gluing local data.
In the smooth case, this culminates in a description
by generators and relations (or generating fields
and OPEs) as seen above.
However, the expressions in (\ref{CDO.VAoid}) and
(\ref{CDO.conformal}) do not seem very inspiring.
Later in \S\ref{CDOP.CDOM}, we will obtain a more
conceptual description of CDOs using semi-infinite
cohomology.

\begin{theorem} \label{thm.globalCDO.iso}
\cite{myCDO}
Let $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$ and $\mc{D}^{\mathrm{ch}}_{M,\nabla,H'}$ be sheaves of CDOs
on a smooth manifold $M$ constructed as in Theorem
\ref{thm.globalCDO}a;
denote by $\bullet$, $\{\;\}$, $\{\;\}_\Omega$
(resp.~$\{\;\}'_\Omega$) the structure maps determined by
$\nabla$ and $H$ (resp.~$H'$) as in (\ref{CDO.VAoid}).

(a) There is a one-to-one correspondence between isomorphisms
$\mc{D}^{\mathrm{ch}}_{M,\nabla,H}\stackrel{\sim}{\rightarrow}\mc{D}^{\mathrm{ch}}_{M,\nabla,H'}$
that restricts to the identity on $C^\infty_M$ and
$\beta\in\Omega^2(M)$ satisfying $d\beta=H'-H$.
Given such $\beta$, the corresponding isomorphism is induced by
an isomorphism of sheaves of vertex algebroids
(see \S\ref{VAoid.VA.mor})
\begin{eqnarray*}
(\mathrm{id},\Delta_\beta):
\big(C^\infty_M,\Omega^1_M,\mc{T}_M,\bullet,\{\;\},\{\;\}_\Omega\big)
\rightarrow
\big(C^\infty_M,\Omega^1_M,\mc{T}_M,\bullet,\{\;\},\{\;\}'_\Omega\big)
\end{eqnarray*}
where $\Delta_\beta:\mc{T}_M\rightarrow\Omega^1_M$ is given by
$\Delta_\beta(X)=\frac{1}{2}\iota_X\beta$.

(b) Every isomorphism described above respects the correspondence
in Theorem \ref{thm.globalCDO}b.
$\qedsymbol$
\end{theorem}

{\it Remarks.}
(i) By Theorems \ref{thm.globalCDO}a and \ref{thm.globalCDO.iso}a,
if $M$ admits sheaves of CDOs, their isomorphism classes form
an $H^3(M)$-torsor.
(ii) Since each sheaf of CDOs $\mc{D}^{\mathrm{ch}}_{M,\nabla,H}$ is fine, for most
purposes it suffices to (and we will) work only with the vertex
algebra of global sections $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$, which will be
called an {\bf algebra of CDOs} on $M$.
The reader should keep in mind that by construction
\begin{eqnarray} \label{CDO.wt01}
\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)_0=C^\infty(M),\qquad
\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)_1=\Omega^1(M)\oplus\mc{T}(M).
\end{eqnarray}
For a description of the higher weights, see
\S\ref{freeVA.PBW}.

\begin{lemma} \label{lemma.1form.0mode}
\cite{myCDO}
Consider an algebra of CDOs $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$ on
a smooth manifold $M$.
For any $\alpha\in\Omega^1(M)$ and $X\in\mc{T}(M)$, we have
$\alpha_0 X=-\iota_X d\alpha$.
Moreover, $\alpha_0=0$ on $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$ if
and only if $d\alpha=0$.
$\qedsymbol$
\end{lemma}

\newpage
\setcounter{equation}{0}
\section{Formal Loop Group Actions on CDOs}
\label{sec.LGaction}

This section investigates the condition under which
a Lie group action on a manifold lifts to a projective
``formal loop group action'' on an algebra of CDOs.
This turns out to be a condition on the equivariant
first Pontrjagin class of the manifold.
The said action provides evidence of a conjectural
interpretation of (smooth) CDOs in terms of
``formal loops'', in the vein of \cite{KV}.

\begin{subsec} \label{sec.Gmfld}
{\bf Setting:~manifold with a Lie group action.}
Throughout this section, let $G$ be a compact connected
Lie group, $\mathfrak{g}$ its Lie algebra, and $\lambda$
an invariant symmetric bilinear form on $\mathfrak{g}$.
Recall the loop algebra
$L\mathfrak{g}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$:
writing $A\otimes t^n$ as $A_n$, the Lie bracket is
given by $[A_n,B_m]=[A,B]_{n+m}$.
Also recall that $\lambda$ determines a central extension
$\hat{\mathfrak{g}}_\lambda=L\mathfrak{g}\oplus\mathbb{C}$ with
\begin{eqnarray*}
[A_n,B_m]=[A,B]_{n+m}+n\lambda(A,B)\delta_{n+m,0},\quad
A,B\in\mathfrak{g},\;\,n,m\in\mathbb{Z}.
\end{eqnarray*}
Let $P$ be a smooth manifold with a smooth right $G$-action.
Later we will specialize to the case of a principal
bundle, but at the moment $P$ can be any right $G$-manifold.
The left $G$-action on $C^\infty(P)$ determines and is
determined by a map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(P)$.
The vector field generated by $A\in\mathfrak{g}$ will be written
as $A^P\in\mc{T}(P)$.
\end{subsec}

\begin{subsec} \label{sec.Cartan}
{\bf The equivariant de Rham complex.}
Recall that $H^*_G(P)=H^*(EG\times_G P)$ can be computed
by the Cartan model $(\Omega^*_G(P),d_G)$.~\cite{GuiSte}
The graded algebra of Cartan cochains is given by
\begin{eqnarray*}
\Omega^*_G(P)=
  \bigoplus_{k\geq 0}\Omega^k_G(P),\qquad
\Omega^k_G(P)=
  \bigoplus_{2i+j=k}\big(\textrm{Sym}^i\mathfrak{g}^\vee
    \otimes\Omega^j(P)\big)^G.
\end{eqnarray*}
Let us adopt a convention:~regard each
$\xi\in\Omega^*_G(P)$ as a $G$-equivariant polynomial map
$\xi:\mathfrak{g}\rightarrow\Omega^*(P)$ and write its value at
$A\in\mathfrak{g}$ as $\xi_A$.
The Cartan differential then reads
\begin{eqnarray*}
(d_G\xi)_A=d\xi_A-\iota_{A^P}\xi_A.
\end{eqnarray*}
The characteristic map $H^*(BG)\rightarrow H^*_G(P)$ is
represented by the inclusion
$(\textrm{Sym}^*\mathfrak{g}^\vee)^G\hookrightarrow\Omega^*_G(P)$.
Given $\eta\in(\textrm{Sym}^*\mathfrak{g}^\vee)^G=H^*(BG)$, its image will be
denoted by $\eta(P)\in H^*_G(P)$.
\end{subsec}

\begin{subsec} \label{sec.inv.data}
{\bf CDOs with a $G$-action.}
Choose a $G$-invariant connection $\nabla$ on $TP$, with
curvature tensor $R$.
This means for $A\in\mathfrak{g}$ we have $L_{A^P}\nabla=0$,
or equivalently
\begin{eqnarray} \label{conn.Ginv}
\nabla(\nabla^t A^P)=-\iota_{A^P}R
\end{eqnarray}
in terms of Definition \ref{tnabla}.
\footnote{
For any $X\in\mc{T}(P)$ we have
$L_X\nabla=\nabla(\nabla^t X)+\iota_X R$.
This is proved as follows:
for $Y,Z\in\mc{T}(P)$
\begin{align*}
(L_X\nabla)_Y Z
&=[X,\nabla_Y Z]-\nabla_{[X,Y]}Z-\nabla_Y[X,Z] \\
&=\nabla_Y\nabla_X Z-\nabla_Y[X,Z]
  -\nabla_X\nabla_Y Z+[X,\nabla_Y Z]
  +\nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z
  -\nabla_{[X,Y]}Z \\
&=\nabla_Y(\nabla^t X)(Z)-(\nabla^t X)(\nabla_Y Z)+R_{X,Y}Z
=\big(\nabla_Y(\nabla^t X)+R_{X,Y}\big)(Z)
\end{align*}
}
Assume $p_1(P)=0$ and choose $H\in\Omega^3(P)^G$ such that
$dH=\mathrm{Tr}\,(R\wedge R)$.
By Theorem \ref{thm.globalCDO}a, $\nabla$ and $H$
determine an algebra of CDOs $\mc{D}^{\mathrm{ch}}_{\nabla,H}(P)$, which
is freely generated by a vertex algebroid
(see also \S\ref{VAoid.VA})
\begin{eqnarray*}
\big(C^\infty(P),\Omega^1(P),\mc{T}(P),
  \bullet,\{\;\},\{\;\}_\Omega\big);
\end{eqnarray*}
when there is no risk of confusion, we simply write
$\mc{D}^{\mathrm{ch}}(P)$.
Clearly, the $G$-invariance of $\nabla$ and $H$
implies $G$-equivariance of the structure maps
$\bullet$, $\{\;\}$, $\{\;\}_\Omega$, so that the
$G$-action on $C^\infty(P)=\mc{D}^{\mathrm{ch}}(P)_0$ extends to
a $G$-action on $\mc{D}^{\mathrm{ch}}(P)$.

Without loss of generality, assume that $\mathrm{Tr}\, R=0$.
(For example, this is true if $\nabla$ is orthogonal
with respect to a Riemannian metric.)
Choose $\omega\in\Omega^1(P)^G$ such that $d\omega=0$.
By Theorem \ref{thm.globalCDO}b, $\omega$ determines
a $G$-invariant conformal structure $\nu^\omega$ on
$\mc{D}^{\mathrm{ch}}(P)$.
The Fourier modes of the associated Virasoro field
will be denoted by $L_n^\omega$, $n\in\mathbb{Z}$.
Soon we will make a more specific choice of $\omega$.
\end{subsec}

{\it Remark.}
The $G$-invariance of $\nabla$ and $H$, as well as that
of other geometric data to appear later, can always be
achieved by averaging over $G$ with respect to the
Haar measure.

\begin{defn} \label{formalLG.action}
Given a vertex algebra $V$,
by an {\bf inner $\hat{\mathfrak{g}}_\lambda$-action} on $V$
we simply mean a map of vertex algebras from
$V_\lambda(\mathfrak{g})$ to $V$, and it is
an {\bf inner $(\hat{\mathfrak{g}}_\lambda,G)$-action}
if its induced $\mathfrak{g}$-action on $V$ (see below)
integrates into a $G$-action.
\end{defn}

{\it Remarks.}
(i) By definition of $V_\lambda(\mathfrak{g})$ (see Example
\ref{VAoid.Lie}), any map $V_\lambda(\mathfrak{g})\rightarrow V$
is determined by its component of weight one, i.e.~a linear map
$\mathfrak{g}=V_\lambda(\mathfrak{g})_1\rightarrow V_1$.
Taking the zeroth modes then yields a map of Lie algebras
from $\mathfrak{g}$ to the inner derivations of $V$.
This is the induced $\mathfrak{g}$-action referred to above.
(ii) Now we may state the goal of this section more precisely:
find the condition under which the given $G$-action on
$\mc{D}^{\mathrm{ch}}(P)_0=C^\infty(P)$ extends to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$.

\begin{subsec} \label{CDO.Cartan}
{\bf Cartan cochains associated to CDOs.}
The $G$-action on $P$ and the vertex algebroid structure associated
to $\mc{D}^{\mathrm{ch}}(P)$ together determine two Cartan cochains of degree $4$,
namely
\begin{eqnarray} \label{Cartan4.VAoid}
\begin{array}{ll}
\chi^{2,2}\in\big(\mathfrak{g}^\vee\otimes\Omega^2(P)\big)^G,
&\chi^{2,2}_A:=\{A^P,-\}_\Omega=
  \mathrm{Tr}\,(\nabla^t A^P\cdot R)-\frac{1}{2}\iota_{A^P}H
\vspace{0.03in}\vss \\
\chi^{4,0}\in\big(\textrm{Sym}^2\mathfrak{g}^\vee\otimes C^\infty(P)\big)^G,
\quad
&\chi^{4,0}_A:=\{A^P,A^P\}
  =-\mathrm{Tr}\,(\nabla^t A^P\cdot\nabla^t A^P)
\end{array}
\end{eqnarray}
Indeed, by (\ref{CDO.VAoid}) and (\ref{conn.Ginv}),
the operator $\{A^P,-\}_\Omega:\mc{T}(P)\rightarrow\Omega^1(P)$
may be viewed as the indicated $2$-form, and
the $G$-invariance of $\nabla$, $H$ implies
the $G$-equivariance of $\chi^{2,2}$, $\chi^{4,0}$.
Moreover, the $G$-action on $P$ and the conformal structure
$\nu^\omega$ on $\mc{D}^{\mathrm{ch}}(P)$ together determine a Cartan cochain
of degree $2$, namely
\begin{eqnarray} \label{Cartan2.conformal}
\chi^{2,0}\in\big(\mathfrak{g}^\vee\otimes C^\infty(P)\big)^G,\qquad
\chi^{2,0}_A:=L_1^\omega A^P=\mathrm{Tr}\,\nabla^t A^P-\omega(A^P)
\end{eqnarray}
in view of (\ref{CDO.L1}).
Indeed, the $G$-invariance of $\nabla$, $\omega$ implies the
$G$-equivariance of $\chi^{2,0}$.
\end{subsec}

\begin{lemma} \label{lemma.Cartan.closed}
(a) The Cartan cochain $2\chi^{2,2}+\chi^{4,0}$ is closed and
represents $8\pi^2 p_1(P)_G\in H^4_G(P)$.
(b) The Cartan cochain $\chi^{2,0}$ is exact.
In fact, it is trivial with a suitable choice of conformal
structure $\nu^\omega$.
\end{lemma}

\begin{proof}
(a) According to \cite{BGV,BT}, the Cartan cochain
$A\mapsto\mathrm{Tr}\,(R-\nabla^t A^P)^2$ is closed and represents
$-8\pi^2 p_1(P)_G$.
Our claim then follows from the calculation
\begin{eqnarray*}
\mathrm{Tr}\,(R-\nabla^t A^P)^2
=dH-2\,\mathrm{Tr}\,(\nabla^t A^P\cdot R)
  +\mathrm{Tr}\,(\nabla^t A^P\cdot\nabla^t A^P)
=(d_G H)_A-2\chi^{2,2}_A-\chi^{4,0}_A
\end{eqnarray*}
where we have used $dH=\mathrm{Tr}\,(R\wedge R)$ and
(\ref{Cartan4.VAoid}).

(b) According to \cite{BGV,BT} again, the Cartan cochain
$A\mapsto\mathrm{Tr}\,(R-\nabla^t A^P)=-\mathrm{Tr}\,\nabla^t A^P$ is exact.
Since we have
$(d_G\omega)_A=d\omega-\omega(A^P)=-\omega(A^P)$, this
proves the exactness of (\ref{Cartan2.conformal}).
This also means there exists some
$\omega'\in\Omega^1(P)^G$ such that
\begin{eqnarray*}
-\mathrm{Tr}\,\nabla^t A^P
=(d_G\omega')_A
=d\omega'-\omega'(A^P)
\end{eqnarray*}
which is equivalent to two equations:
firstly $d\omega'=0$, so that $\omega'$
also determines a conformal structure
$\nu^{\omega'}$;
and secondly $L_1^{\omega'}A^P=0$.
\end{proof}

{\it Remarks.}
(i) The closedness of $2\chi^{2,2}+\chi^{4,0}$ is
equivalent to a pair of equations
\begin{eqnarray} \label{Cartan4.closed}
\phantom{WWWW}
d\chi^{2,2}_A=0,\qquad
d\chi^{4,0}_A=2\iota_{A^P}\chi^{2,2}_A,\qquad
\textrm{for }A\in\mathfrak{g}
\end{eqnarray}
which can also be verified directly from our
assumptions on $\nabla$ and $H$.
(ii) From now on we assume that the conformal
structure $\nu^\omega$ has been chosen such
that $L_1^\omega A^P=0$ for $A\in\mathfrak{g}$, and
write the associated Virasoro operators more
simply as $L_n$.

\vspace{0.08in}
{\it Preparation.}
The following sequence of results all concern
lifting the given map of Lie algebras
$\mathfrak{g}\rightarrow\mc{T}(P)$ to a linear map
of the form
\begin{eqnarray} \label{lift.CDO1}
\mathfrak{g}\,\rightarrow\,
  \mc{D}^{\mathrm{ch}}(P)_1=\mc{T}(P)\oplus\Omega^1(P),\qquad
A\mapsto A^P+h^{2,1}_A
\end{eqnarray}
where $h^{2,1}:\mathfrak{g}\rightarrow\Omega^1(P)$ is
assumed to be $G$-equivariant.
In other words, $h^{2,1}$ is a Cartan cochain.

\begin{prop} \label{prop.lift.Lie}
The linear map (\ref{lift.CDO1}) induces a map of
Lie algebras as follows
\begin{eqnarray} \label{lift.Lie}
\mathfrak{g}\,\rightarrow\,\mathrm{Der}\,\mc{D}^{\mathrm{ch}}(P),\qquad
A\mapsto\big(A^P+h^{2,1}_A\big)_0
\end{eqnarray}
if and only if the closed $2$-form
$\chi^{2,2}_A-dh^{2,1}_A$ is $G$-invariant for
each $A\in\mathfrak{g}$.
\end{prop}

\begin{proof}
For $A,B\in\mathfrak{g}$, we have the following calculation
\begin{align*}
\big[\big(A^P+h^{2,1}_A\big)_0,\big(B^P+h^{2,1}_B\big)_0\big]
&=\Big([A^P,B^P]+\{A^P,B^P\}_\Omega+L_{A^P}h^{2,1}_B
  -L_{B^P}h^{2,1}_A\Big)_0 \\
&=\Big([A,B]^P+h^{2,1}_{[A,B]}\Big)_0
  +\Big(\iota_{B^P}\chi^{2,2}_A-L_{B^P}h^{2,1}_A\Big)_0
\end{align*}
using (\ref{freeVA.comm}), (\ref{Cartan4.VAoid}) and
the $G$-equivariance of $h^{2,1}$.
By Lemma \ref{lemma.1form.0mode}, the second term
vanishes if and only if
\begin{eqnarray*}
0=d\big(\iota_{B^P}\chi^{2,2}_A-L_{B^P}h^{2,1}_A\big)
=L_{B^P}\big(\chi^{2,2}_A-dh^{2,1}_A\big)
\end{eqnarray*}
where we have used (\ref{Cartan4.closed}).
This proves our claim.
\end{proof}

The map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(P)$ extends
in an obvious way to a map of extended Lie algebroids
$i:(\mathbb{C},0,\mathfrak{g})\rightarrow(C^\infty(P),\Omega^1(P),\mc{T}(P))$.
Now we would like to extend it further to a map of vertex
algebroids and hence a map of vertex algebras
(see Example \ref{VAoid.Lie} and \S\ref{VAoid.VA.mor}).

\begin{prop} \label{prop.lift.VA}
The linear map (\ref{lift.CDO1}) determines a map of
vertex algebroids as follows
\begin{eqnarray*}
(i,h^{2,1}):(\mathbb{C},0,\mathfrak{g},0,\lambda,0)\rightarrow
  \big(C^\infty(P),\Omega^1(P),\mc{T}(P),
  \bullet,\{\;\},\{\;\}_\Omega\big)
\end{eqnarray*}
and hence a map of vertex algebras
$V_\lambda(\mathfrak{g})\rightarrow\mc{D}^{\mathrm{ch}}(P)$
if and only if, for each $A\in\mathfrak{g}$, the closed
$2$-form $\chi^{2,2}_A-dh^{2,1}_A$ is $G$-horizontal
and the following equation holds:
\begin{eqnarray} \label{lambda.Cartan40}
\chi^{4,0}_A+2h^{2,1}_A(A^P)
=\lambda(A,A).
\end{eqnarray}
\end{prop}

\begin{proof}
Let $A,B\in\mathfrak{g}$.
According to Definition \ref{VAoid.mor}, the linear map
$h^{2,1}:\mathfrak{g}\rightarrow\Omega^1(P)$ defines a map
between the said vertex algebroids if and only if
\begin{align*}
\{A^P,B^P\}\;\;
&=\lambda(A,B)-h^{2,1}_A(B^P)-h^{2,1}_B(A^P) \\
\{A^P,B^P\}_\Omega
&=-L_{A^P}h^{2,1}_B+L_{B^P}h^{2,1}_A-dh^{2,1}_A(B^P)
  +h^{2,1}_{[A,B]}
\end{align*}
By (\ref{Cartan4.VAoid}) the first equation is equivalent
to (\ref{lambda.Cartan40}).
By (\ref{Cartan4.VAoid}) again and the $G$-equivariance of
$h^{2,1}$, the second equation can be rewritten as
$\iota_{B^P}(\chi^{2,2}_A-dh^{2,1}_A)=0$.
This proves our claim.
In fact, the $G$-horizontality of $\chi^{2,2}_A-dh^{2,1}_A$
always implies that the left side of (\ref{lambda.Cartan40})
is locally constant, since
\begin{eqnarray*}
d\big(\chi^{4,0}_A+2\iota_{A^P}h^{2,1}_A\big)
=2\iota_{A^P}\chi^{2,2}_A-2\iota_{A^P}dh^{2,1}_A
  +2L_{A^P}h^{2,1}_A
=2\iota_{A^P}(\chi^{2,2}_A-dh^{2,1}_A)
\end{eqnarray*}
by virtue of (\ref{Cartan4.closed}) and the $G$-equivariance
of $h^{2,1}$.
Therefore if $P$ is connected, the horizontal condition
guarantees that (\ref{lambda.Cartan40}) must hold for some
$\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
\end{proof}

\begin{prop} \label{prop.Gaction}
The map of Lie algebras (\ref{lift.Lie}) integrates into
a homomorphism $G\rightarrow\mathrm{Aut}\,\mc{D}^{\mathrm{ch}}(P)$
if and only if the closed $2$-form
$\chi^{2,2}_A-dh^{2,1}_A$ vanishes for each $A\in\mathfrak{g}$.
\end{prop}

\begin{proof}
First we compute the one-parameter subgroup of automorphisms
generated by the inner derivation $(A^P+h^{2,1}_A)_0$.
Since $\mc{D}^{\mathrm{ch}}(P)$ is freely generated by a vertex
algebroid, it suffices to compute the automorphisms
at weights $0$ and $1$.
For the following computations, keep (\ref{freeVA.comm})
in mind.
For $f\in C^\infty(P)$ and $\alpha\in\Omega^1(P)$ we
simply have
\begin{align}
\label{Gaction.fcn}
\big(A^P+h^{2,1}_A\big)_0^n f=(A^P)^n f
\qquad&\Rightarrow\qquad
e^{t(A^P+h^{2,1}_A)_0}f=e^{tA}\cdot f \\
\label{Gaction.1form}
\big(A^P+h^{2,1}_A\big)_0^n\alpha=L_{A^P}^n\alpha\;\;
\qquad&\Rightarrow\qquad
e^{t(A^P+h^{2,1}_A)_0}\alpha=e^{tA}\cdot\alpha
\end{align}
where $\cdot$ refers to the given $G$-actions.
For $X\in\mc{T}(P)$ we first have
\begin{align*}
\big(A^P+h^{2,1}_A\big)_0 X
=[A^P,X]+\{A^P,X\}_\Omega-\iota_X dh^{2,1}_A
=[A^P,X]+\iota_X(\chi^{2,2}_A-dh^{2,1}_A)
\end{align*}
by Lemma \ref{lemma.1form.0mode} and
(\ref{Cartan4.VAoid}).
Then it follows by induction and the $G$-equivariance
of $\chi^{2,2}$, $h^{2,1}$ that
\begin{align}
\big(A^P+h^{2,1}_A\big)_0^n X
&=L_{A^P}^n X
  +nL_{A^P}^{n-1}\iota_X(\chi^{2,2}_A-dh^{2,1}_A),\qquad
  n\geq 1 \nonumber\\
\Rightarrow\qquad
e^{t(A^P+h^{2,1}_A)_0}X
&=e^{tA}\cdot X
  +te^{tA}\cdot\iota_X(\chi^{2,2}_A-dh^{2,1}_A)
\label{Gaction.vf}
\end{align}
This finishes the computation of the automorphisms.

If $\chi^{2,2}_A-dh^{2,1}_A$ vanishes for all $A\in\mathfrak{g}$,
it follows from (\ref{Gaction.fcn})--(\ref{Gaction.vf})
that (\ref{lift.Lie}) integrates into the $G$-action described
in \S\ref{sec.inv.data}.
Conversely, assume that (\ref{lift.Lie}) integrates into
a $G$-action.
By (\ref{Gaction.vf}), $\chi^{2,2}_A-dh^{2,1}_A$ must vanish
whenever $e^A=1$.
Since every element of $G$ lies in a torus, the subset
$\{A\in\mathfrak{g}\,|\,e^A=1\}$ spans $\mathfrak{g}$, so that
$\chi^{2,2}_A-dh^{2,1}_A$ must in fact vanish for all
$A\in\mathfrak{g}$.
\end{proof}

{\it Remark.}
Notice that according to the above proof, whenever
(\ref{lift.Lie}) is integrable, the resulting $G$-action
on $\mc{D}^{\mathrm{ch}}(P)$ can only be the one described in
\S\ref{sec.inv.data}.

\begin{subsec}
{\bf Comparing the three conditions.}
Recall the conditions encountered respectively in
Propositions \ref{prop.lift.Lie}, \ref{prop.lift.VA} and
\ref{prop.Gaction}:
\begin{itemize}
\item[(i)\phantom{ii}]
$\chi^{2,2}_A-dh^{2,1}_A$ is $G$-invariant for $A\in\mathfrak{g}$
\vspace{-0.1in}
\item[(ii)\phantom{i}]
$\chi^{2,2}_A-dh^{2,1}_A$ is $G$-horizontal for $A\in\mathfrak{g}$
\vspace{-0.1in}
\item[(iii)]
$\chi^{2,2}_A-dh^{2,1}_A=0$ for $A\in\mathfrak{g}$
\end{itemize}
In general, (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i);
the second implication follows from (\ref{Cartan4.closed}).
From another point of view, a map of vertex algebras as in
Proposition \ref{prop.lift.VA} always induces a map of Lie
algebras as in Proposition \ref{prop.lift.Lie},
but the other implication may seem somewhat
surprising.
In the case $\mathfrak{g}$ is semisimple, so that
$[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$, all three conditions are
equivalent.
\end{subsec}

Here is the main result of this section.

\begin{theorem} \label{thm.formalLG.action}
The $G$-action on $P$ lifts to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$
if and only if
\begin{eqnarray*}
8\pi^2 p_1(P)_G=\lambda(P).
\end{eqnarray*}
Moreover, this action is primary with respect to the chosen
conformal structure $\nu^\omega$.
\end{theorem}

\begin{proof}
Recall Definition \ref{formalLG.action} and
Example \ref{VAoid.Lie}.
First of all, an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
as described is determined by a linear map
$\mathfrak{g}\rightarrow\mc{D}^{\mathrm{ch}}(P)_1$ of the form
$A\mapsto A^P+h^{2,1}_A$, where
$h^{2,1}\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$.
By Propositions \ref{prop.lift.VA} and
\ref{prop.Gaction}, the precise conditions on
$h^{2,1}$ are
\begin{eqnarray} \label{formalLG.h}
\left\{\begin{array}{l}
  \chi^{2,2}_A=dh^{2,1}_A \vspace{0.03in} \\
  \chi^{4,0}_A=\lambda(A,A)-2h^{2,1}_A(A^P)
\end{array}\right.
\quad\iff\quad
2\chi^{2,2}+\chi^{4,0}=\lambda+2d_G h^{2,1}.
\end{eqnarray}
By Lemma \ref{lemma.Cartan.closed}a, this proves the
first claim.
The second claim is simply a restatement of Lemma
\ref{lemma.Cartan.closed}b and a subsequent remark
there.
\end{proof}

{\it Remark.}
In the sequel, $h^{2,1}$ will be called the
{\bf associated Cartan cochain} of the inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action.

\begin{example} \label{CDOG}
{\bf CDOs on a Lie group.}
Let $G$ be a compact simple Lie group.
In this discussion, $A,B$ always mean general elements
of $\mathfrak{g}=T_e G$.
The following notations will be used:
\vspace{-0.1in}
\begin{itemize}
\item[$\cdot$]
$A^\ell$ (resp.~$A^r$) is the left-invariant (resp.~right-invariant)
vector field on $G$ extending $A$;
\vspace{-0.1in}
\item[$\cdot$]
$\theta$ is the Maurer-Cartan form on $G$,
i.e.~$\theta\in\Omega^1(G)\otimes\mathfrak{g}$ such that
$\theta(A^\ell)=A$;
\vspace{-0.1in}
\item[$\cdot$]
$\lambda_{\t{\sf ad}}$ is the Killing form and
$\lambda_0=(2h^\vee)^{-1}\lambda_{\t{\sf ad}}$ the normalized Killing
form on $\mathfrak{g}$, where $h^\vee$ is the dual Coxeter number.
\vspace{-0.1in}
\end{itemize}
There is a canonical isomorphism $H^4(BG)\cong H^3(G)$,
where both spaces are one-dimensional.
The isomorphism is represented by
\begin{eqnarray*}
\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G\quad
\longmapsto\quad
-\frac{1}{6}\,\lambda(\theta\wedge[\theta\wedge\theta])
  \in\Omega^3(G)
\end{eqnarray*}
and we will use $\lambda_0$, or the corresponding
$3$-form $H_0$, as the respective basis.

Now we construct a conformal algebra of CDOs on $G$.
Let $\nabla$ be the flat connection on $TG$ such that
$\nabla A^\ell=0$.
Given $s\in\mathbb{C}$, there is a closed $3$-form $sH_0$.
By Theorems \ref{thm.globalCDO}a and \ref{thm.globalCDO.iso}a,
$\nabla$ and $sH_0$ together define an algebra of CDOs
$\mc{D}^{\mathrm{ch}}_s(G)=\mc{D}^{\mathrm{ch}}_{\nabla,sH_0}(G)$, and every algebra of CDOs on $G$
is up to isomorphism of this form.
Notice that the vertex algebroid structure of
$\mc{D}^{\mathrm{ch}}_s(G)$ includes such special cases as
\begin{align}
\label{CDOG.VAoidl}
&\{A^\ell,A^\ell\}=-2h^\vee\lambda_0(A,A),
&&\hspace{-1in}
\{A^\ell,-\}_\Omega
  =\frac{s}{4}\,\lambda_0(A,[\theta\wedge\theta]) \\
\label{CDOG.VAoidr}
&\{A^r,-\}=0,
&&\hspace{-1in}
\{A^r,-\}_\Omega
  =\frac{s}{4}\,\lambda_0(\theta(A^r),[\theta\wedge\theta])
\end{align}
By Theorem \ref{thm.globalCDO}b, the trivial $1$-form defines
a conformal structure $\nu$ of central charge $2\dim\mathfrak{g}$.
If $J_1,J_2,\cdots$ are a basis of $\mathfrak{g}$ and
$\theta^1,\theta^2,\cdots$ the respective components of
$\theta$, we can write $\nu=J^\ell_{a,-1}\theta^a$.
Also, $\nu$ has the property that $L_1 A^\ell=L_1 A^r=0$.

First consider the action of $G$ on itself by right
multiplication.
The induced map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(G)$
sends $A$ to $A^\ell$.
Both $\nabla$ and $H_0$ are invariant under this action.
Define $h^\ell\in(\mathfrak{g}^\vee\otimes\Omega^1(G))^G$ by
$h^\ell_A=-\frac{s}{2}\lambda_0(A,\theta)$.
In view of (\ref{CDOG.VAoidl}), $h^\ell$ satisfies
(\ref{formalLG.h}) with $\lambda=(-s-2h^\vee)\lambda_0$.
Therefore $h^\ell$ is the Cartan cochain associated to
an inner $(\hat{\mathfrak{g}},G)$-action
\begin{eqnarray} \label{CDOG.formalLGl}
V_{-s-2h^\vee}(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}_s(G)\,;
\end{eqnarray}
denote its image by $V_{-s-2h^\vee}(\mathfrak{g})^\ell$.
This action is primary with respect to $\nu$.

Now consider also the action of $G$ on itself by inverse
left multiplication, i.e.~each $g\in G$ acts via left
multiplication by $g^{-1}$.
The induced map of Lie algebras $\mathfrak{g}\rightarrow\mc{T}(G)$
sends $A$ to $-A^r$.
Since left- and right-invariant vector fields commute,
$\nabla$ and $H_0$ are invariant under this action as well.
Define $h^r\in(\mathfrak{g}^\vee\otimes\Omega^1(G))^G$
by $h^r_A=\frac{s}{2}\lambda_0(\theta(A^r),\theta)$.
In view of (\ref{CDOG.VAoidr}), $h^r$ satisfies
(\ref{formalLG.h}) with $\lambda=s\lambda_0$.
\footnote{
Notice that $(dh^r_A)(B^r,C^r)=\frac{s}{2}\lambda_0(A,[B,C])
=\chi^{2,2}_A(B^r,C^r)$ for $B,C\in\mathfrak{g}$, so that indeed
$dh^r_A=\chi^{2,2}_A$.
}
Therefore $h^r$ is the Cartan cochain associated to
an inner $(\hat{\mathfrak{g}},G)$-action
\begin{eqnarray} \label{CDOG.formalLGr}
V_s(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}_s(G)\,;
\end{eqnarray}
denote its image by $V_s(\mathfrak{g})^r$.
This action is also primary with respect to $\nu$.
\end{example}

{\it Remark.}
For both $G$-actions, since $H^*_G(G)=H^*(\textrm{pt})$,
the condition in Theorem \ref{thm.formalLG.action}
is trivial for any $\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$,
so that in fact there are many other inner
$(\hat{\mathfrak{g}},G)$-actions on $\mc{D}^{\mathrm{ch}}_s(G)$.

\newpage
\setcounter{equation}{0}
\section{CDOs on Principal Bundles}
\label{sec.CDOP}

In this section, the object of interest is an algebra
of CDOs $\mc{D}^{\mathrm{ch}}(P)$ on the total space of a smooth
principal bundle $P\rightarrow M$, equipped
with a projective ``formal loop group action'' in
the sense of \S\ref{sec.LGaction}.
The first goal is to understand the subalgebra of
$\mc{D}^{\mathrm{ch}}(P)$ invariant under the said action.
In subsequent sections we will construct and study
modules over this invariant subalgebra.
The second (and more technical) goal is to give
an alternative description of $\mc{D}^{\mathrm{ch}}(P)$ in the case
$P\rightarrow M$ is a principal frame bundle.
This vertex algebra will play a central role in
the rest of the paper:
it is not merely a particular algebra of CDOs, but
should be regarded as a more fundamental object from
which all algebras of CDOs (as well as other related
constructions) derive.

\begin{subsec} \label{sec.prin.bdl}
{\bf Setting:~principal bundle.}
Let $G$ be again a compact connected Lie group and
$\pi:P\rightarrow M$ a smooth principal $G$-bundle.
Identify $H^*_G(P)$ with $H^*(M)$ via $\pi^*$.
Choose a connection $\Theta$ on $\pi$,
i.e. some $\Theta\in(\Omega^1(P)\otimes\mathfrak{g})^G$
such that $\Theta(A^P)=A$ for $A\in\mathfrak{g}$;
then its curvature is
$\Omega=d\Theta+\frac{1}{2}[\Theta\wedge\Theta]$.
Keep in mind that $\Theta$ is equivalent to
a $G$-equivariant vector bundle decomposition
$TP=T_h P\oplus T_v P$, with
$T_h P=\ker\Theta$ and $T_v P=\ker\pi_*$;
and $\Omega$ measures the non-integrability of
the subbundle $T_h P$.
Given $X\in\mc{T}(M)$, denote its horizontal lift by
$\widetilde X\in\mc{T}_h(P)^G$.
Let us extend the notation $(\cdot)^P$
(see \S\ref{sec.Gmfld}) $C^\infty(P)$-linearly to
represent the isomorphism
$C^\infty(P)\otimes\mathfrak{g}\cong\mc{T}_v(P)$.
For $X,Y\in\mc{T}(M)$ notice that
\begin{eqnarray} \label{horlift.nonint}
[\widetilde X,\widetilde Y]-\widetilde{[X,Y]}=-\Omega(\widetilde X,\widetilde Y)^P.
\end{eqnarray}
Also recall the notations introduced in
\S\ref{sec.Cartan}.

Consider an algebra of CDOs $\mc{D}^{\mathrm{ch}}(P)=\mc{D}^{\mathrm{ch}}_{\nabla,H}(P)$
defined as in \S\ref{sec.inv.data}.
Let $\lambda\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
Recall from Definition \ref{formalLG.action} and the proof
of Theorem \ref{thm.formalLG.action} the meaning of an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$ and its
associated Cartan cochain.
It will be understood without further comment that any inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$ considered
below extends the given $G$-action on $\mc{D}^{\mathrm{ch}}(P)_0=C^\infty(P)$.
\end{subsec}

{\it Preparation.}
Given a representation $\rho:G\rightarrow GL(V)$,
let us denote the induced map of Lie algebras also by
$\rho$ and define an invariant symmetric bilinear form
on $\mathfrak{g}$ by $\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
In particular, $\lambda_{\t{\sf ad}}$ is the Killing form.
If we regard $V$ as a $G$-equivariant vector bundle over
a point and identify $H^4_G(\textrm{pt})$ with
$(\textrm{Sym}^2\mathfrak{g}^\vee)^G$, then
$-8\pi^2 p_1(V)_G=\lambda_\rho$.

\begin{lemma} \label{lemma.Gp1.p1}
$8\pi^2 p_1(P)_G=8\pi^2 p_1(M)-\lambda_{\t{\sf ad}}(P)$.
\end{lemma}

\begin{proof}
Consider the $G$-equivariant decomposition
$TP=T_h P\oplus T_v P$.
On one hand, since $T_h P\cong\pi^*TM$ and $\pi^*$
identifies $H^*_G(P)$ with $H^*(M)$, we have
$p_1(T_h P)_G=p_1(TM)$.
On the other hand, $T_v P\cong P\times\mathfrak{g}$
with $G$ acting on $\mathfrak{g}$ in the adjoint
representation, so that
$-8\pi^2 p_1(T_v P)=\lambda_{\t{\sf ad}}(P)$.
This proves the lemma.
\end{proof}

Hence Theorem \ref{thm.formalLG.action} specializes
to our current setting as follows.

\begin{corollary} \label{cor.formalLG.P}
The $G$-action on $P$ lifts to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$ if
and only if
\begin{eqnarray} \label{formalLG.condition2}
\qquad
8\pi^2 p_1(M)=(\lambda+\lambda_{\t{\sf ad}})(P).
\quad\qedsymbol
\end{eqnarray}
\end{corollary}

{\it Remark.}
For convenience, we will refer to $\mc{D}^{\mathrm{ch}}(P)$ as
a {\bf principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra}
when it is equipped with an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action, or
a {\bf principal $(\hat{\mathfrak{g}},G)$-algebra} if we
do not wish to specify $\lambda$.

\begin{subsec} \label{sec.invCDO}
{\bf The invariant subalgebra.}
Suppose $\mc{D}^{\mathrm{ch}}(P)$ is a principal
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra, i.e.~there is
an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(P)$ defined by
some associated Cartan cochain
$h\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$.
Consider the centralizer subalgebra \cite{Kac,FB-Z}
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}
:=C\big(\mc{D}^{\mathrm{ch}}(P),V_\lambda(\mathfrak{g})\big)
\subset\mc{D}^{\mathrm{ch}}(P).
\end{eqnarray*}
This is the subalgebra whose fields are
$\hat{\mathfrak{g}}_\lambda$-invariant.
The weight-zero component consists of $f\in C^\infty(P)$
such that $(A^P+h_A)_0 f=A^P f=0$ for $A\in\mathfrak{g}$,
namely
\begin{eqnarray} \label{invCDO.0}
\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}_0
=C^\infty(P)^G
=\pi^*C^\infty(M).
\end{eqnarray}
On the other hand, the weight-one component consists of
$\alpha+X\in\Omega^1(P)\oplus\mc{T}(P)$ such that
\begin{align*}
\left\{\begin{array}{l}
(A^P+h_A)_0(\alpha+X)=L_{A^P}\alpha+[A^P,X]=0 \vspace{0.03in} \\
(A^P+h_A)_1(\alpha+X)=\alpha(A^P)+\{A^P,X\}+h_A(X)=0
\end{array}\right.\qquad
\textrm{for }A\in\mathfrak{g}
\end{align*}
where we have used (\ref{freeVA.comm}), Lemma
\ref{lemma.1form.0mode} and (\ref{formalLG.h}).
It follows that there is a pullback square
\begin{eqnarray*}
\xymatrix{
  \mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}_1\ar[r]\ar[d] &
  \mc{T}(P)^G\ar[d]^q \\
  \Omega^1(P)^G\ar[r]^{p\phantom{WW}} &
  (\mathfrak{g}^\vee\otimes C^\infty(P))^G
}
\end{eqnarray*}
where $p$, $q$ are suitable adjoints to
$(\alpha,A)\mapsto-\alpha(A^P)$ and
$(X,A)\mapsto\{A^P,X\}+h_A(X)$ respectively.
Observe that
(i) $p$ is surjective, as contraction with
$\Theta\in(\Omega^1(P)\otimes\mathfrak{g})^G$ furnishes a right inverse,
and
(ii) $\ker p$ is the space of the $G$-invariant horizontal
(i.e.~basic) $1$-forms on $P$, or in other words
$\pi^*\Omega^1(M)$.
This implies the following short exact sequence
\begin{eqnarray}\label{invCDO.1}
\xymatrix{
  0\ar[r] &
  \pi^*\Omega^1(M)\ar[r] &
  \mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}_1\ar[r] &
  \mc{T}(P)^G\ar[r] & 0
}
\end{eqnarray}
In view of (\ref{invCDO.0}) and (\ref{invCDO.1}),
the vertex algebroid associated to
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ (see \S\ref{VA.VAoid}) is
of the form
\begin{eqnarray} \label{invCDO.VAoid}
\big(C^\infty(M),\Omega^1(M),\mc{T}(P)^G,\cdots\big).
\end{eqnarray}
Since $\mc{D}^{\mathrm{ch}}(P)$ is freely generated by its associated
vertex algebroid (see \S\ref{VAoid.VA}),
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ must contain at least
a subalgebra that is freely generated by
(\ref{invCDO.VAoid}).
However, the author does not know if
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ may or may not contain more
elements (see also Example \ref{invCDOG}).
\end{subsec}

{\it Remarks.}
(i) By the above discussion, any module over
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$ contains in its lowest
weight a module over the Atiyah algebroid
$(C^\infty(M),\mc{T}(P)^G)$, which can be e.g.~the space
of sections of a vector bundle with connection
associated to $(P\rightarrow M,\Theta)$.
\footnote{ \label{invVF}
Given a $G$-representation $W$, the space of
sections of the associated vector bundle is
$(C^\infty(P)\otimes W)^G$, which is naturally
a module over $(C^\infty(M),\mc{T}(P)^G)$.
In fact, the action of $\widetilde X\in\mc{T}_h(P)^G$ can be
identified with the covariant derivative along
the corresponding $X\in\mc{T}(M)$, and the action of
$\mc{T}_v(P)^G\cong(C^\infty(P)\otimes\mathfrak{g})^G$ is
induced by the representation
$\mathfrak{g}\rightarrow\textrm{End}\,W$.
}
In subsequent sections we will construct and study
such modules over $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$.
(ii) It is noteworthy that $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$
is different from an algebra of CDOs on the base
manifold $M$ --- compare
(\ref{invCDO.0})--(\ref{invCDO.1}) and
(\ref{CDO.wt01}) --- and by comparison has more
interesting modules.
In \S\ref{CDOP.CDOM} we will see that to recover
an algebra of CDOs on $M$ from $\mc{D}^{\mathrm{ch}}(P)$ requires
the use of semi-infinite cohomology (as well as
a suitable choice of $P\rightarrow M$ plus other
geometric data).

\begin{example} \label{invCDOG}
{\bf The invariant subalgebra on a Lie group.}
This is a continuation of Example \ref{CDOG} and
notations will not be explained again.
Recall the subalgebras
$V_{-s-2h^\vee}(\mathfrak{g})^\ell,\;V_s(\mathfrak{g})^r\subset\mc{D}^{\mathrm{ch}}_s(G)$
generated respectively by $A^\ell+h^\ell_A$ and
$A^r+h^r_A$ for $A\in\mathfrak{g}$.
Consider the centralizer subalgebra
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}
=C\Big(\mc{D}^{\mathrm{ch}}_s(G),V_{-s-2h^\vee}(\mathfrak{g})^\ell\Big).
\end{eqnarray*}
By the discussion in \S\ref{sec.invCDO} and
(\ref{CDOG.VAoidr}),
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}_0=\mathbb{C}$ and
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}_1$ consists of elements
of the form $A^r+\alpha$ where $A\in\mathfrak{g}$ and
$\alpha\in\Omega^1(G)$ satisfies
\begin{eqnarray*}
\phantom{WWW}
\alpha(B^\ell)
=-\{B^\ell,A^r\}-h^\ell_B(A^r)
=\frac{s}{2}\,\lambda_0(B,\theta(A^r))
=h^r_A(B^\ell)\qquad
\textrm{for }B\in\mathfrak{g}
\end{eqnarray*}
i.e.~$\alpha=h^r_A$.
In other words, we have
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}_i=V_s(\mathfrak{g})^r_i$
for $i=0,1$.
Since $V_s(\mathfrak{g})^r$ is freely generated by
its two lowest weights, there is an inclusion
\begin{eqnarray} \label{invCDOG.subalg}
V_s(\mathfrak{g})^r\subset\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}.
\end{eqnarray}
This means the subalgebras
$V_{-s-2h^\vee}(\mathfrak{g})^\ell$ and
$V_s(\mathfrak{g})^r$ commute with each other.

If $s\neq-h^\vee$, then both
$V_{-s-2h^\vee}(\mathfrak{g})^\ell$ and $V_s(\mathfrak{g})^r$
admit Sugawara conformal vectors \cite{Kac,FB-Z}
\begin{eqnarray*}
&\displaystyle
\nu^\ell\in V_{-s-2h^\vee}(\mathfrak{g})^\ell\;
\textrm{ of central charge }
\left(\frac{s+2h^\vee}{s+h^\vee}\right)\dim\mathfrak{g},& \\
&\displaystyle
\nu^r\in V_s(\mathfrak{g})^r\;
\textrm{ of central charge }
\left(\frac{s}{s+h^\vee}\right)\dim\mathfrak{g}.\quad&
\end{eqnarray*}
On the other hand, since $\mc{D}^{\mathrm{ch}}_s(G)$ has a conformal
vector $\nu$ of central charge $2\dim\mathfrak{g}$ and
the subalgebra $V_{-s-2h^\vee}(\mathfrak{g})^\ell$ is
generated by elements that are primary with respect
to $\nu$, the coset construction yields a conformal
vector \cite{Kac,FB-Z}
\begin{eqnarray*}
\nu-\nu^\ell\in\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}\;
\textrm{ of central charge }
\left(\frac{s}{s+h^\vee}\right)\dim\mathfrak{g}.
\end{eqnarray*}
In fact, a computation (which we omit) shows that
$\nu-\nu^\ell=\nu^r$, i.e.~the Sugawara vector of
$V_s(\mathfrak{g})^r$ is also conformal for
$\mc{D}^{\mathrm{ch}}_s(G)^{\hat{\mathfrak{g}},\ell}$.
This seems to suggest that the inclusion
(\ref{invCDOG.subalg}) is in fact surjective, at least
for generic values of $s$, but the author has yet to
find either a proof or a counterexample.
\footnote{
Notice that Lemma \ref{lemma.conf.gen} does not apply
here.
}
\end{example}

\begin{subsec} \label{sec.prin.bdl2}
{\bf Setting refined:~principal frame bundle.}
In addition to the data described in
\S\ref{sec.prin.bdl}, suppose there is also
a representation $\rho:G\rightarrow SO(\mathbb{R}^d)$
and an isomorphism $P\times_\rho\mathbb{R}^d\cong TM$.
This induces a Riemannian metric on $M$ and
an orthogonal connection on $TM$, which (for
simplicity) is assumed to be torsion-free.
Let $[p,v]\in TM$ denote the coset of
$(p,v)\in P\times\mathbb{R}^d$.
For $i=1,\ldots,d$, let $\mathbf{e}_i\in\mathbb{R}^d$ be
the standard basis vectors and $\tau_i\in\mc{T}_h(P)$
the tautological vector fields defined by
\begin{eqnarray*}
\tau_i|_p:=\textrm{horizontal lift of }[p,\mathbf{e}_i]
\end{eqnarray*}
which constitute a framing of $T_h P$.
For $A\in\mathfrak{g}$ and $i,j=1,\ldots,d$, we have
the Lie brackets
\begin{eqnarray} \label{tau.brackets}
[A^P,\tau_i]=\rho(A)_{ji}\tau_j,
\qquad
[\tau_i,\tau_j]=-\Omega(\tau_i,\tau_j)^P
\end{eqnarray}
where $\rho$ here denotes the induced map of Lie
algebras $\mathfrak{g}\rightarrow\mathfrak{so}_d$.
\footnote{
$[\tau_i,\tau_j]$ has no horizontal component precisely
because of our assumption that the induced connection
on $TM$ is torsion-free.
Moreover, the Jacobi identity for any
$\tau_i,\tau_j,\tau_k$ is the same as the two Bianchi
identities combined.
}

Suppose the conformal algebra of CDOs
$\mc{D}^{\mathrm{ch}}(P)=\mc{D}^{\mathrm{ch}}_{\nabla',H'}(P)$ is defined as in
\S\ref{sec.inv.data} using the following data.
(Notice the various change of notations here.)
Let $\nabla'$ be the $G$-invariant \emph{flat}
connection on $TP$ with respect to which all
$A^P$ and $\tau_i$ are parallel, and
$H'$ be a $G$-invariant \emph{closed} $3$-form
on $P$;
together they determine the vertex algebroid
\begin{align} \label{CDOP.VAoid}
\big(C^\infty(P),\Omega^1(P),\mc{T}(P),
  \bullet',\{\;\}',\{\;\}'_\Omega\big)
\end{align}
by which $\mc{D}^{\mathrm{ch}}(P)$ is freely generated
(see Theorem \ref{thm.globalCDO}a).
Also, the trivial $1$-form determines a conformal
vector of $\mc{D}^{\mathrm{ch}}(P)$, namely
\begin{eqnarray} \label{CDOP.conformal}
\nu'=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
\end{eqnarray}
where $t_1,t_2,\cdots$ are a basis of $\mathfrak{g}$;
$\Theta^1,\Theta^2,\cdots$ the corresponding
components of $\Theta$, so that
$\Theta=\Theta^a\otimes t_a$;
$\tau_1,\cdots,\tau_d$ the vector fields defined
above; and
$\tau^1,\cdots,\tau^d$ the $1$-forms given
by $\tau^i(\tau_j)=\delta^i_j$ and $\tau^i(A^P)=0$
(see Theorem \ref{thm.globalCDO}b).
In the rest of this section, we will provide a more
detailed description of this conformal vertex algebra
in the presence of an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action.
\end{subsec}

According to the axioms of a vertex algebroid
(Definition \ref{VAoid}), the entire structure of
(\ref{CDOP.VAoid}) is determined by the special
cases appearing in the following statement.

\begin{lemma} \label{lemma.CDOP.VAoid}
For $f\in C^\infty(P)$, $A,B\in\mathfrak{g}$ and
$i,j=1,\ldots,d$, we have
\begin{itemize}
\item[$\cdot$]
$A^P\bullet'f=\tau_i\bullet'f=0$
\vspace{-0.05in}
\item[$\cdot$]
$\{A^P,B^P\}'=-(\lambda_{\t{\sf ad}}+\lambda_\rho)(A,B)$,\;\;
$\{A^P,\tau_i\}'=0$,\;\;
$\{\tau_i,\tau_j\}'=2\mathrm{Ric}_{ij}$
\vspace{-0.07in}
\item[$\cdot$]
$\{A^P,B^P\}'_\Omega
=\frac{1}{2}\iota_{A^P}\iota_{B^P}H'$,\;\;
$\{A^P,\tau_i\}'_\Omega
=\frac{1}{2}\iota_{A^P}\iota_{\tau_i}H'$,\;\;
$\{\tau_i,\tau_j\}'_\Omega=d\,\mathrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H'$
\end{itemize}
where $\mathrm{Ric}_{ij}=\rho(\Omega(\tau_i,\tau_k))_{jk}
\in C^\infty(P)$.
Moreover, we also have $\nu'_1 A^P=\nu'_1\tau_i=0$.
\end{lemma}

\begin{proof}
Recall (\ref{CDO.VAoid}) and (\ref{CDO.L1}).
For $\bullet$ the claim is clear.
Using (\ref{tau.brackets}) we compute the
following operators
\begin{align*}
&\nabla^{\prime\,t} A^P:\left\{
\begin{array}{l}
  B^P\mapsto\,-[A,B]^P \vspace{0.01in} \\
  \tau_i\;\;\mapsto\,-\rho(A)_{ji}\tau_j
\end{array}\right.,\hspace{-0.35in}
&&\nabla^{\prime\,t}\tau_i:\left\{
\begin{array}{l}
  A^P\mapsto\,\rho(A)_{ji}\tau_j \vspace{0.01in} \\
  \tau_j\;\;\mapsto\,\Omega(\tau_i,\tau_j)^P
\end{array}\right. \\
\Rightarrow\qquad\quad
&\nabla'(\nabla^{\prime\,t} A^P)=0,
&&\nabla'(\nabla^{\prime\,t}\tau_i):\left\{
\begin{array}{l}
  A^P\mapsto\,0 \vspace{0.01in} \\
  \tau_j\;\;\mapsto\,
    (d\Omega(\tau_i,\tau_j))^P
\end{array}\right.
\end{align*}
The rest of the lemma now follows easily from
these calculations and the symmetries of the
Riemannian curvature tensor $\rho(\Omega)$.
\end{proof}

Theorem \ref{thm.formalLG.action} specializes
to our current setting as follows.

\begin{corollary} \label{cor.formalLG.P2}
The $G$-action on $P$ lifts to an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action on $\mc{D}^{\mathrm{ch}}(P)$
if and only if
\begin{eqnarray*}
(\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho)(P)=0.
\end{eqnarray*}
Moreover, this action is primary with respect to
the conformal vector $\nu'$.
\end{corollary}

\begin{proof}
By assumption, $P\rightarrow M$ is a lifting
of the special orthogonal frame bundle
$F_{SO}(TM)\rightarrow M$ along
$\rho:G\rightarrow SO_d$.
Hence we have a commutative diagram
\begin{align*}
\xymatrix{
  &
  H^4(BSO_d)\ar[r]^{\;\;\rho^*}\ar[d] &
  H^4(BG)\ar[d] \\
  H^4(M)\ar[r]^{\cong\phantom{WWW}} &
  H^4_{SO_d}(F_{SO}(TM))\ar[r]^{\phantom{WW}\cong} &
  H^4_G(P)
}
\end{align*}
Also recall the identifications like
$H^4(BG)\cong(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
Since $-8\pi^2 p_1\in H^4(BSO_d)$ can be identified
with the bilinear form $(A,B)\mapsto\mathrm{Tr}\, AB$,
its image under $\rho^*$ is $\lambda_\rho$ and
so by the diagram above we have
$-8\pi^2p_1(M)=\lambda_\rho(P)$.
Now the first claim is just a special case of
Corollary \ref{cor.formalLG.P}.
The other claim is true because $\nu'_1 A^P=0$
for $A\in\mathfrak{g}$ by Lemma \ref{lemma.CDOP.VAoid}.
\end{proof}

{\it Remark.}
In the sequel we will write
$\lambda^*=\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho$.

\begin{subsec} \label{CDOP.FLGaction}
{\bf Inner $(\hat{\mathfrak{g}}_\lambda,G)$-action on CDOs.}
Suppose $\mc{D}^{\mathrm{ch}}(P)$ is a principal
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra, i.e.~there is
an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}(P)$, defined
by some associated Cartan cochain
$h\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$.
Notice that the Cartan cochains in
(\ref{Cartan4.VAoid}) may now be written as
$\chi^{2,2}=\frac{1}{2}d_G H'$ and
$\chi^{4,0}=-\lambda_{\t{\sf ad}}-\lambda_\rho$ by Lemma
\ref{lemma.CDOP.VAoid}.
Hence the condition (\ref{formalLG.h}) on $h$
is now equivalent to the equation
\begin{align} \label{CDOP.h}
d_G(H'-2h)=\lambda^*.
\end{align}
This of course agrees with Corollary
\ref{cor.formalLG.P2}.
The purpose of the next two lemmas is to
modify the description of the vertex algebra
$\mc{D}^{\mathrm{ch}}(P)$ in such a way that the inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action becomes
manifest and the data $(H',h)$ involved are
replaced by a basic $3$-form that trivializes
$\lambda^*(\Omega\wedge\Omega)$.
\end{subsec}

\begin{lemma} \label{lemma.basic.3form}
There is a natural bijection between the
following two sets of data:

(i) $(H',h)\in\Omega^3(P)^G\oplus
(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$ such
that $d_G(H'-2h)=\lambda^*$, and

(ii) $(H,\beta)\in\pi^*\Omega^3(M)\oplus\Omega^2(P)^G$
such that $dH=\lambda^*(\Omega\wedge\Omega)$ and
$\beta|_{T_h P}=0$.
\end{lemma}

\begin{proof}
Once the maps in both directions are described below,
it will be clear they are inverses to each other.
Let $\mathrm{CS}_{\lambda^*}(\Theta)
=\lambda^*(\Theta\wedge\Omega)
-\frac{1}{6}\lambda^*(\Theta\wedge[\Theta\wedge\Theta])$.
Notice that as Cartan cochains
$\mathrm{CS}_{\lambda^*}(\Theta)\in\Omega^3(P)^G$ and
$\lambda^*(-,\Theta)\in(\mathfrak{g}^\vee\otimes\Omega^1(P))^G$
have the following differentials
\begin{align} \label{dCS}
d_G\,\mathrm{CS}_{\lambda^*}(\Theta)
=\lambda^*(\Omega\wedge\Omega)
  -\lambda^*(-,d\Theta),\qquad
d_G\,\lambda^*(-,\Theta)
=\lambda^*(-,d\Theta)-\lambda^*.
\end{align}

{\it From (i) to (ii).}
Let $(H',h)$ be a pair as in (i).
First define a $G$-invariant $2$-form on $P$ by
\begin{align} \label{beta}
\beta(A^P,B^P)=h_A(B^P)-h_B(A^P),\qquad
\beta(A^P,\tau_i)=2h_A(\tau_i),\qquad
\beta(\tau_i,\tau_j)=0
\end{align}
for $A,B\in\mathfrak{g}$ and $i,j=1,\ldots,d$.
Indeed, the $G$-equivariance of $h$ together with
(\ref{tau.brackets}) imply the $G$-invariance of $\beta$.
Observe that the component of $d_G(H'-2h)=\lambda^*$
in $(\textrm{Sym}^2\mathfrak{g}^\vee\otimes C^\infty(P))^G$ allows
us to write
\begin{align} \label{iA.beta}
\iota_{A^P}\beta=2h_A-\lambda^*(A,\Theta).
\end{align}
Then define a $G$-invariant $3$-form on $P$ by the first
expression below
\begin{align}
\label{basic.3form}
H=H'+\mathrm{CS}_{\lambda^*}(\Theta)-d\beta
=H'+\mathrm{CS}_{\lambda^*}(\Theta)
  -2h+\lambda^*(-,\Theta)-d_G\beta
\end{align}
but we also have the second expression thanks to
(\ref{iA.beta}).
It follows from $d_G(H'-2h)=\lambda^*$ and
(\ref{dCS}) that $d_G H=\lambda^*(\Omega\wedge\Omega)$,
or equivalently, $H$ is horizontal (hence basic)
and satisfies $dH=\lambda^*(\Omega\wedge\Omega)$.

{\it From (ii) to (i).}
Use (\ref{basic.3form}) again:
given $(H,\beta)$ as in (ii), define the desired pair
$(H',h)$ by
\begin{align*}
H'-2h
=H-\mathrm{CS}_{\lambda^*}(\Theta)-\lambda^*(-,\Theta)
  +d_G\beta.
\end{align*}
Indeed, it follows from
$d_G H=\lambda^*(\Omega\wedge\Omega)$ and (\ref{dCS})
that $d_G(H'-2h)=\lambda^*$.
\end{proof}

{\it Preparation.}
This continues with the discussion in
\S\ref{CDOP.FLGaction}.
Since the data $(H',h)$ in the definition of $\mc{D}^{\mathrm{ch}}(P)$
as a principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra
satisfy (\ref{CDOP.h}), by Lemma \ref{lemma.basic.3form}
they give rise to a $G$-invariant $2$-form $\beta$ and
a basic $3$-form $H$ satisfying
$dH=\lambda^*(\Omega\wedge\Omega)$.
Let $\Delta:\mc{T}(P)\rightarrow\Omega^1(P)$ be the
$C^\infty(P)$-linear map with
$\Delta(A^P)=-h_A$ for $A\in\mathfrak{g}$ and
$\Delta(\tau_i)=-\frac{1}{2}\iota_{\tau_i}\beta$
for $i=1,\ldots,d$.
By Lemma \ref{lemma.newVAoid}, $\Delta$ determines
an isomorphism $(\textrm{id},\Delta)$ from the vertex
algebroid (\ref{CDOP.VAoid}) to a new vertex
algebroid
\begin{align} \label{CDOP.VAoidnew}
\big(C^\infty(P),\Omega^1(P),\mc{T}(P),
  \bullet,\{\;\},\{\;\}_\Omega\big).
\end{align}
This in turn induces an isomorphism from $\mc{D}^{\mathrm{ch}}(P)$
to the vertex algebra freely generated by
(\ref{CDOP.VAoidnew}), which we denote by
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
By composition, there is an inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action
$V_\lambda(\mathfrak{g})\hookrightarrow\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$
whose associated Cartan cochain is trivial,
i.e.~it is determined by the map
$\mathfrak{g}\hookrightarrow\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$ sending
$A$ simply to $A^P$.

\vspace{0.08in}
According to the axioms of a vertex algebroid
(Definition \ref{VAoid}), the entire structure of
(\ref{CDOP.VAoidnew}) is determined by the
following special cases.

\begin{lemma} \label{lemma.CDOP.VAoidnew}
For $f\in C^\infty(P)$, $A,B\in\mathfrak{g}$ and
$i,j=1,\ldots,d$, we have
\begin{itemize}
\item[$\cdot$]
$A^P\bullet f=\tau_i\bullet f=0$
\vspace{-0.07in}
\item[$\cdot$]
$\{A^P,B^P\}=\lambda(A,B)$,\;\;
$\{A^P,\tau_i\}=0$,\;\;
$\{\tau_i,\tau_j\}=2\mathrm{Ric}_{ij}$
\vspace{-0.07in}
\item[$\cdot$]
$\{A^P,B^P\}_\Omega=\{A^P,\tau_i\}_\Omega=0$,\;\;
$\{\tau_i,\tau_j\}_\Omega=d\,\mathrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H
+\lambda^*(\Omega(\tau_i,\tau_j),\Theta)$
\end{itemize}
\end{lemma}

\begin{proof}
Recall Definition \ref{VAoid.mor} for the meaning of
$(\textrm{id},\Delta):(\ref{CDOP.VAoid})\rightarrow(\ref{CDOP.VAoidnew})$
as well as Lemma \ref{lemma.CDOP.VAoid} for the
structure maps of (\ref{CDOP.VAoid}).
Firstly, the claim for $\bullet$ follows from the
$C^\infty(P)$-linearity of $\Delta$.
Consider again the map of vertex algebroid $(i,h)$ in
Proposition \ref{prop.lift.VA}.
Because of the composition
\begin{align*}
(\textrm{id},\Delta)\circ(i,h)=(i,0):
(\mathbb{C},0,\mathfrak{g},0,\lambda,0)\rightarrow
(\ref{CDOP.VAoidnew})
\end{align*}
it is clear that $\{A^P,B^P\}=\lambda(A,B)$ and
$\{A^P,B^P\}_\Omega=0$.
This is equivalent to the comment above concerning
the inner $(\hat{\mathfrak{g}}_\lambda,G)$-action on
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
The other values of $\{\;\}$ are computed as follows:
\begin{align*}
&\textstyle
\{A^P,\tau_i\}
=\{A^P,\tau_i\}'+h_A(\tau_i)+\frac{1}{2}\beta(\tau_i,A^P)
=0 \\
&\textstyle
\{\tau_i,\tau_j\}
=\{\tau_i,\tau_j\}'+\frac{1}{2}\beta(\tau_i,\tau_j)
  +\frac{1}{2}\beta(\tau_j,\tau_i)
=2\textrm{Ric}_{ij}
\end{align*}
using the definition of $\beta$ in (\ref{beta}).
The other values of $\{\;\}_\Omega$ are computed as
follows:
\begin{align*}
&\textstyle
\{A^P,\tau_i\}_\Omega
=\{A^P,\tau_i\}'_\Omega
+\frac{1}{2}L_{A^P}\iota_{\tau_i}\beta
-L_{\tau_i}h_A
+dh_A(\tau_i)
-\frac{1}{2}\iota_{[A^P,\tau_i]}\beta\\
&\textstyle
\phantom{\{A^P,\tau_i\}'_\Omega}
=\frac{1}{2}\iota_{A^P}\iota_{\tau_i}H'
-\iota_{\tau_i}dh_A=0 \\
&\textstyle
\{\tau_i,\tau_j\}_\Omega
=\{\tau_i,\tau_j\}'_\Omega
+\frac{1}{2}L_{\tau_i}\iota_{\tau_j}\beta
-\frac{1}{2}L_{\tau_j}\iota_{\tau_i}\beta
+\frac{1}{2}d\beta(\tau_i,\tau_j)
+h_{\Omega(\tau_i,\tau_j)}\\
&\textstyle
\phantom{\{\tau_i,\tau_j\}'_\Omega}
=d\,\textrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H'
-\frac{1}{2}\iota_{\Omega(\tau_i,\tau_j)}\beta
-\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}d\beta
+\frac{1}{2}\iota_{\Omega(\tau_i,\tau_j)}\beta
+\frac{1}{2}\lambda^*(\Omega(\tau_i,\tau_j),\Theta) \\
&\textstyle
\phantom{\{\tau_i,\tau_j\}'_\Omega}
=d\,\textrm{Ric}_{ij}
+\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H
+\lambda^*(\Omega(\tau_i,\tau_j),\Theta)
\end{align*}
using, in order:
the $G$-invariance of $\beta$;
the component of (\ref{CDOP.h}) in
$(\mathfrak{g}^\vee\otimes\Omega^2(P))^G$;
the identity
\begin{eqnarray*}
L_{\tau_i}\iota_{\tau_j}
  -L_{\tau_j}\iota_{\tau_i}
  +d\iota_{\tau_j}\iota_{\tau_i}
=L_{\tau_i}\iota_{\tau_j}
  -\iota_{\tau_j}d\iota_{\tau_i}
=L_{\tau_i}\iota_{\tau_j}
  -\iota_{\tau_j}L_{\tau_i}
  +\iota_{\tau_j}\iota_{\tau_i}d
=\iota_{[\tau_i,\tau_j]}
  -\iota_{\tau_i}\iota_{\tau_j}d;
\end{eqnarray*}
the Lie bracket $[\tau_i,\tau_j]=-\Omega(\tau_i,\tau_j)^P$;
equation (\ref{iA.beta});
and finally the definition of $H$ in (\ref{basic.3form}).
\end{proof}

{\it Remark.}
Because of the bijection in Lemma \ref{lemma.basic.3form},
we know that \emph{any} basic $3$-form $H$ that
satisfies $dH=\lambda^*(\Omega\wedge\Omega)$ determines
a vertex algebroid (\ref{CDOP.VAoidnew}) as per Lemma
\ref{lemma.CDOP.VAoidnew}, and hence equivalently
a principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.

\begin{corollary} \label{cor.fund.normal.comm}
In the vertex algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ we have
the following normal-ordered expansions and commutation
relations
\footnote{
The author found it interesting (and comforting)
to verify directly all Jacobi identities in sight.
}
\begin{align*}
&(fA^P)_n
=\sum_{k\geq 0}f_{n-k}A^P_k
+\sum_{k<0}A^P_k f_{n-k},\qquad
(f\tau_i)_n
=\sum_{k\geq 0}f_{n-k}\tau_{i,k}
+\sum_{k<0}\tau_{i,k}f_{n-k} \\
&[A^P_n,B^P_m]=[A,B]^P_{n+m}+n\lambda(A,B)\delta_{n+m,0},
\qquad
[A^P_n,\tau_{i,m}]=\rho(A)_{ji}\tau_{j,n+m}
  \phantom{\rule[-0.05in]{.1pt}{.1pt}} \\
&\textstyle
[\tau_{i,n},\tau_{j,m}]
=-\Omega(\tau_i,\tau_j)^P_{n+m}
+(n-m)(\mathrm{Ric}_{ij})_{n+m}
+\Big(\frac{1}{2}\iota_{\tau_i}\iota_{\tau_j}H
  +\lambda^*(\Omega(\tau_i,\tau_j),\Theta)\Big)_{n+m}
\end{align*}
for $f\in C^\infty(P)$, $A,B\in\mathfrak{g}$, $i,j=1,\ldots,d$
and $n,m\in\mathbb{Z}$.
\end{corollary}

\begin{proof}
This follows immediately from Lemma \ref{lemma.CDOP.VAoidnew}
and Definition \ref{VAoid.VA}.
\end{proof}

\begin{lemma} \label{lemma.CDOP.conformal}
The vertex algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ has a conformal
vector
\begin{eqnarray*}
\nu=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
  -\frac{1}{2}\lambda^*(\Theta_{-1}\Theta)
\end{eqnarray*}
of central charge $2\dim P$.
Moreover, we have $\nu_1 A^P=\nu_1 \tau_i=0$ for
$A\in\mathfrak{g}$ and $i=1,\ldots,d$.
\end{lemma}

\begin{proof}
By construction the isomorphism from $\mc{D}^{\mathrm{ch}}(P)$ to
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ (see {\it Preparation} before
Lemma \ref{lemma.CDOP.VAoidnew}) sends the conformal
vector $\nu'$ in (\ref{CDOP.conformal}) to
\begin{align*}
\nu
&\textstyle
=(t^P_a-h_{t_a})_{-1}\Theta^a
  +(\tau_i-\frac{1}{2}\iota_{\tau_i}\beta)_{-1}\tau^i \\
&\textstyle
=t^P_{a,-1}\Theta^a
  +\tau_{i,-1}\tau^i
  -(h_{t_a})_{-1}\Theta^a
  -\frac{1}{2}(\iota_{\tau_i}\beta)_{-1}\tau^i \\
&\textstyle
=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
  -\Big(h_{t_a}(t_b^P)\Theta^b
  +h_{t_a}(\tau_i)\tau^i\Big)_{-1}\Theta^a
  -\frac{1}{2}\Big(\beta(\tau_i,t^P_a)\Theta^a
  +\beta(\tau_i,\tau_j)\tau^j\Big)_{-1}\tau^i \\
&\textstyle
=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
  -\frac{1}{2}\lambda^*(t_a,t_b)\Theta^b_{-1}\Theta^a
  -h_{t_a}(\tau_i)\tau^i_{-1}\Theta^a
  +h_{t_a}(\tau_i)\tau^i_{-1}\Theta^a+0 \\
&\textstyle
=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
  -\frac{1}{2}\lambda^*(\Theta_{-1}\Theta)
\end{align*}
where we have used, in order:
the component of (\ref{CDOP.h}) in
$(\textrm{Sym}^2\mathfrak{g}^\vee\otimes C^\infty(P))^G$;
the symmetry $\alpha_{-1}\alpha'=\alpha'_{-1}\alpha$
for $1$-forms $\alpha,\alpha'$;
and the definition of $\beta$ in (\ref{beta}).
The said isomorphism also sends $\nu'_1(A^P+h_A)$ to
$\nu_1 A^P$ and $\nu'_1(\tau_i+\iota_{\tau_i}\beta)$ to
$\nu_1\tau_i$.
This proves our claims in view of Lemma
\ref{lemma.CDOP.VAoid} and (\ref{CDO.L1}).
\end{proof}

The following summarizes all our discussions since
\S\ref{sec.prin.bdl2}.

\begin{theorem} \label{thm.CDOP}
Suppose we have the following data:
a principal $G$-bundle $\pi:P\rightarrow M$,
a representation $\rho:G\rightarrow SO(\mathbb{R}^d)$
and an isomorphism $P\times_\rho\mathbb{R}^d\cong TM$;
a connection $\Theta$ on $\pi$ that induces
the Levi-Civita connection on $TM$;
an invariant symmetric bilinear form $\lambda$ on
$\mathfrak{g}$;
and a basic $3$-form $H$ on $P$ satisfying
$dH=\lambda^*(\Omega\wedge\Omega)$, where
$\Omega$ is the curvature of $\Theta$ and
$\lambda^*=\lambda+\lambda_{\t{\sf ad}}+\lambda_\rho$.

Given these data, there is an associated vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ with generating fields
\begin{eqnarray*}
\t{\sf Y}(f,x)\textrm{ for }f\in C^\infty(P),\qquad
\t{\sf Y}(A^P,x)\textrm{ for }A\in\mathfrak{g},\qquad
\t{\sf Y}(\tau_i,x)\textrm{ for }i=1,\ldots,d
\end{eqnarray*}
whose weights are $0$, $1$ and $1$ respectively.
(For the notations see \S\ref{sec.prin.bdl} and
\S\ref{sec.prin.bdl2}.)
Notice that the vacuum is $1\in C^\infty(P)$.
The various OPEs with $\t{\sf Y}(f,x)$ have the following
leading terms
\begin{align*}
&\t{\sf Y}(g,x')\t{\sf Y}(f,x)
=\t{\sf Y}(fg,x)+\t{\sf Y}(fdg,x)(x'-x)+O((x'-x)^2),\quad
g\in C^\infty(P) \\
&\t{\sf Y}(A^P,x')\t{\sf Y}(f,x)
=\frac{\t{\sf Y}(A^P f,x)}{x'-x}+\t{\sf Y}(fA^P,x)+O(x'-x) \\
&\t{\sf Y}(\tau_i,x')\t{\sf Y}(f,x)
=\frac{\t{\sf Y}(\tau_i f,x)}{x'-x}+\t{\sf Y}(f\tau_i,x)+O(x'-x)
\end{align*}
where in fact all the second terms are meant to be taken
(plus linearity) as the definition of fields associated
to arbitrary $1$-forms and vector fields on $P$.
The OPEs between the other generating fields have the
following singular parts
\begin{align*}
&\t{\sf Y}(A^P,x')\t{\sf Y}(B^P,x)\sim
  \frac{\lambda(A,B)}{(x'-x)^2}
  +\frac{\t{\sf Y}([A,B]^P,x)}{x'-x},\qquad
\t{\sf Y}(A^P,x')\t{\sf Y}(\tau_i,x)\sim
  \frac{\rho(A)_{ji}\t{\sf Y}(\tau_j,x)}{x'-x} \\
&\t{\sf Y}(\tau_i,x')\t{\sf Y}(\tau_j,x)\sim
  \frac{2\t{\sf Y}\big(\mathrm{Ric}_{ij},\frac{x+x'}{2}\big)}{(x'-x)^2}
  -\frac{\t{\sf Y}\big(\Omega(\tau_i,\tau_j)^P,x\big)}{x'-x}
  +\frac{\frac{1}{2}\t{\sf Y}\big(\iota_{\tau_i}\iota_{\tau_j}H,x\big)
    +\t{\sf Y}\big(\lambda^*\big(\Omega(\tau_i,\tau_j),
    \Theta\big),x\big)}{x'-x}
\end{align*}
where $\mathrm{Ric}_{ij}=\rho(\Omega(\tau_i,\tau_k))_{jk}$.
Clearly the fields $\t{\sf Y}(A^P,x)$ for $A\in\mathfrak{g}$ represent
an inner $(\hat{\mathfrak{g}}_\lambda,G)$-action
(see Definition \ref{formalLG.action}).
Moreover, there is a Virasoro field of central charge
$2\dim P$ given by
\begin{eqnarray*}
:\t{\sf Y}(t^P_a,x)\t{\sf Y}(\Theta^a,x):
+:\t{\sf Y}(\tau_i,x)\t{\sf Y}(\tau^i,x):
-\frac{1}{2}\,\lambda^*\big(\t{\sf Y}(\Theta,x),\t{\sf Y}(\Theta,x)\big)
\end{eqnarray*}
with respect to which all generating fields are primary.
(For the notations see \S\ref{sec.prin.bdl2}.)
$\qedsymbol$
\end{theorem}

{\it Remark.}
For convenience (and lack of imagination), we will
refer to $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ as
a {\bf principal frame $(\hat{\mathfrak{g}}_\lambda,G)$-algebra}
or, if we do not wish to specify $\lambda$,
a {\bf principal frame $(\hat{\mathfrak{g}},G)$-algebra}.
Even though this vertex algebra arises here as
a particular algebra of CDOs, it should really be regarded
as a more fundamental object for both conceptual and
aesthetic reasons.
In \S\ref{CDOP.CDOM} we will see that any algebra of CDOs
can be recovered in a natural way from a principal frame
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra with $\lambda=-\lambda_{\t{\sf ad}}$;
and this is only a special case of a more general
construction associated to principal
$(\hat{\mathfrak{g}}_\lambda,G)$-algebras with different $\lambda$
(to be introduced in \S\ref{sec.ass}).
Also noteworthy is that, compared to an arbitrary algebra
of CDOs, a principal frame $(\hat{\mathfrak{g}},G)$-algebra
requires a smaller set of generating fields and possesses
arguably more appealing OPEs and Virasoro field.
\footnote{
For generating fields, $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ requires only
a finite number of canonical vector fields (that constitute
a framing of $TP$), as opposed to all vector fields.
For OPEs, compare the vertex algebroids in
Lemma \ref{lemma.CDOP.VAoidnew} and
Theorem \ref{thm.globalCDO}a.
For Virasoro fields, compare the conformal vectors in
Lemma \ref{lemma.CDOP.conformal} and
Theorem \ref{thm.globalCDO}b.
}
For these reasons, it seems desirable to find a more direct
construction of principal frame $(\hat{\mathfrak{g}},G)$-algebras,
perhaps in a future paper.

\begin{subsec} \label{sec.invCDO.can}
{\bf The invariant subalgebra.}
This discussion is a more detailed version of
\S\ref{sec.invCDO} specifically for a principal frame
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
Consider the centralizer subalgebra
\begin{eqnarray*}
\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}
:=C\big(\mc{D}^{\mathrm{ch}}_{\Theta,H}(P),V_\lambda(\mathfrak{g})\big).
\end{eqnarray*}
The weight-zero component is again $C^\infty(P)^G$.
Let us describe the weight-one component below.

Consider an arbitrary element of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}_1$: some
$\alpha+\mathscr{X}+\mathscr{Y}\in\Omega^1(P)\oplus\mc{T}_h(P)\oplus\mc{T}_v(P)$
that is annihilated by $A^P_0$ and $A^P_1$ for $A\in\mathfrak{g}$.
Suppose $t_1,t_2,\cdots$ are a basis of $\mathfrak{g}$ and
$t^1,t^2,\cdots$ the dual basis of $\mathfrak{g}^\vee$.
Let us write $\mathscr{X}=\mathscr{X}^i\tau_i$ and
$\mathscr{Y}=\mathscr{Y}^a t^P_a$.
By Lemma \ref{lemma.CDOP.VAoidnew} and Definition
\ref{VAoid}, both $\{A^P,\mathscr{X}^i\tau_i\}_\Omega$ and
$\{A^P,\mathscr{Y}^a t^P_a\}_\Omega$ vanish;
then by (\ref{freeVA.comm}) the first condition on
$\alpha+\mathscr{X}+\mathscr{Y}$ reads
\begin{align*}
&A^P_0(\alpha+\mathscr{X}+\mathscr{Y})
=L_{A^P}\alpha+[A^P,\mathscr{X}]+[A^P,\mathscr{Y}]=0
\qquad\textrm{for }A\in\mathfrak{g}.
\end{align*}
Hence $\alpha$, $\mathscr{X}$, $\mathscr{Y}$ are
all $G$-invariant.
Notice that the $G$-invariance of $\mathscr{Y}$ means
$A^P\mathscr{Y}^a=-t^a([A,t_b])\mathscr{Y}^b$.
By Lemma \ref{lemma.CDOP.VAoidnew} and Definition
\ref{VAoid} again, we find that
$\{A^P,\mathscr{X}^i\tau_i\}=\rho(A)_{ji}\tau_j\mathscr{X}^i$ and
\begin{align*}
\{A^P,\mathscr{Y}^a t^P_a\}
&=\lambda(A,t_a)\mathscr{Y}^a+[A,t_a]^P\mathscr{Y}^a \\
&=\lambda(A,t_a)\mathscr{Y}^a-t^a([[A,t_a],t_b])\mathscr{Y}^b \\
&=(\lambda+\lambda_{\t{\sf ad}})(A,\mathscr{Y});
\end{align*}
then by (\ref{freeVA.comm}) the second condition on
$\alpha+\mathscr{X}+\mathscr{Y}$ reads
\begin{align*}
A^P_1(\alpha+\mathscr{X}+\mathscr{Y})
=\alpha(A^P)+\rho(A)_{ij}\tau_i\mathscr{X}^j
  +(\lambda+\lambda_{\t{\sf ad}})(A,\mathscr{Y})
=0\qquad\textrm{for }A\in\mathfrak{g}.
\end{align*}
It follows from the two conditions that the $1$-form
$\alpha+(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}
+(\lambda+\lambda_{\t{\sf ad}})(\Theta,\mathscr{Y})$ is $G$-invariant
and horizontal (i.e.~basic).
Therefore $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}_1$ is
the direct sum of the following subspaces:
\begin{align} \label{invCDOP.1}
\pi^*\Omega^1(M),\quad
\big\{\mathscr{X}-(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}:
  \mathscr{X}\in\mc{T}_h(P)^G\big\},\quad
\big\{\mathscr{Y}-(\lambda+\lambda_{\t{\sf ad}})(\Theta,\mathscr{Y}):
  \mathscr{Y}\in\mc{T}_v(P)^G\big\}.
\end{align}
This provides in our current setting an explicit
description of the short exact sequence (\ref{invCDO.1})
together with a splitting.
\end{subsec}

\newpage
\setcounter{equation}{0}
\section{Associated Modules and Algebras over CDOs}
\label{sec.ass}

Consider an algebra of CDOs $\mc{D}^{\mathrm{ch}}(P)$ on the total space
of a principal bundle $P\rightarrow M$, equipped with
a projective ``formal loop group action''.
This section introduces a construction of modules over
the invariant subalgebra of $\mc{D}^{\mathrm{ch}}(P)$ using semi-infinite
cohomology;
it is very much analogous to the construction of associated
vector bundles of $P\rightarrow M$.
In fact, the said modules should have an interpretation
as the spaces of sections of certain vector bundles over
the hypothetical ``formal loop space'' of $M$.
Examples will be studied in \S\ref{CDOP.CDOM} and
\S\ref{sec.FL.spinor}.

The semi-infinite cohomology of a centrally extended loop
algebra can be defined using the Feigin complex, which
is a vertex algebraic analogue of the Chevalley-Eilenberg
complex, but with some important differences.
For more about semi-infinite cohomology, see
e.g.~\cite{Feigin,Voronov,BD}.

\begin{subsec} \label{sec.CE}
{\bf The Chevalley-Eilenberg complex.}
Given a Lie algebra $\mathfrak{g}$, consider the supermanifold
$\Pi\mathfrak{g}$.
Let $t_1,t_2,\ldots$ be a basis of $\mathfrak{g}$;
$t^1,t^2,\ldots$ the dual basis of $\mathfrak{g}^\vee$; and
$(\phi^1,\phi^2,\ldots)$ the corresponding coordinates of
$\Pi\mathfrak{g}$.
By definition $\mathcal{O}(\Pi\mathfrak{g})=\wedge^*\mathfrak{g}^\vee$ and
$\mc{T}(\Pi\mathfrak{g})$ consists of the derivations on
$\wedge^*\mathfrak{g}^\vee$.
Consider
\begin{eqnarray*}
J,\,q\in\mc{T}(\Pi\mathfrak{g}),\qquad
\theta\in\mathcal{O}(\Pi\mathfrak{g})\otimes\mathfrak{g}
\end{eqnarray*}
corresponding respectively to the exterior degree on
$\wedge^*\mathfrak{g}^\vee$, the Chevalley-Eilenberg differential
on $\wedge^*\mathfrak{g}^\vee$, and the Maurer-Cartan form on
$\mathfrak{g}$;
they can be written in coordinates as follows
\begin{eqnarray} \label{JQ.coor}
J=\phi^a\frac{\partial}{\partial\phi^a}\,,\qquad
q=-\frac{1}{2}\,t^a([t_b,t_c])\phi^b\phi^c\frac{\partial}{\partial\phi^a}\,,
  \qquad
\theta=\phi^a\otimes t_a.
\end{eqnarray}
Notice that $J=J\otimes 1$,\; $q=q\otimes 1$ and $\theta$
satisfy
\begin{eqnarray} \label{JQ.relations}
[J,q]=q,\qquad
J\theta=\theta,\qquad
[q,q]=0,\qquad
q\theta+\frac{1}{2}\,[\theta,\theta]=0.
\end{eqnarray}

Let $W$ be a $\mathfrak{g}$-module.
If $J,q,\theta$ are regarded as operators on
$\mathcal{O}(\Pi\mathfrak{g})\otimes W$ and $Q=q+\theta$,
then by (\ref{JQ.relations}) they satisfy
$[J,Q]=Q$ and $[Q,Q]=2Q^2=0$.
The Chevalley-Eilenberg complex of $\mathfrak{g}$ with
coefficients in $W$ can be written as
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{g})\otimes W,\,J,\,Q\Big)
\end{eqnarray*}
where $J$ is the grading operator and $Q$ is
the differential.
\end{subsec}

\begin{subsec} \label{sec.Feigin}
{\bf The Feigin complex.}
Consider the algebra of CDOs $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$, which is
a fermionic version of \S\ref{algCDO};
for details see \cite{myCDO}.
Given an invariant symmetric bilinear form $\lambda$ on
$\mathfrak{g}$, let $\hat{\mathfrak{g}}_\lambda$ be the centrally
extended loop algebra described in \S\ref{sec.Gmfld} and
$V_\lambda(\mathfrak{g})$ the vertex algebra in Example
\ref{VAoid.Lie}.
Now regard
\begin{eqnarray*}
J,q,\theta\textrm{ as elements of }
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes V_\lambda(\mathfrak{g})
\textrm{ of weight }1.
\end{eqnarray*}
The following computations in
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes V_\lambda(\mathfrak{g})$ make use of
(\ref{freeVA.comm}), (\ref{JQ.coor}) and
the super version of (\ref{CDO.VAoid456}):
\begin{align*}
&J_0 q
=[J,q]+\{J,q\}_\Omega=q+0=q \\
&J_0\theta
=(J_0\otimes 1)(\phi^a\otimes t_a)
=J\phi^a\otimes t_a=\phi^a\otimes t_a=\theta \\
&q_0 q
=[q,q]+\{q,q\}_\Omega
=0-t^a([t_b,t_c])\,t^b([t_a,t_d])\,\phi^d d\phi^c
=-\lambda_{\t{\sf ad}}(t_c,t_d)\,\phi^d d\phi^c \\
&\hspace{-0.1in}
\left.\begin{array}{l}
q_0\theta=(q_0\otimes 1)(\phi^a\otimes t_a)
  \rule[-0.1in]{0in}{0in} \\
\theta_0 q=(\phi^a_1\otimes t_{a,-1})(q\otimes 1)
\end{array}\right\}
=q\phi^a\otimes t_a
=-\frac{1}{2}\phi^b\phi^c\otimes[t_b,t_c] \\
&\theta_0\theta
=(\phi^a_0\otimes t_{a,0}+\phi^a_{-1}\otimes t_{a,1})
  (\phi^b\otimes t_b)
=\phi^a\phi^b\otimes[t_a,t_b]
-\lambda(t_a,t_b)\,\phi^b d\phi^a
\end{align*}
where $\lambda_{\t{\sf ad}}$ denotes the Killing form.
Hence $J$ and $Q=q+\theta$ satisfy
\begin{eqnarray*}
\begin{array}{rcc}
&J_0 Q=Q,&
Q_0 Q=-(\lambda_{\t{\sf ad}}+\lambda)(t_a,t_b)\,\phi^a d\phi^b
\phantom{\Big(} \\
\Rightarrow\quad&
[J_0,Q_0]=Q_0,&
\quad
[Q_0,Q_0]
  =(\lambda_{\t{\sf ad}}+\lambda)(t_a,t_b)\cdot
  \sum_{n\in\mathbb{Z}}n\phi^a_{-n}\phi^b_n
\end{array}
\end{eqnarray*}
where we have used (\ref{freeVA.comm}) and
(\ref{freeVA.NOP}).
In particular, $Q_0^2=0$ if and only if
$\lambda=-\lambda_{\t{\sf ad}}$.

Let $W$ be a $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-module.
Notice that $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ is a module over
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes V_{-\lambda_{\t{\sf ad}}}(\mathfrak{g})$.
The Feigin complex of $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$ with
coefficients in $W$ is
\begin{eqnarray*}
\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W,\,J_0,\,Q_0\Big)
\end{eqnarray*}
where $J_0$ is the grading operator and
$Q_0$ is the differential.
\end{subsec}

\begin{subsec} \label{sec.semiinf}
{\bf Semi-infinite cohomology.}
For any $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-module $W$ as above, let
\begin{eqnarray} \label{semiinf}
H^{\frac{\infty}{2}+*}(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},W)
:=H^*\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W,\,Q_0\Big).
\end{eqnarray}
Notice that if $W$ is a vertex algebra such that the
$\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is inner,
i.e. induced by a map of vertex algebras
$V_{-\lambda_{\t{\sf ad}}}(\mathfrak{g})\rightarrow W$,
then (\ref{semiinf}) has the structure of
a $\mathbb{Z}$-graded vertex algebra.
(In this case, $J$ and $Q$ will also denote their own
images in $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$.)
\end{subsec}

{\it Remarks.}
(i) Unlike Lie algebra cohomology, the grading on
semi-infinite cohomology is neither bounded above nor below.
More precisely, the restriction of $J_0$ to
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})_k\otimes W$ takes values between $-k$
and $\dim\mathfrak{g}+k$.
(ii) This is only a special case of semi-infinite cohomology.
For a modern exposition of the notion in the more general
setting of Tate Lie algebras, see \cite{BD}.

\begin{lemma}  \label{lemma.semiinf.ops}
Let $W$ be a $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-module.

(a) The $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-invariant operators on
$W$ induce grading-preserving operators on (\ref{semiinf}).
Moreover, if $W$ is a vertex algebra such that its
$\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is inner, then there
is a map of vertex algebras from the centralizer subalgebra
$W^{\hat{\mathfrak{g}}}=C\big(W,V_{-\lambda_{\t{\sf ad}}}(\mathfrak{g})\big)$
\cite{Kac,FB-Z} to the zeroth gradation of  (\ref{semiinf}).

(b) If there is a Virasoro action on $W$ of central charge $c$
such that the $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is primary,
then it induces a grading-preserving Virasoro action on
(\ref{semiinf}) of central charge $c-2\dim\mathfrak{g}$.
Moreover, if $W$ is in fact a conformal vertex algebra such
that its $\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}$-action is inner and
primary, then (\ref{semiinf}) is also a conformal vertex
algebra with a conformal vector in the zeroth gradation.
\end{lemma}

\begin{proof}
(a) For the first statement, suppose $F\in\textrm{End}\,W$
satisfies $[A_n,F]=0$ for $A\in\mathfrak{g}$ and $n\in\mathbb{Z}$.
Since on $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ we have
\begin{eqnarray*}
[Q_0,1\otimes F]
=[q_0\otimes 1+\phi^a_{-n}\otimes t_{a,n},\,1\otimes F]
=\phi^a_{-n}\otimes[t_{a,n},F]
=0
\end{eqnarray*}
the operator $1\otimes F$ is well-defined on (\ref{semiinf}).
Clearly it preserves the grading.
For the second statement, let $u\in W^{\hat{\mathfrak{g}}}$,
i.e.~$u\in W$ with $A_n u=0$ for $A\in\mathfrak{g}$ and $n\geq 0$.
Since in $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ we have
\begin{eqnarray*}
Q_0(\mathbf{1}\otimes u)
=(q_0\otimes 1+\phi^a_{-n}\otimes t_{a,n})(\mathbf{1}\otimes u)
=\sum_{n\geq 0}\phi^a_{-n}\mathbf{1}\otimes t_{a,n}u
=0
\end{eqnarray*}
the element $\mathbf{1}\otimes u$ represents a class
$[\mathbf{1}\otimes u]$ in (\ref{semiinf}).
Clearly $u\mapsto[\mathbf{1}\otimes u]$ is a map of
vertex algebras and the image is contained in the zeroth
gradation.

(b) The graded vertex algebra $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$ has a conformal
vector $\nu^{\Pi\mathfrak{g}}$ of central charge $-2\dim\mathfrak{g}$
(see \S\ref{sec.ass.C});
denote its Virasoro operators by $L^{\Pi\mathfrak{g}}_n$, $n\in\mathbb{Z}$.
Since $q$ and $\phi^a$ are primary, we have
\begin{eqnarray*}
[L^{\Pi\mathfrak{g}}_n,q_0]=0,\qquad
[L^{\Pi\mathfrak{g}}_n,\phi^a_m]=-(n+m)\phi^a_{n+m},\qquad
n,m\in\mathbb{Z}.
\end{eqnarray*}
Since $\nu^{\Pi\mathfrak{g}}$ belongs to the zeroth gradation,
all $L^{\Pi\mathfrak{g}}_n$ preserve the grading.

For the first statement, suppose $L^W_n\in\textrm{End}\,W$ for
$n\in\mathbb{Z}$ define a Virasoro action of central charge $c$
and satisfy $[L^W_n,A_m]=-mA_{n+m}$ for $A\in\mathfrak{g}$ and
$n,m\in\mathbb{Z}$.
Then $L^{\Pi\mathfrak{g}}_n\otimes 1+1\otimes L^W_n$ for
$n\in\mathbb{Z}$ define a grading-preserving Virasoro action on
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$ of central charge $c-2\dim\mathfrak{g}$.
Since
\begin{align*}
[Q_0,\,L^{\Pi\mathfrak{g}}_n\otimes 1+1\otimes L^W_n]
&=[q_0,L^{\Pi\mathfrak{g}}_n]\otimes 1
  +[\phi^a_{-m},L^{\Pi\mathfrak{g}}_n]\otimes t_{a,m}
  +\phi^a_{-m}\otimes[t_{a,m},L^W_n] \\
&=0+(n-m)\phi^a_{n-m}\otimes t_{a,m}
  +m\phi^a_{-m}\otimes t_{a,n+m} \\
&=0
\end{align*}
the said Virasoro action is well-defined on
(\ref{semiinf}) as well.

For the second statement, suppose $\nu^W\in W$ is
a conformal vector whose Virasoro operators $L^W_n$,
$n\in\mathbb{Z}$, satisfy $[L^W_n,A_m]=-mA_{n+m}$ for
$A\in\mathfrak{g}$ and $n,m\in\mathbb{Z}$.
Then $\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^W$
is a conformal vector of $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes W$
belonging to the zeroth gradation.
Since we have
\begin{align*}
Q_0(\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}
  +\mathbf{1}\otimes\nu^W)
&=Q_0(L^{\Pi\mathfrak{g}}_{-2}\otimes 1
  +1\otimes L^W_{-2})(\mathbf{1}\otimes\mathbf{1}) \\
&=\big[Q_0,\,L^{\Pi\mathfrak{g}}_{-2}\otimes 1
  +1\otimes L^W_{-2}\big](\mathbf{1}\otimes\mathbf{1}) \\
&=0
\end{align*}
by the computation above,
$\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^W$
also represents a conformal vector of (\ref{semiinf}).
\end{proof}

The main purpose of this section is to introduce
the following construction.
As before, $G$ is a compact connected Lie group and
$\mathfrak{g}$ its Lie algebra.
Recall Definition \ref{formalLG.action} and
Corollary \ref{cor.formalLG.P}.

\begin{defn} \label{defn.ass}
Let $\pi:P\rightarrow M$ be a smooth principal $G$-bundle
and $\lambda,\lambda'\in(\textrm{Sym}^2\mathfrak{g}^\vee)^G$.
Assume that $\lambda+\lambda'=-\lambda_{\t{\sf ad}}$ and
$(\lambda+\lambda_{\t{\sf ad}})(P)=8\pi^2 p_1(M)$.
For any principal $(\hat{\mathfrak{g}}_\lambda,G)$-algebra $\mc{D}^{\mathrm{ch}}(P)$
and positive-energy $(\hat{\mathfrak{g}}_{\lambda'},G)$-module $W$,
we define
\begin{align*}
\Gamma^{\mathrm{ch}}(\pi,W)
:=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
  \mc{D}^{\mathrm{ch}}(P)\otimes W\big).
\end{align*}
Notice that if $W$ is a vertex algebra such that the
$(\hat{\mathfrak{g}}_{\lambda'},G)$-action is inner, then
$\Gamma^{\mathrm{ch}}(\pi,W)$ also has the structure of a vertex algebra
(see \S\ref{sec.semiinf}).
\end{defn}

{\it Remarks.}
(i) The positive-energy condition means that there
is a diagonalizable operator $L^W_0$ on $W$ such
that $[L^W_0,A_n]=-nA_n$ for $A\in\mathfrak{g}$, $n\in\mathbb{Z}$,
and the eigenvalues of $L^W_0$ are bounded below.
Whenever $W$ admits a Virasoro action, it will be
understood that $L^W_0$ coincides with the Virasoro
operator that is usually given the same notation.
For consistency, the eigenvalues of $L^W_0$ will
also be called weights.
Let $L^{\Pi\mathfrak{g}}_0$ (resp.~$L^P_0$) be the weight
operator on $\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$ (resp.~$\mc{D}^{\mathrm{ch}}(P)$).
Notice that $L^{\Pi\mathfrak{g}}_0+L^P_0+L^W_0$ commutes
with $Q_0$ (see the proof of Lemma
\ref{lemma.semiinf.ops}b), so that it induces
a weight decomposition of $\Gamma^{\mathrm{ch}}(\pi,W)$.

(ii) The above definition still makes sense without
the integrability and/or the positive-energy conditions.
On the other hand, under these two conditions, we can
say that the lowest-weight component $W_0$ of $W$ is
a $G$-representation and then the lowest-weight component of
$\Gamma^{\mathrm{ch}}(\pi,W)$ is $(C^\infty(P)\otimes W_0)^G$,
i.e. the space of sections of the associated vector
bundle $P\times_G W_0\rightarrow M$.

\begin{lemma} \label{lemma.ass}
Consider again the data in Definition \ref{defn.ass}.

(a) For any $W$ as described, $\Gamma^{\mathrm{ch}}(\pi,W)$ is a module
over $\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}$.
Moreover, if $W$ is a vertex algebra such that
its $(\hat{\mathfrak{g}}_{\lambda'},G)$-action is inner,
then there is a map of vertex algebras
$\mc{D}^{\mathrm{ch}}(P)^{\hat{\mathfrak{g}}}\rightarrow\Gamma^{\mathrm{ch}}(\pi,W)$.

(b) If there is a Virasoro action on $W$ of central charge
$c$ such that the $(\hat{\mathfrak{g}}_{\lambda'},G)$-action
is primary, then it induces a Virasoro action on $\Gamma^{\mathrm{ch}}(\pi,W)$
of central charge $c+2\dim M$.
Moreover, if $W$ is in fact a conformal vertex algebra such
that its $(\hat{\mathfrak{g}}_{\lambda'},G)$-action is inner and
primary, then $\Gamma^{\mathrm{ch}}(\pi,W)$ is also a conformal vertex
algebra.
\end{lemma}

\begin{proof}
Recall \S\ref{sec.invCDO}.
This follows from Lemma \ref{lemma.semiinf.ops}, together
with the fact that $\mc{D}^{\mathrm{ch}}(P)$ is a conformal vertex algebra
with central charge $2\dim P$ and a primary inner
$(\hat{\mathfrak{g}}_\lambda,G)$-action (see Theorem
\ref{thm.formalLG.action}).
\end{proof}

\newpage
\setcounter{equation}{0}
\section{Example: Recovering Algebras of CDOs}
\label{CDOP.CDOM}

Suppose $P\rightarrow M$ is a principal frame bundle and
$\mc{D}^{\mathrm{ch}}(P)$ is an algebra of CDOs on $P$ with a suitable
``formal loop group action''.
In this section, we analyze the zeroth semi-infinite
cohomology of $\mc{D}^{\mathrm{ch}}(P)$ and in the end identify it
as an algebra of CDOs on $M$.
This provides a description of an arbitrary algebra
of CDOs that is conceptually more satisfying than
the original one given in \S\ref{sec.review}.

\begin{subsec} \label{sec.ass.C}
{\bf Goal:~a new description of algebras of CDOs.}
Consider the special case of Definition \ref{defn.ass}
associated to a principal frame
$(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},G)$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ and the trivial
$(\hat{\mathfrak{g}}_0,G)$-module $\mathbb{C}$:
\begin{eqnarray} \label{ass.C}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
  \mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\big)
=H^0\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P),
  Q_0\Big).
\end{eqnarray}
By Lemma \ref{lemma.ass}, $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is a conformal
vertex algebra of central charge $2\dim M$.
For convenience, let us elaborate on all the data
involved and recall some relevant notations.
\begin{itemize}
\item[$\centerdot$]
Let $(t_1,t_2,\cdots)$ be a basis of $\mathfrak{g}$,
$(t^1,t^2,\cdots)$ the dual basis of $\mathfrak{g}^\vee$,
$(\phi^1,\phi^2,\cdots)$ the corresponding coordinates
of the supermanifold $\Pi\mathfrak{g}$, and
$(\partial_1,\partial_2,\cdots)$ their coordinate vector fields.
\vspace{-0.05in}
\item[$\centerdot$]
For the detailed definition of the vertex superalgebra
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})$, see \cite{Kac} or \cite{myCDO}.
Let us mention that, as a fermionic analogue of
\S\ref{algCDO}, it is generated by such elements
as $\phi^1,\phi^2,\cdots$ and $\partial_1,\partial_2,\cdots$,
and has a conformal vector
\begin{eqnarray} \label{oddg.conformal}
\nu^{\Pi\mathfrak{g}}=-\partial_{a,-1}d\phi^a
\end{eqnarray}
of central charge $-2\dim\mathfrak{g}$.
\vspace{-0.05in}
\item[$\centerdot$]
For the detailed definition of the vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$, see Theorem \ref{thm.CDOP} with
$\lambda=-\lambda_{\t{\sf ad}}$ in mind.
Let us mention that it is defined using the following
data:
a smooth principal $G$-bundle $\pi:P\rightarrow M$
together with a representation
$\rho:G\rightarrow SO(\mathbb{R}^d)$ and an isomorphism
$P\times_\rho\mathbb{R}^d\cong TM$;
a connection $\Theta=\Theta^a\otimes t_a$ on $\pi$
that induces the Levi-Civita connection on $TM$;
and a basic $3$-form $H$ on $P$ that satisfies
$dH=\lambda_\rho(\Omega\wedge\Omega)$, where
$\Omega=d\Theta+\frac{1}{2}[\Theta\wedge\Theta]$.
Also, it has a conformal vector
\begin{eqnarray*}
\nu^P=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
  -\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)
\end{eqnarray*}
of central charge $2\dim P$.
(As before, $\rho$ also denotes the induced map
of Lie algebras $\mathfrak{g}\rightarrow\mathfrak{so}_d$
and $\lambda_\rho$ the invariant symmetric
bilinear form on $\mathfrak{g}$ given by
$\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
For the meaning of the other notations, see
\S\ref{sec.prin.bdl2}.)
\vspace{-0.05in}
\item[$\centerdot$]
For details of the Feigin complex appearing in (\ref{ass.C}),
see \S\ref{sec.Feigin}.
Let us mention that the grading operator $J_0$ is
determined by $J_0(\phi^a\otimes u)=\phi^a\otimes u$ and
$J_0(\partial_a\otimes u)=-\partial_a\otimes u$ for any $u$;
and the differential $Q_0=(q+\theta)_0$ is induced by
the elements
\begin{eqnarray} \label{Q.theta}
q=q\otimes\mathbf{1}
=-\frac{1}{2}\,t^a([t_b,t_c])\phi^b\phi^c\partial_a\otimes\mathbf{1},
\qquad
\theta=\phi^a\otimes t_a^P.
\end{eqnarray}
The conformal vector
$\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^P$
belongs to the zeroth gradation and is
$Q_0$-closed.
\vspace{-0.05in}
\end{itemize}
By the assumption on $\pi$, $M$ is an oriented
Riemannian manifold.
Let $\nabla$ be the Levi-Civita connection and $R$
the Riemannian curvature of $M$.
The assumption on $H$ can then be written as
$dH=\mathrm{Tr}\,(R\wedge R)$.
By Theorem \ref{thm.globalCDO}, the data $(\nabla,H)$
determine an algebra of CDOs $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$
with a conformal vector $\nu=\nu^0$ of central
charge $2\dim M$.
The goal of this section is to prove that
\begin{eqnarray*}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
  \mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\big)
\cong\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)
\end{eqnarray*}
as conformal vertex algebras.
In fact, we will analyze $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ without
using any prior knowledge of $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$,
and effectively rediscover the latter in the end.
\end{subsec}

Throughout this section we identify $\Omega^*(M)$ with
the basic subspace of $\Omega^*(P)$.

\begin{subsec} \label{sec.ass.C.0}
{\bf The component of weight zero.}
Consider the Feigin complex that appears in
(\ref{ass.C}):
\begin{eqnarray} \label{Feigin.C}
\Big(\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P),\,
Q_0\Big).
\end{eqnarray}
Since its weight-zero component is simply the
Chevalley-Eilenberg complex (see \S\ref{sec.CE})
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{g})\otimes C^\infty(P),\,
  Q_0\Big)
\end{eqnarray*}
the weight-zero component of (\ref{ass.C}) is
\begin{eqnarray} \label{ass.C.0}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})_0
=H^0\big(\mathfrak{g},C^\infty(P)\big)
=C^\infty(P)^G
=C^\infty(M).
\end{eqnarray}
Understanding the rest of $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ requires
more work, mostly because in higher weights the
Feigin complex (\ref{Feigin.C}) contains elements in
both positive and negative gradations.
\end{subsec}

\begin{lemma} \label{lemma.G}
Let $\gamma=\partial_a\otimes\Theta^a\in
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})_1\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$.
Then we have
\begin{eqnarray*}
Q_0\gamma=\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}
  +\mathbf{1}\otimes t^P_{a,-1}\Theta^a,\qquad
\gamma_0\gamma=0
\end{eqnarray*}
and therefore the anticommutators
\begin{eqnarray*}
\phantom{\Big(}
[Q_0,\gamma_0]=L^{\Pi\mathfrak{g}}_0\otimes 1
  +1\otimes(t^P_{a,-1}\Theta^a)_0,\qquad
[\gamma_0,\gamma_0]=2\gamma_0^2=0.
\end{eqnarray*}
\end{lemma}

\begin{proof}
Recall the element $Q=q+\theta$ from (\ref{Q.theta}).
Let us first write
\begin{eqnarray*}
Q_0\gamma
=\big(q_0\otimes 1+\phi^a_{-1}\otimes t^P_{a,1}
  +\phi^a_0\otimes t^P_{a,0}
  +\phi^a_1\otimes t^P_{a,-1}\big)(\partial_b\otimes\Theta^b).
\end{eqnarray*}
Then each of these terms is computed as follows:
\begin{align*}
&q_0\partial_b\otimes\Theta^b
=\big([q,\partial_b]+\{q,\partial_b\}_\Omega\big)\otimes\Theta^b
=-t^c([t_b,t_d])\phi^d\partial_c\otimes\Theta^b \\
&\phi^a_{-1}\partial_b\otimes t^P_{a,1}\Theta^b
=-\partial_{b,-1}d\phi^a\otimes\Theta^b(t^P_a)
=-\partial_{a,-1}d\phi^a\otimes\mathbf{1} \\
&\phi^a_0\partial_b\otimes t^P_{a,0}\Theta^b
=-\partial_{b,-1}\phi^a\otimes L_{t^P_a}\Theta^b
=-\phi^a\partial_b\otimes t^b([t_a,t_c])\Theta^c \\
&\phi^a_1\partial_b\otimes t^P_{a,-1}\Theta^b
=\mathbf{1}\otimes t^P_{a,-1}\Theta^a
\end{align*}
where we have used (\ref{freeVA.comm}), the super version
of (\ref{Weyl})--(\ref{CDO.VAoid456}) and
the $G$-invariance of $\Theta=\Theta^a\otimes t_a$.
In view of (\ref{oddg.conformal}), this proves
the claimed expression for $Q_0\gamma$.
On the other hand, we have
\begin{eqnarray*}
\gamma_0\gamma
=(\partial_{a,1}\otimes\Theta^a_{-1})(\partial_b\otimes\Theta^b)
=\partial_{a,1}\partial_b\otimes\Theta^a_{-1}\Theta^b
=0
\end{eqnarray*}
thanks to the super version of (\ref{Weyl}).
\end{proof}

\begin{lemma} \label{lemma.Feigin.C.1}
The weight-one component of the Feigin complex
(\ref{Feigin.C}) is quasi-isomorphic to
the following (Chevalley-Eilenberg) subcomplex
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{g})\otimes\mc{D}^{\mathrm{ch}}_h(P)_1,
  \,Q_0\Big)
\end{eqnarray*}
where $\mc{D}^{\mathrm{ch}}_h(P)_1\subset\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$ is
the direct sum of the subspaces
\begin{eqnarray*}
\Omega^1_h(P)
\quad\textrm{and}\quad
\Big\{\mathscr{X}-(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}:
  \mathscr{X}=\mathscr{X}^i\tau_i\in\mc{T}_h(P)\Big\}.
\end{eqnarray*}
\end{lemma}

\begin{proof}
First we compute the operator $(t^P_{a,-1}\Theta^a)_0$ on
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_k$, $k=0,1$.
Notice that (\ref{freeVA.comm})--(\ref{freeVA.NOP})
will be used repeatedly.
For $f\in C^\infty(P)$ and $\alpha\in\Omega^1(P)$, we have
\begin{align*}
&(t^P_{a,-1}\Theta^a)_0 f
=\Theta^a_0\,t^P_{a,0}f
=0 \\
&(t^P_{a,-1}\Theta^a)_0\alpha
=\big(\Theta^a_0\,t^P_{a,0}+\Theta^a_{-1}t^P_{a,1}\big)\alpha
=0+\alpha(t^P_a)\Theta^a
=\alpha_v
\end{align*}
where $\alpha_v$ means the vertical part of $\alpha$.
For $\mathscr{X}=\mathscr{X}^i\tau_i\in\mc{T}_h(P)$, we have
\begin{align*}
(t^P_{a,-1}\Theta^a)_0\mathscr{X}
&=\big(t^P_{a,-1}\Theta^a_1+\Theta^a_0\,t^P_{a,0}
  +\Theta^a_{-1}t^P_{a,1}\big)\mathscr{X} \\
&=0-\iota_{[t^P_a,\mathscr{X}]}d\Theta^a
  +\rho(t_a)_{ij}(\tau_i\mathscr{X}^j)\Theta^a \\
&=-\iota_{[t^P_a,\mathscr{X}]}\Omega^a
  +(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}
\end{align*}
using Lemma \ref{lemma.1form.0mode} and the computation
of $\{A^P,\mathscr{X}\}$ in \S\ref{sec.invCDO.can}.
For $\mathscr{Y}=\mathscr{Y}^a t^P_a\in\mc{T}_v(P)$, we have
\begin{align*}
(t^P_{a,-1}\Theta^a)_0\mathscr{Y}
&=\big(t^P_{a,-1}\Theta^a_1
  +\Theta^a_0\,t^P_{a,0}
  +\Theta^a_{-1}t^P_{a,1}\big)\mathscr{Y} \\
&=\Theta^a(\mathscr{Y})t^P_a
  -\iota_{[t^P_a,\mathscr{Y}]}d\Theta^a
  +\big([t_a,t_b]^P\mathscr{Y}^b-\lambda_{\t{\sf ad}}(t_a,\mathscr{Y})\big)
    \Theta^a \\
&=\mathscr{Y}
  -\Theta^b([t^P_a,\mathscr{Y}])\iota_{t^P_b}d\Theta^a
  +\Theta^b\big([[t_a,t_b]^P,\mathscr{Y}]\big)\Theta^a \\
&=\mathscr{Y}
  +\Theta^b([t^P_a,\mathscr{Y}])t^a([t_b,t_c])\Theta^c
  +t^c([t_a,t_b])\Theta^b([t^P_c,\mathscr{Y}])\Theta^a \\
&=\mathscr{Y}
\end{align*}
using Lemma \ref{lemma.CDOP.VAoidnew},
Lemma \ref{lemma.1form.0mode} and
the computation of $\{A^P,\mathscr{Y}\}$ in
\S\ref{sec.invCDO.can} (with $\lambda=-\lambda_{\t{\sf ad}}$ but
without the assumption that $\mathscr{Y}$ is $G$-invariant).

Now consider the null-homotopic operator on the Feigin
complex (\ref{Feigin.C}) that is defined by the
first expression below but is also equivalent to the
other expressions by Lemma \ref{lemma.G}:
\begin{align*}
e=\big[Q_0,\,\gamma_0 Q_0 \gamma_0\big]
=\big[Q_0,\gamma_0\big]^2
=\Big(L_0^{\Pi\mathfrak{g}}\otimes 1
  +1\otimes(t^P_{a,-1}\Theta^a)_0\Big)^2.
\end{align*}
Let $e_1$ denote its restriction to weight one.
By the calculations above, we have
\begin{eqnarray*}
\begin{array}{ll}
e_1(u_1\otimes f)=u_1\otimes f, &
u_1\in\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{g})_1,\; f\in C^\infty(P) \\
e_1(u_0\otimes\alpha)=u_0\otimes\alpha_v, &
u_0\in\mathcal{O}(\Pi\mathfrak{g}),\;\alpha\in\Omega^1(P) \\
e_1(u_0\otimes\mathscr{X})=u_0\otimes(\tau_i\mathscr{X}^j)
  \rho(\Theta)_{ij},\quad &
u_0\in\mathcal{O}(\Pi\mathfrak{g}),\;\mathscr{X}\in\mc{T}_h(P) \\
e_1(u_0\otimes\mathscr{Y})=u_0\otimes\mathscr{Y}, &
u_0\in\mathcal{O}(\Pi\mathfrak{g}),\;\mathscr{Y}\in\mc{T}_v(P)
\end{array}
\end{eqnarray*}
Notice that $e_1$ is an idempotent.
Therefore the image of $1-e_1$ is a quasi-isomorphic
subcomplex.
This is the subcomplex stated in the lemma.
\end{proof}

\begin{subsec} \label{sec.ass.C.1}
{\bf The component of weight one.}
By Lemma \ref{lemma.Feigin.C.1}, the weight-one component
of (\ref{ass.C}) is
\begin{align}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})_1
&\cong H^0\big(\mathfrak{g},\mc{D}^{\mathrm{ch}}_h(P)_1\big)
=\mc{D}^{\mathrm{ch}}_h(P)_1^G \nonumber \\
&=\Omega^1_h(P)^G\oplus
  \big\{\mathscr{X}-(\tau_i\mathscr{X}^j)\rho(\Theta)_{ij}:
  \mathscr{X}\in\mc{T}_h(P)^G\big\} \nonumber \\
&=\Omega^1(M)\oplus
  \big\{\widetilde X-(\tau_i\widetilde X^j)\rho(\Theta)_{ij}:
  X\in\mc{T}(M)\big\} \label{ass.C.1}
\end{align}
\footnote{
By Lemma \ref{lemma.ass}, there is a map of vertex algebras
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\hat{\mathfrak{g}}}\rightarrow\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$.
Comparing (\ref{invCDOP.1}) and (\ref{ass.C.1}), we see
that the weight-one component of the said map is surjective
and its kernel is $\mc{T}_v(P)^G$ (since $\lambda=-\lambda_{\t{\sf ad}}$).
}
Keep in mind that we identify $\Omega^*(M)$ with the basic
subspace of $\Omega^*(P)$, and denote the horizontal lift
of any $X\in\mc{T}(M)$ by $\widetilde X\in\mc{T}_h(P)^G$.
Notice that by (\ref{tau.brackets}) the $G$-invariance
of $\widetilde X=\widetilde X^i\tau_i$ means
\begin{align} \label{lift.inv}
A^P\widetilde X^i=-\rho(A)_{ij}\widetilde X^j\quad\textrm{for }A\in\mathfrak{g}.
\end{align}
Also notice that the operator
$\nabla X\in\Gamma(\textrm{End}\,TM)$ lifts horizontally to
\begin{align} \label{DX.lift}
\widetilde{\nabla X}
=\big(d\widetilde X^j+\rho(\Theta)_{jk}\widetilde X^k\big)\otimes\tau_j
=(\tau_i\widetilde X^j)\,\tau^i\otimes\tau_j
\end{align}
where the first equality simply expresses the relation
between $\nabla$ and $\Theta$, and the second equality
follows from (\ref{lift.inv}).
The unusual-looking term in (\ref{ass.C.1}) can be given
a global interpretation using (\ref{DX.lift}).
\end{subsec}

\begin{subsec} \label{sec.ass.C.VAoid}
{\bf The associated vertex algebroid.}
In view of (\ref{ass.C.0}) and (\ref{ass.C.1}),
the extended Lie algebroid associated to $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
is $(C^\infty(M),\Omega^1(M),\mc{T}(M))$ with the
usual structure maps (see \S\ref{VA.VAoid}).
More precisely, we are identifying each $X\in\mc{T}(M)$
with
\begin{align*}
\phantom{WWWW}
\ell X:=\,\textrm{class of }\widetilde X-(\tau_i\widetilde X^j)\rho(\Theta)_{ij}
\;\;\in\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})_1.
\end{align*}
\footnote{
For example, it follows from (\ref{horlift.nonint}) that
$[\ell X,\ell Y]=\ell[X,Y]$, i.e.~the Lie bracket in the
said extended Lie algebroid indeed agrees with the usual
Lie bracket on $\mc{T}(M)$.
}
Consider the vertex algebroid associated to
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ (see \S\ref{VA.VAoid} again):
\begin{eqnarray*}
\big(
  C^\infty(M),\Omega^1(M),\mc{T}(M),
  \bullet,\{\;\},\{\;\}_\Omega
\big).
\end{eqnarray*}
The explicit expressions of $\bullet,\{\;\},\{\;\}_\Omega$
may not seem very meaningful, but we will obtain
them below for the sole purpose of recovering the
corresponding known expressions for $\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$.
Let $V^f$ be the vertex algebra freely generated by
the above vertex algebroid, and
\begin{eqnarray*}
\Phi:V^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
\end{eqnarray*}
the resulting universal map (see \S\ref{VAoid.VA}).
By construction, $\Phi$ is an isomorphism in the two
lowest weights.
In fact, we will see that $\Phi$ is an isomorphism in
all weights, which means $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is freely
generated by its associated vertex algebroid.
\end{subsec}

\begin{prop} \label{prop.ass.C.VAoid}
For $f\in C^\infty(M)$ and $X,Y\in\mc{T}(M)$, we have
\begin{align*}
\ell X\bullet f\quad
&=(\nabla X)f \\
\{\ell X,\ell Y\}\;\;
&=-\mathrm{Tr}\,(\nabla X\cdot\nabla Y) \\
\{\ell X,\ell Y\}_\Omega
&\textstyle
=\mathrm{Tr}\,\Big(-\nabla(\nabla X)\cdot\nabla Y
  +\nabla X\cdot\iota_Y R
  -\iota_X R\cdot\nabla Y\Big)
  +\frac{1}{2}\iota_X\iota_Y H
\end{align*}
\end{prop}

\begin{proof}
Let us verify these expressions at the level of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$.
Recall the normal-ordered expansions and commutation
relations in Corollary \ref{cor.fund.normal.comm},
as well as those in the larger collection
(\ref{freeVA.comm})--(\ref{freeVA.NOP}).
By definition (\ref{VA.VAoid.456}), $\ell X\bullet f$
is represented in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ by
\begin{align*}
\big(\widetilde X-\tau_i\widetilde X^j\,&\rho(\Theta)_{ij}\big)_{-1}f
-\big(\widetilde{fX}-\tau_i(\widetilde{fX})^j\rho(\Theta)_{ij}\big) \\
&=\big(\tau_{i,-1}\widetilde X^i_0+\widetilde X^i_{-1}\tau_{i,0}\big)f
  -f\widetilde X
  +\big(\tau_i(f\widetilde X^j)-f\tau_i\widetilde X^j\big)\rho(\Theta)_{ij}
\qquad\qquad \\
&=f\widetilde X+(\tau_i f)d\widetilde X^i
  -f\widetilde X
  +(\tau_i f)\widetilde X^j\rho(\Theta)_{ij} \\
&=(\tau_i f)\big(d\widetilde X^i+\rho(\Theta)_{ij}\widetilde X^j\big)
\end{align*}
which by (\ref{DX.lift}) proves the claim.
Now keep (\ref{lift.inv}) in mind.
By (\ref{VA.VAoid.456}), $\{\ell X,\ell Y\}$ is represented by
\begin{align*}
\big(\widetilde X-\tau_i\widetilde X^j\,&\rho(\Theta)_{ij}\big)_1
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\rho(\Theta)_{k\ell}\big) \\
&=\big(\widetilde X^i_1\tau_{i,0}+\widetilde X^i_0\tau_{i,1}\big)
  \tau_{k,-1}\widetilde Y^k \\
&=-[\tau_{k,-1},\widetilde X^i_1]\,\tau_{i,0}\widetilde Y^k
  -[[\tau_{i,0},\tau_{k,-1}],\widetilde X^i_1]\,\widetilde Y^k
  +\widetilde X^i_0\,[\tau_{i,1},\tau_{k,-1}]\,\widetilde Y^k \\
&=-(\tau_k\widetilde X^i)(\tau_i\widetilde Y^k)
  +\widetilde Y^k\Omega(\tau_i,\tau_k)^P\widetilde X^i
  -\widetilde X^i\Omega(\tau_i,\tau_k)^P\widetilde Y^k
  +2\textrm{Ric}_{ik}\widetilde X^i\widetilde Y^k \\
&=-(\tau_k\widetilde X^i)(\tau_i\widetilde Y^k)
\end{align*}
which by (\ref{DX.lift}) again proves the claim.
The computation of $\{\ell X,\ell Y\}_\Omega$ is more
tedious and will only be sketched here.
First of all, by (\ref{VA.VAoid.456}) and Lemma
\ref{lemma.1form.0mode}, $\{\ell X,\ell Y\}_\Omega$ is
represented in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ by
\begin{align}
&\quad\;
\big(\widetilde X-\tau_i\widetilde X^j\,\rho(\Theta)_{ij}\big)_0
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\rho(\Theta)_{k\ell}\big)
-\big(\widetilde{[X,Y]}-\tau_i\widetilde{[X,Y]}{}^j\rho(\Theta)_{ij}\big)
\nonumber \\
&=\big(\tau_{i,-1}\widetilde X^i_1
  +\widetilde X^i_0\tau_{i,0}
  +\widetilde X^i_{-1}\tau_{i,1}\big)\tau_{k,-1}\widetilde Y^k
  -\widetilde{[X,Y]}
\nonumber \\
&\qquad\qquad\qquad\qquad
  +\iota_{\widetilde Y}d\big(\tau_i\widetilde X^j\,\rho(\Theta)_{ij}\big)
  -L_{\widetilde X}\big(\tau_i\widetilde Y^j\,\rho(\Theta)_{ij}\big)
  +\tau_i\widetilde{[X,Y]}{}^j\rho(\Theta)_{ij}
\label{ass.C.VAoid3}
\end{align}
With some work, it turns out the first line in
(\ref{ass.C.VAoid3}) can be rewritten as
\begin{align*}
\textstyle
-\Omega(\widetilde X,\widetilde Y)^P
-\tau_i\widetilde Y^k\,d(\tau_k\widetilde X^i)
+\frac{1}{2}\iota_{\widetilde X}\iota_{\widetilde Y}H
+\lambda^*(\Omega(\widetilde X,\widetilde Y),\Theta)
\end{align*}
while the second line in (\ref{ass.C.VAoid3}),
thanks to the first Bianchi identity, equals
\begin{align*}
\big((\tau_i\widetilde X^k)(\tau_k\widetilde Y^j)
  -(\tau_i\widetilde Y^k)(\tau_k\widetilde X^j)\big)\rho(\Theta)_{ij}
+\tau_i\widetilde X^j\,\iota_{\widetilde Y}\rho(\Omega)_{ij}
-\tau_i\widetilde Y^j\,\iota_{\widetilde X}\rho(\Omega)_{ij}
-\lambda_\rho(\Omega(\widetilde X,\widetilde Y),\Theta).
\end{align*}
Notice that $\lambda^*=\lambda_\rho$ in our current
setting (see \S\ref{sec.ass.C}) and by the proof of
Lemma \ref{lemma.Feigin.C.1} we may ignore the term
$-\Omega(\widetilde X,\widetilde Y)^P$.
Then it follows from (\ref{tau.brackets}) and (\ref{lift.inv})
that $\{\ell X,\ell Y\}_\Omega$ is also represented by
\begin{align*}
\textstyle
-(\tau_\ell\tau_k\widetilde X^i)(\tau_i\widetilde Y^k)\tau^\ell
+\tau_i\widetilde X^j\,\iota_{\widetilde Y}\rho(\Omega)_{ij}
-\tau_i\widetilde Y^j\,\iota_{\widetilde X}\rho(\Omega)_{ij}
+\frac{1}{2}\iota_{\widetilde X}\iota_{\widetilde Y}H.
\end{align*}
By (\ref{DX.lift}) again this proves our last claim.
\end{proof}

{\it Remark.}
This result recovers the vertex algebroid described in
Theorem \ref{thm.globalCDO}a (as $\nabla$ is now
torsion-free).
In other words, we have $V^f=\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$.

\begin{subsec} \label{sec.ass.C.conformal}
{\bf The conformal vector.}
According to \S\ref{sec.ass.C} and Lemma \ref{lemma.G},
the vertex algebra $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ has a conformal
vector $\nu^M$ of central charge $2d=2\dim M$,
represented by
\begin{align}
\nu^{\Pi\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\nu^P
  -Q_0\gamma
&=\mathbf{1}\otimes\Big(\nu^P-t^P_{a,-1}\Theta^a\Big)
\nonumber \\
&\textstyle
=\mathbf{1}\otimes\Big(\tau_{i,-1}\tau^i
  -\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)\Big)
\label{ass.C.conformal}
\end{align}
The key to understanding the whole structure of
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ is the observation that $\nu^M$ is
generated by the associated vertex algebroid (in a certain way).
In fact, as we will see, its representative
(\ref{ass.C.conformal}) can be expressed entirely
in terms of elements of (\ref{ass.C.1}).
\end{subsec}

For the following statement, notice that by assumption
$\pi:P\rightarrow M$ is a lifting of the special orthogonal
frame bundle $F_{SO}(TM)\rightarrow M$
(see \S\ref{sec.ass.C}).

\begin{prop} \label{prop.ass.C.conformal}
The vertex algebra $V^f$ has a conformal vector $\nu^f$
with $\Phi(\nu^f)=\nu^M$ (see \S\ref{sec.ass.C.VAoid}).
Given an open subset $U\subset M$ and a smooth
section $\sigma:U\rightarrow\pi^{-1}(U)\subset P$ of
$\pi$, there is a local expression
\begin{align*}
\textstyle
\nu^f|_U
=(\ell\sigma_i)_{-1}\sigma^i
+\frac{1}{2}\mathrm{Tr}\,\big(\Gamma^\sigma_{-1}\Gamma^\sigma)
+\sigma^i([\sigma_j,\sigma_k])\,
  \sigma^k_{-1}\Gamma^\sigma_{ji}
\end{align*}
\footnote{ \label{underlying.sheaf}
Recall from \S\ref{sec.review} that, like most
constructions in this paper, $V^f$ is the space of global
sections of an underlying sheaf.
}
where $(\sigma_1,\ldots,\sigma_d)$ is the
$C^\infty(U)$-basis of $\mc{T}(U)$ induced by $\sigma$;
$(\sigma^1,\ldots,\sigma^d)$ the dual basis of
$\Omega^1(U)$;
and $\Gamma^\sigma=\rho(\sigma^*\Theta)
\in\Omega^1(U)\otimes\mathfrak{so}_d$.
In particular, $\nu^f\in\mathcal{F}_{\preceq(-1;-1)}V^f$
(see \S\ref{freeVA.PBW}).
\end{prop}

\begin{proof}
Consider the smooth map $g:\pi^{-1}(U)\rightarrow G$
defined by $\sigma(\pi(p))=p\cdot g(p)$ for
$p\in\pi^{-1}(U)$.
To ease notations, we also write
$\rho(g):\pi^{-1}(U)\rightarrow SO(\mathbb{R}^d)$ simply
as $g$.
Then we have
\begin{eqnarray} \label{frame.lift}
\widetilde\sigma_i=g_{ri}\tau_r,\qquad
\sigma^i=g_{ri}\tau^r,\qquad
\tau_r=g_{ri}\widetilde\sigma_i,\qquad
\tau^r=g_{ri}\sigma^i.
\end{eqnarray}
Since $\nabla\sigma_i=\Gamma^\sigma_{ji}\otimes\sigma_j$,
it follows from (\ref{DX.lift}) and (\ref{frame.lift}) that
\begin{eqnarray} \label{dg}
dg+\rho(\Theta)\cdot g
=(\tau_k g)\tau^k
=g\cdot\Gamma^\sigma
\end{eqnarray}
where $\cdot$ denotes matrix multiplication.

The main task here is to express (\ref{ass.C.conformal})
entirely in terms of $\sigma$.
Let us first write down
\begin{align*}
\tau_{r,-1}\tau^r
&=\tau_{r,-1}g_{ri,0}\sigma^i
=(\tau_{r,-1}g_{ri})_{-1}\sigma^i
-\big(g_{ri,-1}\tau_{r,0}+g_{ri,-2}\tau_{r,1}\big)
  g_{si}\tau^s \\
&\textstyle
=\widetilde\sigma_{i,-1}\sigma^i
-(\tau_r g_{si})\tau^s_{-1}dg_{ri}
-\mathrm{Tr}\,\big(g^{-1}\cdot\rho(\Theta)_{-1}dg\big)
-\frac{1}{2}\mathrm{Tr}\,\big(g^{-1}\cdot Tdg\big)
\end{align*}
using (\ref{frame.lift}),
Corollary \ref{cor.fund.normal.comm} and
the Lie derivative $L_{\tau_r}\tau^s=\rho(\Theta)_{sr}$
implied by (\ref{tau.brackets}).
Then we work on each term separately, using
(\ref{frame.lift}) and (\ref{dg}) a number of times:
\begin{align*}
\textrm{2nd term}
&=(\tau_r g_{si})\tau^s_{-1}
  \big(\rho(\Theta)_{rt}g_{ti}
  -g_{rj}\Gamma^\sigma_{ji}\big) \\
&=-g_{si}(\tau_r g_{ti})\tau^s_{-1}\rho(\Theta)_{rt}
  +g_{si} g_{rj}(\tau_r g_{sk})\sigma^k_{-1}
    \Gamma^\sigma_{ji} \\
&=-(\tau_r\widetilde\sigma_i^t)\sigma^i_{-1}\rho(\Theta)_{rt}
  +g_{si} g_{rk}(\tau_r g_{sj})\sigma^k_{-1}
    \Gamma^\sigma_{ji}
  +\sigma^i([\widetilde\sigma_j,\widetilde\sigma_k])\sigma^k_{-1}
    \Gamma^\sigma_{ji} \\
&=-(\tau_r\widetilde\sigma_i^t)\sigma^i_{-1}\rho(\Theta)_{rt}
  +\mathrm{Tr}\,\big(\Gamma^\sigma_{-1}\Gamma^\sigma\big)
  +\sigma^i([\sigma_j,\sigma_k])\sigma^k_{-1}
    \Gamma^\sigma_{ji}  \\
\textrm{3rd term}
&=\lambda_\rho(\Theta_{-1}\Theta)
-\mathrm{Tr}\,\big(g^{-1}\cdot\rho(\Theta)_{-1}\cdot
  g\cdot\Gamma^\sigma\big) \\
\textrm{4th term}
&\textstyle
=-\frac{1}{2}\,T\big(\mathrm{Tr}\,(g^{-1}dg)\big)
-\frac{1}{2}\mathrm{Tr}\,\big(g^{-1}\cdot dg_{-1}
  \cdot g^{-1}\cdot dg\big) \\
&\textstyle
=0-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)
+\mathrm{Tr}\,\big(g^{-1}\cdot\rho(\Theta)_{-1}\cdot
  g\cdot\Gamma^\sigma\big)
-\frac{1}{2}\mathrm{Tr}\,(\Gamma^\sigma_{-1}\Gamma^\sigma)
\end{align*}
These calculations together yield an identity
in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(\pi^{-1}(U))$:
\begin{align} \label{ass.C.conformal.local}
\textstyle
\tau_{r,-1}\tau^r
-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta)
=\big(\widetilde\sigma_i
  -\tau_r\widetilde\sigma^t_i\,\rho(\Theta)_{rt}\big)_{-1}
  \sigma^i
+\frac{1}{2}\mathrm{Tr}\,(\Gamma^\sigma_{-1}\Gamma^\sigma)
  +\sigma^i([\sigma_j,\sigma_k])\sigma^k_{-1}
    \Gamma^\sigma_{ji}.
\end{align}
The left hand side, as observed in
\S\ref{sec.ass.C.conformal}, represents
$\nu^M\in\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$;
and it is defined globally on $P$.
Now that the right hand side is manifestly generated by
the subspace (\ref{ass.C.1}), it defines an element
$\nu^f\in V^f$ such that $\Phi(\nu^f)=\nu^M$.

It remains to show that $\nu^f$ is a conformal
vector of $V^f$.
According to \cite{Kac,FB-Z}, this amounts to
checking:
(i) $\nu^f_{-1}=T$,
(ii) $\nu^f_0=L_0$, and
(iii) $\nu^f_2\nu^f\in\mathbb{C}$.
For (i) and (ii) it suffices to check them on
\begin{eqnarray*}
C^\infty(U)\cup\{\ell\sigma_1,\ldots,\ell\sigma_d\}
\end{eqnarray*}
because $V^f$ is locally generated by these elements.
In fact, since $\Phi(\nu^f)=\nu^M$ is known to be conformal
and $\Phi$ is an isomorphism in weights $0$ and $1$,
everything we need to check automatically holds except
for the equation $\nu^f_{-1}\ell\sigma_i=T(\ell\sigma_i)$
in weight $2$.
This equation can be verified by a straightforward calculation,
which we omit.
\end{proof}

{\it Remark.}
Since we have already observed that $V^f=\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$,
this result recovers the conformal vector $\nu^0$ described
in Theorem \ref{thm.globalCDO}b (as $\Gamma^\sigma$ is
now traceless).

\begin{corollary} \label{cor.ass.C.freeVA}
The map of vertex algebras
$\Phi:V^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$
(see \S\ref{sec.ass.C.VAoid}) is an isomorphism.
\end{corollary}

\begin{proof}
By Proposition \ref{prop.ass.C.conformal},
the conformal vector $\nu^M\in\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ belongs to
$\mathcal{F}_{\preceq(-1;-1)}$.
Then Lemma \ref{lemma.conf.gen} applies so that
$\Phi$ is surjective.
By construction, the ideal $\ker\Phi\subset V^f$ is
trivial in weights $0$ and $1$.
Then by Proposition \ref{prop.ass.C.conformal} again
and Lemma \ref{lemma.conf.ideal}, $\ker\Phi$ is
in fact trivial in all weights.
\end{proof}

Propositions \ref{prop.ass.C.VAoid},
\ref{prop.ass.C.conformal} and
Corollary \ref{cor.ass.C.freeVA} together show
that $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})\cong\mc{D}^{\mathrm{ch}}_{\nabla,H}(M)$
as conformal vertex algebras.
This fulfills the goal of this section.
Let us summarize our work in the form of a new
description of algebras of CDOs.

\begin{theorem} \label{thm.CDO.semiinf}
Suppose $\pi:P\rightarrow M$ is a smooth prinicpal
$G$-bundle and $\rho:G\rightarrow SO(\mathbb{R}^d)$ is
a representation such that there is an isomorphism
$P\times_\rho\mathbb{R}^d\cong TM$.
Given a principal frame $(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},G)$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ (see Theorem \ref{thm.CDOP}), the zeroth
semi-infinite cohomology
\begin{eqnarray*}
\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})
=H^{\frac{\infty}{2}+0}\big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}},
  \mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\big)
\end{eqnarray*}
is an algebra of CDOs on $M$.
Up to isomorphism, every algebra of CDOs on $M$ arises
this way.
$\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ as a vertex algebra is freely generated
by its weight-zero and weight-one components, which are
represented by the following subspaces of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ bijectively:
\begin{align*}
\pi^*C^\infty(M),\qquad
\pi^*\Omega^1(M)\oplus
  \big\{\widetilde X-\tau_i\widetilde X^j\,\rho(\Theta)_{ij}:
  X\in\mc{T}(M)\big\}.
\end{align*}
(For details of the latter representation, see the proof
of Lemma \ref{lemma.Feigin.C.1}.)
Moreover, $\Gamma^{\mathrm{ch}}(\pi,\mathbb{C})$ has a conformal vector
of central charge $2d=2\dim M$, represented in
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ by
\begin{align*}
\textstyle
\qquad
\tau_{i,-1}\tau^i
-\frac{1}{2}\lambda_\rho(\Theta_{-1}\Theta).
\qquad\qedsymbol
\end{align*}
\end{theorem}

To conclude this section, let us sketch an extension of
Theorem \ref{thm.CDO.semiinf}.

\begin{subsec} \label{CDO.cs}
{\bf Generalization to supermanifolds.}
Suppose $\pi:P\rightarrow M$ and
$\rho:G\rightarrow SO(\mathbb{R}^d)$ are the same as above;
also let $\rho':G\rightarrow U(\mathbb{C}^r)$ be another
representation, $E=P\times_{\rho'}\mathbb{C}^r$ the associated
vector bundle and $\Pi E$ the corresponding cs-manifold.
\cite{QFS.susy}
The $G$-action on
$\mathcal{O}(\Pi\mathbb{R}^r)\otimes\mathbb{C}=\wedge^*(\mathbb{C}^r)^\vee$
induced by $\rho'$ lifts to an inner
$(\hat{\mathfrak{g}}_{\lambda_{\rho'}},G)$-action on
$\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r)$, a fermionic analogue of
\S\ref{algCDO} (see also Definition \ref{formalLG.action}).
By Theorem \ref{thm.CDOP}, there exists a principal frame
$(\hat{\mathfrak{g}}_\lambda,G)$-algebra $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ with
$\lambda+\lambda_{\rho'}=-\lambda_{\t{\sf ad}}$ if and only if
\begin{eqnarray*}
\lambda^*(P)=(\lambda_\rho-\lambda_{\rho'})(P)=0
\qquad\iff\qquad
p_1(M)-ch_2(E)=0.
\end{eqnarray*}
In this case, we can apply Definition \ref{defn.ass}
to construct a vertex superalgebra
\begin{eqnarray*}
\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r))
=H^{\frac{\infty}{2}+0}\Big(\hat{\mathfrak{g}}_{-\lambda_{\t{\sf ad}}}\,,\,
  \mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r)\Big).
\end{eqnarray*}
Moreover, the $(\hat{\mathfrak{g}}_{\lambda_{\rho'}},G)$-action
on $\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r)$ is primary if and only if
\begin{eqnarray*}
\rho'(\mathfrak{g})\subset\mathfrak{su}_r
\qquad\iff\qquad
c_1(E)=0.
\end{eqnarray*}
In this case, $\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r))$
has a conformal vector of central charge $2(d-r)$.
By a similar analysis, $\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^r))$ is
an algebra of CDOs on $\Pi E$ in the sense of \cite{myCDO}.
In particular, in the case $\rho'=\rho_{\mathbb{C}}$
(i.e. $E=TM_{\mathbb{C}}$), both obstructions are trivial
and $\Gamma^{\mathrm{ch}}(\pi,\mc{D}^{\mathrm{ch}}(\Pi\mathbb{R}^d))$ is the smooth
chiral de Rham algebra of $M$.
\end{subsec}

\newpage
\setcounter{equation}{0}
\section{Example: Spinor Module over CDOs}
\label{sec.FL.spinor}

Given a spin manifold $M$ with a trivialization of its first
Pontrjagin form, we are interested in the space of sections
of the hypothetical ``spinor bundle on the formal loops of $M$''
as a module over its ``differential operators''.
The object of study in this section \emph{should} have the
above geometric meaning, but is more precisely an example
of the construction introduced in \S\ref{sec.ass} defined
using semi-infinite cohomology.
Following a similar strategy as in \S\ref{CDOP.CDOM}, we
analyze this spinor module so as to obtain a more explicit
description in terms of generating data and relations.
It is hoped that a deeper understanding, including the
identification of an appropriate Dirac operator, will lead to
a useful geometric theory of the Witten genus.
(This was in fact the original motivation of the paper.)

In this section, $G=\textrm{Spin}_d$ and
$\mathfrak{g}=\mathfrak{spin}_d=\mathfrak{so}_d$ where $d=2d'$ is even
(except in \S\ref{sec.Cl}).
Since $\mathfrak{so}_d$ is simple, we use the more common notation
$(\widehat{\mathfrak{so}}_d)_k$ in place of
$(\widehat{\mathfrak{so}}_d)_{k\lambda_0}$ (see \S\ref{sec.Gmfld}),
where $\lambda_0$ is the normalized Killing form,
i.e. $\lambda_0(A,B)=\frac{1}{2}\mathrm{Tr}\, AB$\, for $A,B\in\mathfrak{so}_d$.

\begin{subsec} \label{sec.Cl}
{\bf The Ramond Clifford algebra.}
Let $C\ell$ be the unital $\mathbb{Z}/2\mathbb{Z}$-graded
associative $\mathbb{C}$-algebra with the following generators
and relations
\begin{eqnarray} \label{Cl}
e_{i,n}\textrm{ odd},\;\;i=1,\ldots,d,\;\;n\in\mathbb{Z},
\qquad
[e_{i,n},e_{j,m}]
=e_{i,n}e_{j,m}+e_{j,m}e_{i,n}
=-2\delta_{ij}\delta_{n+m,0}.
\end{eqnarray}
(The notations are chosen to resemble those in \cite{spin}.)
Suppose $W$ is a $C\ell$-module with the property that
for each $w\in W$, there exists $N\in\mathbb{Z}$ such that
$e_{i,n}w=0$ for $n>N$.
Then the operators
\begin{eqnarray} \label{Cl.so}
A^{C\ell}_n
=\frac{1}{4}A_{ji}\,e_{i,n-r}e_{j,r}
\quad\textrm{for }A\in\mathfrak{so}_d,\;n\in\mathbb{Z}
\end{eqnarray}
define an $(\widehat{\mathfrak{so}}_d)_1$-action on $W$.
On the other hand, the operators
\begin{eqnarray} \label{Cl.Virasoro}
L^{C\ell}_n
=-\frac{1}{8}\sum_{r\geq 0}(2r-n)e_{i,n-r}e_{i,r}
  +\frac{1}{8}\sum_{r<0}(2r-n)e_{i,r}e_{i,n-r}
  +\frac{d}{16}\delta_{n,0}\quad
\textrm{for }n\in\mathbb{Z}
\end{eqnarray}
define a Virasoro action on $W$ of central charge
$d/2$.
This is in fact the Sugawara construction associated to
the above $(\widehat{\mathfrak{so}}_d)_1$-action, and
accordingly satisfies
\begin{eqnarray*}
[L^{C\ell}_n,A^{C\ell}_m]=-mA^{C\ell}_{n+m}\quad
\textrm{for }n,m\in\mathbb{Z}.
\end{eqnarray*}
The eigenvalues of $L^{C\ell}_0$ are called weights
(as in previous sections) and each $e_{i,n}$ changes
weights by $-n$.
For more details, see e.g. \cite{Fuchs}
\end{subsec}

For the rest of the section, $d=2d'$ is even.

\begin{subsec} \label{sec.S}
{\bf The spinor representation of $\widehat{\mathfrak{so}}_{2d'}$.}
Let $C\ell_0$ (resp.~$C\ell_{\geq 0}$) be the
subalgebra of $C\ell$ generated by those $e_{i,n}$ with
$n=0$ (resp.~$n\geq 0$).
The finite-dimensional Clifford algebra $C\ell_0$ has a unique
(up to isomorphism) irreducible $\mathbb{Z}/2\mathbb{Z}$-graded
representation $S_0$.
Let us also regard $S_0$ as a $C\ell_{\geq 0}$-module on which
$\{e_{i,n}\}_{n>0}$ act trivially.
Then define a $C\ell$-module by
\begin{eqnarray*}
S=C\ell\otimes_{C\ell_{\geq 0}}S_0.
\end{eqnarray*}
By (\ref{Cl}), $S$ as a vector space is spanned by elements of
the form
\begin{eqnarray*}
e_{i_p,n_p}\cdots e_{i_1,n_1}s,\qquad
n_1<0,\;\;
(n_p,i_p)<\cdots<(n_1,i_1),\;\;
s\in S_0
\end{eqnarray*}
where the indicated pairs are ordered lexicographically.
According to \S\ref{sec.Cl}, $S$ admits
a $((\widehat{\mathfrak{so}}_{2d'})_1,\textrm{Spin}_{2d'})$-action
as well as a Virasoro action of central charge $d'$, such that
the former is primary.
Notice that the element displayed above has weight
\begin{eqnarray*}
\frac{d'}{8}+|n_1|+\ldots+|n_p|.
\end{eqnarray*}
For convenience, we will write $S_k\subset S$ for
the component of weight $(d'/8)+k$.
\end{subsec}

Given a representation
$\rho:\mathfrak{so}_{2d'}\rightarrow\mathfrak{gl}(V)$, let $\lambda_\rho$
denote the invariant symmetric bilinear form on $\mathfrak{so}_{2d'}$
defined by $\lambda_\rho(A,B)=\mathrm{Tr}\,\rho(A)\rho(B)$.
In particular, notice that $\lambda_\rho=2\lambda_0$ for
the standard representation $\rho$, and
$\lambda_{\t{\sf ad}}=(4d'-4)\lambda_0$.

\begin{subsec} \label{sec.ass.S}
{\bf The spinor module over CDOs.}
Consider the special case of Definition \ref{defn.ass}
associated to a principal frame
$((\hat{\mathfrak{so}}_{2d'})_{3-4d'},\textrm{Spin}_{2d'})$-algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ and the
$((\hat{\mathfrak{so}}_{2d'})_1,\textrm{Spin}_{2d'})$-module $S$:
\begin{align}
\Gamma^{\mathrm{ch}}(\pi,S)
&=H^{\frac{\infty}{2}+0}\Big(
  (\widehat{\mathfrak{so}}_{2d'})_{4-4d'}\,,
  \,\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S\Big) \nonumber \\
&=H^0\Big(
  \mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)
  \otimes S\,,\,Q_0
\Big) \label{ass.S}
\end{align}
By Lemma \ref{lemma.ass} and \S\ref{sec.S},
$\Gamma^{\mathrm{ch}}(\pi,S)$ is a module over the vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$ and
admits a Virasoro action of central charge
$5d'=\frac{5}{2}\dim M$.
For convenience, let us elaborate on all the data involved
and recall some relevant notations.
\begin{itemize}
\item[$\centerdot$]
Let $(t_1,t_2,\ldots)$ be a basis of $\mathfrak{so}_{2d'}$,
$(t^1,t^2,\ldots)$ the dual basis of $(\mathfrak{so}_{2d'})^\vee$,
$(\phi^1,\phi^2,\ldots)$ the corresponding
coordinates of the supermanifold $\Pi\mathfrak{so}_{2d'}$, and
$(\partial_1,\partial_2,\ldots)$ their coordinate vector fields.
\vspace{-0.05in}
\item[$\centerdot$]
For comments on the conformal vertex superalgebra
$\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})$, see \S\ref{sec.ass.C}.
\vspace{-0.05in}
\item[$\centerdot$]
For the detailed definition of the vertex algebra
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$, see Theorem \ref{thm.CDOP} with
$\lambda=(3-4d')\lambda_0$ in mind.
Let us mention that it is defined using the following
data:
a principal $\textrm{Spin}_{2d'}$-frame bundle
$\pi:P\rightarrow M$;
the Levi-Civita connection $\Theta=\Theta^a\otimes t_a$
on $\pi$;
and a basic $3$-form $H$ on $P$ that satisfies
$dH=\lambda_0(\Omega\wedge\Omega)$, where
$\Omega=d\Theta+\frac{1}{2}[\Theta\wedge\Theta]$.
Also, there is a conformal vector
\begin{eqnarray*}
\nu^P=t^P_{a,-1}\Theta^a+\tau_{i,-1}\tau^i
  -\frac{1}{2}\lambda_0(\Theta_{-1}\Theta)
\end{eqnarray*}
of central charge $2\dim P$.
(For the notations see \S\ref{sec.prin.bdl2}.)
\vspace{-0.05in}
\item[$\centerdot$]
For details on the Feigin complex in (\ref{ass.S}),
see \S\ref{sec.Feigin}.
Let us mention that its differential is
\begin{align} \label{ass.S.Q.theta}
Q_0
=q_0\otimes 1\otimes 1
+\phi^a_{-r}\otimes t^P_{a,r}\otimes 1
+\phi^a_{-r}\otimes 1\otimes t^{C\ell}_{a,r}
\end{align}
where $q$ is the odd vector field shown in (\ref{JQ.coor}).
Also the Virasoro operators
\begin{align*}
\qquad\qquad
L^{\Pi{\mathfrak{so}}}_n\otimes 1\otimes 1
+1\otimes L^P_n\otimes 1
+1\otimes 1\otimes L^{C\ell}_n,
\qquad n\in\mathbb{Z}
\end{align*}
preserve the gradation and commute with $Q_0$.
\vspace{-0.05in}
\end{itemize}
Let us repeat the geometric ingredients in
slightly different words:
$M$ is a Riemannian manifold with a spin structure
and a $3$-form $H$ that satisfies
$dH=\frac{1}{2}\mathrm{Tr}\,(R\wedge R)$, where $R$ is the
Riemannian curvature.
This can be viewed as the de Rham version of
a \emph{string structure}.
In the sequel, we will describe $\Gamma^{\mathrm{ch}}(\pi,S)$ more
explicitly in terms of generating data (i.e. a subspace and
some fields) and their relations.
\end{subsec}

Throughout this section we identify $\Omega^*(M)$ with
the basic subspace of $\Omega^*(P)$.

\begin{subsec} \label{sec.ass.S.0}
{\bf The component of the lowest weight.}
Consider the Feigin complex appearing in (\ref{ass.S}):
\begin{eqnarray} \label{Feigin.S}
\Big(
  \mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)
  \otimes S\,,\,Q_0
\Big).
\end{eqnarray}
Since its component of weight $d'/8$ is simply
the Chevalley-Eilenberg complex (see \S\ref{sec.CE})
\begin{eqnarray*}
\Big(\mathcal{O}(\Pi\mathfrak{so}_{2d'})\otimes C^\infty(P)
  \otimes S_0\,,\,Q_0\Big)
\end{eqnarray*}
the component of (\ref{ass.S}) of weight $d'/8$ is
\begin{eqnarray} \label{ass.S.0}
\Gamma^{\mathrm{ch}}(\pi,S)_0
=H^0\big(\mathfrak{so}_{2d'},C^\infty(P)\otimes S_0\big)
=(C^\infty(P)\otimes S_0)^{\mathfrak{so}_{2d'}}
=S(M)
\end{eqnarray}
i.e.~the space of sections of the spinor bundle on $M$.
Understanding the rest of $\Gamma^{\mathrm{ch}}(\pi,S)$ requires more work,
just as in \S\ref{CDOP.CDOM}.
\end{subsec}

\begin{lemma} \label{lemma.G.S}
Let $\gamma=\partial_a\otimes\Theta^a\in
\mc{D}^{\mathrm{ch}}(\Pi\mathfrak{so}_{2d'})_1\otimes\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$.
For $n\in\mathbb{Z}$ we have
\begin{eqnarray*}
[Q_0,\,\gamma_n\otimes 1]
=L^{\Pi\mathfrak{so}}_n\otimes 1\otimes 1
+1\otimes(t^P_{a,-1}\Theta^a)_n\otimes 1
+1\otimes\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}\,.
\end{eqnarray*}
Moreover, for $n\in\mathbb{Z}$ and
$\mathscr{Y}=\mathscr{Y}^a t^P_a\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}
\subset\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)_1$ we also have
\begin{eqnarray*}
[Q_0,\,
  (\gamma_0(1\otimes\mathscr{Y}))_n\otimes 1]
=1\otimes\mathscr{Y}_n\otimes 1
+1\otimes\mathscr{Y}^a_{n-r}\otimes t^{C\ell}_{a,r}\,.
\end{eqnarray*}
\end{lemma}

\begin{proof}
Recall the differential $Q_0$ from (\ref{ass.S.Q.theta}).
The first anticommutator is computed as follows
\begin{align*}
[Q_0,\,\gamma_n\otimes 1]
&=[(q\otimes 1+\phi^a\otimes t^P_a)_0,\,\gamma_n]\otimes 1
+[\phi^a_{-r},\partial_{b,n-s}]\otimes\Theta^b_s
  \otimes t^{C\ell}_{a,r} \\
&=L^{\Pi\mathfrak{so}}_n\otimes 1\otimes 1
+1\otimes(t^P_{a,-1}\Theta^a)_n\otimes 1
+1\otimes\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}
\end{align*}
using Lemma \ref{lemma.G} and the fermionic analogue of
(\ref{Weyl}).
On the other hand, since
$\gamma_0(1\otimes\mathscr{Y})=\partial_b\otimes\mathscr{Y}^b$,
the second anticommutator can be written as
\begin{align*}
[Q_0,(\partial_b\otimes\mathscr{Y}^b)_n\otimes 1]
&=[(q\otimes 1+\phi^a\otimes t^P_a)_0,
  (\partial_b\otimes\mathscr{Y}^b)_n]\otimes 1
+[\phi^a_{-r},\partial_{b,n-s}]\otimes\mathscr{Y}^b_s
  \otimes t^{C\ell}_{a,r} \\
&=\big((q\otimes 1+\phi^a\otimes t^P_a)_0
  (\partial_b\otimes\mathscr{Y}^b)\big)_n\otimes 1
+1\otimes\mathscr{Y}^a_{n-r}\otimes t^{C\ell}_{a,r}
\end{align*}
using again the analogue of (\ref{Weyl}).
It remains to compute the element
$(q\otimes 1+\phi^a\otimes t^P_a)_0
(\partial_b\otimes\mathscr{Y}^b)$, which is the sum of the
following:
\begin{align*}
&(q_0\partial_b)\otimes\mathscr{Y}^b
=[q,\partial_b]\otimes\mathscr{Y}^b
=-t^c([t_b,t_d])\phi^d\partial_c\otimes\mathscr{Y}^b \\
&(\phi^a_0\otimes t^P_{a,0})(\partial_b\otimes\mathscr{Y}^b)
=\phi^a\partial_b\otimes t^P_a\mathscr{Y}^b
=-t^b([t_a,t_c])\phi^a\partial_b\otimes\mathscr{Y}^c \\
&(\phi^a_1\otimes t^P_{a,-1})(\partial_b\otimes\mathscr{Y}^b)
=\delta^a_b\otimes t^P_{a,-1}\mathscr{Y}^b
=1\otimes\mathscr{Y}
\end{align*}
Indeed, the first line is similar to a computation
in the proof of Lemma \ref{lemma.G};
the second follows from the
$\mathfrak{so}_{2d'}$-invariance of $\mathscr{Y}$;
and the last follows from the analogue of
(\ref{Weyl}) and Corollary
\ref{cor.fund.normal.comm}.
\end{proof}

{\it Preparation.}
Let $C\ell(M)=(C^\infty(P)\otimes
C\ell_0)^{\mathfrak{so}_{2d'}}$ which is the same as the algebra
of $C^\infty(M)$-linear endomorphisms of $S(M)$.
Given $X\in\mc{T}(M)$, let us denote its horizontal lift by
$\widetilde X=\widetilde X^i\tau_i\in\mc{T}_h(P)^{\mathfrak{so}_{2d'}}$ (as usual) and
its Clifford action by $cX=\widetilde X^i\otimes e_{i,0}\in C\ell(M)$.
Notice that $\{cX:X\in\mc{T}(M)\}$ generates $C\ell(M)$
as an algebra.
On the other hand, each
$\mathscr{Y}=\mathscr{Y}^a t^P_a\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$
satisfies
\begin{align*}
\mathscr{Y}\otimes 1+\mathscr{Y}^a\otimes t_a=0
\quad\textrm{on}\quad
(C^\infty(P)\otimes S_0)^{\mathfrak{so}_{2d'}}=S(M)
\end{align*}
and hence represents an endomorphism
$c\mathscr{Y}=-\mathscr{Y}^a\otimes t_a\in C\ell(M)$.
\footnote{
To be precise, we are regarding $\mathfrak{so}_d$ as a Lie
subalgebra of $C\ell_0$ via the standard inclusion
$A\mapsto\frac{1}{4}A_{ji}e_{i,0}e_{j,0}$.
}

\begin{subsec} \label{sec.ass.S.fields}
{\bf Fields of low weights.}
Recall Lemma \ref{lemma.semiinf.ops}a for the special case
(\ref{ass.S}):
the $\widehat{\mathfrak{so}}_{2d'}$-invariant fields
on $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$, including
the vertex operators of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$,
induce fields on $\Gamma^{\mathrm{ch}}(\pi,S)$.
Let us now describe a collection of such fields
(or rather their Fourier modes);
we will later see that their actions on the subspace
$\Gamma^{\mathrm{ch}}(\pi,S)_0=S(M)$ generate the entire space
$\Gamma^{\mathrm{ch}}(\pi,S)$.

Given $f\in C^\infty(M)$ and $X\in\mc{T}(M)$, there are fields
on $\Gamma^{\mathrm{ch}}(\pi,S)$ whose Fourier modes are
\begin{align}
f_n
&:=\textrm{the operator induced by }f_n\otimes 1
\nonumber \\
\ell X_n
&:=\textrm{the operator induced by }
  (\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij})_n\otimes 1
\label{ass.S.gen.fields} \\
c X_n
&:=\textrm{the operator induced by }
  \widetilde X^i_{n-r}\otimes e_{i,r}
\nonumber
\end{align}
Indeed, the first two are well-defined because both $f$ and
$\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij}$ belong to
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$
according to \S\ref{sec.invCDO.can};
and so is the last one because for $A\in\mathfrak{so}_{2d'}$ and
$m\in\mathbb{Z}$ we have
\begin{align*}
\big[A^P_m\otimes 1+1\otimes A^{C\ell}_m,\,
  \widetilde X^i_{n-r}\otimes e_{i,r}\big]
&=[A^P_m,\widetilde X^i_{n-r}]\otimes e_{i,r}
  +\widetilde X^i_{n-r}\otimes[A^{C\ell}_m,e_{i,r}] \\
&=(A^P\widetilde X^i)_{m+n-r}\otimes e_{i,r}
  +\widetilde X^i_{n-r}\otimes(A_{ji}e_{j,m+r}) \\
&=-A_{ij}\widetilde X^j_{m+n-r}\otimes e_{i,r}
  +A_{ji}\widetilde X^i_{n-r}\otimes e_{j,m+r} \\
&=0
\end{align*}
where we have used (\ref{freeVA.comm}), the
$\mathfrak{so}_{2d'}$-invariance of $\widetilde X$ and
(\ref{Cl})--(\ref{Cl.so}).
These three types of fields have weights
$0$, $1$ and $\frac{1}{2}$ respectively.

It will be convenient to also consider some other
fields generated by those in (\ref{ass.S.gen.fields}).
Given $\alpha\in\Omega^1(M)$ and
$\mathscr{Y}\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$, there are fields on
$\Gamma^{\mathrm{ch}}(\pi,S)$ whose Fourier modes are
\begin{align} \label{ass.S.other.fields}
\begin{array}{rl}
\alpha_n
&:=\textrm{the operator induced by }\alpha_n\otimes 1 \\
c\mathscr{Y}_n
&:=\textrm{the operator induced by }
  (\mathscr{Y}+\lambda_0(\Theta,\mathscr{Y}))_n\otimes 1
\phantom{\Big)}
\end{array}
\end{align}
These are well-defined because both $\alpha$ and
$\mathscr{Y}+\lambda_0(\Theta,\mathscr{Y})$ belong to
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)^{\widehat{\mathfrak{so}}_{2d'}}$ according to
\S\ref{sec.invCDO.can}.
To see that $\alpha_n$ is determined by
(\ref{ass.S.gen.fields}), it suffices to consider $\alpha=fdg$
and notice that
\begin{eqnarray} \label{ass.S.1formfield}
(fdg)_n=-\sum_{r\in\mathbb{Z}}r f_{n-r}g_r\quad
\textrm{for }f,g\in C^\infty(M)
\end{eqnarray}
by (\ref{freeVA.comm})--(\ref{freeVA.NOP}).
On the other hand, by Lemma \ref{lemma.G.S} we may represent
$c\mathscr{Y}_n$ alternatively by
\begin{align}
-\mathscr{Y}^a_{n-r}\otimes t^{C\ell}_{a,r}
+\lambda_0(\Theta,\mathscr{Y})_n\otimes 1.
\label{vertical.Cl}
\end{align}
The next lemma explains how $c\mathscr{Y}_n$ is determined by
(\ref{ass.S.gen.fields}).
\end{subsec}

\begin{lemma} \label{lemma.verfield}
For $\mathscr{Y}\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$, the operator
$c\mathscr{Y}_n$ on $\Gamma^{\mathrm{ch}}(\pi,S)$ is a sum of operators of the form
\begin{align} \label{cX.cY.normal}
\sum_{r\geq 0}cX_{n-r}\,cY_r
-\sum_{r<0}cY_r\,cX_{n-r}
-2\langle X,\nabla Y\rangle_n
+\langle X,Y\rangle_n,\quad
X,Y\in\mc{T}(M)
\end{align}
where $\langle\;\rangle$ and $\nabla$ denote the Riemannian
metric and Levi-Civita connection on $TM$.
\end{lemma}

\begin{proof}
Let $\mathscr{Y}_{ij}=\mathscr{Y}^a(t_a)_{ij}\in C^\infty(P)$.
As explained above, $c\mathscr{Y}_n$ is represented in
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ by (\ref{vertical.Cl}),
which by (\ref{Cl.so}) and (\ref{freeVA.NOP}) can be
written as
\begin{align*}
\textstyle
\frac{1}{4}\,\big(\mathscr{Y}_{ij,n-r}\otimes 1\big)
  \big(1\otimes e_{i,r-s}e_{j,s}
  -2\Theta_{ij,r}\otimes 1\big).
\end{align*}
Since $c\mathscr{Y}=\frac{1}{4}\,\mathscr{Y}_{ij}\otimes e_{i,0}e_{j,0}$
corresponds to a $2$-vector field under the isomorphism
$C\ell(M)\cong\Gamma(\wedge^* TM)$, (\ref{vertical.Cl}) can
be expressed further as a sum of operators of the form
\begin{align} \label{Cl2.field}
\textstyle
\qquad
\frac{1}{2}\,
\big((\widetilde X^i\widetilde Y^j-\widetilde Y^i\widetilde X^j)_{n-r}
  \otimes 1\big)
\big(1\otimes e_{i,r-s}e_{j,s}
  -2\Theta_{ij,r}\otimes 1\big),\quad
X,Y\in\mc{T}(M)
\end{align}
where the factor of $\frac{1}{2}$ is included for
convenience.
Let us show that (\ref{Cl2.field}) represents
(\ref{cX.cY.normal}), which will then prove the lemma.
For the calculations below, keep in mind
(\ref{freeVA.comm})--(\ref{freeVA.NOP}).

Consider the first two terms in (\ref{cX.cY.normal}) together.
By definition (\ref{ass.S.gen.fields}) they are represented by
\begin{align*}
\sum_{r\geq 0}(\widetilde X^i_{n-r-s}\otimes e_{i,s})
  (\widetilde Y^j_{r-t}\otimes e_{j,t})
-\sum_{r<0}(\widetilde Y^j_{r-t}\otimes e_{j,t})
  (\widetilde X^i_{n-r-s}\otimes e_{i,s}).
\end{align*}
In view of (\ref{Cl}), let us split this sum into three parts.
The first part consists of terms with $i\neq j$:
\begin{align*}
\sum_r\sum_{i\neq j}\sum_{s,t}
  \widetilde X^i_{n-r-s}\widetilde Y^j_{r-t}\otimes e_{i,s}e_{j,t}
=\sum_{i\neq j}\sum_{s,t}
  (\widetilde X^i\widetilde Y^j)_{n-s-t}\otimes e_{i,s}e_{j,t}\,.
\end{align*}
The second part consists of terms with $i=j$ and
$s+t\neq 0$, which vanishes by symmetry:
\begin{align*}
\sum_r\sum_i\sum_{s+t\neq 0}
  \widetilde X^i_{n-r-s}\widetilde Y^i_{r-t}\otimes e_{i,s}e_{i,t}
=\sum_i\sum_{s+t\neq 0}
  (\widetilde X^i\widetilde Y^i)_{n-s-t}\otimes e_{i,s}e_{i,t}
=0\,.
\end{align*}
The last part consists of terms with $i=j$ and $s+t=0$,
which is computed as follows:
\begin{align*}
\sum_{r\geq 0}\sum_{i,s}
  \widetilde X^i_{n-r-s}&\widetilde Y^i_{r+s}\otimes e_{i,s}e_{i,-s}
-\sum_{r<0}\sum_{i,s}
  \widetilde X^i_{n-r-s}\widetilde Y^i_{r+s}\otimes e_{i,-s}e_{i,s} \\
&=\sum_{i,u}\widetilde X^i_{n-u}\widetilde Y^i_u\otimes
  \Big(\sum_{s\geq -u}e_{i,-s}e_{i,s}
  -\sum_{s>u}e_{i,-s}e_{i,s}\Big) \\
&=-\sum_{i,u}(2u+1)\widetilde X^i_{n-u}\widetilde Y^i_u\otimes 1 \\
&=-(\widetilde X^i\widetilde Y^i)_n\otimes 1
  +2(\widetilde X^i d\widetilde Y^i)_n\otimes 1 \\
&=-\langle X,Y\rangle_n\otimes 1
  +2\langle X,\nabla Y\rangle_n\otimes 1
  -2\big(\widetilde X^i\widetilde Y^j\Theta_{ij}\big)_n\otimes 1
\end{align*}
where we have used (\ref{DX.lift}) in the last equality.
It follows from these calculations and a little reorganization
that (\ref{cX.cY.normal}) is indeed represented by
(\ref{Cl2.field}) as claimed.
\end{proof}

{\it Remark.}
Since we have
$cX\cdot cY=-\langle X,Y\rangle\otimes 1+\frac{1}{2}
(\widetilde X^i\widetilde Y^j-\widetilde Y^i\widetilde X^j)\otimes e_{i,0}e_{j,0}$ in
$C\ell(M)$, the proof above shows that if we are to associate
a field to $cX\cdot cY$, it (or rather its Fourier modes)
should be given by the first three terms in
(\ref{cX.cY.normal}).
Perhaps this defines the correct normal-ordered product of the
fields associated to $cX$ and $cY$, where the term
$-2\langle X,\nabla Y\rangle_n$ is analogous to a normal-order
constant.

\vspace{0.08in}
Now we record the relations between the lowest-weight
component $\Gamma^{\mathrm{ch}}(\pi,S)_0=S(M)$ and the fields introduced in
\S\ref{sec.ass.S.fields}, as well as the relations between
the fields themselves.

\begin{prop} \label{prop.S.fields.ground}
For $f\in C^\infty(M)$, $X\in\mc{T}(M)$ and $s\in S(M)$, we have
\begin{eqnarray*}
&f_n s=cX_n s=\ell X_n s=0\;\;\;\textrm{for}\;\;n>0 &\\
&f_0 s=fs,\qquad
cX_0 s=cX\cdot s,\qquad
\ell X_0 s=\nabla_X s&
\end{eqnarray*}
where $\cdot$ denotes Clifford multiplication and
$\nabla$ the Levi-Civita connection.
\end{prop}

\begin{proof}
The first line is true simply because $S(M)\subset\Gamma^{\mathrm{ch}}(\pi,S)$
has the lowest weight.
The second line follows immediately from (\ref{ass.S.0})
and (\ref{ass.S.gen.fields}).
\end{proof}

\begin{prop} \label{prop.S.fields.relations}
Recall the maps $\bullet$, $\{\;\}$, $\{\;\}_\Omega$ in
Proposition \ref{prop.ass.C.VAoid}.
For $f,g\in C^\infty(M)$, $X,Y\in\mc{T}(M)$ and
$n,m\in\mathbb{Z}$, we have the normal-ordered
expansions
\begin{align*}
(fg)_n=f_{n-r}g_r,\quad
c(fX)_n=f_{n-r}\,cX_r,\quad
\ell(fX)_n=\sum_{r\geq 0} f_{n-r}\,\ell X_r
  +\sum_{r<0}\ell X_r f_{n-r}
  -(\ell X\bullet f)_n
\end{align*}
as well as the supercommutation relations
\begin{align*}
&[f_n,g_m]=0,\qquad
[cX_n,f_m]=0,\qquad
[\ell X_n,f_m]=(Xf)_{n+m} \\
&[\ell X_n,cY_m]=c(\nabla_X Y)_{n+m},\qquad
[cX_n,cY_m]=-2\langle X,Y\rangle_{n+m} \\
&[\ell X_n,\ell Y_m]=\ell[X,Y]_{n+m}
  -c\,\Omega(\widetilde X,\widetilde Y)^P_{n+m}
  +(\{\ell X,\ell Y\}_\Omega)_{n+m}
  +n\{\ell X,\ell Y\}_{n+m}
\end{align*}
where $\langle\;\;\rangle$ and $\nabla$ denote
the Riemannian metric and Levi-Civita connection on $TM$.
Notice that the canonical isomorphism
$\mc{T}_v(P)^{\mathfrak{so}_{2d'}}\cong\Gamma(\mathrm{End}\,TM)$
identifies $-\Omega(\widetilde X,\widetilde Y)^P$ with the Riemannian curvature
operator $R_{X,Y}=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$.
\end{prop}

\begin{proof}
Let us verify the relations on the level of
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$.
For the calculations below, keep in mind
(\ref{freeVA.comm})--(\ref{freeVA.NOP}) and
(\ref{ass.S.gen.fields}).
The first two normal-ordered expansions are
easy, e.g.
\begin{align*}
(\widetilde{fX})^i_{n-s}\otimes e_{i,s}
=f_{n-s-t}\widetilde X^i_t\otimes e_{i,s}
=(f_{n-r}\otimes 1)(\widetilde X^i_{r-s}\otimes e_{i,s}).
\end{align*}
The remaining normal-ordered expansion follows immediately from
the equation
\begin{align*}
\widetilde{fX}-\tau_i(\widetilde{fX})^j\Theta_{ij}
=(\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij})_{-1}f
-\ell X\bullet f
\end{align*}
obtained from the first calculation in the proof of Proposition
\ref{prop.ass.C.VAoid}.

For the supercommutators, the first three are easy.
The next two are computed as follows, using (\ref{DX.lift})
and (\ref{Cl}) respectively:
\begin{align*}
&\big[(\widetilde X-\tau_i\widetilde X^j\Theta_{ij})_n\otimes 1,\,
  \widetilde Y^k_{m-r}\otimes e_{k,r}\big] \\
&\qquad\qquad
=\big[\widetilde X_n,\widetilde Y^k_{m-r}]
  \otimes e_{k,r}
=(\widetilde X^i\tau_i\widetilde Y^k)_{n+m-r}\otimes e_{k,r}
=(\widetilde{\nabla_X Y})^k_{n+m-r}\otimes e_{k,r} \\
&\big[\widetilde X^i_{n-r}\otimes e_{i,r},\,
  \widetilde Y^j_{m-s}\otimes e_{j,s}\big] \\
&\qquad\qquad
=\widetilde X^i_{n-r}\widetilde Y^j_{m-s}\otimes[e_{i,r},e_{j,s}]
=-2\widetilde X^i_{n-r}\widetilde Y^i_{m+r}\otimes 1
=-2\langle X,Y\rangle_{n+m}\otimes 1
\end{align*}
The last (super)commutator follows immediately from the
equations
\begin{align*}
&\big(\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij}\big)_1
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\Theta_{k\ell}\big)
=\{\ell X,\ell Y\} \\
&\big(\widetilde X-\tau_i\widetilde X^j\,\Theta_{ij}\big)_0
\big(\widetilde Y-\tau_k\widetilde Y^\ell\,\Theta_{k\ell}\big)\\
&\qquad\qquad
=\big(\widetilde{[X,Y]}-\tau_i\widetilde{[X,Y]}{}^j\Theta_{ij}\big)
-\Omega(\widetilde X,\widetilde Y)^P
-\lambda_0(\Omega(\widetilde X,\widetilde Y),\Theta)
+\{\ell X,\ell Y\}_\Omega
\end{align*}
together with (\ref{ass.S.other.fields});
these equations are the results of the second and third
calculations in the proof of Proposition
\ref{prop.ass.C.VAoid},
applied to our current setting where $\lambda^*=\lambda_0$ and
$\lambda_\rho=2\lambda_0$ (see \S\ref{sec.ass.S}).
\end{proof}

The following construction is manufactured using
precisely the information about $\Gamma^{\mathrm{ch}}(\pi,S)$ we have
gathered so far:
the subspace $S(M)$,
the fields in \S\ref{sec.ass.S.fields} and
their relations in Propositions \ref{prop.S.fields.ground}
and \ref{prop.S.fields.relations}.

\begin{subsec} \label{sec.ass.S.free}
{\bf Comparing with a generators-and-relations construction.}
Let $\mathcal{U}$ be a unital $\mathbb{Z}/2\mathbb{Z}$-graded associative
algebra with generators
\begin{align*}
\textrm{even}:\quad
f_n,\;\,
\ell X_n,\;\,
\alpha_n,\;\,
c\mathscr{Y}_n;\qquad\quad
\textrm{odd}:\quad
cX_n
\end{align*}
for $f\in C^\infty(M)$, $X\in\mc{T}(M)$,
$\alpha\in\Omega^1(M)$, $\mathscr{Y}\in\mc{T}_v(P)^{\mathfrak{so}_{2d'}}$
and $n\in\mathbb{Z}$, such that
(i) $1_n=\delta_{n,0}$,
(ii) $f\mapsto f_n$, $X\mapsto\ell X_n$, $\cdots$ are linear and
(iii) they satisfy the supercommutation relations in Proposition
\ref{prop.S.fields.relations}.
The subalgebra $\mathcal{U}_+\subset\mathcal{U}$ generated by
$\{f_n,\,\ell X_n,\,cX_n\}_{n\geq 0}$ has an action on $S(M)$ as
described in Proposition \ref{prop.S.fields.ground}.
Let $\widetilde W^f=\mathcal{U}\otimes_{\mathcal{U}_+}S(M)$ and
$W^f=\widetilde W^f/\sim$ be the quotient obtained by imposing the
normal-ordered expansions in (\ref{ass.S.1formfield}),
Lemma \ref{lemma.verfield} and Proposition
\ref{prop.S.fields.relations}.
Define an operator $L^f_0$ on $W^f$ by
\begin{align} \label{freemod.wt}
L^f_0|_{S(M)}=\frac{d'}{8},\qquad
[L^f_0,f_n]=-nf_n,\qquad
[L^f_0,\ell X_n]=-n\,\ell X_n,\qquad
[L^f_0,cX_n]=-n\,cX_n
\end{align}
which are consistent with all the above relations;
its eigenvalues are called weights.

By construction, there is a unique linear map
\begin{eqnarray*}
\Phi:W^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,S)
\end{eqnarray*}
that restricts to the identity on $S(M)$
\footnote{
Since the composition
$\widetilde W^f\twoheadrightarrow W^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,S)$ restricts
to the identity on $1\otimes S(M)\subset\widetilde W^f$, the quotient
map must be injective there.
Hence we may indeed identify $S(M)$ as a subspace of $W^f$.
}
and respects all the above operators.
Clearly $\Phi$ is weight-preserving.
As we will see later, $\Phi$ is in fact an isomorphism and
thus provides a generators-and-relations description
of $\Gamma^{\mathrm{ch}}(\pi,S)$.
\end{subsec}

{\it Remark.}
The relations imposed in the construction of $W^f$ are not
subject to any higher relations.
For example, given $X,Y\in\mc{T}(M)$ and $s\in S(M)$,
the expression $[\ell X_0,cY_0]s\,$ can be evaluated by
either taking the commutator first or applying the action
on $S(M)$ first, but the results are identical automatically
due to the fact that the Levi-Civita connection is compatible
with Clifford multiplication.~\cite{spin}
Similarly, other composites of relations (e.g.~Jacobi identities)
are already implied by the geometric data.

\begin{subsec} \label{sec.ass.S.Virasoro}
{\bf The Virasoro action.}
According to \S\ref{sec.ass.S} and Lemma \ref{lemma.G.S},
the following operators
\begin{align}
\textstyle
(\tau_{i,-1}\tau^i)_n\otimes 1
+1\otimes L^{C\ell}_n
-\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}
-\frac{1}{2}\lambda_0(\Theta_{-1}\Theta)_n\otimes 1,\qquad
n\in\mathbb{Z}
\label{ass.S.Virasoro}
\end{align}
on $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ induce a Virasoro action
on $\Gamma^{\mathrm{ch}}(\pi,S)$ of central charge $5d'$;
denote the induced operators by $L^M_n$.
The key to understanding the entire structure of
$\Gamma^{\mathrm{ch}}(\pi,S)$ is the observation that $L_0^M$ is
generated by the fields described in
\S\ref{sec.ass.S.fields} (in a certain way).
\end{subsec}

For the following statement, recall that $\pi:P\rightarrow M$
is by assumption a lifting of the orthogonal frame bundle of $TM$
(see \S\ref{sec.ass.S}).

\begin{prop} \label{prop.ass.S.wt}
Let $U\subset M$ be an open subset and
$\sigma:U\rightarrow\pi^{-1}(U)\subset P$ a smooth
section of $\pi$.
Both weight operators $L^f_0$ on $W^f$
(see \S\ref{sec.ass.S.free}) and $L^M_0$ on $\Gamma^{\mathrm{ch}}(\pi,S)$
have the local expression:
\footnote{
Recall from \S\ref{sec.review} that, like most
constructions in this paper, both $W^f$ and $\Gamma^{\mathrm{ch}}(\pi,S)$
are the spaces of global sections of underlying sheaves.
}
\begin{align}
&\sum_{r\geq 0}\sigma^i_{-r}(\ell\sigma_i)_r
+\sum_{r<0}(\ell\sigma_i)_r\sigma^i_{-r}
+\frac{3}{4}\,\mathrm{Tr}\,(\Gamma^\sigma_{-r}\Gamma^\sigma_r)
+\sigma^i([\sigma_j,\sigma_k])_{-r}\,\sigma^k_{-s}
  (\Gamma^\sigma_{ji})_{r+s} \nonumber \\
&\qquad\quad
-\frac{1}{2}\sum_{r>0}r(c\sigma_i)_{-r}(c\sigma_i)_r
+\frac{1}{4}(c\sigma_i)_{-r}(c\sigma_j)_{-s}
  (\Gamma^\sigma_{ij})_{r+s}
+\frac{d'}{8}
\label{ass.S.wt.fields}
\end{align}
where $(\sigma_1,\ldots,\sigma_{2d'})$ is the
$C^\infty(U)$-basis of $\mc{T}(U)$ induced by $\sigma$;
$(\sigma^1,\ldots,\sigma^{2d'})$ the dual basis of
$\Omega^1(U)$;
and $\Gamma^\sigma=\sigma^*\Theta\,\in
\Omega^1(U)\otimes\mathfrak{so}_{2d'}$.
\end{prop}

\begin{proof}
First consider $L^M_0$.
As explained above, $L^M_0$ is represented in
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ by
$(\ref{ass.S.Virasoro})_{n=0}$, which we would
like to split into two parts:
\begin{align*}
\textstyle
\ell'
=\big(\tau_{i,-1}\tau^i
  -\frac{1}{2}\,\mathrm{Tr}\,(\Theta_{-1}\Theta)\big)_0\otimes 1,
\qquad
\ell''
=1\otimes L^{C\ell}_0
-\Theta^a_{-r}\otimes t^{C\ell}_{a,r}
+\frac{1}{4}\,\mathrm{Tr}\,(\Theta_{-1}\Theta)_0\otimes 1.
\end{align*}
Since the calculation in the proof of Proposition
\ref{prop.ass.C.conformal} is valid for any principal frame
algebra, $\ell'$ can be expressed as the zeroth mode of
(\ref{ass.C.conformal.local}).
Our main task is to express $\ell''$ entirely in terms of
$\sigma$ as well.

For the calculations below, keep in mind
(\ref{freeVA.comm})--(\ref{freeVA.NOP}) and adopt again the
notations in the proof of Proposition \ref{prop.ass.C.conformal}.
Let $\theta=g^{-1}dg$.
Also let us introduce a (slight abuse of) notation:
\begin{align} \label{frame.Cl}
(c\sigma_i)_r
=\widetilde\sigma_{i,r-s}^j\otimes e_{j,s}
=g_{ji,r-s}\otimes e_{j,s},\qquad
i=1,\ldots,2d',\;\;r\in\mathbb{Z}.
\end{align}
By (\ref{Cl}) these operators satisfy the anticommutation
relations
\begin{align} \label{frame.Cl.comm}
[(c\sigma_i)_r,(c\sigma_j)_s]
=-2\delta_{ij}\delta_{r+s,0}.
\end{align}
Now the first term of $\ell''$ can be written as the first
line below by (\ref{Cl.Virasoro}) and (\ref{frame.Cl}), and
the subsequent rearrangement is valid because every sum in sight
is finite when applied to an arbitrary element:
\begin{align*}
1\otimes L^{C\ell}_0
&=\frac{1}{4}\bigg(
  -\sum_{r\geq 0}r\,(c\sigma_j)_s(c\sigma_k)_t
  +\sum_{r<0}r\,(c\sigma_k)_t(c\sigma_j)_s\bigg)
  \big(g_{ij,-r-s}g_{ik,r-t}\otimes 1\big)
+\frac{d'}{8} \\
&=\frac{1}{4}\bigg(
  -\sum_{t\geq 0}r\,(c\sigma_j)_s(c\sigma_k)_t
  +\sum_{t<0}r\,(c\sigma_k)_t(c\sigma_j)_s
  -\sum_{r\geq 0>t}
    r\,[(c\sigma_j)_s,(c\sigma_k)_t] \\
&\qquad\qquad\qquad
+\sum_{r<0\leq t}
  r\,[(c\sigma_j)_s,(c\sigma_k)_t]\bigg)
  \big(g_{ij,-r-s}g_{ik,r-t}\otimes 1\big)
+\frac{d'}{8}
\end{align*}
For the first two sums above, write $r=t+(r-t)$ and sum
over $r$:
\begin{align*}
\textrm{1st sum}\;
&=-\frac{1}{4}\sum_{t\geq 0}
  t\,(c\sigma_j)_{-t}(c\sigma_j)_t
+\frac{1}{4}\sum_{t\geq 0}
  (c\sigma_j)_s(c\sigma_k)_t
  \big(\theta_{jk,-s-t}\otimes 1\big) \\
\textrm{2nd sum}
&=\frac{1}{4}\sum_{t<0}
  t\,(c\sigma_j)_t(c\sigma_j)_{-t}
-\frac{1}{4}\sum_{t<0}
  (c\sigma_k)_t(c\sigma_j)_s
  \big(\theta_{jk,-s-t}\otimes 1\big)
\end{align*}
For the other two sums above, apply (\ref{frame.Cl.comm}) and
rearrange:
\begin{align*}
\textrm{3rd\,+\,4th sums}
&=\frac{1}{2}\bigg(
  \sum_{r\geq 0>t}-\sum_{r<0\leq t}\bigg)\,
  r\,g_{ij,-r+t}g_{ij,r-t}\otimes 1
=\frac{1}{4}\sum_u(u^2-u)\,
  g_{ij,-u}g_{ij,u}\otimes 1 \\
&=-\frac{1}{4}\big(\mathrm{Tr}\,(dg^{-1})_{-1}(dg)\big)_0\otimes 1
=\frac{1}{4}\,\mathrm{Tr}\,(\theta_{-1}\theta)_0
  \otimes 1
\end{align*}
These calculations together yield the following expression
for the first term of $\ell''$:
\begin{align} \label{ClL0.local}
1\otimes L^{C\ell}_0
=-\frac{1}{2}\sum_{t>0}t(c\sigma_j)_{-t}(c\sigma_j)_t
+\frac{1}{4}(c\sigma_j)_s(c\sigma_k)_t
  (\theta_{jk,-s-t}\otimes 1)
+\frac{1}{4}\mathrm{Tr}\,(\theta_{-1}\theta)_0\otimes 1
+\frac{d'}{8}\,.
\end{align}
On the other hand, the second term of $\ell''$ can be written
as follows by (\ref{Cl.so}) and (\ref{frame.Cl}), where
the normal ordering of $e_{i,r-s}e_{j,s}$ guarantees
the validity of the subsequent step:
\begin{align*}
-\Theta^a_{-r}\otimes t^{C\ell}_{a,r}
&=\frac{1}{4}\Theta_{ij,-r}\otimes e_{i,r-s}e_{j,s}
=\frac{1}{4}\sum_{s\geq 0}
  \Theta_{ij,-r}\otimes e_{i,r-s}e_{j,s}
-\frac{1}{4}\sum_{s<0}
  \Theta_{ij,-r}\otimes e_{j,s}e_{i,r-s} \\
&=\frac{1}{4}\bigg(
  \sum_{s\geq 0}(c\sigma_k)_t(c\sigma_\ell)_u
  -\sum_{s<0}(c\sigma_\ell)_u(c\sigma_k)_t\bigg)
  \big((g^{-1}\Theta)_{kj,-s-t}\,g_{j\ell,s-u}
    \otimes 1\big)
\end{align*}
These sums can be handled in a similar way as in the calculation
of $1\otimes L_0^{C\ell}$.
Let us omit the details and just write down the result:
\begin{align} \label{Theta.Cl.local}
-\Theta^a_{-r}\otimes t^{C\ell}_{a,r}
=\frac{1}{4}\,(c\sigma_k)_t(c\sigma_\ell)_u
  \big((g^{-1}\Theta\,g)_{k\ell,-t-u}\otimes 1\big)
+\frac{1}{2}\,\mathrm{Tr}\,\big((g^{-1}\Theta\,g)_{-1}\theta\big)_0
  \otimes 1.
\end{align}
Then it follows from (\ref{ClL0.local}), (\ref{Theta.Cl.local})
and (\ref{dg}) that
\begin{align*}
\ell''
=-\frac{1}{2}\sum_{t>0}
  t(c\sigma_j)_{-t}(c\sigma_j)_t
+\frac{1}{4}(c\sigma_j)_s(c\sigma_k)_t
  (\Gamma^\sigma_{jk,-s-t}\otimes 1)
+\frac{1}{4}\mathrm{Tr}\,(\Gamma^\sigma_{-1}\Gamma^\sigma)_0
  \otimes 1
+\frac{d'}{8}\,.
\end{align*}
Combining this with the earlier comment on $\ell'$ and keeping
in mind (\ref{ass.S.gen.fields})--(\ref{ass.S.other.fields}),
we see that $\ell'+\ell''$ represents the operator
(\ref{ass.S.wt.fields}).
This proves the claim for $L^M_0$.

Since the expression in (\ref{ass.S.wt.fields}) is generated
by the operators in
(\ref{ass.S.gen.fields})--(\ref{ass.S.other.fields}),
it determines an operator $L'_0$ on $W^f$.
To show $L'_0=L^f_0$, we need to verify that $L'_0$ satisfies
(\ref{freemod.wt}) using only the relations in
(\ref{ass.S.1formfield}), Lemma \ref{lemma.verfield}
and Propositions
\ref{prop.S.fields.ground}--\ref{prop.S.fields.relations}.
This is straightforward (but uninteresting).
\end{proof}

\begin{corollary} \label{cor.ass.S.free}
The linear map $\Phi:W^f\rightarrow\Gamma^{\mathrm{ch}}(\pi,S)$
(see \S\ref{sec.ass.S.free}) is an isomorphism.
\end{corollary}

\begin{proof}
Let $f\in C^\infty(M)$, $\alpha\in\Omega^1(M)$
and $X\in\mc{T}(M)$.
By definition, $\Phi$ is an isomorphism on
the lowest weight $d'/8$.
Assume that $\Phi$ is an isomorphism on all weights up to
$(d'/8)+k-1$ for some $k>0$.
Let $u\in\Gamma^{\mathrm{ch}}(\pi,S)_k$.
For reason of weight as well as
(\ref{ass.S.1formfield}), elements of the form
\begin{align} \label{lowered}
f_n u,\;\ell X_n u,\;cX_n u\;\textrm{ for }n>0,\qquad
\alpha_n u\;\textrm{ for }n\geq 0
\end{align}
belong to the image of $\Phi$;
then by Proposition \ref{prop.ass.S.wt} so does
$ku=(L^M_0-d'/8)u$.
This proves the surjectivity of $\Phi$ on weight $(d'/8)+k$.
On the other hand, let $u\in W^f_k\cap\ker\Phi$.
Since the elements (\ref{lowered}) are also in the kernel of
$\Phi$, for reason of weight and (\ref{ass.S.1formfield}) again
they must be trivial;
then by Proposition \ref{prop.ass.S.wt} so is $ku=(L^f_0-d'/8)u$.
This proves the injectivity of $\Phi$ on weight $(d'/8)+k$.
By induction, $\Phi$ is an isomorphism on all weights.
\end{proof}

The following summarizes our analysis of $\Gamma^{\mathrm{ch}}(\pi,S)$.

\begin{theorem} \label{thm.ass.S}
Suppose $M^{2d'}$ is a Riemannian manifold with
a spin structure $\pi:P\rightarrow M$ and
a $3$-form $H$ satisfying $dH=\frac{1}{2}\mathrm{Tr}\,(R\wedge R)$,
where $R$ is the Riemannian curvature.
Let $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)$ be the associated principal
frame $(\widehat{\mathfrak{so}}_{2d'},\mathrm{Spin}_{2d'})$-algebra
(see Theorem \ref{thm.CDOP}), $S$ the spinor representation
of $\widehat{\mathfrak{so}}_{2d'}$, and
\begin{align*}
\Gamma^{\mathrm{ch}}(\pi,S)
=H^{\frac{\infty}{2}+0}\big(
  \widehat{\mathfrak{so}}_{2d'}\,,
  \,\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S\big).
\end{align*}
Then $\Gamma^{\mathrm{ch}}(\pi,S)$ contains $S(M)$ as a subspace
(see \S\ref{sec.ass.S.0}) and admits three particular types
of fields $\t{\sf Y}(f,x)$, $\t{\sf Y}(\ell X,x)$, $\t{\sf Y}(cX,x)$ associated to
$f\in C^\infty(M)$ and $X\in\mc{T}(M)$, whose Fourier modes
are represented in $\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$ by
the following operators: (see \S\ref{sec.ass.S.fields})
\begin{align*}
f_n\otimes 1,\qquad
(\widetilde X-\tau_i\widetilde X^j\Theta_{ij})_n\otimes 1,\qquad
\widetilde X^i_{n-r}\otimes e_{i,r},\qquad
n\in\mathbb{Z}.
\end{align*}
In fact, $\Gamma^{\mathrm{ch}}(\pi,S)$ is generated from $S(M)$
by the actions of these fields,
subject only to the relations stated in Propositions
\ref{prop.S.fields.ground}--\ref{prop.S.fields.relations}
(see also (\ref{ass.S.1formfield}) and Lemma
\ref{lemma.verfield}).
Moreover, there is a Virasoro action on $\Gamma^{\mathrm{ch}}(\pi,S)$ of central
charge $5d'$, induced by the following operators on
$\mc{D}^{\mathrm{ch}}_{\Theta,H}(P)\otimes S$: (see \S\ref{sec.ass.S.Virasoro})
\begin{align*}
\textstyle
\qquad
(\tau_{i,-1}\tau^i)_n\otimes 1
+1\otimes L^{C\ell}_n
-\Theta^a_{n-r}\otimes t^{C\ell}_{a,r}
-\frac{1}{4}\mathrm{Tr}\,(\Theta_{-1}\Theta)_n\otimes 1,\qquad
n\in\mathbb{Z}.
\qquad
\qedsymbol
\end{align*}
\end{theorem}

The generators-and-relations description of $\Gamma^{\mathrm{ch}}(\pi,S)$
allows us to define a filtration and identify the
associated graded space.

\begin{subsec} \label{ass.S.PBW}
{\bf PBW filtration.}
Given an increasing sequence of negative integers
$\mf{n}=\{n_1\leq\cdots\leq n_s<0\}$, possibly empty,
let us write
\begin{align*}
|\mf{n}|=|n_1|+\cdots+|n_s|\quad (0\textrm{ if }\mf{n}=\{\}),\qquad
\mf{n}(i)=\textrm{number of times }i\textrm{ appears in }\mf{n}
\end{align*}
and regard $\mf{n}$ as a partition of $-|\mf{n}|$.
For any nonnegative integer $w$, let $\mathscr{I}_w$ be the set
of triples $(\mf{n};\mf{m};\mf{p})$ of such sequences with
$|\mf{n}|+|\mf{m}|+|\mf{p}|=w$ and $\mf{m}(i)\leq 2d'$ for all $i<0$.
Define a partial ordering on $\mathscr{I}_w$ by declaring
that $(\mf{n};\mf{m}\;\mf{p})\prec(\mf{n}';\mf{m}';\mf{p}')$ if one of the
following is true:
\vspace{-0.05in}
\begin{itemize}
\item[$\cdot$]
$|\mf{n}|<|\mf{n}'|$, or $|\mf{n}|=|\mf{n}'|$ and $|\mf{m}|<|\mf{m}'|$
\vspace{-0.08in}
\item[$\cdot$]
$\mf{n}'$ is a proper subpartition of $\mf{n}$, $\mf{m}=\mf{m}'$
and $\mf{p}=\mf{p}'$
\vspace{-0.08in}
\item[$\cdot$]
$\mf{n}=\mf{n}'$, $\mf{m}=\mf{m}'$ and $\mf{p}$ is a proper subpartition
of $\mf{p}'$
\vspace{-0.05in}
\end{itemize}
For example in $\mathscr{I}_3$, a particular ascending chain is
\begin{align*}
(;;-2,-1)
\prec(;;-3)
\prec(;-1,-1;-1)
\prec(-2;-1;)
\prec(-1,-1;-1;)
\end{align*}
while $(;;-1,-1,-1)$ and $(-1,-1,-1;;)$ are
the unique minimal and maximal elements.

Given a sequence $\mf{n}=\{n_1\leq\cdots\leq n_s<0\}$
as above and an $s$-tuple
$\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_s)$ in
$\Omega^1(M)$, or an $s$-tuple
$\boldsymbol{X}=(X_1,\ldots,X_s)$ in $\mc{T}(M)$, let us introduce
the notations
\begin{align*}
\boldsymbol{\alpha}_\mf{n}=\alpha_{1,n_1}\cdots\alpha_{s,n_s},\quad
\boldsymbol{cX}_\mf{n}=(cX_1)_{n_1}\cdots(cX_s)_{n_s},\quad
\boldsymbol{\ell X}_\mf{n}=(\ell X_1)_{n_1}\cdots(\ell X_s)_{n_s}\quad
(1\textrm{ if }\mf{n}=\{\})
\end{align*}
which are operators on $\Gamma^{\mathrm{ch}}(\pi,S)$ (see
\S\ref{sec.ass.S.fields}).
It follows from Theorem \ref{thm.ass.S} that for $w\geq 0$
we have
\begin{align*}
\Gamma^{\mathrm{ch}}(\pi,S)_w
=\textrm{span}\Big\{
  \boldsymbol{\ell X}_\mf{n}\boldsymbol{cY}_\mf{m}\boldsymbol{\alpha}_\mf{p}\, s:
  (\mf{n};\mf{m}\;\mf{p})\in\mathscr{I}_w;\,
  \textrm{all suitable }\boldsymbol{\alpha},\boldsymbol{X},\boldsymbol{Y};\,
  s\in S(M)
\Big\}.
\end{align*}
Indeed, as $f_n=-\frac{1}{n}(df)_n$ for
$f\in C^\infty(M)$ and $n\neq 0$, Propositions
\ref{prop.S.fields.ground} and \ref{prop.S.fields.relations}
allow us to express every element of $\Gamma^{\mathrm{ch}}(\pi,S)$ in the
indicated form.
For $(\mf{n};\mf{m};\mf{p})\in\mathscr{I}_w$ consider the subspaces
\begin{align*}
\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}
&=\textrm{span}\big\{
  \boldsymbol{\ell X}_{\mf{n}'}\boldsymbol{cY}_{\mf{m}'}\boldsymbol{\alpha}_{\mf{p}'} s:
  (\mf{n}';\mf{m}';\mf{p}')\preceq(\mf{n};\mf{m};\mf{p})
\big\}\subset\Gamma^{\mathrm{ch}}(\pi,S)_w \\
\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}
&=\textrm{span}\big\{
  \boldsymbol{\ell X}_{\mf{n}'}\boldsymbol{cY}_{\mf{m}'}\boldsymbol{\alpha}_{\mf{p}'} s:
  (\mf{n}';\mf{m}';\mf{p}')\prec(\mf{n};\mf{m};\mf{p})
\big\}\subset\Gamma^{\mathrm{ch}}(\pi,S)_w
\end{align*}
For the next statement, let $O$ (and $O'$)
stand for one of the operators of the form
$\alpha_n$, $cX_n$ or $\ell X_n$ with $n<0$, and
$fO$ the corresponding operator $(f\alpha)_n$,
$c(fX)_n$ or $\ell(fX)_n$, where
$f\in C^\infty(M)$.
The subspaces $\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}$ and
$\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}$ have the following
properties:
\begin{align*}
\begin{array}{crcrl}
\textrm{(i)}&\;\;
\cdots OO'\cdots\in\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}&
\;\;\Rightarrow&
\cdots[O,O']\cdots\in\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}\phantom{,}& \\
\textrm{(ii)}&
\cdots Os\in\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}&
\;\;\Rightarrow& \phantom{\Big(}
\cdots\big((fO)s-O(fs)\big)\in\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})},&
\quad s\in S(M)
\end{array}
\end{align*}
Indeed (i) follows from the supercommutation
relations in Proposition \ref{prop.S.fields.relations}
and (ii) from the normal-ordered expansions there as
well as Proposition \ref{prop.S.fields.ground}.
Consequently, there is a natural isomorphism
\begin{align*}
&\mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}/\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})} \\
&\qquad\qquad
\cong
\bigg(\bigotimes_{i<0}\textrm{Sym}^{\mf{p}(i)}\Omega^1(M)\bigg)\otimes
\bigg(\bigotimes_{i<0}\textrm{Sym}^{\mf{n}(i)}\mc{T}(M)\bigg)\otimes
\bigg(\bigotimes_{i<0}\wedge^{\mf{m}(i)}\mc{T}(M)\bigg)\otimes S(M)
\end{align*}
where all tensor, symmetric and exterior products
are over $C^\infty(M)$.
Let $q$ be a formal variable.
When all $(\mf{n};\mf{m}\;\mf{p})\in\mathscr{I}_w$ and all $w\geq 0$
are considered, we obtain
\begin{align}
&\bigoplus_{w\geq 0}\bigg(
  q^w\bigoplus_{(\mf{n};\mf{m};\mf{p})\in\mathscr{I}_w}
  \mathcal{F}_{\preceq(\mf{n};\mf{m};\mf{p})}/\mathcal{F}_{\prec(\mf{n};\mf{m};\mf{p})}
\bigg) \nonumber \\
&\qquad\qquad\quad\cong
\bigg(\bigotimes_{k\geq 1}\textrm{Sym}_{q^k}\Omega^1(M)\bigg)
\otimes\bigg(\bigotimes_{k\geq 1}\textrm{Sym}_{q^k}\mc{T}(M)\bigg)
\otimes\bigg(\bigotimes_{k\geq 1}\wedge_{q^k}\mc{T}(M)\bigg)
\otimes S(M) \label{ass.S.AG}
\end{align}
where $\textrm{Sym}_t=\sum_{n=0}^\infty t^n\textrm{Sym}^n$
and $\wedge_t=\sum_{n=0}^{2d'}t^n\wedge^n$ as usual.
\footnote{
If the partial ordering on $\mathscr I_w$ is extended to
a total ordering, the latter will induce
a filtration on $\Gamma^{\mathrm{ch}}(\pi,S)_w$ whose associated
graded space is the coefficient of $q^w$ in
(\ref{ass.S.AG}).
}
\end{subsec}

To conclude this section, let us use an example to
illustrate a speculation about $\Gamma^{\mathrm{ch}}(\pi,S)$ that we plan
to investigate in a future paper.

\begin{subsec} \label{ass.S.sconformal}
{\bf The case of $\mathbb{R}^{2d'}$:~(0,1) superconformal structure.}
When $M=\mathbb{R}^{2d'}$ with the standard spin structure and
$H$ is the trivial $3$-form, $\Gamma^{\mathrm{ch}}(\pi,S)$ can be identified
with $\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})\otimes S$.
Recall the definition of $\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})$ from
\S\ref{sec.CDO.Rd}.
Then it is easy to check that the following operators on
$\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})\otimes S$
\begin{align*}
&\textstyle
L_n=\frac{1}{4}
  \big((\partial_i+db^i)_{-1}(\partial_i+db^i)\big)_n
  \otimes 1 \\
&\textstyle
\bar L_n=-\frac{1}{4}
  \big((\partial_i-db^i)_{-1}(\partial_i-db^i)\big)_n
  \otimes 1
  +1\otimes L_n^{C\ell},
\qquad n\in\mathbb{Z} \\
&\textstyle
\bar G_n=\frac{1}{2}
  (\partial_i-db^i)_{n-r}\otimes e_{i,r}
\end{align*}
satisfy the $(0,1)$ (Ramond) superconformal algebra
of central charges $(2d',3d')$, namely
\begin{align*}
&\textstyle
[L_n,L_m]=(n-m)L_{n+m}
  +\frac{d'}{6}(n^3-n)\delta_{n,-m} \\
&[L_n,\bar L_m]=[L_n,\bar G_m]=0 \phantom{\big(}\\
&\textstyle
[\bar L_n,\bar L_m]=(n-m)\bar L_{n+m}
  +\frac{d'}{4}(n^3-n)\delta_{n,-m} \\
&\textstyle
[\bar L_n,\bar G_m]=\big(\frac{n}{2}-m\big)\bar G_{n+m},
\qquad
[\bar G_n,\bar G_m]=2\bar L_{n+m}
  +\frac{d'}{4}(4n^2-1)\delta_{n,-m}
\end{align*}
Notice that the diagonal Virasoro action given by
$L_n+\bar L_n$ is precisely the one described in
\S\ref{sec.ass.S.Virasoro}.

Let us consider the operator $\slashed D=2\bar G_0$ and its kernel.
Since $\slashed D^2=4(\bar L_0-d'/8)$, on $\ker\slashed D$
we have $\bar L_0=d'/8$.
Notice that $\mc{D}^{\mathrm{ch}}(\mathbb{R}^{2d'})\otimes S$ can be generated from
$S(\mathbb{R}^{2d'})=C^\infty(\mathbb{R}^{2d'})\otimes S_0$ by the
actions of $(\partial_i+db^i)_n$, $(\partial_i-db^i)_n$ and $e_{i,n}$ for $n<0$.
Also notice that $\bar L_0\geq d'/8$ on $S(\mathbb{R}^{2d'})$ and
\begin{align*}
[\bar L_0,(\partial_i+db^i)_n]=0,\qquad
[\bar L_0,(\partial_i-db^i)_n]=-n(\partial_i+db^i)_n,\qquad
[\bar L_0,e_{i,n}]=-ne_{i,n}.
\end{align*}
All these imply that $\ker\slashed D$ must be contained in the
subspace generated from $S(\mathbb{R}^{2d'})$ by $(\partial_i+db^i)_n$
only.
This subspace can be identified with
\begin{align} \label{spinors.Rd.subspace}
\bigg(
  \bigotimes_{k\geq 1}\textrm{Sym}_{q^k}\mc{T}(\mathbb{R}^{2d'})
\bigg)\otimes S(\mathbb{R}^{2d'})
\end{align}
where the various products are taken over
$C^\infty(\mathbb{R}^{2d'})$ and the power of $q$ corresponds to
the eigenvalue of $L_0+\bar L_0-d'/8$.
Since $\slashed D$ also commutes with $(\partial_i+db^i)_n$,
it follows that the restriction of $\slashed D$
to (\ref{spinors.Rd.subspace}) is simply a classical
twisted Dirac operator on $\mathbb{R}^{2d'}$.
Moreover, all $L_n$ commute with $\slashed D$ and hence induce
a Virasoro action on $\ker\slashed D$ of central charge $2d'$.
\end{subsec}

{\it Remark.}
The author speculates that, in general, the Virasoro action
on $\Gamma^{\mathrm{ch}}(\pi,S)$ described in \S\ref{sec.ass.S.Virasoro}
always extends to an action of the $(0,1)$
superconformal algebra (which includes a Dirac-Ramond operator
$\slashed D=2\bar G_0$);
and if $M$ is closed, a more sophisticated version of the argument
above would prove that
\begin{align*}
\textrm{Virasoro character of }\ker\slashed D
&:=q^{-d'/12}\times
\textrm{supertrace of }q^{L_0}\textrm{ on }\ker\slashed D \\
&=q^{-d'/12}\times
\textrm{supertrace of }q^{L_0+\bar L_0-d'/8}\textrm{ on }
  \ker\slashed D \\
&=q^{-d'/12}\times\hat{A}\bigg(M,\,
  \bigotimes_{k\geq 1}\textrm{Sym}_{q^k}TM\otimes{\mathbb{C}}
  \bigg) \\
&=\eta(q)^{-2d'}\times\textrm{Witten genus of }M
\end{align*}
where $\eta(q)$ the Dedekind $\eta$-function.
This would provide at least a partial mathematical
formulation of the original, physical interpretation
of the Witten genus.
\cite{Witten.ell,Witten.index}
It is believed that such a description of the Witten genus
may lead to various geometric applications as well as a better
understanding of its family version in elliptic cohomology.
\cite{Hopkins.ICM}

\newpage

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  {\sc\large\filcenter}
  {Appendix \S\thesection.}{.5em}{}

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