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\begin{titlepage}
\begin{flushright}
FTUV/95-24 IFIC/95-24 \\
\end{flushright}

\vspace{0.7cm}
\begin{center}
{\Large
What can we learn from the $Z \rightarrow b\bar{b} $ vertex?
}
\vspace{1.2cm}

{\Large Denis Comelli} \footnote{E-mail: Comelli@evalvx.ific.uv.es
   supported by the "Ministerio de la scientia y de la instruccion  de
Espana"}\\
\vspace{0.2cm}
{\em University of Valencia 46100 Burjassot (Valencia),
Spain}\\

\vspace{0.4cm}

\end{center}
\begin{abstract}
I stress the fact
that a complete study of possible New Physics effects in the
$Z \rightarrow b\bar{b}$ vertex
requires a combined analysis of the ratio
$R_b= \Gamma(Z \rightarrow b\bar{b})/ \Gamma(Z \rightarrow hadrons)$
and the longitudinally polarized forward backward asymmetry, $A_b$.

This is illustrated with a number of models such as the two Higgs-doublet
model, the MSSM, Technicolour and models with an extra $Z'$.
The use of effective lagrangean techniques is briefly discussed.
\vspace{0.4cm}

\end{abstract}

\end{titlepage}
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\newpage
{\bf \large }

The remarkable experimental effort done by the LEP collaboration has
brought some hadronic processes into the realm of high precision
measurements.
The best example arises in the ratio
$R_b= \Gamma(Z \rightarrow b\bar{b})/ \Gamma(Z \rightarrow hadrons)$
which is now known with an error of the order of 1$\%$
($R_b= 0.2204 \pm 0.0020$).
This quantity is a direct measurement of the non oblique vertex
correction, since the oblique and QCD corrections cancel in the ratio.

The effective vertex $Z \rightarrow b\bar{b}$ can be parametrized in terms of
two form factors, the left handed $g_L$ and the right handed $g_R$:
\begin{equation}
\frac{g}{2 \cos \theta_W} Z^{\mu}  \bar{\psi}_b
(\gamma_{\mu}(g_R\gamma_R +g_L\gamma_L ))\psi_b\ ,
\end{equation}
where $\gamma_{R,L}=(1\pm \gamma_5)$. A third form factor,
the so called magnetic moment, is neglected.

The effective couplings may be written as
$g_{L,R}= g_{L,R}^0 +\delta g_{L,R}$
with
$g_{L}^0=- 1/2+ 1/3
\sin^2 \theta_W $ and $ g_{L}^0=1/3
\sin^2 \theta_W $.
The shifts $\delta g_{L,R}$ contain the
effect of the non oblique  Radiative Corrections (RC) of the Standard Model
(SM), due to the would be goldstone boson exchange
and QCD effects due to the non negligible $b$ mass,
as well as
possible New Physics (NP) effects.
In order to determine the two form factors separately we need two
independent measurements of the same vertex.
One is provided by $R_b$ and the other by the longitudinally polarized
forward backward asymmetry defined as :
\begin{equation}
A_b= \frac{
\sigma(e_L\rightarrow b_F)-\sigma(e_R \rightarrow b_F)-\sigma(e_L \rightarrow
b_B)+\sigma(e_R \rightarrow b_B)}
{\sigma(e_L \rightarrow b_F)+\sigma(e_R \rightarrow b_F)+\sigma(e_L \rightarrow
b_B)+\sigma(e_R \rightarrow b_B)}\ ,
\end{equation}
which is directly measured at SLAC \cite{slac} as well as in the
forward backward asymmetry at LEP \cite{lep}.
Remarkably  $A_b$  depends  only on the $Z$
coupling of the $b$ quark  \cite{blondel}.
\begin{equation}
A_b= \frac{g_L^2-g_R^2}{g_L^2+g_R^2}
\end{equation}

The effect of  the vertex corrections
can be outlined through the ratios  of
the $Z\rightarrow b \bar{b}$ rate and asymmetry to those
of the decay $Z \rightarrow s \bar{s}$,
\begin{equation}
\frac{\Gamma_b}{\Gamma_s}=1+\delta_{bV} \;\; {\rm and}
\;\; \frac{A_b}{A_s}=1+\eta_b
\end{equation}
We may write
$\delta_{bV}$ and $\eta_b$
in terms of the shift $\delta g_{L,R}$ as,
\begin{equation}
\delta_{bV}= -\frac{4(1+b)}{(1+b^2)} [ \delta g_L
-\frac{(1-b)}{(1+b)}\delta g_R]
\end{equation}
\begin{equation}
\eta_b=-\frac{4(1-b)(1+b)^2}{b(1+b^2)}[\delta
g_R+\frac{(1-b)}{(1+b)}\delta g_L]\ ,
\end{equation}
where $b=1-4/3 \sin^2 \theta_W$.
Thus,
$\delta_{bV}$ and $\eta_b$
are naturally orthogonal in the ($\delta g_L,\delta g_R$) basis,
so that the effects that would not contribute to one observable will
be revealed by the other one and vice-versa.
In particular,
the NP effects contributing only to the left handed coupling
($\delta g_R=0$) imply the correlation $ \eta_b/\delta_{bV}=0.068$.
Conversely,
for the physics that contributes only to the right handed
($\delta g_L=0$) one finds $ \eta_b/\delta_{bV}=-2.068$.
This means that $ \eta_b$ is very
sensitive to $\delta g_R$ whereas $\delta_{bV}$ is very sensitive
to $\delta g_L$.

The leading SM corrections to the $Z \rightarrow b\bar{b}$ vertex are
very important.
They are due  to  the exchange of the would be
Goldstone boson in the t'Hooft-Feynman gauge,
and,
in the limit of large top mass ($m_t>>m_W$),
are quadratic in $m_t$ contributing only to
$\delta g_L$ \cite{akundov}.
As a consequence  any  deviation of $ \eta_b$ from zero is a signal of
NP.

Using the present  value  of $m_t=174 \pm20$ GeV \cite{fermilab},
the LEP data  indicate that the SM prediction for $R_b$ might be
inconsistent with such a heavy top up to 2.5 $\sigma$,
and that NP contribution, $\delta_{bV}^{NP}$,
must be positive.
This initial hint seems very interesting and justifies a thorough
investigation of possible NP models providing $\delta_{bV}^{NP}>0$,
together with an analysis of their predictions for $\eta_b$.

\bigskip

$\Box$   The simplest extension of the SM is the Two Higgs Doublet
Model (THDM) in which the electroweak symmetry breaking sector
involves two fundamental scalar doublets $H_{1,2}$.
The physical spectrum contains an extra pair of charged scalar
particles $H^{\pm}$ , a pseudoscalar $A^0$,
and two neutral CP even scalars $h^0$ and $H^0$.

The parameter $\tan \beta$, defined as the ratio of the two vevs
$<H_1>/<H_2>$, allows us to distinguish two different regions for the RC
\cite{two}:

1) $\tan \beta \sim 1$: for which the only important contributions
come from the $H^{\pm}$ loops. One finds $\delta g_R=0$, $\delta g_L \sim
0(m_t^2/M_H^2)$ and consequently
$\delta_{bV} <0$ and $\eta_b\simeq 0$ , in contradiction with the previous
expectation.

2) $\tan \beta \geq m_t/m_b$ : in this region the Yukawa coupling of
the $b$ quark can be comparable to that of the $t$ quark so
that the process involving intermediate $b$ quark and neutral scalars
become important.
For the charged sector we find $\delta g_L =0$, $\delta g_R=0(m_b^2
\tan ^2\beta/M_H^2)$  with $\delta_{bV} >0$ and $\eta_b\simeq
-2\delta_{bV}$.
For the neutral sector $\delta g_R$ and $\delta g_L$ are $0(m_b^2
\tan ^2\beta/M_H^2)$ with $\delta_{bV} >0$ and $\eta_b\geq 0$.
In particular, for a light
pseudoscalar $A^0$,
these last contributions can be very large.

\bigskip

$\Box$ The next models that I discuss are the SUSY theories .
The Higgs sector of the Minimal SUSY
model coincides with that of THDM,
once the mass relations required by SUSY are imposed.
In addition to these contributions discussed above,
we also have contributions coming from intermediate states of the
superpartners of the SM particles .
For light $M_{A^0}$ and large $\tan \beta$ the effect can be important
and goes in the right direction, namely, $\delta_{bV} >0$
 and $\eta_b\geq 0$.\cite{two}.
These superparticle contributions introduce a lot of new free
parameters (the masses of the stop-sbottom left and right and the
respective mixing angles, the $\mu$ term, and the gaugino
masses)\cite{susy}.

The contribution from the strong interactions involving the gluino are
practically negligible.
Sizeable corrections can come instead from the stop-chargino sector
with

$\delta g_L\simeq 0(m_t^2/(\sin^2 \beta M_{SUSY}^2))$  and
$\delta_{bV} >0$
for small $\tan \beta$; and

$\delta g_R
\simeq 0(m_b^2 \tan^2\beta/ M_{SUSY}^2)$
for large $\tan \beta$ ,
with $\delta_{bV}\simeq-1/2 \eta_b<0$.

For large $\tan \beta$ there are also  sizeable
contributions from the neutral
fermionic supersector (neutralino-sbottom loop)
which act along the right experimental direction ,

 $\delta g_L\simeq
-\delta g_R\simeq 0(m_b^2 \tan^2\beta/ M_{SUSY}^2)$
with $\eta_b\simeq-0.3 \delta_{bV}<0$.

In general it is possible to state that, if the
discrepancy of $R_b$ is to be explained by the MSSM,
then direct observation of superpartners at LEPII or
at an upgraded Tevatron can be expected
\cite{kane}.

\bigskip

$\Box$  These vertex corrections come out  to be
 very sensitive also to completely
different models, like Technicolor, in which the dynamical electroweak
symmetry breaking is induced by the vev of bilinear fermion (instead of a
fundamental scalar particle).

To provide simultaneously the masses of the W and Z bosons and of the
ordinary fermions, additional gauge interactions (Extended
Technicolor (ETC)) are  introduced.

The effective low energy theory provides a sizeable coupling between
the third quark family and the Technifermions inducing a big shift in
$\delta g_L\sim O(m_t/4\pi v)$.
This
implies a negative contribution to $\delta_{bV}$,
which is about 8 $\sigma$
away from the experimental value \cite{technicolor}.
Such a negative result, added to  the presence of FCNC, puts
this models in serious trouble.

Also, in the more popular walking technicolor the effect can be at most
a factor 2 smaller \cite{etc}.
\bigskip

$\Box$  Another interesting class of models that can affect this
vertex are theories containing extra gauge bosons.
When this extra bosons mix with the SM one, the rotation to the mass
basis induces a change in the coupling of the physical Z to the
fermions which is proportional to $\delta g_{LR}^{f}\simeq -x
m_Z^2/M_{Z'}^2 g_{Z' LR}^{f}  $, where $x$ is a model dependent
quantity and $g_{Z' LR}^{f}$ are the coupling of the extra $Z'$ with
the $f$-fermions.

The limits on the mixing angle
of the extra gauge bosons,
$\theta= x m_Z^2/M_{Z'}^2 $,
coupling universally to the fermions
(as in Superstring inspired and Gut models)
are so strong that the mixing effect on the
$Z\rightarrow b \bar{b}$ vertex cannot exceed the 1 $\%$.

Recently,
some Technicolor inspired models have been explored
in which the extra $Z'$ responsible for the generation of the top quark mass
couples more strongly to the $b_L$, $t_L$ and $t_R$
than to the other fermions.
The implications of these models are studied in ref.\cite{holdom}  in which
it is found that the existence of contributions proportional to $\delta
g_L=-\delta g_R\;\;(\eta_b/\delta_{bV}\simeq -0.3)$  open a possible
reconciliation of the ETC with the experiments.

\bigskip

$\Box$  Finally, I would like to discuss
an alternative method to investigate the
effects of possible NP on the $Zb \bar{b}$ vertex.
I will mention only the results of a general investigation
that can be found in ref \cite{verza},
which exploits effective lagrangean techniques.
In this approach the Lagrangean is written as
$L=L_{SM}+ \sum_i f_i O_i/\Lambda^2$,
where $L_{SM}$ is the SM Lagrangean,
$\Lambda$ is the scale at which NP appears and $ O_i$ are all the
gauge and CP invariant
operators of dimension 6 which contribute to the RC.

Using the ansatz that the NP generates only operators containing gauge
boson fields (11 independent operators) and operators  with
at least one $t_R$ field (the $t_R$ field is related to the $m_t$ mass)
(14 operators) they find that the leading order RC to
$Zb \bar{b}$ [of order $0(m_t^2/\Lambda^2 \log \Lambda^2$)]
obey the following pattern:

1)only two $O_i$ of the gauge sector
and four of the fermionic-$t_R$ sector affect
$\delta g_L$, thus yielding $\eta_b\simeq 0$;

2)only one operator with one $t_R$ field affects $\delta g_R$
and, hence, affecting also $\eta_b$.

In practice in this formalism a sizeable non zero value of $\eta_b$
is directly related to the presence of one operator $O_{t_R}$.
However, this technique must be used with care, due to the possible
presence of large finite contributions of order O($m_t^2/\Lambda^2$)
\cite{zheng}.

\vspace{1cm}

{\bf In conclusion}
I have discussed the possible "directions" of NP on the
plane ($\delta_{bV}, \eta_b$)  .

In the first quadrant we find the contributions of the THDM,
of most part of the parameter space of SUSY models,
and of modest Z' contributions;
in the fourth quadrant there are instead the extra Z' models
inspired by Technicolor and the presence of effects from the $O_{t_R}$
operator.

The next improved measurements of the forward backward asymmetry as
well as $A_b$ in SLAC  can be of help to disentangle indications of
NP.

\vspace{0.5cm}

I would like to take this opportunity to thank the organizers for
the excellent environment and interesting discussions they
have extended to us at this conference.
I would also like to thank A.\ Rossi and J.P Silva for reading the manuscript.

\vspace{0.5cm}


text:
\section*{Introduction}

Let $\Gamma \subseteq \n_0$ be a numerical semigroup, generated by positive
integers $a_1, \dots, a_r$ without common factors, such that $H = \n_0
\setminus \Gamma$ is a finite set. We consider its semigroup algebra $A =
k[\Gamma]$ over the field $k = \cc$ of complex numbers. Since $X = \spec(A)$
is the affine monomial curve $X = \{ (t^{a_1}, t^{a_2}, \dots, t^{a_r}): t
\in k \} \subseteq \aff^r_k$, we call $A$ a \emph{monomial curve}.

We described the ring $D = \diff(A)$ of differential operators on a monomial
curve $A = k[\Gamma]$ in Eriksen \cite{erik03}, using the graded structure.
For any degree $w$, there is a homogeneous differential operator $P_w$ of
minimal order in $D_w$, given by
    \[ P_w = t^w \prod_{\gamma \in \Omega(w)} \, (E - \gamma) \]
where $E = t \partial$ is the Euler derivation, and $\Omega(w) = \{ \gamma
\in \Gamma: \gamma + w \not \in \Gamma \}$. Moreover, $E$ together with $\{
P_w: |w| \in \{ a_1, \dots, a_r \} \text{ or } |w| \in H \}$ is a set of
homogeneous generators for $D$, and $D_w = P_w \cdot k[E]$. We also showed
that the associated graded ring $\gr D$ is the finitely generated semigroup
algebra $\gr D = k[\Gamma'] \subseteq k[t,\xi]$, with
    \[ \Gamma' = \{ (m,n) \in \n_0^2: n \ge \sigma(m-n) \} \subseteq \n_0^2
    \]
where we write $\sigma(w)$ for the cardinality of $\Omega(w)$, which
satisfies $\sigma(w) = d(P_w)$.

In this paper, we use these results to study the holonomic $D$-modules on a
monomial curve. Note that $\gr D$ is a positively graded $k$-algebra via the
Bernstein filtration, but it is not generated by homogeneous elements of
degree one. We use the results of Shukla \cite{shuk80} to prove that for any
good filtration of $M$, there is a period $m \ge 1$ and polynomials $P_0,
\dots, P_{m-1} \in \q[t]$ such that $\dim_k \left( M_{nm+r} \right) = P_r(n)$
for $0 \le r < m$ and for all sufficiently large integers $n$. Moreover, the
polynomials
    \[ P_r(t) = \frac{e}{d!} \, t^d + \text{ terms of lower degree} \]
all have the same leading term. We define $d(M) = d$ and $e(M) = e$ to be the
dimension and multiplicity of $M$, and prove the Bernstein inequality $d(M)
\ge 1$.

We define a $D$-module to be holonomic if $d(M) = 1$, and use the properties
of dimension and multiplicity to study holonomic $D$-modules, generalizing
the standard results; see Coutinho \cite{cout95}. In particular, we show that
a $D$-module is holonomic if and only if it has finite length; this means
that we can study the length category of holonomic $D$-modules via its simple
modules and their extensions. The simplest example of a monomial curve is $A
= k[t]$, with the first Weyl algebra $D = \weyl$ as its ring of differential
operators. The simple modules over the first Weyl algebra were classified in
Block \cite{bloc81}, and form a very large set. We therefore restrict our
attention to the graded case.

The simple graded modules over the graded ring $D$ are the simple objects in
the length category $\grdhol$ of graded holonomic $D$-modules. We classify
the simple graded $D$-modules, up to graded isomorphisms and twists:

\begin{ithm}
If $A = k[\Gamma]$ is a monomial curve, then the simple graded left modules
over the ring $D$ of differential operators on $A$, up to graded isomorphisms
and twists, are
    \[ \{ M_0 \} \cup \{ M_{\alpha}: \alpha \in J^* \} \cup \{ M_{\infty} \}
    \]
where $J^* = \{ \alpha \in k: 0 \le \re(\alpha) < 1, \; \alpha \neq 0 \}$.
Moreover, we have that $M_0 = A$ and that $M_{\alpha} = D/D \cdot (E -
\alpha)$ for all $\alpha \in J^*$.
\end{ithm}

Any ring of differential operators on a monomial curve is Morita equivalent
to the first Weyl algebra $\weyl$, which is itself the ring of differential
operators on the monomial curve $A = k[t]$. We obtain the following special
case:

\begin{icor}
The simple graded left modules over the first Weyl algebra $D = \weyl$, up to
graded isomorphisms and twists, are given by $M_0 = D/D\partial$, $M_{\infty}
= D/Dt$, and $M_{\alpha} = D/D(E-\alpha)$ for all $\alpha \in J^*$.
\end{icor}

The Krull-Schmidt Theorem holds in $\grdhol$ since it is a length category.
In Eriksen \cite{erik18-itext}, we describe a constructive method for finding
the indecomposable objects in $\grdhol$ when the simple objects and their
extensions are given. We have therefore computed the extensions of the simple
graded $D$-modules:

\begin{iprop}
If $D$ is the ring of differential operators on a monomial curve, then the
extensions of $M_{\beta}$ by $M_{\alpha}$ in $\grdhol$ are given by
    \[ \ext^1_D(M_{\alpha},M_{\beta})_0 = \begin{cases} k \xi \cong k, &
    (\alpha,\beta) = (0,\infty), (\infty,0) \\ k \xi \cong k, & \alpha =
    \beta \in J^* \\ 0, & \text{ otherwise} \end{cases} \]
for all simple graded $D$-modules $M_{\alpha}, M_{\beta}$ with $\alpha,
\beta \in J^* \cup \{ 0, \infty \}$.
\end{iprop}

For example, when $D = \weyl$ is the first Weyl algebra, the proposition
shows that $\ext^1_D(M_{\infty}, M_0)_0 = k \xi \cong k$ for a non-split
graded extension $\xi$. We represent $\xi$ by the graded extension
    \[ 0 \to Dt/D\partial t \to D/D\partial t \to D/Dt \to 0 \]
where $D/Dt = M_{\infty}$ and $Dt/D\partial t \cong M_{0}[-1]$. Hence $M =
D/D \partial t$ is the unique indecomposable graded holonomic $D$-module of
length two with composition series $M \supseteq M_0[-1] \supseteq 0$ and
simple factors $M_{\infty}$ and $M_0[-1]$.

It turns out that this example can be generalized. In fact, we use the
results of this paper to classify all graded holonomic $D$-modules in
Eriksen \cite{erik18-itext}, where we prove the following result:

\begin{ithm}
Let $D = \weyl$ be the first Weyl algebra. The indecomposable $D$-modules
in $\grdhol$ are, up to graded isomorphisms and twists, given by
    \[ M(\alpha,n) = D/D \, (E-\alpha)^n, \quad M(\beta,n) = D/D \; \mathbf
    w(\beta,n) \]
where $n \ge 1$, $\alpha \in J^* = \{ \alpha \in k: 0 \le \re(\alpha) < 1,
\; \alpha \neq 0 \}$, $\beta \in \{ 0, \infty \}$, and $\mathbf
w(\beta,n)$ is the alternating word on $n$ letters in $t$ and $\partial$,
ending with $\partial$ if $\beta = 0$, and ending in $t$ if $\beta = \infty$.
\end{ithm}

We remark that the assumption that $k = \cc$ is one of convenience; the
results would hold over any algebraically closed field $k$ of characteristic
$0$, and the methods would be applicable if $k$ is any field of
characteristic $0$.

\section{Differential operators on affine monomial curves}
\label{s:diff-ring}

Let $\Gamma \subseteq \n_0$ be a numerical semigroup, and let $\{ a_1, a_2,
\dots, a_r \}$ be a minimal set of generators of $\Gamma$. Without loss of
generality, we may assume that these generators are without common factors,
such that $H = \n_0 \setminus \Gamma$ is a finite set. We consider the
semigroup algebra $A = k[\Gamma] \cong k[t^{a_1}, t^{a_2}, \dots, t^{a_r}]
\subseteq k[t]$ over the field $k = \cc$ of complex numbers, and the algebra
$D = \diff(A)$ of $k$-linear differential operators on $A$. Notice that $A =
k[\Gamma]$ is the coordinate ring of an affine monomial curve $X \subseteq
\aff^r$. By abuse of language, we shall call $A = k[\Gamma]$ a monomial
curve.

We notice that $A = k[\Gamma]$ is a positively graded ring. Let $S = \{ t^n:
n \in \Gamma \} \subseteq A$ be the multiplicatively closed subset consisting
of the homogeneous elements in $A$, and consider the graded localization $A
\subseteq S^{-1}A = k[t,t^{-1}] = T$. By general localization results for
rings of differential operators, there is a localization
    \[ D = \diff(A) \to S^{-1}D = S^{-1}A \otimes_A D = k[t,t^{-1}] \otimes_A
    D = \diff(T) \]
and we may identify $D$ with the subring $\{ P \in \diff(T): P(A) \subseteq A
\} \subseteq \diff(T)$; see for instance Smith, Stafford \cite{smit-staf88}.

Let us write $B = k[t]$ for the normalization of $A$, and $\diff(B)$ for its
algebra of $k$-linear differential operators. We remark that $B$ is itself
the coordinate ring of an affine monomial curve, the affine line $\aff^1$,
and that $\diff(B) = \weyl = k\langle t,\partial \rangle$ is the first Weyl
algebra, generated by $t$ and $\partial = d/dt$ and with relation $[\partial,
t] = 1$. In particular, it follows that
    \[ \diff(T) \cong k[t,t^{-1}] \otimes_{k[t]} \weyl \cong k[t,t^{-1}]
    \langle \partial \rangle \]
There is a grading on $\diff(T)$ induced by the grading on $T$, such that $P
\in \diff(T)$ is homogeneous of degree $w$ if and only if $P(T_i) \subseteq
T_{i+w}$ for all integers $i$. In concrete terms, $t^n \partial^m \in
\diff(T)$ is homogeneous of degree $n-m$. Moreover, $D = \diff(A)$ is the
graded subalgebra
    \[ D = \{ P \in \diff(T): P(A) \subseteq A \} \subseteq k[t,t^{-1}]
    \langle \partial \rangle \]
We write $D = \osum_w \, D_w$, where $D_w$ is the linear subspace of
differential operators of degree $w$.

Based on these results, we gave an explicit description of the algebra $D$ of
differential operators on an affine monomial curve in Eriksen \cite{erik03}:
For any degree $w \in \z$, we have that $D_w = P_w \cdot k[E]$, where $E = t
\partial$ is the Euler derivation of degree zero, and $P_w$ is the
homogeneous differential operator of degree $w$ given by
    \[ P_w = t^w \prod_{\gamma \in \Omega(w)} \; (E - \gamma) \]
determined by the set $\Omega(w) = \{ \gamma \in \Gamma: \gamma + w \not \in
\Gamma \}$. In particular, we proved the following result:

\begin{theorem}
If $A = k[\Gamma]$ is a monomial curve, then the $k$-algebra $D$ of
differential operators on $A$ is generated by the Euler derivation $E$ and
the differential operators $P_w$ for all degrees $w$ such that $|w| \in \{
a_1, a_2, \dots, a_r \}$ or $|w| \in H$.
\end{theorem}

The following structural results on the algebra $D$ of differential operators
on $A$ is a consequence of the work of Smith, Stafford \cite{smit-staf88}:

\begin{theorem} \label{t:diff-str}
If $A = k[\Gamma]$ is a monomial curve, then the ring $D$ of differential
operators on $A$ is Morita equivalent with the first Weyl algebra $\diff(B)
= \weyl$, and $D$ has the following properties:
\begin{enumerate}
\item $D$ is a simple Noetherian ring
\item $A$ is a simple left $D$-module
\item $D$ has Krull dimension $1$ and Gelfand-Kirillov dimension $2$
\item $D$ is a hereditary ring
\end{enumerate}
\end{theorem}
\begin{proof}
Since the normalization of $X = \spec(A)$ is the affine line $\spec(B) =
\aff^1$, and the normalization map $\aff^1 \to X$ is injective, it follows
from Proposition 3.3, Theorem 3.4, Theorem 3.7 and Proposition 4.2 in Smith,
Stafford \cite{smit-staf88} that $D$ is Morita equivalent with the Weyl
algebra $\weyl$, that $D$ is a simple ring, and that $A$ is a simple left
$D$-module. The rest follows from the fact that the properties are preserved
under Morita equivalence, and hold for $\weyl$. \qed
\end{proof}

Let $D^p \subseteq D$ be the set of differential operators in $D$ of order at
most $p$. We call $\{ D^p \}$ the order filtration of $D$, and consider the
associated graded ring
    \[ \gr D = \osum_{p \ge 0} D^p/D^{p-1} \]
It is well-known that $\gr D$ is a commutative ring, and that $\gr \diff(B) =
k[t,\xi]$, where $\xi$ is the image of $\partial$. In Eriksen \cite{erik03},
we proved that, for $\Gamma \neq \n_0$, the ring $\gr D = k[\Gamma'] \subseteq
k[t,\xi]$ is a semigroup algebra with minimal set of generators
    \[ \{ t^{\sigma(-w)} \xi^{\sigma(w)}: |w| \in \{ a_1, \dots, a_r \}
    \text{ or } |w| \in H \} \cup \{ t \xi \} \]
In the proof, we use the fact that $P_w$ has leading term $t^{w + \sigma(w)}
\partial^{\sigma(w)}$, where $\sigma(w)$ is the cardinality of $\Omega(w)$,
and that $\sigma(-w) = \sigma(w) + w$ for all integers $w$. In particular,
$\gr D = k[\Gamma'] $ is a finitely generated $k$-algebra of Krull dimension
two, and $\n_0^2 \setminus \Gamma'$ is a finite set.

Let us write $D^p_w = D^p \cap D_w \subseteq D$ for the linear subspace of
differential operators of degree $w$ and order at most $p$ in $D$, and define
    \[ B^n = \osum_{2p+w \le n} \; D^p_w \]
for all integers $n$. Then $\{ B^n \}$ is a filtration of $D$, which we call
the \emph{Bernstein filtration}. Note that any differential operator $P \in
B^n$ is of the form
    \[ P = \sum_{i+j \le n} \, c_{ij} \, t^i \partial^j \]
Clearly, the associated graded ring $\osum_n \, B^n/B^{n-1}$ is naturally
isomorphic to $\gr D$. Moreover, since $\gr D \subseteq k[t,\xi]$, it follows
that $B^0 = k$, that $B^n = 0$ for $n < 0$, and that $\dim_k B^n$ is finite
for all integers $n$.

\section{Holonomic D-modules on monomial curves}

Let $M$ be a left $D$-module. A \emph{filtration} of $M$ is a chain $M_0
\subseteq M_1 \subseteq \dots$ of $k$-linear subspaces of $M$ such that
$\cup_n \, M_n = M$ and $B^m \cdot M_n \subseteq M_{m+n}$ for all integers
$m,n \ge 0$. By convention, we let $M_n = 0$ for $n < 0$. We say that the
filtration is \emph{good} if the associated graded module $\gr M = \osum_n \,
M_n/M_{n-1}$ is a finitely generated module over $\gr D$.

We recall that there is a good filtration of $M$ if and only if $M$ is a
finitely generated $D$-module. Moreover, if $\{ M_n \}$ and $\{ M'_n \}$ are
two good filtrations of $M$, then there is an integer $N$ such that
    \[ M_{n-N} \subseteq M'_n \subseteq M_{n+N} \]
for all integers $n$. This is a standard result; see for instance Chapter 1
in Bj\"{o}rk \cite{bjor79}.

We fix a good filtration $\{ M_n \}$ of a left $D$-module $M$, and remark
that $\dim_k M_n$ is finite for all integers $n \ge 0$. In fact, we have that
$(\gr M)_n = M_n/M_{n-1}$ is finitely generated over $B_0 = k$, since $\gr M$
is a finitely generated module over $\gr D$, and $\gr D$ is positively graded
with $(\gr D)_0 = B_0 = k$. The Hilbert function of $\gr M$ is the function
$\hb(\gr M,-)$ given by
    \[ \hb(\gr M,n) = \dim_k \left( M_n/M_{n-1} \right) = \dim_k M_n - \dim_k
    M_{n-1} \]
and $\hb^1(\gr M,n) = \dim_k M_n$ is the first iterated Hilbert function of
$\gr M$.

\begin{proposition} \label{p:hilb-qp}
Let $M$ be a non-zero, finitely generated left $D$-module and let $\{ M_n \}$
be a good filtration of $M$. Then there is a positive integer $m \ge 1$ and
polynomials $P_r(t) \in \q[t]$ for $0 \le r < m$, such that $\dim_k \left(
M_{nm+r} \right) = P_r(n)$ for all sufficiently large integers $n$. Moreover,
we have that
    \[ P_r(t) = \frac{e}{d!} \cdot t^d + \text{ terms of lower degree} \]
for $0 \le r < d$, where $d$ is the Krull dimension of $\gr M$, and $e > 0$
is a positive integer.
\end{proposition}
\begin{proof}
To simplify notation, we write $R = \gr D$, and $N = \gr M$.  We consider the
Hilbert series $\hb_{N}(t) = \sum_n \hb(N,n) \cdot t^n$ of $N$ in $\z[[t]]$,
which can be written as a rational function
    \[ \hb_{N}(t) = \frac{Q(t)}{(1-t^l)^d} \]
where $Q(t) \in \z[t]$ with $Q(1) > 0$, and $l \ge 1$ is a positive integer.
This follows from Proposition 4.4.1 and the following remarks in Bruns,
Herzog \cite{brun-herz93}, since there is a homogeneous system of parameters
for $N$ of common degree $l > 0$. Let
    \[ m = \min \{ l \ge 1: \hb_{N}(t) \cdot (1-t^l)^d \in \z[t] \} \]
and let $a(t) = \hb_{N}(t) \cdot (1-t^m)^d \in \z[t]$. Then $a(1) > 0$, and
by Proposition 2.3 and Remark 2.4 in Campbell et al. \cite{cghsw00}, it
follows that there are polynomials $p_r(t) \in \q[t]$ for $0 \le r < m$ such
that $\deg p_r(t) \le d-1$ and $p_r(n)=\hb(N,nm+r)$ for all $n$ large enough.
Furthermore, since $a(1) > 0$, it follows that $p_r(t)$ has degree $d-1$ for
at least one integer $r$ with $0 \le r < m$. Let us write $a_r(t)$ for the
polynomial
    \[ a_r(t) = \sum_{n \ge 0} \, \alpha_{nm+r} \cdot t^n \]
in $\z[t]$, where the coefficients $\alpha_i$ for $i \ge 0$ are given by
$a(t) = \sum \, \alpha_i t^i$. We obtain polynomials $a_0(t), \dots, a_{m-1}
(t) \in \z[t]$ with $a(1) = a_0(1) + \dots + a_{m-1}(1)$. Furthermore, the
polynomial $p_r(t)$ has the form
    \[ p_r(t)= \frac{a_r(1)}{(d-1)!} \; t^{d-1} + \text{ terms of lower
    degree} \]
for $0 \le r < m$, where $a_r(1) \ge 0$ for all $r$. To compute the first
iterated Hilbert function, we choose a positive integer $n_0$ such that
$\hb(N,nm+r) = p_r(n)$ for $n > n_0$ and $0 \le r < m$. If we let $C_r =
\hb(N,nm+r+1) + \dots + \hb(N,nm+m-1)$, then we have, for $n > n_0$, that
\begin{align*}
\hb^1(N,nm + r) &= \sum_{s = 0}^{m-1} \sum_{t = 0}^{n} \hb(N,tm+s) - C_r \\
&= \sum_{s = 0}^{m-1} \sum_{t=n_0+1}^n p_s(t) - C_r + \hb^1(N,n_0 m + m-1) \\
&= \sum_{s = 0}^{m-1} \sum_{t = n_0+1}^n \frac{a_s(1)}{(d-1)!} \, t^{d-1} +
\text{ terms of lower degree } \\
&= \sum_{t = n_0 + 1}^{n} \frac{a(1)}{(d-1)!} \, t^{d-1} + \text{ terms of
lower degree } \\
&= \frac{a(1)}{d!} \, t^d + \text{ terms of lower degree }
\end{align*}
since $C_r$ and $\hb^1(N,n_0 m + m-1)$ are constants. Hence, there are
polynomials $P_r(t) \in \q[t]$ for $0 \le r < m$ of the form
    \[ P_r(t) = \frac{e}{d!} \, t^d + \text{ terms of lower degree } \]
such that $\hb^1(nm+r) = P_r(n)$ for all $n > n_0$, where $d$ is the Krull
dimension of $N$ and $e = a(1) > 0$ is a positive integer. \qed
\end{proof}

We remark that $d$ and $e$ in Proposition \ref{p:hilb-qp} are independent of
the chosen good filtration $\{ M_n \}$ of $M$. We define the \emph{dimension}
of $M$ to be $d(M) = d$, and the \emph{multiplicity} of $M$ to be $e(M) = e$
for any finitely generated left $D$-module $M \neq 0$. These invariants have
the following properties: If $0 \to M' \to M \to M'' \to 0$ is a short exact
sequence of $D$-modules, then $d(M) = \max \{ d(M'), d(M'') \}$. If moreover
$d(M') = d(M) = d(M'')$, then $e(M) = e(M') + e(M'')$. This follows from the
fact that a good filtration of $M$ induces good filtrations of $M'$ and $M''$.

\begin{proposition}[Bernstein's inequality]
For any finitely generated left $D$-module $M \neq 0$, we have $d(M) \ge 1$.
\end{proposition}
\begin{proof}
From Theorem \ref{t:diff-str}, it follows that $D$ is a simple ring, and
the algebra homomorphism  $D \to \enm_k(M)$ is therefore injective. This
implies that $M$ cannot be finite dimensional over $k$, hence $d(M) \ge 1$.
\qed
\end{proof}

We say that a finitely generated left $D$-module $M$ is \emph{holonomic} if
$M \neq 0$ and $d(M) = 1$. By convention, $M = 0$ is also considered to be
holonomic. If $M$ is holonomic, then it has finite length. In fact, if $M'
\subseteq M$ is a non-zero submodule of $M$, then $M'$ is holonomic and $e(M')
< e(M)$ if $M' \neq M$. Hence the length of $M$ is at most $e(M)$.

\begin{proposition} \label{p:dmod-hol}
Let $M$ be a finitely generated left $D$-module. Then the following
conditions are equivalent:
\begin{enumerate}
\item $M$ is holonomic
\item $M$ has finite length
\item $M$ is cyclic and not isomorphic to $D$
\end{enumerate}
\end{proposition}
\begin{proof}
If $M$ is holonomic, then it has finite length by the comment above. If $M$
has finite length, then $M$ is cyclic by Theorem 1.8.18 in Bj\"{o}rk
\cite{bjor79}, since $D$ is a simple ring by Theorem \ref{t:diff-str}.
Moreover, $M$ is not isomorphic to $D$ since $d(M) = 1$ and $d(D) = 2$. In
fact, the Bernstein filtration is a good filtration of $D$, and we have
\begin{align*}
\dim_k B^n &= 1 + 2 + \dots + (n+1) - s = \frac{(n+1)(n+2)}{2} - s \\
&= \frac12 n^2 + \text{ terms of lower degree}
\end{align*}
for $n$ sufficiently large, where $\gr D = k[\Gamma'] \subseteq k[t,\xi]$ and
$s$ is the cardinality of $\n_0^2 \setminus \Gamma'$. Finally, we show that
if $M$ is cyclic and not isomorphic to $D$, then it is holonomic. In this
case, we may assume that $M = D/I$ with $I \neq 0$, and there is a non-zero
element $P \in I$ and a principal left ideal $J = D \cdot P \subseteq I$. The
short exact sequence
    \[ 0 \to D \xrightarrow{\cdot P} D \to D/J \to 0 \]
shows that $d(D/J) = 1$. In fact, if $d(D/J) = 2$, then $e(D) = e(D) +
e(D/J)$, and this is a contradiction since $e(D) = 1$ from the computation
above. The short exact sequence $0 \to I/J \to D/J \to D/I \to 0$ gives that
$d(M) = d(D/I) \le d(D/J) = 1$, and this implies that $M$ is holonomic. \qed
\end{proof}

\section{Graded holonomic D-modules}

Let $D$ be the algebra of differential operators on a monomial curve $A =
k[\Gamma]$, with the $\z$-grading $D = \osum_w \, D_w$ described in Section
\ref{s:diff-ring}, and consider the category $\grdmod$ of $\z$-graded left
$D$-modules. An object of $\grdmod$ is a left $D$-module $M$ with a grading
    \[ M = \osum_w \, M_w \]
such that $D_v \cdot M_w \subseteq M_{v+w}$, and a morphism $\phi: M \to N$
in $\grdmod$ is a $D$-module homomorphism which is homogeneous of degree
zero, such that $\phi(M_w) \subseteq N_w$. For any graded $D$-module $M$ in
$\grdmod$ and any integer $n$, we denote by $M[n]$ the $n$'th twisted
$D$-module of $M$, with grading given by $M[n]_i = M_{n+i}$.

We wish to study and classify the graded holonomic $D$-modules, up to graded
isomorphisms in $\grdmod$ and twists. The full subcategory $\grdhol \subseteq
\grdmod$ of graded holonomic $D$-modules consists of graded $D$-modules $M$
of finite length, with composition series
    \[ M = M_0 \supseteq M_1 \supseteq \dots \supseteq M_{n-1} \supseteq M_n
    = 0 \]
in $\grdmod$ of length $n \le e(M)$; see Proposition \ref{p:dmod-hol}. We are
interested in the simple objects in $\grdmod$, since these are the simple
factors $M_i/M_{i+1}$ in the composition series.

\begin{lemma}
A graded left $D$-module $M$ is a simple object in $\grdmod$ if and only if
it is simple considered as a left $D$-module.
\end{lemma}
\begin{proof}
It is clear that if $M$ is simple as a left $D$-module, then it is simple
in $\grdmod$. To prove the converse, assume that $M$ is simple in $\grdmod$.
Then it follows from Theorem II.7.5 in N\u{a}st\u{a}sescu, van Oystaeyen
\cite{nast-oys79} that it is either simple or $1$-critical considered as a
left $D$-module. We claim that it cannot be $1$-critical. In fact, we may
choose a homogeneous element $m \neq 0$ of degree $w$ in $M$, and consider
the short exact sequence
    \[ 0 \to I \to D(-w) \xrightarrow{\cdot m} M \to 0 \]
where $D(-w)$ is a twist of $D$ and $I = \{ P \in D(-w): P \cdot m = 0 \}$
is the annihilator of $m$. Since $I \neq 0$, it follows from Proposition
\ref{p:dmod-hol} that $D/I$ is a holonomic $D$-module, and therefore of
finite length. This implies that $M$ is Artinian, of Krull dimension zero,
and it is therefore not $1$-critical. It follows that any simple object in
$\grdmod$ is simple considered as a left $D$-module. \qed
\end{proof}

The simple modules over the Weyl algebra $\weyl$ were classified in Block
\cite{bloc81}. We shall classify the simple objects in $\grdmod$ by adapting
Block's results to the graded situation. For any graded left $D$-module $M$,
we define
    \[ T_S(M) = \{ m \in M: sm = 0 \text{ for an element } s \in S \}
    \subseteq M \]
It follows from the fact that $S$ is an Ore set for $D$ that $T_S(M)$ is a
left $D$-module. We say that $M$ is an $S$-torsion module if $T_S(M) = M$,
and that it is an $S$-torsion free module if $T_S(M) = 0$. Notice that any
simple left $D$-module is either an $S$-torsion module or an $S$-torsion
free module, and that it is an $S$-torsion module if and only if $S^{-1}M =
0$.

There is a graded division algorithm in the ring $\diff(T)$ in the following
sense: For any homogeneous differential operators $P,Q \in \diff(T)$ with $Q
\neq 0$, we have that
    \[ P = L \cdot Q + R \]
for unique homogeneous differential operators $L,R \in \diff(T)$, where the
order $d(R) < d(Q)$. Therefore, any homogeneous left ideal $I \subseteq
\diff(T)$ is principal. In fact, if $P \neq 0$ is an element in $I$ with
minimal order, then $I = \diff(T) \cdot P$. This implies that a graded left
$\diff(T)$-module is simple considered as a $\diff(T)$-module if and only if
it is a simple object in the category of graded left $\diff(T)$-modules, and
we can talk about simple graded left $\diff(T)$-modules without ambiguity.

\begin{proposition} \label{p:grdmod-loc}
The assignment $M \mapsto S^{-1}M$ defines a bijective correspondence
    \[ S^{-1}: \grsimp_D[\text{S-torsion free}] \to \grsimp_{\diff(T)} \]
from the set of isomorphism classes of simple graded left $D$-modules that
are $S$-torsion free, and the set of isomorphism classes of simple graded
left $\diff(T)$-modules.
\end{proposition}
\begin{proof}
If $M$ is a simple graded left $D$-module that is $S$-torsion free, then
$S^{-1}M$ is a simple graded left $\diff(T)$-module, since $S$ consists of
homogeneous elements and localization is exact; see also Lemma 2.2.1 in
Block \cite{bloc81}. This defines the map $S^{-1}$, and that it is injective
follows from the proof of Lemma 2.2.1 in Block \cite{bloc81}. To show that
$S^{-1}$ is surjective, let us consider a graded simple left
$\diff(T)$-module $N$. We choose a non-zero homogeneous element $n \in N$
of degree $w$, which gives a short exact sequence
    \[ 0 \to I \to \diff(T)[-w] \xrightarrow{\cdot n} N \to 0 \]
where $I = \{ P \in \diff(T): P \cdot n = 0 \} = \ann(n)$. Let $J = I \cap
D[-w] \subseteq D[-w]$. This is a non-zero homogeneous ideal in $D[-w]$, and
$M = D[-w]/J \subseteq \diff(T)[-w]/I \cong N$ is a graded $D$-submodule.
Since $J \neq 0$, $M$ is a graded $D$-module of finite length by Proposition
\ref{p:dmod-hol}, hence it contains a graded simple left $D$-module $K$.
Then it follows from Lemma 2.2.1 in Block \cite{bloc81} and its proof that
$N \cong S^{-1}K$ since $K \subseteq M \subseteq N$. \qed
\end{proof}

Next, we classify the simple graded left $\diff(T)$-modules, up to graded
isomorphisms and twists. Any simple graded left $\diff(T)$-module $N$ must
be of the form $N \cong \diff(T)[-w]/I$, where $I = \diff(T) \cdot P$ is a
homogeneous principal left ideal that is maximal. Hence, we must have that
$P = t^w (E-\alpha)$ for $\alpha \in k$, and $t^w$ is a unit in $\diff(T)$.
We obtain the cases $\alpha = 0$, which gives the graded simple module
    \[ N_0 = \diff(T) / \diff(T) \cdot E = \diff(T) / \diff(T) \cdot \partial
    \cong T \]
and $\alpha \neq 0$, which gives the graded simple module
    \[ N_{\alpha} = \diff(T) / \diff(T) \cdot (E-\alpha) \]
It is not difficult to see that $N_{\alpha} \cong T[\alpha]$ if $\alpha \in
\z$, with isomorphism given by $1 \mapsto t^{\alpha}$. Similarly, we have
that $N_{\alpha} \cong N_{\beta}[\alpha-\beta]$ when $\alpha-\beta \in \z$,
with isomorphism given by $1 \mapsto t^{\alpha-\beta}$.

\begin{lemma}
Let $J = \{ \alpha \in k: 0 \le \re(\alpha) < 1 \}$. Then the set of simple
graded $\diff(T)$-modules, up to graded isomorphisms and twists, are given
by $\{ N_{\alpha}: \alpha \in J \}$.
\end{lemma}
\begin{proof}
We claim that if $\alpha - \beta \not \in \z$, then $N_{\alpha} \not \cong
N_{\beta}$ as left $\diff(T)$-modules. In light of the comments above, this
is enough to prove the lemma, since $J$ is a fundamental domain for $k/\z$.
To prove the claim, we consider the Weyl algebra $\weyl$ as a special case of
$D = \diff(A)$ with $A = k[\Gamma]$ and $\Gamma
= \n_0$. By Proposition \ref{p:grdmod-loc}, the left module $M_{\alpha} =
\weyl/\weyl \cdot (E-\alpha)$ corresponds to $N_{\alpha} = S^{-1}M_{\alpha}$
under localization when $\alpha \not \in \z$, and $M_0 = \weyl / \weyl \cdot
\partial$ corresponds to $N_0$. By results of Dixmier on the Weyl algebra, we
have that $M_{\alpha} \not \cong M_{\beta}$ when $\alpha -\beta \neq \z$; see
Lemma 24 in Dixmier \cite{dixm63} and Proposition 4.4 in Dixmier
\cite{dixm70}, and this proves the claim. \qed
\end{proof}

Let $A = k[\Gamma]$ be a monomial curve, let $D$ be the ring of differential
operators on $A$, and let $\weyl$ be the ring of differential operators on $B
= k[t]$. Then $D$ is Morita equivalent with the Weyl algebra $\weyl$, and by
Section 3.14 of Smith, Stafford \cite{smit-staf88}, the equivalence $\mcat D
\to \mcat{\weyl}$ of module categories is given by $M \mapsto D(A,B)
\otimes_D M$, where $D(A,B)$ is the $\weyl$-$D$ bimodule
    \[ D(A,B) = \{ P \in \diff(T): P(A) \subseteq B \} \subseteq \diff(T) \]
Notice that $D(A,B)$ is a graded $\weyl$-$D$ bimodule, with $S^{-1}D(A,B) =
\diff(T)$. Hence, there is an induced equivalence $F: \grdmod \to \grmod{
\weyl}$ of categories of graded modules, and it commutes with localization:
    \[ \xymatrix{ & \grmod{\diff(T)} & \\ \grdmod \ar[ur]^{S^{-1}}
    \ar[rr]_F & & \grmod{\weyl} \ar[ul]_{S^{-1}} } \]
We know that $\grsimp_D[\text{S-torsion free}] \cong \grsimp_{\diff(T)}$, and
we write $M_{\alpha}$ for the unique simple graded left $D$-module $M$ such
that $S^{-1}M = N_{\alpha}$ for $\alpha \in J$. In fact, it follows from the
proof of Proposition \ref{p:grdmod-loc} that $M_{\alpha}$ is the unique
simple submodule of $D/D \cdot (E - \alpha)$ if $\alpha \neq 0$, and that
$M_0$ is the unique simple submodule of $D/D \cdot \partial = A$. We claim
that $M_{\alpha} = A$ if $\alpha = 0$, and that $M_{\alpha} = D/D \cdot (E -
\alpha)$ if $\alpha \neq 0$. If $\alpha = 0$, this is clear since $A$ is a
simple left $D$-module. If $\alpha \neq 0$, then the claim follows from the
following result, since simple modules are preserved by Morita equivalence:

\begin{lemma}
If $A = k[t]$, then $M_{\alpha} = D/D \cdot (E - \alpha)$ for all $\alpha \in
J$ with $\alpha \neq 0$.
\end{lemma}
\begin{proof}
It is sufficient, in light of the comments above, to show that $D/D \cdot (E -
\alpha)$ is a simple module over $D = \weyl$. This follows from Lemma 24 in
Dixmier \cite{dixm63}. \qed
\end{proof}

Finally, we classify the simple graded $D$-modules that are $S$-torsion
modules. We first consider the case $A = k[t]$, and the $S'$-torsion modules
over the Weyl algebra $D = \weyl$, where $S' = k[t]^*$. In this case, it
follows from Proposition 4.1 and Corollary 4.1 in Block \cite{bloc81} that
the simple $S'$-torsion $D$-modules are given by
    \[ V(\beta) = D \otimes_A A/(t-\beta) \cong D / D \cdot (t - \beta) \]
for $\beta \in k$. Moreover, $V(\beta)$ is a graded left $D$-module, or an
$S$-torsion module, if and only if $\beta = 0$. Hence, $V(0) = D/D \cdot t$ is
the unique simple graded left module over the Weyl algebra $D = \weyl$ that is
$S$-torsion.

Let $D$ be the ring of differential operators on $A$ when $A = k[\Gamma]$ is
any monomial curve. Since $D$ is Morita equivalent to the Weyl algebra and the
property of being $S$-torsion is preserved under Morita equivalence, there is
a unique simple graded left $D$-module $M_{\infty}$ that is $S$-torsion.
Moreover, we have that
    \[ M_{\infty} = D(B,A) \otimes_{\weyl} V(0) \]
where $D(B,A) = \{ P \in \diff(T): P(B) \subseteq A \} \subseteq \diff(T)$ by
Section 3.14 in Smith, Stafford \cite{smit-staf88}. We summarize these
results as follows:

\begin{theorem}
If $A = k[\Gamma]$ is a monomial curve, then the simple graded left modules
over the ring $D$ of differential operators on $A$, up to graded isomorphisms
and twists, are given by
    \[ \{ M_0 \} \cup \{ M_{\alpha}: \alpha \in J^* \} \cup \{ M_{\infty} \}
    \]
where $J^* = \{ \alpha \in k: 0 \le \re(\alpha) < 1, \; \alpha \neq 0 \}$.
Moreover, we have that $M_0 = A$ and that $M_{\alpha} = D/D \cdot (E -
\alpha)$ for all $\alpha \in J^*$.
\end{theorem}

\begin{corollary}
The simple graded left modules over the first Weyl algebra $D = \weyl$, up to
graded isomorphisms and twists, are given by $M_0 = D/D\partial$, $M_{\infty}
= D/Dt$, and $M_{\alpha} = D/D(E-\alpha)$ for all $\alpha \in J^*$.
\end{corollary}

Let $M_{\alpha},M_{\beta}$ be simple graded $D$-modules in $\grdhol$, with
$\alpha, \beta \in J \cup \{ \infty \}$. Their extensions in $\grdhol$ are
given by $\ext^1_D(M_{\alpha},M_{\beta})_0$ and can be computed using free
resolutions. Moreover, extensions are preserved under Morita equivalence,
and it is therefore enough to consider the case $D = \weyl$.

\begin{proposition}
If $D$ is the ring of differential operators on a monomial curve, then the
extensions of $M_{\beta}$ by $M_{\alpha}$ in $\grdhol$ are given by
    \[ \ext^1_D(M_{\alpha},M_{\beta})_0 = \begin{cases} k \xi \cong k, &
    (\alpha,\beta) = (0,\infty), (\infty,0) \\ k \xi \cong k, & \alpha =
    \beta \in J^* \\ 0, & \text{ otherwise} \end{cases} \]
for all simple graded $D$-modules $M_{\alpha}, M_{\beta}$ with $\alpha,
\beta \in J^* \cup \{ 0, \infty \}$.
\end{proposition}
\begin{proof}
By the comments above, we may assume that $D = \weyl$ is the Weyl algebra.
We show the computation of the extensions in the case $\alpha = \infty$ and
$\beta = 0$; all other cases can be done in a similar way. The $D$-module
$M_{\infty} = D/Dt$ over the Weyl algebra $D = \weyl$ has the free resolution
    \[ 0 \gets M_{\infty} \gets D \xleftarrow{\cdot t} D \gets 0 \]
When we apply $\hmm_D(-,M_0)$ to this exact seqence, we obtain
    \[ \hmm_D(D,M_0) \cong M_0 \xrightarrow{t \cdot} \hmm_D(D,M_0) \cong M_0
    \to 0 \]
and $\ext^1_D(M_{\infty},M_0) \cong M_0/tM_0 \cong A/tA = k[t]/(t) \cong
k$, which is concentrated in degree zero. \qed
\end{proof}

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