text:
\section{Introduction}
The Pure Spinor formalism \cite{Berkovits2000} is a formulation of the superstring that allows for quantization with manifest super-Poincar\'e symmetry. In flat space, the action describes a free theory of matter fields $(Z^M) = (x^m, \t^\a, \th^{\widehat\alpha})$ and bosonic spinors (ghosts) $\l^\a$, ${\widehat\lambda}^{\widehat\alpha}$ subject to the pure spinor constraints $\l^\a {\gamma}^m_{\a\b} \l^\b = {\widehat\lambda}^{\widehat\alpha} {\gamma}^m_{{\widehat\alpha}{\widehat\beta}} {\widehat\lambda}^{\widehat\beta} = 0$. And also, there exist BRST charges $Q$ and ${\widehat Q}$ that encode the physical states of the theory in their cohomology.

It was shown in \cite{Berkovits2001} that given the Pure Spinor string in a general curved background, the conditions of nilpotency and holomorphicity of the BRST currents imply the Type II Supergravity equations of motion. One characteristic of the Pure Spinor sigma model is that the matter and ghost fields enter the action in different ways; this is,  while the matter fields have a second order kinetic term, the ghosts $\l^\a$ and ${\widehat\lambda}^{\widehat\alpha}$ are in first order form. Thus, the computations leading to the Type II SUGRA equations of motion did not show a clear geometrical interpretation.

In this paper we fill this gap by analyzing the Pure Spinor action in Hamiltonian form. The existence of this form of the action will initially depend on the invertibility of the Ramond-Ramond background field $P^{\a{\widehat\alpha}}$ that appears in the original action but we will see later that this condition can be dropped and still describe a general Type II background. This version of the action will have as target space a graded supermanifold $M$ parametrized by matter and ghosts $(x^m, \t^{\mu}, \th^{\widehat\mu}, \l^\a, {\widehat\lambda}^{\widehat\alpha})$ where both type of fields are treated on the same footing.

In understanding the action of sigma models in Hamiltonian form, we need to make use of the concept of a generalized metric that first appeared in \cite{Hitchin2002} in the context of Generalized Complex Geometry \cite{Gualtieri2004}. The generalized metric was one of the elements used to extend Kahler geometry. Although, in the Physics literature, an equivalent structure had already appeared in the work of Gates, Hull and Rocek \cite{Gates1984} when studying $N=(2,2)$ supersymmetric sigma models (See \cite{Zabzine2006} for a review). In essence, for sigma models with target space $M$, a generalized metric gives a way of encoding both a Riemann metric $G$ and a two-form $B$ in a single tensor structure defined on $TM\oplus T^*M$ known as generalized tangent space. However, this ``generalized geometry" is limited in the sense that only allows to describe systems that posses a second order formulation leaving aside the ones with only a first order action or a mixture of both such as the pure spinor action. We will show that these sigma models with different formulations can be treated in a unified manner if we extend the original definition of a generalized metric to a one with a non-definite signature. Such an extension was given in \cite{Severa2018} but was only used to study sigma models with second order formulations having pseudo-Riemannian metrics.

When the symmetries of sigma models are studied, as well as their algebra of currents, there naturally appears the notion of Dorfman brackets $[\![\cdot,\cdot]\!]$ \cite{Alekseev2004,Nekrasov2005}. It was shown in \cite{Severa2017} that given $S = \int (G+B)_{mn}\partial x^m \bar\partial x^n $, the conditions for $\d x^m = V^m(x)$ to be a symmetry (i.e. ${\mathcal L}_V G = 0$ and ${\mathcal L}_V B = dF$ for some one-form $F_m(x)$) become equivalent to imposing the preservation of the subbundles $graph(\pm G+B)\subset TM\oplus T^*M$ under the action of $[\![ (V, F),\ \cdot\ ]\!]$. In the context of generalized geometry, these subbundles $graph(\pm G+B)$ are the $(\pm 1)$-eigenspaces $(TM\oplus T^*M)_\pm$ of a generalized metric. We will show that a similar situation occurs when we deal with the generalized metric with non-definite signature where these eigenspaces cannot be solved to be the graphs of some tensors anymore. Though still, the conditions of $(V,F)$ generating a symmetry will be equivalent to either the preservation of these eigenspaces or the fact that the Lie derivative of $(V,F)$ \cite{Kotov2010} on the generalized metric gives zero.

This paper is organized as follows. In section 2, we start with the sigma model of the bosonic string in curved space $M$ and study its action in Hamiltonian form. We see there is a matrix that characterizes the theory known as a {\it generalized metric} $\AA$ on $TM\oplus T^*M$. Then, we propose this form of the action as a starting point such as to include theories that do not admit a second order form. Lastly, for this generalization we obtain that the conditions to have holomorphic and nilpotent currents can be written roughly in terms of the Lie derivative of $\AA$ and the nilpotency of a section of $TM\oplus T^*M$ w.r.t $[\![\cdot,\cdot]\!]$.

In section 3, we take the action of the Pure Spinor string in a general curved background and then transform it into its Hamiltonian form. Initially we assume that this is only valid for backgrounds with an invertible Ramond-Ramond field $P^{\a{\widehat\alpha}}$ but later we show that this assumption can be dropped (since every term containing $P^{-1}_{{\widehat\alpha}\a}$ gets cancelled). Moreover, we verify that the matrix $\AA_{PS}$ appearing in the pure spinor action (that encodes the vielbein and background fields $(\O, {\widehat\Omega}, C, {\widehat C}, P, S)$) satisfy the properties of a generalized metric. This section finishes by applying the nilpotency and holomorphicity conditions from the section prior to this one and rederiving the Type II SUGRA constraints. We finish discussing the implications that the ghost grading $(\l^\a,{\widehat\lambda}^{\widehat\alpha})$ has on the generalized metric.

In section 4 we give our conclusions. We leave for the appendices the following subjects. A discussion on the relation between worldsheet conformal symmetry of the action and the fact that a generalized metric $\AA$ belongs to the orthosymplectic supergroup (when written as a super-matrix). Some theorems and proofs on about generalized metrics. Also, our conventions for super-geometry are given at the end.

\section{Holomorphic and nilpotent currents}
In this section we study the mathematical structures needed to write the conditions of holomorphicity and nilpotency of currents in terms of associated sections on $TM\oplus T^*M$.

\subsection{Sigma models in Hamiltonian form}
Let's start our discussion with the sigma model action of a string moving in a target space $M$ parametrized by coordinates $(x^m)$
  \begin{equation}
  S[x] = \frac12 \int \bar\partial x^n \partial x^m (G_{mn} + B_{mn})
  \end{equation}
It is known that a transformation $\d x^m = {\varepsilon} V^m(x)$ is a symmetry of the action when ${\mathcal L}_V G = 0$ and ${\mathcal L}_V B = dF$ for some one-form $F_m(x)$. Such a symmetry provide us with a conserved current $(j_z,j_{\bar z})$, and the expression for each component can be computed using Noether's procedure
  \begin{align}
  j_z & =  \frac12 \partial x^n (G_{nm} + B_{nm})V^m + \frac12 \partial x^n F_n \label{jz} \\
  j_{\bar z} & = \frac12 \bar\partial x^n (G_{nm} - B_{nm})V^m - \frac12 \bar\partial x^n F_n \label{jzb}
  \end{align}

We can also use the Hamiltonian form of the action which breaks explicit worldsheet conformal symmetry. The action in Hamiltonian form becomes
  \begin{equation}\label{hamil}
  S = \int \partial_t x^m p_m - \frac12 (\partial_{\sigma} x^m\ p_m)\left(
    \begin{array}{cc}
    (G -BG^{-1}B)_{mn} & -(BG^{-1})_m{}^n \\
    (G^{-1}B)^m{}_n & (G^{-1})^{mn}
    \end{array}
  \right) \left(
    \begin{array}{c}
    \partial_{\sigma} x^n \\
    p_n
    \end{array}
  \right)
  \end{equation}
and the components of the Noether current (\ref{jz}) and (\ref{jzb}) are now
  \begin{align}
  j_z & = \frac12 (V\ F)\left[\left(
    \begin{array}{cc}
    I & 0 \\
    0 & I
    \end{array}
  \right) +  \left(
    \begin{array}{cc}
    -BG^{-1} & G-BG^{-1} B \\
    G^{-1} & G^{-1}B
    \end{array}
  \right) \right] \left(
    \begin{array}{c}
    p \\
    \partial_{\sigma} x
    \end{array}
  \right) \label{current-1} \\
  j_{\bar z} & = \frac12  (V\ F) \left[ \left(
    \begin{array}{cc}
    I & 0 \\
    0 & I
    \end{array}
  \right) - \left(
    \begin{array}{cc}
    -BG^{-1} & G-BG^{-1} B \\
    G^{-1} & G^{-1}B
    \end{array}
  \right) \right] \left(
    \begin{array}{c}
    p \\
    \partial_{\sigma} x
    \end{array}
  \right) \label{current-2}
  \end{align}
The matrices in the action and Noether current composed of the background fields $G,B,G^{-1}$ can be understood in terms of the so-called {\it generalized geometry}. Intuitively it can be understood as follows. Just in the same sense that Riemannian geometry deals with manifolds and metrics defined on their tangent spaces, {\it generalized} geometry will care about a manifold $M$ with a {\it generalized} metric defined on its {\it generalized} tangent space $TM\oplus T^*M$.

The matrix appearing in the action (\ref{hamil}) is what we will call the matrix representation of the generalized metric
  \begin{equation}\label{gen-met-2nd}
  A := \left(
    \begin{array}{cc}
    (G -BG^{-1}B)_{mn} & -(BG^{-1})_m{}^n \\
    (G^{-1}B)^m{}_n & (G^{-1})^{mn}
    \end{array}
  \right)
  \end{equation}
while the actual generalized metric is (in an abstract sense) a symmetric bilinear form on $TM\oplus T^*M$.
This matrix has the following defining%
\footnote{
  These are defining in the sense that any bilinear form $A$ on $TM\oplus T^*M$ with those properties will necessarily be of the form (\ref{gen-met-2nd}). This is, it will be associated to tensors $G_{mn}$ and $B_{mn}$.
}\label{footnote}
properties: it is symmetric and positive-definite, and it satisfies the $O(d,d)$ condition
  \begin{equation}\label{odd-condition}
  A^t \left(
    \begin{array}{cc}
    0 & 1 \\
    1 & 0
    \end{array}
  \right) A = \left(
    \begin{array}{cc}
    0 & 1 \\
    1 & 0
    \end{array}
  \right)
  \end{equation}
where $d$ is the dimension of the target space $M$. Furthermore, to the generalized metric we can associate an endomorphism $U_A$ on $TM\oplus T^*M$ with matrix representation given by
  \begin{equation}\label{invo-operator}
  U_A := A \left(
    \begin{array}{cc}
    0 & 1 \\
    1 & 0
    \end{array}
  \right) = \left(
    \begin{array}{cc}
    -BG^{-1} & G-BG^{-1} B \\
    G^{-1} & G^{-1}B
    \end{array}
  \right)
  \end{equation}
This matrix is the one appearing in both components of the Noether current (\ref{current-1}-\ref{current-2}). The operator $U_A$ has the property of being an involution i.e.
  \begin{equation}
  U_A^2 = \left(
    \begin{array}{cc}
    1 & 0 \\
    0 & 1
    \end{array}
  \right)
  \end{equation}
which is a direct consequence of the $O(d,d)$ condition (\ref{odd-condition}) that $A$ satisfies.

So far what we have done is to take a sigma model in second order formalism and transform its action into its Hamiltonian form. This process can be reversed by solving the equations of motion for $P_m$. However, we are interested in theories that do not admit a second order formulation (i.e. the ones that are purely first order or a mixture of both which the Pure Spinor string is an example of) i.e. theories where the generalized metric $A$ is not of the form (\ref{gen-met-2nd}) anymore. To study these types of theories we need to modify the definition of generalized metric.

The original definition of a generalized metric \cite{Hitchin2002} assumes that $A$ is of the form (\ref{gen-met-2nd}). This assumption can be understood in the following way. When we consider the associated operator to the generalized metric $A$
  \begin{equation}\label{associated}
  U_A := A \left(
    \begin{array}{cc}
    0 & 1 \\
    1 & 0
    \end{array}
  \right) = \left(
    \begin{array}{cc}
    A_m{}^n & A_{mn} \\
    A^{mn} & A^m{}_n
    \end{array}
  \right)
  \end{equation}
we have that is an involution i.e. $(U_A)^2 = 1$, and so we have $(\pm 1)$-eigenspaces which are defined as
  \begin{equation}\label{bundles}
  (TM\oplus T^*M)_\pm := \left\{ (V^m\ F_m)\ |\ (V^m\ F_m) \left(
    \begin{array}{cc}
    A_m{}^n & A_{mn} \\
    A^{mn} & A^m{}_n
    \end{array}
  \right) = \pm (V^n\ F_n) \right\}
  \end{equation}
providing a way to express the bundle $(TM\oplus T^*M)$ as a direct sum $(TM\oplus T^*M) = (TM\oplus T^*M)_+ \oplus (TM\oplus T^*M)_-$. It turns out that these eigenspaces can always be solved as the graphs of some operators $(\pm G+B)$ i.e.
  \begin{equation}
  (TM\oplus T^*M)_\pm = graph(\pm G+B):=\{ (\xi^m\ \a_m)\ |\ \a_m = \xi^n (\pm G_{nm} + B_{nm}) \}
  \end{equation}
for $G$ a metric tensor and $B$ a two-form. This ultimately implies that the action possesses a second order formulation (See Appendix \ref{gen-met} for a proof). We will see that in the case of the Pure Spinor sigma model, due to the target space being a graded supermanifold, it becomes impossible to represent these eigenspaces as graphs of some operators $\pm G +B$ without some entries in $G$ and $B$ having negative powers of $\l^\a$ and ${\widehat\lambda}^{\widehat\alpha}$. The easiest way to see this is to consider the case of flat space which is analyzed in section \ref{flat}.

In particular we must relax the condition of positivity of $A$ to admit an arbitrary signature. Thus, we define%
  \footnote{
  There are three different but equivalent definitions of a generalized metric. These equivalencies are also extended for the generalized metric with indefinite signature. We refer the reader to Appendix \ref{gen-met}.
  } %
the generalized metric with signature $(d+a,d-a)$ \cite{Severa2018} as a bilinear form in $TM\oplus T^*M$ such that its matrix representation $A$ satisfies
  \begin{enumerate}
  \item being symmetric
  \item having signature $(d+a, d-a)$
  \item the $O(d,d)$ property
  \end{enumerate}
Notice that for the value $a = d$ the signature becomes $(2d, 0)$ which is the same as positivity, and we are back to the original definition. However, unlike in the positive definite case, the eigenspaces $(TM\oplus T^*M)_\pm$ are not graphs of some tensors in general. For this to happen, the following two conditions must be satisfied. First of all, the dimension of $(TM\oplus T^*M)_\pm$ should be equal to $d = dim(TM) = dim(T^*M)$; and secondly, their intersections with the tangent $TM$ and contangent bundle $T^*M$ should be trivial. Equivalently,
  \begin{align}
  (TM\oplus T^*M)_\pm = graph(\pm G_{\bullet\bullet} + B_{\bullet\bullet}) \quad & \Leftrightarrow\quad A^{mn}\ \text{invertible}\\
  (TM\oplus T^*M)_\pm = graph(\pm R^{\bullet\bullet} + S^{\bullet\bullet}) \quad & \Leftrightarrow \quad A_{mn}\ \text{invertible}
  \end{align}
where $G$ and $R$ are symmetric while $B$ and $S$ anti-symmetric. From the sigma model point of view, we are only interested in the first case since this is the one that determines if the action has a second order formulation.

Finally, we claim that, for the bosonic case, the following setting is the correct generalization to study in a unified manner sigma models that mix second and first order formulations. We start with an action already in Hamiltonian form characterized by a generalized metric $A$ of indefinite signature
  \begin{equation}
  S[x,p] = \int \partial_t x^m P_m - \frac12 (\partial_{\sigma} x^m\ P_m)\left(
    \begin{array}{cc}
    A_{mn} & A_m{}^n \\
    A^m{}_n & A^{mn}
    \end{array}
  \right) \left(
    \begin{array}{c}
    \partial_{\sigma} x^n \\
    P_n
    \end{array}
  \right)
  \end{equation}
Included in the definition of generalized metric is the $O(d,d)$ property (\ref{odd-condition}) that implies that this action is $2d$ conformally invariant. Furthermore, depending on the rank of the matrix $A^{mn}$, the action can be rewritten as a manifestly conformal invariant theory that mixes a first and second order formulation
  \begin{equation}
  S \sim \int \Pi_z^k \Pi_{\bar z}^\ell (\eta_{k\ell} + b_{k\ell} ) + \frac12 \Pi_z^\ell p_{{\bar z} \ell} + \frac12 \Pi_{\bar z}^\ell p_{z \ell}\ ,\quad k,\ell=1,\ldots,rk[A^{mn}]
  \end{equation}
where $A^{mn} = e^m_k \eta^{k\ell} e^n_\ell$ and $\Pi_{z,{\bar z}}^\ell = \partial_{z,{\bar z}} x^m e_m^\ell$, $(p_{z,{\bar z}})_\ell=e^m_\ell(P_{t m}\pm P_{{\sigma} m})$. See Appendix \ref{app-ortho} for details on the relation of the $O(d,d)$ condition and conformal invariance.

It also happens that the group $O(d,d)$ acts%
  \footnote{
  This action of $O(d,d)$ on the space of generalized metrics is not in general a symmetry of the worldsheet theory.
  }
on the space of generalized metrics. Given $\mathcal O\in O(d,d)$, the generalized metric $A$ transforms as
  \begin{equation}
  A \quad \longrightarrow \quad \mathcal O^t A \mathcal O
  \end{equation}
In particular, it can be shown that the space of positive-definite generalized metrics coincides with the orbit of the unit matrix under this action i.e. $A = \mathcal O^t \mathcal O$. And, for cases where $A$ has indefinite signature, we need to consider other orbits.

To deal with the symmetries of this action in Hamiltonian form, we define a current $(j_z,j_{\bar z})$ associated to a section $(V^m\ F_m)$ of $TM\oplus T^*M$ following the form of (\ref{current-1}-\ref{current-2})
  \begin{equation}
    \begin{aligned}
    j_z & := (V\ F) {\mathcal P}_+ \left(
      \begin{array}{c}
      P \\
      \partial_{\sigma} x
      \end{array}
    \right) \\
    j_{\bar z} & := (V\ F) {\mathcal P}_- \left(
      \begin{array}{c}
      P \\
      \partial_{\sigma} x
      \end{array}
    \right)
    \end{aligned}
  \end{equation}
where ${\mathcal P}_\pm$ are the projectors%
\footnote{
  This projectors are acting on the section $(V\ F)$ from the right. We chose this convention to emulate the one that we will use for the super-case in the next section.
}
onto the ($\pm 1$)-eigenspaces (\ref{bundles}) of $U_A$
  \begin{equation}\label{projectors}
  {\mathcal P}_\pm := \frac12( I \pm U_A) = \frac12 \left[ \left(
    \begin{array}{cc}
    \d_m{}^n & 0 \\
    0 & \d^m{}_n
    \end{array}
  \right) \pm \left(
    \begin{array}{cc}
    A_m{}^n & A_{mn} \\
    A^{mn} & A^m{}_n
    \end{array}
  \right) \right]
  \end{equation}
The conservation of $(j_z,j_{\bar z})$ produce some conditions on the generalized metric $A$ and section $(V,\ F)$ as we will show in a following section.

\subsection{Generalized metrics on supermanifolds}
Let's generalize our proposal to admit fermionic fields. This is, the target space is now a supermanifold $M$ with coordinates $({\mathbb Z}^i)$ $i=1,\ldots,d|1,\ldots,s$ collectively denoting bosonic and fermionic ones. Then we take the action to be
  \begin{equation}\label{action}
  S[{\mathbb Z},{\mathbb P}] = \int \left\{\partial_t {\mathbb Z}^i {\mathbb P}_i - \frac12 (\partial_{\sigma} {\mathbb Z}^i\ {\mathbb P}_i) \left(
    \begin{array}{cc}
    \AA_{ij} & \AA_i{}^j \\
    \AA^i{}_j & \AA^{ij}
    \end{array}
  \right) \left(
    \begin{array}{c}
    \partial_{\sigma} {\mathbb Z}^j (-)^j \\
    {\mathbb P}_j
    \end{array}
  \right)\right\}
  \end{equation}
where $({\mathbb P}_i)$ is the conjugate momentum to $({\mathbb Z}^i)$ and the matrix $\AA$ is the generalized metric on $TM\oplus T^*M$ characterizing the theory. In this super-case, the condition of $\AA$ being symmetric implies that its supermatrix blocks satisfy
  \begin{equation}
  \AA_{ij} = (-)^{ij} \AA_{ji}\ ,\quad \AA_i{}^j = (-)^{ij + j} \AA^j{}_i\ ,\quad \AA^{ij} = (-)^{ij + i +j} \AA^{ji}
  \end{equation}
while the $O(d,d)$ condition for the bosonic case generalizes to be
  \begin{equation}\label{osp-condition}
  \left(
    \begin{array}{cc}
    \AA_{ik} & \AA_i{}^k \\
    \AA^i{}_k & \AA^{ik}
    \end{array}
  \right) \left(
    \begin{array}{cc}
    0 & \d^k{}_l \\
    \d_k{}^l & 0
    \end{array}
  \right) \left(
    \begin{array}{cc}
    \AA_{lj} & \AA_l{}^j \\
    \AA^l{}_j & \AA^{lj}
    \end{array}
  \right) = \left(
    \begin{array}{cc}
    0 & \d_i{}^j \\
    \d^i{}_j & 0
    \end{array}
  \right)
  \end{equation}
This condition, as in the bosonic case, guarantees that the action (\ref{action}) has conformal invariance although not manifest. However, notice that on supermanifolds there is no notion of signature of a biliner form $\AA$, so the signature condition on $\AA$ gets dropped from the definition of generalized metric. And, to determine if the theory admits or not a second order formulation i.e. if the $(\pm 1)$-eigenspaces of ${\mathbb U}_{\AA}$ can be written as the graphs of some metric and two-form tensors $(\pm{\mathbb G}_{ij} + {\mathbb B}_{ij})$, we need to directly analyze the invertibility of $\AA^{ij}$.

As an example we can consider
  \begin{equation}
  \AA^{ij} = E_{a,\a}{}^i \left(
    \begin{array}{cc}
    \eta^{ab} & 0 \\
    0 & \eta^{\a\b}
    \end{array}
  \right) E_{b,\b}{}^j
  \end{equation}
where the labels $a,b,\a,\b$ are thought as flat indices, and the blocks $\eta^{ab}$, $\eta^{\a\b}$ are respectively invertible and non-invertible. Then, of course, because of a non-invertible part in $\AA^{ij}$ there is not a second order formulation but using (\ref{osp-condition}) this action could still be rewritten in a manifestly conformal invariant way
  \begin{equation}
  S\sim \int \Pi_z^a (\eta_{ab} + A_a{}^c\eta_{cb})\Pi_{\bar z}^b + \frac12\Pi^\a_z d_{{\bar z} \a} + \frac12\Pi^\a_{\bar z} d_{z \a} + \frac12\Pi^\a_z A_\a{}^b \eta_{ba} \Pi^a_{\bar z} - \frac12\Pi^\a_{\bar z} A_\a{}^b\eta_{ba}\Pi^a_z
  \end{equation}
where $\Pi_{z,{\bar z}}^a = \partial_{z,{\bar z}} {\mathbb Z}^i E_i{}^a$, $\Pi_{z,{\bar z}}^\a = \partial_{z,{\bar z}} {\mathbb Z}^i E_i{}^\a$ and $(d_{z,{\bar z}})_\a = E_\a{}^i({\mathbb P}_{t i} \pm {\mathbb P}_{{\sigma} i})$. Furthermore, we can also show that when $\AA$ is written as a supermatrix (diagonal bosonic and off-diagonal fermionic blocks), equation (\ref{osp-condition}) becomes equivalent to the defining property of the orthosymplectic supergroup $OSp(d,d|2s)$. See Appendix \ref{app-ortho} for more details.

Just like in the bosonic case, we also define the components of a super current $(j_z, j_{\bar z})$ as
  \begin{align}
  j_z & :=  ({\mathbb V}^i\ {\mathbb F}_i) \left(
    \begin{array}{cc}
    ({\mathcal P}_+)_i{}^j & ({\mathcal P}_+)_{ij} \\
    ({\mathcal P}_+)^{ij} & ({\mathcal P}_+)^i{}_j
    \end{array}
  \right) \left(
    \begin{array}{c}
    {\mathbb P}_j \\
    \partial_{\sigma} {\mathbb Z}^j(-)^j
    \end{array}
  \right) \\
  j_{\bar z} & := ({\mathbb V}^i\ {\mathbb F}_i) \left(
    \begin{array}{cc}
    ({\mathcal P}_-)_i{}^j & ({\mathcal P}_-)_{ij} \\
    ({\mathcal P}_-)^{ij} & ({\mathcal P}_-)^i{}_j
    \end{array}
  \right) \left(
    \begin{array}{c}
    {\mathbb P}_j \\
    \partial_{\sigma} {\mathbb Z}^j(-)^j
    \end{array}
  \right)
  \end{align}
where ${\mathcal P}_\pm$ are the projectors onto the $(\pm 1)$-eigenspaces of ${\mathbb U}_\AA$ denoted by $(TM\oplus T^*M)_\pm$ and defined exactly as in (\ref{projectors})
  \begin{equation}
  {\mathcal P}_\pm := \frac12 \left[ \left(
    \begin{array}{cc}
    \d_i{}^j & 0 \\
    0 & \d^i{}_j
    \end{array}
  \right) \pm \left(
    \begin{array}{cc}
    \AA_i{}^j & \AA_{ij} \\
    \AA^{ij} & \AA^i{}_j
    \end{array}
  \right) \right]
  \end{equation}
And, again we see that this current is associated to a super vector field ${\mathbb V}^i$ and super $1$-form ${\mathbb F}_i$, or equivalently, to a section $({\mathbb V}\ {\mathbb F})$ of $TM\oplus T^*M$.

\subsection{Holomorphicity}
Now we can proceed to analyze the holomorphicity and anti-holomorphicity conditions on currents. First of all, we notice that when the section $({\mathbb V}\ {\mathbb F})$ associated to a current belongs to either eigenspace $(TM\oplus T^*M)_\pm$, one of the components of the current becomes zero. This is,
  \begin{equation}\label{hol}
  ({\mathbb V},{\mathbb F}) \in (TM\oplus T^*M)_+ \quad \Leftrightarrow \quad j_z = {\mathbb V}^i {\mathbb P}_i + {\mathbb F}_i \partial_{\sigma} {\mathbb Z}^i(-)^i\ ,\quad j_{\bar z} = 0
  \end{equation}
or
  \begin{equation}\label{antihol}
  ({\mathbb V},{\mathbb F}) \in (TM\oplus T^*M)_- \quad \Leftrightarrow \quad j_z = 0 \ ,\quad j_{\bar z} = {\mathbb V}^i {\mathbb P}_i + {\mathbb F}_i \partial_{\sigma} {\mathbb Z}^i (-)^i
  \end{equation}
The implication of a section satisfying any of these is that the current conservation condition, i.e. $\bar\partial j_z + \partial j_{\bar z} = 0$, actually becomes the condition of holomorphicity $\bar\partial j_z = 0$ or antiholomorphicity $\partial j_{\bar z} = 0$ for the current.

Finally, to obtain the symmetry conditions (and hence the holomorphiciy conditions), we compute the divergence of the current $(\bar\partial j_z + \partial j_{\bar z})$ using equations of motion
  \begin{align}
  \partial_t {\mathbb Z}^i & = \frac12 \left[ ((-)^i \AA^i{}_j\ (-)^i \AA^{ij}) \left(
    \begin{array}{c}
    \partial_{\sigma} {\mathbb Z}^j(-)^j \\
    {\mathbb P}_j
    \end{array}
  \right) +  (\partial_{\sigma} {\mathbb Z}^j\ {\mathbb P}_j) \left(
    \begin{array}{c}
    \AA_j{}^i \\
    \AA^{ji}
    \end{array}
  \right) \right] \\
  \partial_t {\mathbb P}_i & = -\frac12\left[ (\partial_{\sigma} {\mathbb Z}^k\ {\mathbb P}_k)(-)^{ki}\left(
    \begin{array}{cc}
    \partial_i \AA_{kj} & \partial_i \AA_k{}^j \\
    \partial_i \AA^k{}_j & \partial_i \AA^{kj}
    \end{array}
  \right) \right. \left(
    \begin{array}{c}
    \partial_{\sigma} {\mathbb Z}^j (-)^j \\
    {\mathbb P}_j
    \end{array}
  \right) \nonumber \\
  & \qquad \left. - \partial_{\sigma} \left( (\AA_{ij}\ \AA_i{}^j) \left(
    \begin{array}{c}
    \partial_{\sigma}{\mathbb Z}^j(-)^j \\
    {\mathbb P}_j
    \end{array}
  \right) \right) - \partial_{\sigma}\left( (\partial_{\sigma}{\mathbb Z}^j\ {\mathbb P}_j)\left(
    \begin{array}{c}
    \AA_{ji} \\
    \AA^j{}_i
    \end{array}
  \right) \right) \right]
  \end{align}
This computation results in
  \begin{align}
  \partial j_{\bar z} + \bar\partial j_z & = -\frac12 (\partial_{\sigma} {\mathbb Z}^i\ {\mathbb P}_i) \left[(-)^{i{\mathbb V}} \left(
    \begin{array}{cc}
    {\mathcal L}_{\mathbb V} \AA_{ij} & {\mathcal L}_{\mathbb V} \AA_i{}^j \\
    {\mathcal L}_{\mathbb V} \AA^i{}_j & {\mathcal L}_{\mathbb V} \AA^{ij}
    \end{array}
  \right) \right.\nonumber \\
  & \quad - \left. (-)^{i{\mathbb F}} \left(
    \begin{array}{cc}
    -(d{\mathbb F})_{(i|k} \AA^k{}_{|j)} & -(d{\mathbb F})_{ik} \AA^{kj} \\
    (-)^{(i+k){\mathbb F}} \AA^{ik}(d{\mathbb F})_{kj} & 0
    \end{array}
  \right) \right] \left(
    \begin{array}{c}
    \partial_{\sigma} {\mathbb Z}^j(-)^j \\
    {\mathbb P}_j
    \end{array}
  \right)
  \end{align}
And, since the parities of ${\mathbb V}$ and ${\mathbb F}$ are the same, we can extract the conditions to have a conserved current
  \begin{equation}\label{holomorphicity}
    \begin{aligned}
    ({\mathcal L}_{\mathbb V} \AA)_{ij} & = -[(d{\mathbb F})_{ik} \AA^k{}_j + (-)^{ij}(d{\mathbb F})_{jk} \AA^k{}_i] \\
    ({\mathcal L}_{\mathbb V} \AA)_i{}^j & = -(d{\mathbb F})_{ik}\AA^{kj} \\
    ({\mathcal L}_{\mathbb V} \AA)^{ij} & = 0
    \end{aligned}
  \end{equation}
In terms of the associated ${\mathbb U}_\AA$, these equation can be written as
  \begin{equation}\label{act-operator}
  {\mathcal L}_{({\mathbb V},\ {\mathbb F})} {\mathbb U} := {\mathcal L}_{\mathbb V} {\mathbb U} + \left[ \left(
    \begin{array}{cc}
    0 & d{\mathbb F} \\
    0 & 0
    \end{array}
  \right) ,{\mathbb U} \right] = 0
  \end{equation}
where $[\cdot, \cdot]$ is a commutator. The operation defined in (\ref{act-operator}) can be understood as the Lie derivative of ${\mathbb U}$ along the section $({\mathbb V},\ {\mathbb F})$, and it can be shown that it also behaves as a derivation i.e. ${\mathcal L}_{({\mathbb V},\ {\mathbb F})} ({\mathbb U}_1 {\mathbb U}_2) = ({\mathcal L}_{({\mathbb V},\ {\mathbb F})}{\mathbb U}_1) {\mathbb U}_2 + {\mathbb U}_1 ({\mathcal L}_{({\mathbb V},\ {\mathbb F})} {\mathbb U}_2)$.

As we already mentioned, this set of equations that determine if a section generates a symmetry becomes the holomorphicity (or antiholomorphicity) conditions when $({\mathbb V}\ {\mathbb F})$ associated to our current belongs to the subbundle $(TM\oplus T^*M)_+$ or $(TM\oplus T^*M)_-$ i.e. when the section is an eigenvector of ${\mathbb U}_\AA$.

\subsection{Nilpotency}
In this part we generalize the results of \cite{Alekseev2004,Nekrasov2005} to the case where the target space is a supermanifold. This is, given currents of the form
  \begin{equation}
  j_{({\mathbb V}, {\mathbb F})} := {\mathbb V}^i {\mathbb P}_i + {\mathbb F}_i \partial_{\sigma} {\mathbb Z}^i (-)^i
  \end{equation}
we can compute their Poisson brackets to obtain
  \begin{align}
  \left\{j_{({\mathbb V}_1, {\mathbb F}_1)}({\sigma}_1), j_{({\mathbb V}_2, {\mathbb F}_2)}({\sigma}_2) \right\}_{PB} & = j_{[\![({\mathbb V}_1, {\mathbb F}_1),({\mathbb V}_2,{\mathbb F}_2)]\!]}\d({\sigma}_1-{\sigma}_2) \nonumber \\
  & \qquad - \langle({\mathbb V}_1,{\mathbb F}_1),({\mathbb V}_2,{\mathbb F}_2)\rangle\d'({\sigma}_1-{\sigma}_2)
  \end{align}
where the brackets $[\![\cdot,\cdot]\!]$ and $\langle\cdot,\cdot\rangle$ are the Dorfman bracket and canonical inner product in $TM\oplus T^*M$. Explicitly these brackets are
  \begin{align}
  [\![({\mathbb V}_1, {\mathbb F}_1),({\mathbb V}_2,{\mathbb F}_2)]\!] & = \left([{\mathbb V}_1, {\mathbb V}_2],\ {\mathcal L}_{{\mathbb V}_1} {\mathbb F}_2 - (-)^{{\mathbb V}_2 {\mathbb F}_1}\i_{{\mathbb V}_2}d{\mathbb F}_1\right) \\
  \langle({\mathbb V}_1,{\mathbb F}_1),({\mathbb V}_2,{\mathbb F}_2)\rangle & = \i_{{\mathbb V}_1} {\mathbb F}_2 + (-)^{{\mathbb V}_2 {\mathbb F}_1} \i_{{\mathbb V}_2}{\mathbb F}_1
  \end{align}
We can also see that this Dorfman bracket is complementary to the Lie derivative of an operator defined in (\ref{act-operator}) since it can be verified that
  \begin{equation}\label{distributive}
  [\![ ({\mathbb V}_1,\ {\mathbb F}_1), {\mathbb U}({\mathbb V}_2,\ {\mathbb F}_2) ]\!] = ({\mathcal L}_{({\mathbb V}_1,\ {\mathbb F}_1)} {\mathbb U})({\mathbb V}_2,\ {\mathbb F}_2 ) + {\mathbb U}([\![({\mathbb V}_1,{\mathbb F}_1),({\mathbb V}_2,{\mathbb F}_2)]\!])
  \end{equation}
or more compactly, if we denote $\SS = ({\mathbb V},\ {\mathbb F})$, $\SS\cdot {\mathbb U} = {\mathcal L}_\SS {\mathbb U}$ and $\SS_1\cdot\SS_2 = [\![ \SS_1,\SS_2]\!]$, the previous equation becomes $\SS_1\cdot({\mathbb U}(\SS_2)) = (\SS_1\cdot{\mathbb U})(\SS_2) + {\mathbb U}(\SS_1\cdot\SS_2)$.

The Dorfman bracket give us another way of characterizing symmetries. It was shown in \cite{Severa2017} that for a sigma model $S=\int (G+B)\bar\partial x^m \partial x^n$, a section $(V, F)$ generates a symmetry (i.e. ${\mathcal L}_V G =0$ and ${\mathcal L}_V G = dF$) if and only if $[\![ (V, F), \cdot]\!]$ preserves the eigenspaces $graph(\pm G + B)$. Using (\ref{distributive}) this result can be generalized to the following. A section $({\mathbb V}, {\mathbb F})$ generates a symmetry (i.e. ${\mathcal L}_{({\mathbb V},{\mathbb F})} {\mathbb U}_\AA = 0$) of the action (\ref{action}) if and only if the operation $[\![({\mathbb V},\ {\mathbb F}), \cdot]\!]$ preserves the eigenspaces $(TM\oplus T^*M)_\pm$.
  \begin{equation}
  {\mathcal L}_{(V,F)} {\mathbb U} = 0 \quad \Leftrightarrow \quad [\![ (V,\ F),\ \cdot\ ]\!]\ \text{preserves}\ (TM\oplus T^*M)_\pm
  \end{equation}

Now we proceed to investigate the nilpotency of a fermionic current $j_{({\mathbb Q}, {\mathbb F})}$ associated to a section $({\mathbb Q},{\mathbb F})$. Taking its Poisson bracket with itself we get
  \begin{equation}
  \{j_{({\mathbb Q},{\mathbb F})}, j_{({\mathbb Q},{\mathbb F})}\}_{PB} = 0\quad \Longrightarrow \quad j_{[\![({\mathbb Q},{\mathbb F}),({\mathbb Q},{\mathbb F})]\!]} = 0
  \end{equation}
since the inner product $\langle ({\mathbb Q},{\mathbb F}),({\mathbb Q},{\mathbb F})\rangle$ is trivially zero. This condition implies that the Dorfman bracket needs to be zero $[\![ ({\mathbb Q},{\mathbb F}),({\mathbb Q},{\mathbb F}) ]\!] = 0$ which in terms of the vector and $1$-form components reads
  \begin{equation}\label{nilpotency}
  [{\mathbb Q},{\mathbb Q}] = 0\ ,\quad {\mathcal L}_{\mathbb Q} {\mathbb F} + \i_{\mathbb Q} d{\mathbb F} = 0
  \end{equation}
Notice that we are requiring the nilpotency of the currents since for fermionic ones, this is equivalent to nilpotency of its charge $Q = \int d{\sigma} j_{({\mathbb Q}, {\mathbb F})}({\sigma})$.

\section{Pure spinor string}
In this section we will show that the Pure Spinor action in curved background admits a Hamiltonian form, and that its matrix $\AA_{PS}$ satisfies the properties of a generalized metric. Also we will see that $\AA_{PS}$ possesses two eigenvectors that generate the BRST currents of the theory.

The Pure Spinor action in a curved background \cite{Berkovits2001} is written in terms of worldsheet matter fields $(Z^M) = (X^m,\t^{\mu},\th^{\widehat\mu})$ and ghost fields $(\l^\a,{\widehat\lambda}^{\widehat\alpha})$ and momenta $(\o_\a,{\widehat\omega}_{\widehat\alpha})$. Plus, the ghost variables satisfy the pure spinor constraint $(\l {\gamma}^a \l) = ({\widehat\lambda}{\gamma}^a{\widehat\lambda}) = 0$
  \begin{align}
  S_{PS} = \int d^2z &\ \frac12 \bar\partial Z^N \partial Z^M(G_{MN} + B_{MN}) + \bar\partial Z^M E^\a_M d_\a + \partial Z^M E^{\widehat\alpha}_M {\widehat d}_{\widehat\alpha} \nonumber \\
  & \quad + \bar\partial Z^M \O_{M\a}{}^\b \l^\a \o_\b + \partial Z^M \widehat\O_{M{\widehat\alpha}}{}^{\widehat\beta} {\widehat\lambda}^\a{\widehat\omega}_{\widehat\beta} \nonumber \\
  & \quad + P^{\a{\widehat\alpha}} d_\a {\widehat d}_{{\widehat\alpha}} + C_\a^{\b{\widehat\beta}} \l^\a \o_\b {\widehat d}_{\widehat\beta} + {\widehat C}_{\widehat\alpha}^{{\widehat\beta}\b} {\widehat\lambda}^{\widehat\alpha} {\widehat\omega}_{\widehat\beta} {d}_\b \nonumber \\
  & \quad + S_{\a{\widehat\alpha}}^{\b{\widehat\beta}} \l^\a \o_\b {\widehat\lambda}^{\widehat\alpha} {\widehat\omega}_{\widehat\beta} + \o_\a\bar\partial \l^\a + {\widehat\omega}_{\widehat\alpha} \partial {\widehat\lambda}^{\widehat\alpha} \label{ps-action}
  \end{align}
The background fields $E_M{}^A$, $G_{MN}$ and $B_{MN}$ are the vielbein, super-metric and two-form, $\O_{M\a}{}^\b$ and ${\widehat\Omega}_{M{\widehat\alpha}}{}^\b$ are the spin connections, the fields $C^{\a{\widehat\alpha}}_\b$, $\widehat C^{{\widehat\alpha}\a}_{\widehat\beta}$ are related to the gravitino and dilatino field strengths while $P^{\a{\widehat\alpha}}$ encodes the Ramond-Ramond fields in its gamma matrix expansion and $S^{\a{\widehat\alpha}}_{\b{\widehat\beta}}$ is related to the curvature of the manifold.

The other elements of the Pure Spinor formalism are BRST currents given by $j_B = \l^\a d_\a$ and $\widetilde{j}_B = {\widehat\lambda}^{\widehat\alpha} {\widehat d}_{\widehat\alpha}$. To transform it into its Hamiltonian form, we first need to assume that $P^{\a{\widehat\alpha}}$ is invertible%
\footnote{
A backgrounds satisfying this requirement is AdS \cite{Berkovits2010}.
}%
, so we can solve for $d_\a$ and ${\widehat d}_{\widehat\alpha}$ using their equations of motion. The EOM for $d_\a$ and $\widehat d_{\widehat\alpha}$ are
  \begin{align}
  d_\a & = \left( \partial Z^M E_M{}^{\widehat\alpha} + (\l C\o)^{\widehat\alpha} \right) P^{-1}_{{\widehat\alpha}\a} \\
  \hat d_{\widehat\alpha} & = - P^{-1}_{{\widehat\alpha}\a} \left( \bar\partial Z^M E_M{}^\a + ({\widehat\lambda} \widehat C \o)^\a\right)
  \end{align}
and we can replace these expressions back into the action and BRST currents. This will result in the field $P^{-1}_{{\widehat\alpha}\a}$ appearing everywhere; however, when we use the Legendre transform on the matter fields $(Z^M)$ to compute the Hamiltonian form of the action (i.e. solving for $\partial_t Z^M$ in the relation $P_M = \frac{\d L }{\d \partial_t Z^M}$) all terms containing $P^{-1}_{{\widehat\alpha}\a}$ vanish in the action
  \begin{equation}\label{raw-action}
  \begin{aligned}
  S_{PS} = \int\partial_t Z^M P_M + \o_\a & \bar\partial \l^\a + {\widehat\omega}_{\widehat\alpha} \partial {\widehat\lambda}^{\widehat\alpha} \\
  - \frac12 \partial_{\sigma} Z^N \partial_{\sigma} Z^M \Big( G_{MN} - B_{ML}A^{LP} B_{PN} & - E_{(M|}{}^{\widehat\alpha} E_{\widehat\alpha}{}^P B_{P|N)} + E_{(M|}{}^\a E_\a{}^P B_{P|N)}\Big) \\
  -\frac12 (P_M - \l\O_M\o - {\widehat\lambda}{\widehat\Omega}_M{\widehat\omega}) & A^{MN} (P_N - \l\O_N\o - {\widehat\lambda}{\widehat\Omega}_N{\widehat\omega})  \\
  + (\l C\o^{\widehat\alpha} E_{\widehat\alpha}{}^M + {\widehat\lambda}{\widehat C}{\widehat\omega}^\a E_\a{}^M)(P_M - & \l\O_M\o - {\widehat\lambda}{\widehat\Omega}_M{\widehat\omega}) + (\l{\widehat\lambda} S)^{\a{\widehat\alpha}}\o_\a {\widehat\omega}_{\widehat\alpha} \\
  + \partial_{\sigma} Z^N \Big( B_{NP}A^{PM} + E_N{}^{\widehat\alpha} E_{\widehat\alpha}{}^M - & E_N{}^\a E_\a{}^M \Big) (P_M - \l\O_M\o - {\widehat\lambda}{\widehat\Omega}_M{\widehat\omega}) \\
  + \partial_{\sigma} Z^N (\l C\o)^{\widehat\alpha} E_{\widehat\alpha}{}^M B_{MN} + \partial_{\sigma} Z^N ({\widehat\lambda}{\widehat C}{\widehat\omega})^\a E_\a{}^M & B_{MN} + \partial_{\sigma} Z^M ({\widehat\lambda}{\widehat\Omega}_M{\widehat\omega}) - \partial_{\sigma} Z^M (\l\O_M\o)
  \end{aligned}
  \end{equation}
where $A^{MN} = \left( (-)^M E_a{}^M E_b{}^N\eta^{ab} + E_\a{}^M E_{\widehat\alpha}{}^N P^{\a{\widehat\alpha}} - E_{\widehat\alpha}{}^M E_\a{}^N P^{\a{\widehat\alpha}}\right)$. The same occurs when the BRST currents are computed, obtaining expressions free from $P^{-1}_{{\widehat\alpha}\a}$
  \begin{align}
  j_B & = \l^\a E_\a{}^M P_M - \l^\a E_\a{}^M(\l\O_M\o) - \l^\a E_\a{}^M({\widehat\lambda}{\widehat\Omega}_M{\widehat\omega}) - \partial_{\sigma} Z^M B_{MN} \l^\a E_\a{}^N \label{brst-z}\\
  {\widetilde j}_B & = {\widehat\lambda}^{\widehat\alpha} E_{\widehat\alpha}{}^M P_M - {\widehat\lambda}^{\widehat\alpha} E_{\widehat\alpha}{}^M(\l\O_M\o) - {\widehat\lambda}^{\widehat\alpha} E_{\widehat\alpha}{}^M({\widehat\lambda}{\widehat\Omega}_M{\widehat\omega}) - \partial_{\sigma} Z^M B_{MN} {\widehat\lambda}^{\widehat\alpha} E_{\widehat\alpha}{}^N \label{brst-zb}
  \end{align}
Notice that these expressions for the BRST currents are exactly the same as the ones appearing in \cite{Berkovits2001}.

This action (\ref{raw-action}) can be put in the form of (\ref{action}) as we will see in next section, and more importantly we claim that the condition on the invertibility of $P^{\a{\widehat\alpha}}$ can be dropped since even the most degenerate case such as Pure Spinor in flat space (where $P^{\a{\widehat\alpha}}=0$) is described by (\ref{raw-action}).

\subsection{Generalized metric and BRST sections for the Pure Spinor string}
The Pure Spinor action in Hamiltonian form (\ref{raw-action}) can be written as in (\ref{action}) where there is a generalized metric $\AA$ containing the all background fields that characterize the theory. We consider the target space to be parametrized by
  \begin{equation}
  ({\mathbb Z}^i) = (Z^M, \l^\a, {\widehat\lambda}^{\widehat\alpha})
  \end{equation}
matter and ghost fields, while their momentum variables are $({\mathbb P}_i) = (P_M, \o_\a, {\widehat\omega}_{\widehat\alpha})$. Here the bosonic coordinates are $(x^m, \l^\a, {\widehat\lambda}^{\widehat\alpha})$ and the fermionic ones $(\t^{\mu},\th^{\widehat\mu})$. With these considerations, the generalized metric in the Pure Spinor case $\AA_{PS}$ can be read to be
  \begin{equation}\label{ps-met}
  \AA_{PS} = \left(
    \begin{array}{ccc|ccc}
    \AA_{MN}    & 0          & 0             & \AA_M{}^N   & \AA_M{}^\b   & \AA_M{}^{\widehat\beta}  \\
    0           & 0          & 0             & 0           & \d_\a{}^\b   & 0            \\
    0           & 0          & 0             & 0           & 0            & -\d_{\widehat\alpha}{}^{\widehat\beta} \\
    \hline
    \AA^M{}_N   & 0          & 0             & \AA^{MN}    & \AA^{M\b}    & \AA^{M{\widehat\beta}}   \\
    \AA^\a{}_N  & \d^\a{}_\b & 0             & \AA^{\a N}  & \AA^{\a\b}   & \AA^{\a{\widehat\beta}} \\
    \AA^{\widehat\alpha}{}_N & 0          & -\d^{\widehat\alpha}{}_{\widehat\beta} & \AA^{{\widehat\alpha} N} & \AA^{{\widehat\alpha} \b} & \AA^{{\widehat\alpha}{\widehat\beta}}
    \end{array}
  \right)
  \end{equation}
where the expressions for each component block are left to the Appendix \ref{app}.

Next we would like to have some fermionic sections $({\mathbb Q}\ {\mathbb F})$ and $({\widetilde {\mathbb Q}}\ {\widetilde {\mathbb F}})$ that reduce to the BRST currents (\ref{brst-z}-\ref{brst-zb}) after showing that they are eigenvectors of ${\mathbb U}_{(\AA_{PS})}$. By looking at the BRST currents we see that there is a natural guess, and they are given by  $({\mathbb Q}^i , {\mathbb F}_i ) = (Q^M, Q^\a, Q^{\widehat\alpha};\ F_M, F_\a, F_{\widehat\alpha})$ where
  \begin{equation}\label{brst-sect1}
  ({\mathbb Q}^i) = \left(
    \begin{array}{c}
    Q^M \\
    Q^\a \\
    Q^{\widehat\alpha}
    \end{array}
  \right) = \left(
    \begin{array}{c}
    \l^\a E_\a{}^M \\
    - \l^\b E_\b{}^M (\l \O_M)^\a \\
    -\l^\b E_\b{}^M ({\widehat\lambda} {\widehat\Omega}_M)^{\widehat\alpha}
    \end{array}
  \right)\ ,\quad ({\mathbb F}_i) = \left(
    \begin{array}{c}
    F_M \\
    F_\a \\
    F_{\widehat\alpha}
    \end{array}
  \right) = \left(
    \begin{array}{c}
    Q^N B_{NM} \\
    0 \\
    0
    \end{array}
  \right)
  \end{equation}
and similarly for $({\widetilde {\mathbb Q}}^i , {\widetilde{\mathbb F}}_i ) = ({\widetilde Q}^M, {\widetilde Q}^\a, {\widetilde Q}^{\widehat\alpha};\ {\widetilde F}_M, {\widetilde F}_\a, {\widetilde F}_{\widehat\alpha})$, we have
  \begin{equation}\label{brst-sect2}
  ({\widetilde {\mathbb Q}}^i) = \left(
    \begin{array}{c}
    \widetilde Q^M \\
    \widetilde Q^\a \\
    \widetilde Q^{\widehat\alpha}
    \end{array}
  \right) = \left(
    \begin{array}{c}
    {\widehat\lambda}^{\widehat\alpha} E_{\widehat\alpha}{}^M \\
    - {\widehat\lambda}^{\widehat\beta} E_{\widehat\beta}{}^M (\l \O_M)^\a \\
    -{\widehat\lambda}^{\widehat\beta} E_{\widehat\beta}{}^M ({\widehat\lambda} {\widehat\Omega}_M)^{\widehat\alpha}
    \end{array}
  \right)\ ,\quad ({\widetilde {\mathbb F}}_i) = \left(
    \begin{array}{c}
    \widetilde F_M \\
    \widetilde F_\a \\
    \widetilde F_{\widehat\alpha}
    \end{array}
  \right) = \left(
    \begin{array}{c}
    \widetilde Q^N B_{NM} \\
    0 \\
    0
    \end{array}
  \right)
  \end{equation}

Next we state our main results concerning the description of the pure spinor sigma model in terms of generalized geometry. These results can be proven with a lengthy but straightforward computation
  \begin{itemize}
  \item The generalized metric $\AA_{PS}$ corresponding to the Pure Spinor action satisfies the property (\ref{osp-condition})
    \begin{equation}
    \AA_{PS} \left(
      \begin{array}{cc}
      0 & 1 \\
      1 & 0
      \end{array}
    \right) \AA_{PS} = \left(
      \begin{array}{cc}
      0 & 1 \\
      1 & 0
      \end{array}
    \right)
    \end{equation}
    or equivalently, its associated ${\mathbb U}_{(\AA_{PS})}$ satisfies ${\mathbb U}_{(\AA_{PS})}^2 = 1$. This means that the Pure Spinor action $S_{PS}$ in Hamiltonian form still possesses $SO(1,1)$ worldsheet symmetry, and it can be rewritten with manifest conformal invariance which is the original sigma model (\ref{ps-action}). Plus, we also have that when $\AA_{PS}$ is reordered as a super-matrix, it belongs to the orthosymplectic supergroup $OSp(d,d|2s)$.

    \item The sections $({\mathbb Q}^i\ {\mathbb F}_i)$ and $({\widetilde {\mathbb Q}}^i\ {\widetilde {\mathbb F}}_i)$ that we defined in (\ref{brst-sect1}) and (\ref{brst-sect2}) satisfy the property of being eigenvectors of $\mathbb U_{\AA}$ with eigenvalues $(+1)$ and $(-1)$ respectively. In turn this implies that the components of the currents associated to these sections coincide with the BRST currents
      \begin{align}
      ({\mathbb Q}\ {\mathbb F}) & \quad \longrightarrow \quad j_z = j_B\ ,\quad j_{\bar z} = 0 \\
      ({\widetilde {\mathbb Q}}\ {\widetilde {\mathbb F}}) & \quad \longrightarrow \quad j_z = 0\ ,\quad j_{\bar z} = \widetilde j_B
      \end{align}
    Then, as was discussed, the symmetry conditions for each case become holomorphicity and antiholomorphicity conditions for the BRST currents. These equations will ultimately imply the Type II SUGRA constraints (See section \ref{sect-sugra}).

    \item The target space of the pure spinor sigma model is a graded supermanifold (grading given by ghost number). This fact implies that there cannot exist tensors $({\mathbb G}_{ij})_{i,j=M,\a}$ and $({\mathbb B}_{ij})_{i,j=M,\a}$ such that $(TM\oplus T^*M)_\pm$ are solved to be the graph of $(\pm{\mathbb G}+{\mathbb B})$. In other words, the pure spinor sigma model does not admit a second order formulation (See section \ref{grading} for details).
  \end{itemize}

\subsection{Type II SUGRA constraints}\label{sect-sugra}
We now show that for the Pure Spinor case the conditions of nilpotency (\ref{nilpotency}) and holomorphicity (\ref{holomorphicity}) that we found actually imply the Type II SUGRA constraints which of course agrees with the results of \cite{Berkovits2001}.

The constraints of nilpotency (\ref{nilpotency}) in the Pure Spinor case can be simplified further. To avoid a cumbersome computation let's define a $2$-form field
  \begin{equation}
  [{\mathbb B}_{ij}] := \left(
    \begin{array}{ccc}
    B_{MN} & 0 & 0 \\
    0 & 0 & 0 \\
    0 & 0 & 0
    \end{array}
  \right)
  \end{equation}
with field strength
  \begin{equation}
  {\mathbb H}_{ijk} = \left\{
    \begin{array}{ll}
    H_{MNP} := (dB)_{MNP} & ; i=M,\ j=N,\ k=P \\
    0 & ;\ \text{any other case}
    \end{array}
  \right.
  \end{equation}
This means that we can write the $1$-form simply as ${\mathbb F} = \i_{\mathbb Q} {\mathbb B}$, and then the second equation in (\ref{nilpotency}) becomes
  \begin{align}
  {\mathcal L}_{\mathbb Q} (\i_{\mathbb Q}{\mathbb B}) + \i_{\mathbb Q} d(\i_{\mathbb Q}{\mathbb B}) & = \i_{[{\mathbb Q},{\mathbb Q}]} {\mathbb B} - (\i_{\mathbb Q} {\mathcal L}_{\mathbb Q}{\mathbb B}) + \i_{\mathbb Q} d(\i_{\mathbb Q}{\mathbb B}) \nonumber \\
  & = \i_{[{\mathbb Q},{\mathbb Q}]}{\mathbb B} - \i_{\mathbb Q} \i_{\mathbb Q} d{\mathbb B}
  \end{align}
where in the last line we used Cartan's formula ${\mathcal L}_{\mathbb Q} = \i_{\mathbb Q} d + d \i_{\mathbb Q}$. Thus, together with the Lie bracket $[{\mathbb Q}, {\mathbb Q}] = 0$, nilpotency of the BRST current $j_B$ implies the following constraints
  \begin{equation}\label{nil1}
  [{\mathbb Q},{\mathbb Q}] = 0 \ ,\quad \i_{\mathbb Q}\i_{\mathbb Q} {\mathbb H} = 0
  \end{equation}
In a completely analogous way we obtain the rest of constraints for the relations $\{{\widetilde j}_B, {\widetilde j}_B\} = 0$ and $\{j_B, {\widetilde j}_B\} = 0$
  \begin{align}
  [{\widetilde {\mathbb Q}},{\widetilde {\mathbb Q}}] = 0\ ,\quad \i_{\widetilde {\mathbb Q}}\i_{\widetilde {\mathbb Q}} {\mathbb H} = 0 \label{nil2} \\
  [{\mathbb Q}, {\widetilde {\mathbb Q}}] = 0\ ,\quad \i_{\mathbb Q}\i_{\widetilde {\mathbb Q}} {\mathbb H} = 0 \label{nil3}
  \end{align}
Then, by actually computing the Lie brackets we arrive at
  \begin{align}
  \l^\a\l^\b T_{\a\b}{}^C = \l^\a\l^\b\l^{\gamma} R_{\a\b{\gamma}}{}^{\sigma} = \l^\a\l^\b \widehat R_{\a\b{\widehat \gamma}}{}^{\widehat\sigma} = 0 & \ ,\quad \l^\a\l^\b H_{\a\b C} = 0 \label{sugra-nil1} \\
  {\widehat\lambda}^{\widehat\alpha}{\widehat\lambda}^{\widehat\beta} T_{{\widehat\alpha}{\widehat\beta}}{}^C = {\widehat\lambda}^{\widehat\alpha}{\widehat\lambda}^{\widehat\beta} R_{{\widehat\alpha}{\widehat\beta}{\gamma}}{}^{\sigma} = {\widehat\lambda}^{\widehat\alpha}{\widehat\lambda}^{\widehat\beta}{\widehat\lambda}^{\widehat \gamma} \widehat R_{{\widehat\alpha}{\widehat\beta}{\widehat \gamma}}{}^{\widehat\sigma} = 0 & \ ,\quad {\widehat\lambda}^{\widehat\alpha}{\widehat\lambda}^{\widehat\beta} H_{{\widehat\alpha}{\widehat\beta} C} = 0 \label{sugra-nil2}\\
  T_{\a{\widehat\beta}}{}^C = \l^\a\l^{\gamma} R_{\a{\widehat\beta}{\gamma}}{}^{\sigma} = {\widehat\lambda}^{\widehat\alpha}{\widehat\lambda}^{\widehat \gamma} \widehat R_{{\widehat\alpha}\b{\widehat \gamma}}{}^{\widehat\sigma} = 0 & \ ,\quad H_{\a{\widehat\beta} C} = 0 \label{sugra-nil3}
  \end{align}

Finally, using (\ref{holomorphicity}) in the Pure Spinor case, we have that the holomorphicity conditions $\bar\partial j_B = 0$ become
  \begin{equation}\label{hol1}
  ({\mathcal L}_{\mathbb Q} \AA)_{ij} + (d\i_{\mathbb Q} {\mathbb B})_{(i|k} \AA^k{}_{|j)} = 0\ ,\quad ({\mathcal L}_{\mathbb Q} \AA)_i{}^j + (d\i_{\mathbb Q} {\mathbb B})_{ik} \AA^{kj} = 0 \ ,\quad ({\mathcal L}_{\mathbb Q} \AA)^{ij} = 0
  \end{equation}
and the antiholomorphicity ones $\partial {\widetilde j}_B = 0$ become
  \begin{equation}\label{hol2}
  ({\mathcal L}_{\widetilde {\mathbb Q}} \AA)_{ij} + (d\i_{\widetilde {\mathbb Q}} {\mathbb B})_{(i|k} \AA^k{}_{|j)} = 0\ ,\quad ({\mathcal L}_{\widetilde {\mathbb Q}} \AA)_i{}^j + (d\i_{\widetilde {\mathbb Q}} {\mathbb B})_{ik} \AA^{kj} = 0 \ ,\quad ({\mathcal L}_{\widetilde {\mathbb Q}} \AA)^{ij} = 0
  \end{equation}
Using the explicit form of the generalized metric and sections, we can compute these equations. Schematically, the Lie derivative ${\mathcal L}_{({\mathbb Q},{\mathbb F})}$ will act on the vielbein $E_A{}^M$, spin connection $\O_{M\a}{}^\b$ and two-form $B_{MN}$ producing the torsion $\l^{\gamma} T_{{\gamma} A}{}^M$, curvature tensor $\l^{\gamma} R_{{\gamma} M\a}{}^\b$ and the three-form field strength $\l^{\gamma} H_{{\gamma} M N}$. For the first set of equations (\ref{hol1}) we get
  \begin{equation}\label{sugra-hol1}
    \begin{aligned}
    & T_{\a(bc)} = H_{\a bc} = H_{\a {\widehat\beta} {\gamma}} = T_{\a\b c} + H_{\a\b c} = T_{\a{\widehat\beta} c} - H_{\a{\widehat\beta} c} = 0  \\
    & T_{\a b}{}^{\gamma} - T_{\a{\widehat \gamma} b}P^{{\gamma}{\widehat \gamma}} = T_{\a b}{}^{\widehat \gamma} + T_{\a{\gamma} b}P^{{\gamma}{\widehat \gamma}} = T_{\a\b}{}^{\widehat \gamma} + \frac12 H_{\a\b{\gamma}}P^{{\gamma}{\widehat \gamma}} = T_{\a{\widehat\beta}}{}^{\gamma} = 0 \\
    & {\nabla}_\a P^{\b{\widehat\beta}} + T_{\a\rho}{}^\b P^{\rho{\widehat\beta}} - C^{\b {\widehat\beta}}_\a = \widehat R_{a\b {\widehat \gamma}}{}^{\widehat\sigma} - T_{\b\rho a}\widehat C^{{\widehat\sigma}\rho}_{\widehat \gamma} = \widehat R_{\a\b{\widehat \gamma}}{}^{\widehat\sigma} + \frac12 H_{\a\b\bullet} \widehat C^{{\widehat\sigma} \bullet}_{\widehat \gamma} = 0 \\
    & - S_{\a{\widehat\alpha}}^{\b{\widehat\beta}} + \widehat R_{\a{\widehat \gamma}{\widehat\alpha}}{}^{\widehat\beta} P^{\b{\widehat \gamma}} + {\nabla}_\a \widehat C^{{\widehat\beta}\b}_{\widehat\alpha} + T_{\a{\gamma}}{}^\b \widehat C^{{\widehat\beta}{\gamma}}_{\widehat\alpha} = 0 \\
    & \l^\a \l^\b \left( R_{a\a\b}{}^{\gamma} - T_{\a{\widehat\beta} a} C^{{\gamma} {\widehat\beta}}_\b \right) = \l^\a\l^\b R_{{\widehat\beta} \a\b}{}^{\gamma} = 0 \\
    & \l^\a \l^\b \left( {\nabla}_\a C_\b^{{\gamma}{\widehat \gamma}} - R_{\a{\sigma}\b}{}^{\gamma} P^{{\sigma}{\widehat \gamma}}\right) = \l^\a\l^\b \left( {\nabla}_\a S_{\b{\widehat\beta}}^{{\gamma}{\widehat \gamma}} - \widehat R_{\a\bullet{\widehat\beta}}{}^{\widehat \gamma} C_\b^{{\gamma}\bullet} - R_{\a\bullet \b}{}^{{\gamma}} \widehat C^{{\widehat \gamma}\bullet}_{\widehat\beta} \right) = 0
    \end{aligned}
  \end{equation}
and for the other (\ref{hol2})
  \begin{equation}\label{sugra-hol2}
    \begin{aligned}
    & T_{{\widehat\alpha}(bc)} = H_{{\widehat\alpha} bc} = H_{{\widehat\alpha} \b {\widehat \gamma}} = T_{{\widehat\alpha}{\widehat\beta} c} - H_{{\widehat\alpha}{\widehat\beta} c} = T_{{\widehat\alpha}\b c} + H_{{\widehat\alpha}\b c} = 0  \\
    & T_{{\widehat\alpha} b}{}^{\widehat \gamma} - T_{{\widehat\alpha}{\gamma} b}P^{{\gamma}{\widehat \gamma}} = T_{{\widehat\alpha} b}{}^{\gamma} + T_{{\widehat\alpha}{\widehat \gamma} b}P^{{\gamma}{\widehat \gamma}} = T_{{\widehat\alpha}{\widehat\beta}}{}^{\gamma} + \frac12 H_{{\widehat\alpha}{\widehat\beta}{\widehat \gamma}}P^{{\gamma}{\widehat \gamma}} = T_{{\widehat\alpha}\b}{}^{\widehat \gamma} = 0 \\
    & {\nabla}_{\widehat\alpha} P^{\b{\widehat\beta}} + T_{{\widehat\alpha}\widehat\rho}{}^{\widehat\beta} P^{\b\widehat\rho} - \widehat C^{{\widehat\beta} \b}_{\widehat\alpha} = R_{a{\widehat\beta} {\gamma}}{}^{\sigma} - T_{{\widehat\beta}\widehat\rho a} C^{{\sigma}\widehat\rho}_{\gamma} = R_{{\widehat\alpha}{\widehat\beta}{\gamma}}{}^{\sigma} + \frac12 H_{{\widehat\alpha}{\widehat\beta}\bullet} C^{{\sigma} \bullet}_{\gamma} = 0 \\
    & - S_{\a{\widehat\alpha}}^{\b{\widehat\beta}} + R_{{\widehat\alpha}{\gamma}\a}{}^\b P^{{\gamma}{\widehat\beta}} + {\nabla}_{\widehat\alpha} C^{\b{\widehat\beta}}_\a + T_{{\widehat\alpha}{\widehat \gamma}}{}^{\widehat\beta} C^{\b{\widehat \gamma}}_\a = 0 \\
    & {\widehat\lambda}^{\widehat\alpha} {\widehat\lambda}^{\widehat\beta} \left( \widehat R_{a{\widehat\alpha}{\widehat\beta}}{}^{\widehat \gamma} - T_{{\widehat\alpha}\b a} \widehat C^{\b{\widehat \gamma}}_{\widehat\beta} \right) = {\widehat\lambda}^{\widehat\alpha}{\widehat\lambda}^{\widehat\beta} \widehat R_{\b {\widehat\alpha}{\widehat\beta}}{}^{\widehat \gamma} = 0 \\
    & {\widehat\lambda}^{\widehat\alpha} {\widehat\lambda}^{\widehat\beta} \left( {\nabla}_{\widehat\alpha} \widehat C_{\widehat\beta}^{{\widehat \gamma}{\gamma}} - \widehat R_{{\widehat\alpha}{\widehat\sigma}{\widehat\beta}}{}^{\widehat \gamma} P^{{\gamma}{\widehat\sigma}}\right) = {\widehat\lambda}^{\widehat\alpha}{\widehat\lambda}^{\widehat\beta} \left( {\nabla}_{\widehat\alpha} S_{\b{\widehat\beta}}^{{\gamma}{\widehat \gamma}} - R_{{\widehat\alpha}\bullet\b}{}^{\gamma} \widehat C_{\widehat\beta}^{{\widehat \gamma}\bullet} - \widehat R_{{\widehat\alpha}\bullet {\widehat\beta}}{}^{{\widehat \gamma}} C^{{\gamma}\bullet}_\b \right) = 0
    \end{aligned}
  \end{equation}
These set of equations are the same%
\footnote{
The difference in relative signs come from our use of different conventions for the contraction of vectors and tensors.
}
 as in \cite{Berkovits2001}, and it was proven there that they imply the correct Type II SUGRA constraints.

\subsection{Flat space}\label{flat}
So far we showed how it all works in the most general setting but it may help to do some explicit computations in the simplest case.

The flat space background corresponds to setting the vielbein fields to
  \begin{equation}
  [E_M{}^A] = \left(
    \begin{array}{ccc}
    \d_m^a & 0 & 0 \\
    -\frac12 ({\gamma}^a\t)_{\mu} & \d_{\mu}^\a & 0 \\
    -\frac12 ({\gamma}^a\th)_{\widehat\mu} & 0 & \d_{\widehat\mu}^{\widehat\alpha}
    \end{array}
  \right)\ ,\quad [E_A{}^M] = \left(
    \begin{array}{ccc}
    \d_a^m & 0 & 0 \\
    \frac12 ({\gamma}^m\t)_\a & \d_\a^{\mu} & 0 \\
    \frac12 ({\gamma}^m\th)_{\widehat\alpha} & 0 & \d_{\widehat\alpha}^{\widehat\mu}
    \end{array}
  \right)
  \end{equation}
the $B$-field to
  \begin{equation}
  [B_{AB}] = -\frac12 \left(
    \begin{array}{ccc}
    0 & -({\gamma}_a\t)_\b & ({\gamma}_a\th)_{\widehat\beta} \\
    ({\gamma}_b\t)_\a & 0 & \frac12 ({\gamma}^a\t)_\a({\gamma}_a\th)_{\widehat\beta} \\
    -({\gamma}_b\th)_{\widehat\alpha} & -\frac12 ({\gamma}^a\th)_{\widehat\alpha}({\gamma}_a\t)_\b & 0
    \end{array}
  \right)
  \end{equation}
and the rest of background fields $\O_{M\a}{}^\b,{\widehat\Omega}_{M{\widehat\alpha}}{}^{\widehat\beta}, P^{\a{\widehat\alpha}}, C_\a^{\b{\widehat\beta}} , \widehat C_{\widehat\alpha}^{{\widehat\beta} \b}$ to zero. Then the block matrix $\AA$ simplifies greatly. In $[\AA^{ij}]$ and $[\AA_{ij}]$ the only non-zero blocks are $\AA^{MN}$ and $\AA_{MN}$ respectively, where
  \begin{equation}
  [\AA^{MN}] = \left(
    \begin{array}{ccc}
    \eta^{mn} & 0 & 0 \\
    0 & 0 & 0 \\
    0 & 0 & 0
    \end{array}
  \right)\ , \qquad [\AA_{MN}] = \left(
    \begin{array}{ccc}
    \eta_{mn} & 0 & 0 \\
    0 & 0 & 0 \\
    0 & 0 & 0
    \end{array}
  \right)
  \end{equation}
Likewise, $[\AA_i{}^j]$ reduces to
  \begin{equation}
  [\AA_i{}^j] = \left(
    \begin{array}{ccc}
    \AA_M{}^N & 0 & 0 \\
    0 & \d_\a{}^\b & 0 \\
    0 & 0 & -\d_{\widehat\alpha}{}^{\widehat\beta}
    \end{array}
  \right)\quad\text{where}\quad [\AA_M{}^N] = \left(
    \begin{array}{ccc}
    0 & 0 & 0 \\
    0 & \d_{\mu}{}^{\nu} & 0 \\
    0 & 0 & -\d_{\widehat\mu}{}^{\widehat\nu}
    \end{array}
  \right)
  \end{equation}
This implies that the action takes the following form
  \begin{align}
  S & = \int \partial_t x^m P_m + \partial_t \t^\a p_\a + \partial_t \th^{\widehat\alpha} \widehat p_{\widehat\alpha} + \partial_t \l^\a \o_\a + \partial_t{\widehat\lambda}^{\widehat\alpha} {\widehat\omega}_{\widehat\alpha}  \nonumber \\
  & \qquad - \left( \frac12 P_m \eta^{mn} P_n + \frac12 \partial_{\sigma} x^n\partial_{\sigma} x^m \eta_{mn} - (\partial_{\sigma}\th^{\widehat\alpha} \widehat p_{\widehat\alpha} - \partial_{\sigma}\t^\a p_\a) - (\partial_{\sigma}{\widehat\lambda}^{\widehat\alpha} {\widehat\omega}_{\widehat\alpha} - \partial_{\sigma}\l^\a \o_\a)\right) \nonumber \\
  & = \int (\partial_t x^m P_m -\frac12 P_m P^m -\frac12 \partial_{\sigma} x^n\partial_{\sigma} x^m\eta_{mn}) + \bar\partial\t^\a p_\a + \partial\th^{\widehat\alpha} \widehat p_{\widehat\alpha} + \bar\partial\l^\a \o_\a + \partial{\widehat\lambda}^{\widehat\alpha} {\widehat\omega}_{\widehat\alpha} \label{flat-action}
  \end{align}
and after solving for $P_m$ using its equations of motion we end up with the Pure Spinor string in flat space
  \begin{equation}
  S = \int \frac12 \bar\partial x^m \partial x_m + \bar\partial\t^\a p_\a + \partial\th^{\widehat\alpha} \widehat p_{\widehat\alpha} + \bar\partial\l^\a \o_\a + \partial{\widehat\lambda}^{\widehat\alpha} {\widehat\omega}_{\widehat\alpha}
  \end{equation}

The eigenspaces $(TM\oplus T^*M)_\pm$ in flat space can be easily computed since the operator ${\mathbb U}_{\AA_{flat}}$ is simply
  \begin{equation}
  {\mathbb U}_{\AA_{flat}} = \left(
    \begin{array}{ccccc|ccccc}
    0 & & & & & \eta_{mn} & & & &\\
    & \d^{\mu}_{\nu} & & & & &  0 & & & \\
    & & -\d^{\widehat\mu}_{\widehat\nu} & & & & & 0 & & \\
    & & & \d^\a_\b & & & & & 0 & \\
    & & & & -\d^{\widehat\alpha}_{\widehat\beta} & & & & & 0 \\
    \hline
    \eta^{mn} & & & & & 0 & & & & \\
    & 0 & & & & &  \d^{\mu}_{\nu} & & & \\
    & & 0 & & & & & -\d^{\widehat\mu}_{\widehat\nu} & & \\
    & & & 0 & & & & & \d^\a_\b & \\
    & & & & 0 & & & & & -\d^{\widehat\alpha}_{\widehat\beta}
    \end{array}
  \right)
  \end{equation}
Then, the $(+1)$-eigenvectors are
  \begin{equation}
  ({\mathbb V}, {\mathbb F}) = \left( {\mathbb V}^m \frac{\partial}{\partial x^m} + {\mathbb V}^{\mu} \frac{\partial}{\partial\t^{\mu}} + {\mathbb V}^\a \frac{\partial}{\partial\l^\a},\ {\mathbb V}^n \eta_{nm} \partial_{\sigma} x^m + {\mathbb F}_{\mu} \partial_{\sigma} \t^{\mu} + {\mathbb F}_\a \partial_{\sigma} \l^\a \right)
  \end{equation}
and the $(-1)$-eigenvectors are
  \begin{equation}
  (\widetilde{\mathbb V}, {\widetilde {\mathbb F}}) = \left( \widetilde{\mathbb V}^m \frac{\partial}{\partial x^m} + \widetilde{\mathbb V}^{\widehat\mu} \frac{\partial}{\partial\th^{\widehat\mu}} + \widetilde{\mathbb V}^{\widehat\alpha} \frac{\partial}{\partial{\widehat\lambda}^{\widehat\alpha}},\ -\widetilde{\mathbb V}^n \eta_{nm} \partial_{\sigma} x^m + {\widetilde {\mathbb F}}_{\widehat\mu} \partial_{\sigma} \th^{\widehat\mu} + {\widetilde {\mathbb F}}_{\widehat\alpha} \partial_{\sigma} {\widehat\lambda}^{\widehat\alpha} \right)
  \end{equation}
for ${\mathbb V}^{m,{\mu},\a}$, $\widetilde{\mathbb V}^{m,{\widehat\mu},{\widehat\alpha}}$, ${\mathbb F}_{{\mu},\a}$, ${\widetilde {\mathbb F}}_{{\widehat\mu},{\widehat\alpha}}$ functions of $(x^m, \t^{\mu}, \th^{\widehat\mu},\l^\a, {\widehat\lambda}^{\widehat\alpha})$. Thus, we can see both eigen\-spaces have the same dimension. Furthermore, it becomes clear that these eigen\-spaces $(TM\oplus T^*M)_\pm$ have non-trivial intersections with $TM$ and $T^*M$
  \begin{equation}
  \begin{aligned}
  (TM\oplus T^*M)_+ \cap TM & = \left\{ {\mathbb V}^{\mu} \frac{\partial}{\partial\t^{\mu}} + {\mathbb V}^\a \frac{\partial}{\partial\l^\a} \right\} \\
  (TM\oplus T^*M)_+ \cap T^*M & = \left\{ {\mathbb F}_{\mu} \partial_{\sigma} \t^{\mu} + {\mathbb F}_\a \partial_{\sigma} \l^\a \right\} \\
  (TM\oplus T^*M)_- \cap TM & = \left\{ \widetilde{\mathbb V}^{\widehat\mu} \frac{\partial}{\partial\th^{\widehat\mu}} + \widetilde{\mathbb V}^{\widehat\alpha} \frac{\partial}{\partial{\widehat\lambda}^{\widehat\alpha}} \right\} \\
  (TM\oplus T^*M)_- \cap T^*M & = \left\{ {\widetilde {\mathbb F}}_{\widehat\mu} \partial_{\sigma} \th^{\widehat\mu} + {\widetilde {\mathbb F}}_{\widehat\alpha} \partial_{\sigma} {\widehat\lambda}^{\widehat\alpha} \right\}
  \end{aligned}
  \end{equation}
In the generic case, these intersections reduce but never become trivial. We will see in the next section that because of the ghost number grading on the target space, the intersections between $(TM\oplus T^*M)_\pm$ and $T^*M$ may be lifted to be zero but the directions $\frac{\partial}{\partial \l^\a}$ and $\frac{\partial}{\partial{\widehat\lambda}^{\widehat\alpha}}$ will always be in $(TM\oplus T^*M)_+ \cap TM$ and $(TM\oplus T^*M)_-\cap TM$ respectively.

Special cases of sections belonging to these eigenspaces are the ones that generate the BRST currents that for flat space become
  \begin{align}
  j_B & = \frac12(\l{\gamma}^m\t) P_m + \l^\a p_\a + \frac12 (\l{\gamma}_m\t)\partial_{\sigma} x^m + \frac14 (\l{\gamma}^m\t)({\gamma}_m\t)_{\mu} \partial_{\sigma} \t^{\mu} \\
  {\widetilde j}_B & = \frac12({\widehat\lambda}{\gamma}^m\th) P_m + {\widehat\lambda}^{\widehat\alpha} {\widehat p}_{\widehat\alpha} + \frac12 ({\widehat\lambda}{\gamma}_m\th)\partial_{\sigma} x^m + \frac14 ({\widehat\lambda}{\gamma}^m\th)({\gamma}_m\th)_{\widehat\mu} \partial_{\sigma} \th^{\widehat\mu}
  \end{align}
and after solving for $P_m$ in (\ref{flat-action}) will take their usual form in flat space
  \begin{align}
  j_B & = \l^\a \left( p_\a  + \frac12 ({\gamma}_m\t)_\a \partial x^m + \frac18 ({\gamma}^m\t)_\a({\gamma}_m\t)_{\mu} \partial \t^{\mu} \right) \\
  j_B & = {\widehat\lambda}^{\widehat\alpha} \left( {\widehat p}_{\widehat\alpha}  + \frac12 ({\gamma}_m\th)_{\widehat\alpha} \bar\partial x^m + \frac18 ({\gamma}^m\th)_{\widehat\alpha}({\gamma}_m\th)_{\widehat\mu} \bar\partial \th^{\widehat\mu} \right)
  \end{align}

Since the flat case is the simplest solution to Type II supergravity equations, we can verify that the conditions of holomorphicity and nilpotency are satisfied. We take as an example the holomorphic case, and compute the Lie derivatives w.r.t. ${\mathbb Q}$. In the case of $[{\mathcal L}_{\mathbb Q} \AA_{ij}]$ the only non-zero components are
  \begin{equation}
  [({\mathcal L}_{\mathbb Q}\AA)_{MN}] = \left(
    \begin{array}{ccc}
    0 & -\frac12 (\l{\gamma}_m)_{\nu} & 0 \\
    -\frac12 (\l{\gamma}_n)_{\mu} & 0 & 0 \\
    0 & 0 & 0
    \end{array}
  \right)\ , \quad [({\mathcal L}_{\mathbb Q} \AA)_{\a N}] = \left( \frac12 ({\gamma}_n\t)_\a,\ 0,\ 0 \right)
  \end{equation}
and $({\mathcal L}_{\mathbb Q}\AA)_{M\b} = ({\mathcal L}_{\mathbb Q}\AA)_{\b M}$. For the matrix $[({\mathcal L}_{\mathbb Q}\AA)_i{}^j]$, the only non-zero components are
  \begin{equation}
  [({\mathcal L}_{\mathbb Q}\AA)_M{}^N] = \left(
    \begin{array}{ccc}
    0 & 0 & 0 \\
    \frac12(\l{\gamma}^n)_{\mu} & 0 & 0 \\
    0 & 0 & 0
    \end{array}
  \right)\ ,\quad [({\mathcal L}_{\mathbb Q}\AA)_\a{}^N] = \left( -\frac12({\gamma}^n\t)_\a,\ 0,\ 0 \right)
  \end{equation}
And, for $[({\mathcal L}_{\mathbb Q}\AA)^{ij}]$ we encounter that all of its components vanish. Furthermore, to verify the holomorphicity equations we still to compute $[(d{\mathbb F})_{ij}]$ which has non-zero components
  \begin{align}
  [(d{\mathbb F})_{MN}] & = \left(
    \begin{array}{ccc}
    0 & \frac12 (\l{\gamma}_m)_{\nu} & 0 \\
    -\frac12 (\l{\gamma}_n)_{\mu} & \frac14 (\l{\gamma}_m)_{({\mu}}({\gamma}^m\t)_{{\nu})} -\frac12 (\l{\gamma}_m\t){\gamma}^m_{{\mu}{\nu}} & 0 \\
    0 & 0 & 0
    \end{array}
  \right) \\
  [(d{\mathbb F})_{\a N}] & = \left(\frac12 ({\gamma}_n\t)_\a,\ -\frac14({\gamma}^m\t)_\a({\gamma}_m\t)_{\nu},\ 0\right)
  \end{align}
and $(d{\mathbb F})_{M\b} = - (d{\mathbb F})_{\b M}$. With all this, it is simple to verify that the holomorphicity equations (\ref{hol1}) are satisfied in the flat space case i.e. $\bar\partial j_B = 0$ as expected.

Finally, to verify the nilpotency conditions (\ref{nil1}), we compute the Lie bracket $[{\mathbb Q},{\mathbb Q}]$ where the only non-zero component turns out to be
  \begin{equation}\label{eq1}
  ([{\mathbb Q},{\mathbb Q}])^m = \l{\gamma}^m \l
  \end{equation}
Meanwhile, the field strength $H_{MNP}$ in flat space is
  \begin{equation}
    \begin{aligned}
    H_{m{\nu}\rho} = ({\gamma}_m)_{{\nu}\rho}\ &,\quad H_{m{\widehat\nu}{\widehat\rho}} = -({\gamma}_m)_{{\widehat\nu}\widehat\rho} \ , \\
    H_{{\mu}{\nu}\widehat\rho} = -\frac12 ({\gamma}_a)_{{\mu}{\nu}}({\gamma}^a\th)_{\widehat\rho}\ &,\quad H_{{\widehat\mu}{\widehat\nu}\rho} = \frac12 ({\gamma}_a)_{{\widehat\mu}{\widehat\nu}}({\gamma}^a\t)_\rho
    \end{aligned}
  \end{equation}
and zero for any other mix of indices. This implies that the only non-zero components of $(\i_{\mathbb Q}\i_{\mathbb Q} {\mathbb H})$ are $(\i_Q\i_Q H)_M = (-)^P Q^P Q^N H_{NPM}$ that explicitly become
  \begin{equation}\label{eq2}
    \begin{aligned}
    (\i_Q \i_Q H)_m & = - (\l{\gamma}_m\l) \\
    (\i_Q \i_Q H)_{\mu} & = -(\l{\gamma}^m\t)(\l{\gamma}_m)_{\mu} \\
    (\i_Q \i_Q H)_{\widehat\mu} & = \frac12 (\l{\gamma}_m\l)({\gamma}^m\th)_{\widehat\mu}
    \end{aligned}
  \end{equation}
Then the values in both equations (\ref{eq1}) and (\ref{eq2}) vanish because of the pure spinor constraint $(\l{\gamma}^m\l) = 0$ and the Fierz identity $({\gamma}_m)_{{\sigma}({\mu}}({\gamma}^m)_{{\nu}\rho)} = 0$. Thus, we have verified the nilpotency of the BRST current $j_B$. In an analogous way the antiholomorphicity and nilpotency of ${\widetilde j}_B$ in flat case can be proven.

\subsection{Grading of the target space}\label{grading}

We will consider that the target space for the pure spinor sigma model is a graded manifold \cite{Voronov2001}. The grading is given by the bosonic coordinates $(\l^\a, {\widehat\lambda}^{\widehat\alpha})$ and is conventionally called ghost number.

We also require that the generalized metric $\AA$ has grading $(0,0)$ or in other words of ghost number zero
  \begin{equation}
  \AA = (d{\mathbb Z}^j\otimes d{\mathbb Z}^i)\AA_{ij} + (\frac{\partial}{\partial{\mathbb Z}^j}\otimes d{\mathbb Z}^i) \AA_i{}^j + (d{\mathbb Z}^j\otimes \frac{\partial}{\partial{\mathbb Z}^i}) \AA^i{}_j + (\frac{\partial}{\partial{\mathbb Z}^j}\otimes\frac{\partial}{\partial{\mathbb Z}^i})\AA^{ij}
  \end{equation}
Thus, all terms proportional to objects such as $d\l^\a\otimes d\l^\b$, $dZ^M\otimes d\l^\a$, $d\l^\a\otimes \frac{\partial}{\partial Z^M}$,$d\l^\a\otimes \frac{\partial}{\partial{\widehat\lambda}^{\widehat\beta}}$ need to vanish since their coefficient functions are not allowed to have negative ghost number i.e. no inverse powers of $\l^\a$, ${\widehat\lambda}^{\widehat\alpha}$. Also, remember that all coefficients such as $\AA_{ij}$, $\AA^{ij}$, $\AA_i{}^j$ are functions on the base manifold $M$, in other words, they only depend on variables $(Z^M, \l^\a, {\widehat\lambda}^{\widehat\alpha})$ and not on the momentum variables $(P_M, \o_\a,{\widehat\omega}_{\widehat\alpha})$ which may account for negative ghost number. Then the following coefficients (entries in the generalized metric) are zero
  \begin{equation}\label{zeroes}
  \begin{aligned}
  \AA_{M\a} = \AA_{M{\widehat\alpha}} = \AA_{\a\b} = \AA_{\a{\widehat\beta}} = \AA_{{\widehat\alpha}{\widehat\beta}} = 0 \\
  \AA_\a{}^M = \AA_\a{}^{\widehat\alpha} = \AA_{\widehat\alpha}{}^M = \AA_{\widehat\alpha}{}^\a = 0
  \end{aligned}
  \end{equation}
as well as their transposes. This assertion can be readily verified in the expression for the generalized metric of the Pure Spinor string (\ref{ps-met}).

As a consequence of the ghost number grading, we have (\ref{zeroes}) which together with the fact that for the Pure Spinor case $\AA^\a{}_\b = \d^\a_\b$ and $\AA^{\widehat\alpha}{}_{\widehat\beta} = -\d^{\widehat\alpha}_{\widehat\beta}$ imply that the vector fields
  \begin{equation}
  {\mathbb V} = {\mathbb V}^\a(Z,\l,{\widehat\lambda}) \frac{\partial}{\partial \l^\a}\ ,\quad \widetilde{\mathbb V} = \widetilde{\mathbb V}^{\widehat\alpha}(Z,\l,{\widehat\lambda}) \frac{\partial}{\partial{\widehat\lambda}^{\widehat\alpha}}
  \end{equation}
are respectively $(+1)$ and $(-1)$ eigenvectors of ${\mathbb U}_{(\AA_{PS})}$ for any curved background. This means the eigenspaces $(TM\oplus T^*M)_\pm$ have a non-trivial intersection with the tangent bundle $TM$. Equivalently we could say that the $[\AA_{ij}]$ part of the generalized metric $\AA_{PS}$ is not invertible.

On the other hand, if we were to try to compute the intersections of $(\pm 1)$-eigenspaces of ${\mathbb U}_{\AA_{PS}}$ with $T^*M$ just as we did for the flat case, we would see that there exist a set of equations that may not have solutions in the generic case, meaning that the intersections $(TM\oplus T^*M)_\pm \cap T^*M$ could lifted to be zero. Then, it is better to study directly the invertibility of the matrix $[\AA^{ij}]$. In the pure spinor case it can be written as $[\AA^{ij}] = [{\mathbb E}_{\underline i}{}^i]^T [\AA^{\underline i \underline j}] [{\mathbb E}_{\underline j}{}^j]$ where the underlined labels can be thought as flat indices and ${\mathbb E}_{\underline i}{}^j$ as vielbein. Thus, $[\AA^{ij}]$ is congruent to $[\AA^{\underline i \underline j}]$
  \begin{equation}\label{congruent}
  [\AA^{ij}] \quad \sim \quad [\AA^{\underline i \underline j}] = \left(
    \begin{array}{c|cccc}
    \eta^{ab} & 0 & 0 & 0 & 0 \\
    \hline
    0 & 0 & P^{\a{\widehat\beta}} & 0 & ({\widehat\lambda} \widehat C)^{{\widehat\beta}\a} \\
    0 & - P^{\b{\widehat\alpha}} & 0 & (\l C)^{\b{\widehat\alpha}} & 0 \\
    0 & 0 & (\l C)^{\a{\widehat\beta}} & 0 & (\l{\widehat\lambda} S)^{\a{\widehat\beta}} \\
    0 & ({\widehat\lambda}\widehat C)^{{\widehat\alpha} \b} & 0 & (\l{\widehat\lambda} S)^{\b{\widehat\alpha}} & 0
    \end{array}
  \right)
  \end{equation}
and the vielbein ${\mathbb E}_{\underline i}{}^j$ is
  \begin{equation}
  [{\mathbb E}_{\underline i}{}^j] = \left(
    \begin{array}{ccc}
    E_A{}^N & E_A{}^N(\l\O_N)^\b & E_A{}^N({\widehat\lambda}{\widehat\Omega}_N)^{\widehat\beta} \\
    0 & \d_\a{}^\b & 0 \\
    0 & 0 & \d_{\widehat\alpha}{}^{\widehat\beta}
    \end{array}
  \right)
  \end{equation}
We can now see that the second diagonal block in (\ref{congruent}) is not invertible since its inverse would need to have entries with inverse powers of $\l^\a$ and ${\widehat\lambda}^{\widehat\alpha}$ which is forbidden by construction on graded manifolds. This means there is not possible to have a second order formulation for the pure spinor sigma model in terms on some tensor ${\mathbb G}_{ij}$ and ${\mathbb B}_{ij}$.

\section{Summary and future directions}
In order to obtain a formulation of the Pure Spinor string that treats symmetrically matter and ghost fields, we have studied sigma model actions in their Hamiltonian form. We also showed that by extending the definition of a generalized metric we can treat in a unified manner theories that mix first and second order actions. Then, we verified that the Pure Spinor sigma model action (\ref{ps-action}) can be expressed in Hamiltonian form for any curved background, and the appearing matrix $\AA_{PS}$ satisfies the conditions of a generalized metric. Furthermore, there exist fermionic sections $({\mathbb Q},\ {\mathbb F})$ and $({\widetilde {\mathbb Q}},\ {\widetilde {\mathbb F}})$ that are eigenvectors of $\AA_{PS}$ and generate the BRST currents $j_B$ and $\widetilde j_B$.

Finally, we were able to describe the conditions of holomorphicity and nilpotency of currents in terms of constraints on the associated sections and the generalized metric (\ref{holomorphicity}, \ref{nilpotency}). And, ultimately we used this in the Pure Spinor case to deduce that the background fields $(E_\a{}^M,E_{\widehat\alpha}{}^M,\O_{M\a}{}^\b, {\widehat\Omega}_{M{\widehat\alpha}}{}^{\widehat\beta}, C^{\a{\widehat\alpha}}_\b, {\widehat C}^{{\widehat\alpha}\a}_{\widehat\beta}, S^{\a{\widehat\alpha}}_{\b{\widehat\beta}})$ satistfy the Type II supergravity constraints (\ref{sugra-nil1}-\ref{sugra-nil3}, \ref{sugra-hol1}-\ref{sugra-hol2}).

A future direction is to take the Pure Spinor sigma model and study the simpler case where the "matter" part of the target space is a complex supermanifold. With the additional supposition that the 2-form $B_{MN}$ is the Kahler form, we are able to consistently associate holomorphic variables in the worldsheet and target space. Thus, we land on a theory that is an honest first order formulation much like in \cite{Losev2005}
  \begin{equation}
  S = \int p_M \bar\partial z^M + \bar p_{\bar M} \partial \bar z^{\bar M} + \o_\a\bar\partial\l^\a + {\widehat\omega}_{\widehat\alpha} \partial{\widehat\lambda}^{\widehat\alpha} -\frac12 (p_M\ \o_\a) \left(
    \begin{array}{cc}
    A^{M\bar M} & A^{M{\widehat\alpha}} \\
    A^{\a \bar M} & A^{\a{\widehat\alpha}}
    \end{array}
  \right) \left(
    \begin{array}{c}
    \bar p_{\bar M} \\
    {\widehat\omega}_{\widehat\alpha}
    \end{array}
  \right) \nonumber
  \end{equation}
where the matrix $A$ contains all background fields. And, the $BRST$ currents simplify to be generated only by vector fields
  \begin{equation}
    \begin{aligned}
    {\mathbb Q}_B & = \l^\a E_\a{}^M \frac{\partial}{\partial z^M} - \l^\a E_\a{}^M \l^\b \O_{M \b}{}^{\gamma} \frac{\partial}{\partial \l^{\gamma}} \\
    {\widetilde {\mathbb Q}}_B & = {\widehat\lambda}^{\widehat\alpha} E_{\widehat\alpha}{}^{\bar M} \frac{\partial}{\partial {\bar z}^{\bar M}} - {\widehat\lambda}^{\widehat\alpha} E_{\widehat\alpha}{}^{\bar M} {\widehat\lambda}^{\widehat\beta} {\widehat\Omega}_{{\bar M} {\widehat\beta}}{}^{\widehat \gamma} \frac{\partial}{\partial {\widehat\lambda}^{\widehat \gamma}}
    \end{aligned}
  \end{equation}
A particular example of this is the Pure Spinor string in a Calabi-Yau background.

\vspace{1cm}

\noindent {\bf Acknowledgements.}
The author would like to thank Andrei Mikhailov for suggesting the problem and useful discussions. This project was financially supported by FAPESP grant 2016/22579-9.

\newpage

redpajama_set_name: RedPajamaArXiv


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We've taken our popular Red and dry-hopped it with a single hop for a unique & tasty variation. This well-balanced amber ale has caramel and toffee malt notes and a subtle spicy hop character. It won 1st Place in the category of American Amber Ale at the 12th Great International Beer Competition. It's been cask aged and dryhopped with Citra. The Citra hops lend a fruity, citrusy note to the nose and adds to the complexity of this brew.


text:
Q: bash - `read` cannot read from a redirect, `cat` works fine? I've been trying to send some AT commands to my modem and want to capture response into a variable. Here's my code:
exec 3<>/dev/ttyUSB3

echo -e "AT+CGSN\n" >&3

cat <&3
#read -r RESPONSE <&3
#echo "Response was $RESPONSE"

exec 3<&-
exec 3>&-

Results:
$ ./imei_checker.sh
AT+CGSN

356538041935676

OK

AT+CGSN

356538041935676

OK

But if I change cat to read, it doesn't work:
$ ./imei_checker.sh
Response was AT+CGSN

2 more questions:

*

*Why it show the dupplicate output?

*How do I close the file handle properly? exec 3<&- and exec 3>&-
seems doesn't work. I have to press Ctrl+C to
get control of the Terminal back.

A: read will only read a single line, unlike cat which will basically read and echo until end of file.
For a read version, you're best off reading with a timeout up until the point you get the OK (and storing any line that contains a lot of digits).
I think you'll find that it's not the closing of the number 3 file handle that's stopping things - it's more likely to be the cat which will continue to read/echo until an end-of-file event that isn't happening.
You can be certain of this if you just put:
echo XYZZY

immediately before the closing exec statements. If it's still in the cat, you'll never see it.
So, using a looping read version will probably fix that as well.

By way of example, here's how you can use read to do this with standard input:
#!/bin/bash

NUM=
while true ; do
    read -p "> " -t 10 -r RESP <&0
    if [[ $? -ge 128 ]] ; then RESP=OK ; fi
    echo "Entered: $RESP"
    if [[ $RESP = OK ]] ; then break ; fi

    if [[ $RESP =~ ^[0-9] ]] ; then NUM=$RESP ; fi
done

echo "Finished, numerics were: '$NUM'"

It uses the timeout feature of read to detect if there's no more input (setting the input to OK so as to force loop exit). If you do get an OK before then, it exits normally anyway, the timeout simply caters for the possibility that the modem doesn't answer as expected.
The number is set initially to nothing but overwritten by any line from the "modem" that starts with a number.
Two sample runs, with and without an OK response from the "modem":
pax> ./testprog.sh
> hello
Entered: hello
> 12345
Entered: 12345
> OK
Entered: OK
Finished, numerics were: '12345'

pax> ./testprog.sh
> hello
Entered: hello
> now we wait 10 secs
Entered: now we wait 10 secs
> Entered: OK
Finished, numerics were: ''

It wouldn't be too hard to convert that to something similar with you modem device (either read <&3 or read -u3 will work just fine).

That would basically translate to your environment as:
exec 3<>/dev/ttyUSB3
echo -e "AT+CGSN\n" >&3
NUM=
while true ; do
    read -t 10 -r RESP <&3
    if [[ $? -ge 128 ]] ; then RESP=OK ; fi
    echo "Entered: $RESP"
    if [[ $RESP = OK ]] ; then break ; fi
    if [[ $RESP =~ ^[0-9] ]] ; then NUM=$RESP ; fi
done
echo "Finished, numerics were: '$NUM'"
exec 3<&-
exec 3>&-

Now I haven't tested that since I don't have a modem hooked up (been on broadband for quite a while now) but it should be close to what you need, if not exact.

A: read takes the descriptor to read from as the argument following -u. See help read for details.

A: If you want to get the individual lines into variables I'd suggest to wrap read into a while:
while read -r RESPONSE <&3; do
  echo "Response was $RESPONSE"
  ## e.g.:
  [ "$RESPONSE" = "OK" ] && break
done

However, if you want "everything" that is sent back to you to reside in $RESPONSE you could do it like this:
RESPONSE="$(cat <&3)"

redpajama_set_name: RedPajamaStackExchange


After more than a decade of professional writing, a writer is well-versed in a variety of industries. Jessica Mehta is highly experienced a variety of projects, and welcomes queries from any industry. One of the greatest perks of writing is the ability to quickly and efficiently research new subjects and discover an entirely new world in the process.


President Donald Trump and First Lady Melania Trump are spending the weekend at their home on Mar-a-Lago.
The president has no public events scheduled, but check back as this page will be updated should events warrant.
President Donald Trump's Schedule for Sunday, February 17, 2019 is original content from Conservative Daily News – Where Americans go for news, current events and commentary they can trust – Conservative News Website for U.S. News, Political Cartoons and more.


text:
Conventions in Tennessee
Scifi and game convention.
ConNooga
Anime and game convention.
MTAC
Apr 7-9, 2023
Cosplay convention.
Metrotham Con Episode IV
Multi-genre convention.
ICCCon
FarleyCon Comic Book Expo
Aug 12-13, 2023
Scifi and literature convention.
Chattacon
Jan 12-14, 2024
What are fan conventions?
If you haven't been to a fan convention yet then you are missing out! Hundreds of thousands of people go to fan conventions every year. In fact, some of the largest fan conventions like San Diego Comic Con in San Diego, California and Japan Expo in Paris, France have over 100,000 people attend! Conventions include all kinds of fandoms, ranging from furry fans to people who love anime.
How to prepare for a convention in Tennessee.
First thing to do is get your tickets. Tickets for conventions often go on-sale far in advance of the event, so check the official website for your event to find out how much tickets cost. Prices can vary widely between conventions, with major festivals charging far more than small and intimate local shows.
After you have your tickets you need to plan transportation and lodgings. Many people attend fan conventions for several days. These conventions often publish a hotel room block that allows you to book hotel rooms at a significant discount. Most conventions take place over a weekend and so this makes for a nice retreat from the mundane world of the day-to-day.

redpajama_set_name: RedPajamaCommonCrawl


SAN JUAN, Puerto Rico - In the mountains located in the western interior of Puerto Rico is a small general hospital that not only cares for its community's health but is the economic center of the remote town of Castañer.
With the closest medical centers more than an hour away, the Castañer General Hospital is the lifeline to more than 6,000 residents in the mountainous region. Additionally the hospital is the town's main source of employment and water.
After Hurricane Irma hit Puerto Rico September 5, the hospital lost one of two back-up generators. They quickly rushed to install and connect their second generator prior to Hurricane Maria's landfall, and that generator has been powering the hospital ever since. Industry recommends complete maintenance every 240 hours and the generator at the hospital was extremely overdue and showed signs of weakening.
With little time to spare, on November 10, a FEMA temporary generator arrived at the remote facility. The 250 kilowatt generator is the 500th generator installed by the USACE Emergency Temporary Power Planning and Response Teams from Albuquerque and Pittsburgh Districts.
"The generator arrived just in time," said Adrian Gonzalez, Castañer General Hospital chief operating officer. "Our current generator had been working non-stop for 1,300 hours and was showing signs of failure."
For more than 60 years the hospital has served the cities of Lares, Adjuntas, Yauco and Las Marias. Gonzalez said the generator not only allows them to provide emergency medical support and clinics to its patients but it also runs a water pump that has been providing water from the facilities well to residents.
"Without that generator, we could not provide emergency medical services, we would lose medicines that require refrigeration and the local residents would have no clean water," said Gonzalez.
As the contracted team of technicians worked to install the new generator, another USACE team arrived to assess the soon to be off-line generator. Castañer General Hospital was also approved by FEMA as a Non-Fed generator to be maintained, repaired and fueled by USACE until commercial power is restored.
"This generator will allow for the hospital to perform complete maintenance on their generator, and then both generators can be used alternately to power the hospital," said Puerto Rico Recovery Field Office Commander Col. James DeLapp.
On October 28th the Albuquerque District Emergency Temporary Power PRT installed its 311th temporary generator in support of Hurricane Maria response and recovery efforts, surpassing the record set during Hurricane Katrina. As the temporary power mission continues the Pittsburgh District has taken the baton and continues the task of providing FEMA generators to communities like Castañer.
With a new FEMA generator and a guarantee of support for their hard working generator, the hospital staff and Castañer. General Hospital can breathe a little easier. According to Gonzalez they can relax now knowing they have a new working generator on site. "This will allow us to focus on other things we haven't been able to for a while."
By days end a total of generators have been installed throughout Puerto Rico providing more than 154 megawatts of power to more than 49 hospitals, 44 waste water facilities, 242 water pumps, 30 emergency services facilities, 10 airports and many other critical facilities.
"With the Puerto Rico's power grid operating at approximately 43 percent of pre-storm capacity, the ability to provide generators for critical facilities continues to be a high priority for FEMA and the U.S. Army Corps of Engineers," said DeLapp.


Welcome to Lisa.B.Designs : Watch Out For Icebergs!
The sentiments are from the Hobby Art Christmas Frosting stamp set.
This coming Friday is class day at Fant Hall, Maidstone, we'll be using Imagination Craft's beautiful Christmas Art Stamps to make two cards. More information is in the October newsletter, those of you who subscribe will receive it sometime tonight. It will also be posted here on Wednesday, or you can visit the October Class event on the Lisa B Designs Facebook page.
There are still a few places available in both the afternoon & evening class. If you would like to join us, please contact Lisa no later than Wednesday evening, this gives us time to prepare the correct number of class projects.
That's all from me for now, have a good week & I'll see lots of you on Friday.


Jamaica is a constitutional monarchy with two spheres of government, national and local. Local government was entrenched in the constitution by the passing of the Constitution (Amendment) (Local Government) Act in July 2015. The main governing legislation for local authorities comprises the Parish Councils Act 1887, the Kingston City Corporation Act 1923 and the Municipalities Act 2003. The Ministry of Local Government and Community Development provides oversight for the 14 local authorities. The current configuration of local authorities in Jamaica consists of 12 parish councils, one municipal corporation – the Kingston and St Andrew Corporation (KSAC) – and a second tier municipal council, Portmore Municipal Council (PMC), which falls within the parish of St Catherine. Local authorities are empowered to raise revenue via user fees, by-law penalties, trade licences and property taxes. They also benefit from ad-hoc, conditional transfers from central government. Local government is responsible for parish infrastructure and poor relief and shares responsibility with central government for a range of services including water supply, environmental health and local economic development.
The Association of Local Government Authorities of Jamaica (ALGAJ) is the representative organisation responsible for the interests and welfare of local government authorities. It also seeks to advance local government in general. Its membership is drawn from councillors and the mayors of all 14 local authorities.
There are three laws in various stages of development which will strategically impact the local government system in Jamaica. The Local Government Unified Services and Employment Bill and the Local Government Financing and Financial Management Bill are with the Attorney General's Office, while the Local Governance Bill is still being developed. Legislation is also being developed that will provide for a unified service in which one commission will be responsible for local government personnel.
Read the profile of the local government system in Jamaica.
CLGF will continue to work with The Ministry of Local Government and Community Development, The Association of Local Government Authorities of Jamaica (ALGAJ) and other partners as it takes forward the EC funded project "Strengthening local government's role as a partner in development" focused on strengthening the role of local government and other local and national actors in localising and implementing the new Sustainable Development Goals (SDGs).


text:
Q: Query Restful API Passing Username and Password I am building a Restful API for my users.
I have these:
1. GET          -> Users/{id}                    This is to get by Id
2. POST         -> Users/{DTO}                   This is to add
3. PATCH/PUT    -> Users/{DTO}                   This is for updating the info
4. DELETE       -> Users/{id}                    Removing
5. DELETE       -> Users/{username}/reset_pass   Reset Password
6. POST         -> Users/?username={userName}&password={password}
So I am a little unsure about number 5 and 6 (specially 6).
I would like to know what you think about this ? sending my user's password in a query string in the Body.
I think Uber uses the pattern that I used for resetting password. What do you think about that ?
Thanks

A: 4. DELETE -> Users/{id} Removing
5. DELETE -> Users/{username}/reset_pass Reset Password

*

*This is not consistent. You either identify by user IDs or user names. If you want both, use different URL naming schemes.

*reset_pass is verb like. Consider using DELETE Users/{id}/password. Depending on what happens to the password, DELETE may or may not suit the use case.

6. POST -> Users/?username={userName}&password={password}
Again, this is not consistent with your URL format. If you'd like to create a new password for a user, use POST Users/{id}/password.

I would like to know what you think about this ? sending my user's password in a query string in the Body.

It is neater to model it into a request body, however it makes no difference in terms of security. Unless you use HTTPS anyone will be able to sniff a HTTP POST regardless of where in the request you decide to pass your data.

A: This is just my opinion, but on #6 maybe putting the password in the JSON body of the POST request would be better. That way it isn't directly in the URL (and I think the REST principles say that when you POST you should have the body be the object that is being posted).
Of course, these passwords should already be hashed so it's not too big a deal, but it just seems safer/nicer.


text:
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Focused on Machining
By Eric Peterson | Feb 05, 2017
Aerospace & Electronics Bioscience & Medical Industrial & Equipment Supply Chain Colorado
Louviers, Colorado
www.focusedonmachining.com
Industry: Industrial & Contract
Products: Machining services
New owner Justin Quinn is setting his sights on diversification and geographic expansion for his machine shop in Douglas County.
After six years in the Air Force working as a mechanic on cargo planes, Quinn studied finance at University of Northern Colorado before joining Centennial Bank in 2010. After working in credit and commercial lending for six years, he acquired Focused on Machining in Oct. 2016.
"What attracted me to this industry: I have three uncles who are machinists in Longmont," says Quinn. "I also really missed being mechanical and building things."
But it wasn't easy to find the right manufacturer to buy. "There's not a lot of companies doing this," he says. "I was actually on the hunt for a company for a year and a half."
During the search, he met Focused on Machining founder Rich Pollmiller. "Rich and I had lunch a year into my search and we really hit it off," says Quinn.
While Pollmiller wasn't thinking of selling his machine shop at that point, he soon changed his mind. In Sept. 2016, Quinn told him what he'd value the company at based on the financials and they made a deal. "In 30 minutes, we ironed out the details," says Quinn.
Pollmiller started the company in englewood in 2008 and moved it to Louviers to be closer to his brother's shop, Superior Metal Products. Revenue grew to about $1.4 million in 2014, but dipped to "right under $1 million" in 2016. "Our customer base is primarily aerospace," says Quinn. Medical manufacturers and the valve industry are secondary markets. The majority of customers are located on the Front Range.
Quinn has an aggressive growth plan. He wants to hit $1.5 million in sales in 2017 and $5 million by 2022. "I think it's doable. We've just got to put the systems in place to get there."
With nine CNC milling centers, and both horizontal and vertical machines, the company is focused on "higher-volume parts," he adds. "We have purchase orders that extend many months at a time, and we deliver 5,000 to 10,000 parts over several months."
READ: Colorado's Top 5 manufacturing communities
Quinn is hoping to diversify into new markets. "We are branching out into other states," he says. The company recently landed its first customer in New Mexico.
He also wants to become a direct supplier to aerospace (the company is currently a second-tier supplier) by updating the company's ISO 9001 certification, and move into high-value, low-volume work in robotics and automation as well.
Focusing on both low- and high-volume work is a differentiator. "A lot of shops in town do one or the other," says Quinn. "There aren't a lot of shops that do both."
Quinn, 32, says he's part of an informal "Under 40 Club" of young machine shop owners on the Front Range with Mike Swartzlander of Cutter Innovations and Tony Feltman of Spire Manufacturing.
"As this industry continues to age, there's going to be a big transition as the 60- to 70-year-olds retire," says Quinn.
Challenges: "My challenge right now is continuing to convert from Rich's ownership into mine," says Quinn. "That's going very well."
He says "building out the team" is another challenge. "I've made three hires since I've been here."
Opportunities: "Becoming a direct supplier to high-end aerospace companies" is a prime opportunity, and Quinn sees Colorado's cannabis industry as another potential market. "We're targeting the marijuana industry to make food-grade components for automation."
Beyond Colorado, he's looking for customers in the sister Four Corners states of Arizona, Utah, and New Mexico.
Needs: "Our biggest need is customers," says Quinn. "Finding new customers to meet growth demands."
Another need: "We're actually looking for an ERP system right now," he says. "We're doing everything with spreadsheets right now. It's working pretty well, but I have ERP on my must-do list for 2017."
Connect with Focused on Machining on SCoP
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