# Geometrization of the TUY/WZW/KZ connection

Indranil Biswas  Mathematics Department, Shiv Nadar University, NH91, Tehsil
Dadri, Greater Noida, Uttar Pradesh 201314, India<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Swarnava Mukhopadhyay  School of Mathematics, Tata
Institute of Fundamental Research, Mumbai-400005, India
<EMAIL_ADDRESS>and  Richard Wentworth  Department of Mathematics,
University of Maryland, College Park, MD 20742, USA<EMAIL_ADDRESS>
###### Abstract.

Given a simple, simply connected, complex algebraic group $G$, a flat
projective connection on the bundle of nonabelian theta functions on the
moduli space of semistable parabolic $G$-bundles over any family of smooth
projective curves with marked points was constructed by the authors in an
earlier paper. Here, it is shown that the identification between the bundle of
nonabelian theta functions and the bundle of WZW conformal blocks is flat with
respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. As
an application, we give a geometric construction of the Knizhnik-Zamolodchikov
connection on the trivial bundle over the configuration space of points in the
projective line whose typical fiber is the space of invariants of tensor
product of representations.

###### 2020 Mathematics Subject Classification:

Primary: 14H60, 32G34, 53D50; Secondary: 81T40, 14F08

I.B. is supported in part by a J.C. Bose fellowship and both I.B. and S.M. are
also partly supported by DAE, India under project no.
1303/3/2019/R&D/IIDAE/13820. S.M. received additional funding from the Science
and Engineering Research Board, India (SRG/2019/000513). R.W. is supported in
part by National Science Foundation grants DMS-1906403 and DMS-2204346.

###### Contents

  1. 1 Introduction
  2. 2 Conformal Blocks
    1. 2.1 Affine Lie algebras and integrable modules
    2. 2.2 Sheaf of Conformal Blocks
    3. 2.3 Coordinate free construction
  3. 3 Energy momentum tensor and the Sugawara construction
    1. 3.1 Construction of the WZW/TUY connection
    2. 3.2 WZW/TUY connection
  4. 4 Projective connections via heat operators
    1. 4.1 Heat operators and the Hitchin-van Geemen-de Jong equation
    2. 4.2 Heat operators and metaplectic quantization
      1. 4.2.1 Rewriting the deformation equation
      2. 4.2.2 Metaplectic correction d’après Scheinost-Schottenloher
  5. 5 Parabolic Hitchin symbol as in Biswas-Mukhopadhyay-Wentworth
    1. 5.1 The Hitchin symbol
    2. 5.2 Parabolic Hitchin connection via heat operators
  6. 6 Proof of Theorem 1.1
    1. 6.1 Twisted $\mathcal{D}$-modules via quasi-section of Drinfeld-Simpson
    2. 6.2 Projective heat operator from Sugawara
    3. 6.3 The parabolic duality pairing and the Hitchin symbol
    4. 6.4 Proof of the Main theorem (Theorem 1.1)
  7. 7 Geometrization of the KZ equation on invariants
    1. 7.1 KZ connection
    2. 7.2 Invariants as global sections
    3. 7.3 Differential operators
  8. A Moduli spaces of parabolic bundles
    1. A.1 Parabolic $G$ bundles
    2. A.2 Parabolic bundles as equivariant bundles
    3. A.3 Parabolic determinants as equivariant determinants
  9. B Uniformization of moduli spaces and conformal blocks
    1. B.1 The line bundle on the universal moduli stack
      1. B.1.1 The relative affine flag varieties
      2. B.1.2 The central extension
    2. B.2 The relative uniformization and parabolic theta functions
      1. B.2.1 Parabolic determinant as the Borel-Weil-Bott line bundle
      2. B.2.2 Parabolic theta functions and conformal blocks

##  1\. Introduction

The Wess-Zumino-Witten (WZW) model [48, 69] is a cornerstone of two
dimensional rational conformal field theories [10, 44]. The WZW conformal
blocks were constructed mathematically by Tsuchiya-Ueno-Yamada [66]. Let
$\widehat{\mathfrak{g}}$ be an affine Lie algebra and $(C,\,\bm{p})$ a smooth
curve $C$ with $n$-distinct marked points
$\bm{p}\,=\,(p_{1},\,\cdots,\,p_{n})$. Choose formal coordinates
$\bm{\xi}\,=\,(\xi_{1},\,\cdots,\,\xi_{n})$ around $\bm{p}$, and using these
coordinates assign a copy of $\widehat{\mathfrak{g}}$ to each point $p_{i}$.
Fix a positive integer $\ell$. Then for any choice of $n$-tuple of integrable
highest weights $\vec{\lambda}\,=\,(\lambda_{1},\,\cdots,\,\lambda_{n})$ of
level $\ell$, the construction in [66] associates a finite dimensional vector
space
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(C,\bm{p},\bm{\xi},\mathfrak{g},\ell)$
to the data $(C,\,\bm{p},\,\bm{\xi})$. For a family of smooth curves
$\pi\,:\,\mathcal{C}\,\to\,S$ with $n$-distinct sections $\bm{p}$, these
vector spaces patch together to produce a coherent sheaf
$\mathcal{V}^{\dagger}_{\vec{\lambda}}(\mathfrak{g},\ell)\,\to\,S$. The
Sugawara construction [66] endows this sheaf with the structure of a twisted
$\mathcal{D}$-module, and hence
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$ is actually a
holomorphic vector bundle. The authors of [66] show that this vector bundle
extends to the Deligne-Mumford-Knudsen compactification
$\overline{\mathcal{M}}_{g,n}^{\prime}$ of the moduli spaces of $n$-pointed
curves $\mathcal{M}_{g,n}^{\prime}$ with chosen formal coordinates. Moreover,
the flat projective connection on the interior $\mathcal{M}_{g,n}^{\prime}$
extends to a flat projective connection with logarithmic singularities over
$\overline{\mathcal{M}}_{g,n}^{\prime}$. The bundle
$\mathcal{V}^{\dagger}_{\vec{\lambda}}(\mathfrak{g},\ell)\,\to\,S$ of
conformal blocks is sometimes called the _Friedan-Shenker bundle_. We refer to
the above mentioned flat projective connection on
$\mathcal{V}^{\dagger}_{\vec{\lambda}}(\mathfrak{g},\ell)\,\to\,S$ as the
WZW/TUY connection.

Later, Tsuchimoto [64] gave a coordinate free construction of the bundle of
conformal blocks and showed that it descends to a vector bundle
$\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$ on the Deligne-
Mumford-Knudsen moduli space $\overline{\mathcal{M}}_{g,n}$ of $n$-pointed
stable nodal curves (cf. Fakhruddin [25]). The flat projective connection also
descends to a projective connection with logarithmic singularities. In other
words, there is a projectively flat isomorphism between the conformal blocks
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$ and the pullback
$F^{*}\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$ under the
natural forgetful map
$F\,:\,\overline{\mathcal{M}}_{g,n}^{\prime}\,\to\,\overline{\mathcal{M}}_{g,n}$.
We refer the reader to Section 2 for a construction of conformal blocks and to
Section 3 for the construction of the WZW/TUY connection.

We now discuss how conformal blocks are related to moduli spaces of bundles on
curves. The moduli space $M_{G}(C)$ of principal bundles, with a reductive
structure group $G$, on a smooth projective curve $C$, provides a natural
nonabelian generalization of the Jacobian variety $J(C)$, which parametrizes
line bundles of degree zero on $C$. The moduli space of (semistable) principal
$G$-bundles on a smooth projective algebraic curve is itself a projective
variety. It was originally constructed through Geometric Invariant Theory. Its
smooth locus parametrizes isomorphism classes of stable bundles with minimal
automorphism groups (see [18]), also known as the regularly stable loci. There
are various important variations on this construction. One can choose marked
points $\bm{p}\,=\,(p_{1},\,\cdots,\,p_{n})$ on the algebraic curve $C$ and
decorate a principal $G$-bundle $\mathcal{P}$ with a generalized flag
structure over $\bm{p}$, leading to the notion of quasi-parabolic bundles.
Additionally, one can choose weight data
$\bm{\tau}\,=\,(\tau_{1},\,\cdots,\,\tau_{n})$ in the Weyl alcoves, or
equivalently weights $\vec{\lambda}\,=\,(\lambda_{1},\,\cdots,\,\lambda_{n})$,
and use them to define a suitable notion of stability and semistablity. The
corresponding moduli spaces $M^{par,ss}_{G,\bm{\tau}}(C,\bm{p})$ can, in turn,
be understood as the space of representations of the fundamental group of the
corresponding punctured surface $C\setminus\\{\bm{p}\\}$, where the loops
around the marked points go to fixed conjugacy classes determined by
$\bm{\tau}$ [43, 57]. This generalizes the classical results of Narasimhan-
Seshadri [47] and Ramanathan [51], proved in the non-parabolic case.

The moduli space $M_{G,\bm{\tau}}^{par,ss}(C,\bm{p})$ is equipped with a
natural ample determinant of cohomology line bundle
$\operatorname{Det}_{par,\phi}(\bm{\tau})$ associated to a choice of faithful
linear representation $\phi$ of $G$. This generalizes the theta line bundle on
the Jacobian variety $J(C)$. Therefore, the global sections of this line
bundle on $M_{G,\bm{\tau}}^{par,ss}(C,\bm{p})$ can thus be thought of as a
nonabelian generalization of the classical theta functions. We refer the
reader to Section A.1 for more details on the constructions of the moduli
space and the parabolic determinant line bundle on it.

Via the uniformization theorems of Harder and Drinfeld-Simpson [24, 31],
moduli spaces of parabolic bundles also have an adèlic description that
directly connects to the representation theory of affine Lie algebras via the
work of several authors (see [8, 27, 37, 40, 49, 59]). Using this, the
corresponding moduli stack of principal $G$-bundles and its parabolic analog
$\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})$ can be expressed as a double quotient

$\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})\,=\,G(\Gamma(C,\mathcal{O}_{C}(*\bm{p}))\backslash\prod_{i=1}^{n}G(\mathbb{C}((\xi_{i})))/\mathcal{P}_{i}\
,$

where the $\mathcal{P}_{i}$ are parahoric subgroups of
$G(\mathbb{C}[[\xi_{i}]])$ determined by the weights $\tau_{i}$. The weights
also determine a homogeneous
$G(\Gamma(C,\,\mathcal{O}_{C}(*\bm{p})))$-equivariant line bundle
$\mathscr{L}_{\vec{\lambda}}$ on $\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})$. The
line bundle $\operatorname{Det}^{\otimes a}_{par,\phi}(\bm{\tau})$ coincides
with $\mathscr{L}_{\vec{\lambda}}$, where $a$ is a rational number determined
by the Dynkin index of the representation $\phi$. Generalizations (see Kumar
[36], Mathieu [42]) of the Borel-Weil theorems (see (B.3)) for affine flag
varieties $G(\mathbb{C}((\xi)))/\mathcal{P}_{i}$, coupled with the adèlic
description, give a canonical isomorphism (see (B.2)) with conformal blocks

$\mathcal{V}^{\dagger}_{\vec{\lambda}}(C,\bm{p},\bm{\xi},\mathfrak{g},\ell)\,\cong\,H^{0}(M^{par,ss}_{G,\bm{\tau}}(C,\bm{p}),\,\operatorname{Det}^{\otimes
a}_{par,\phi}(\bm{\tau})).$

This isomorphism can be reinterpreted as the Chern-Simon/WZW correspondence.
More details are given in Section B.1.

Using differential geometric methods, Hitchin, [32], generalizes a
construction of Mumford-Welters [68] to obtain a flat projective connection on
the Friedan-Shenker bundle with fibers
$H^{0}(M_{G}(C),\,\operatorname{Det}^{\otimes\ell})$, from the viewpoint of
geometric quantization  in the sense of Kostant-Souriau. This connection also
appears in Witten’s [70] interpretation of Jones polynomial link invariants as
3-manifold invariants. Hitchin’s construction was reinterpreted by van Geemen-
de Jong [67] sheaf theoretically in terms of the existence of a “heat
operator”, which in the relative setting is a differential operator that is a
combination of a first order operator with one that is second order on the
fibers (see Section 4). We recall the details of the general methods of
Hitchin-van Geemen-de Jong [67] in Section 4. We also refer to the several
complementary approaches of Andersen [1], Axelrod-Witten-della Pietra [3],
Baier-Bolognesi-Martens-Pauly [4], Faltings [26], Ginzburg [30], Ran [52],
Ramadas [50], Sun-Tsai [61] and for generalizations to reductive groups,
Belkale [11].

In [39], Laszlo showed that the connection constructed by Hitchin and the one
in [66] coincide under the natural identification of
$H^{0}(M_{G}(C),\,\operatorname{Det}^{\otimes\ell})$ with
$\mathcal{V}_{0}^{\dagger}(C,\mathfrak{g},\ell)$. A similar result for twisted
Spin groups was also proved by Mukhopadhyay-Wentworth [45]. The following
questions are natural in the context of parabolic moduli spaces:

  1. (1)

Is there a projective heat operator (see Section 4 and Definition 4.1) on the
line bundle $\operatorname{Det}^{\otimes a}_{par,\phi}(\bm{\tau}))$ that
induces a flat projective connection on the vector bundle over
$\mathcal{M}_{g,n}$ with fibers
$H^{0}(M^{par,ss}_{G,\bm{\tau}}(C,\bm{p}),\,\operatorname{Det}^{\otimes
a}_{par,\phi}(\bm{\tau}))$?

  2. (2)

If such a connection exists, is the identification of conformal blocks with
nonabelian parabolic theta functions flat with respect to this connection and
the WZW/TUY connection?

For $g(C)\,\geq\,2$, Scheinost-Schottenloher, [55], constructed a parabolic
Hitchin connection for $G\,=\,\mathsf{SL}_{r}$ under the assumption that the
canonical bundle of $M^{par,ss}_{\mathsf{SL}_{r},\bm{\tau}}(C,\bm{p})$ admits
a square-root. Bjerre [21] removed the “restriction” in [55] for
$G\,=\,\mathsf{SL}_{r}$ by working on a different parabolic moduli space with
full flags. In both [21, 55], the authors construct a connection on the push-
forward “metaplectically corrected” line bundles of the form
$\operatorname{Det}^{\otimes a}_{par,\phi}(\bm{\tau})\otimes
K_{M^{par,ss}_{\mathsf{SL}_{r},\bm{\tau}}(C,\bm{p})}^{1/2}$. We also refer the
reader to Remark 6.3. In [19], we constructed a projective heat operator on
$\operatorname{Det}^{\otimes a}_{par,\phi}(\bm{\tau})$ in general. This was
produced from a candidate parabolic Hitchin symbol (see (5.6)) satisfying a
Hitchin-van Geemen-de Jong type equation (see (5.7))111Subsequent to the
submission of our paper [19], in May 2023 a draft of the thesis of Zakaria
Ouaras appeared in which the author proves the existence of a unique flat
projective connection in the case of moduli spaces of parabolic vector bundles
with arbitrary fixed determinant and genus $g\geq 2$ , 222We have been
informed [2] that in the case of genus zero, $\mathsf{SL}_{2}$, and equal
weights $\lambda$ sufficiently small so that conformal blocks are invariants,
the Hitchin connection constructed in [1] agrees with the KZ equation..

The following result answers the above question (2) and thus generalizes the
result of Laszlo, proved in the non-parabolic case.

######  Theorem 1.1 (Main Theorem).

Let $S$ parametrize a smooth family of $n$-pointed curves. Let
$\pi_{e}\,:\,M^{par,rs}_{G,\bm{\tau}}\,\to\,S$ be the relative moduli space
parametrizing regularly stable parabolic $G$ bundles, i.e., stable parabolic
bundles with minimal automorphisms. The natural isomorphism

$\mathbb{P}\mathbb{V}^{\dagger}_{\vec{\lambda}}(\mathfrak{g},\ell)\,\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\,\mathbb{P}\pi_{e*}\operatorname{Det}_{par,\phi}^{\otimes
a}(\bm{\tau})\ ,$

constructed via the uniformization theorem, between the projectivizations of
the bundles of conformal blocks and nonabelian parabolic theta functions, is
flat for the WZW/TUY connection on
$\mathbb{P}\mathbb{V}^{\dagger}_{\vec{\lambda}}(\mathfrak{g},\ell)$ and the
parabolic Hitchin connection on
$\mathbb{P}\pi_{e\ast}\operatorname{Det}_{par,\phi}^{\otimes a}(\bm{\tau})$.

We are guided by a fundamental observation that if an algebraic group $G$ acts
on a smooth variety $X$ and $\mathscr{L}$ is a $G$-equivariant line bundle on
$X$, then the map induced by the Beilinson-Bernstein localization functor
$\operatorname{Loc}:\mathcal{U}\mathfrak{g}\rightarrow\Gamma(X,\mathcal{D}(\mathscr{L}))$
is a quantum analog of the moment map for the $G$ action on $X$, and the
corresponding graded map
$\operatorname{Sym}\mathfrak{g}\rightarrow\Gamma(X,gr(\mathcal{D}(\mathscr{L})))$
is dual to the moment map. Hence, it is “independent” of the line bundle
$\mathscr{L}$. Thus, an essential point in the proof of this theorem is the
fact that the symbols of the Sugawara operators coming from affine Lie
algebras do not depend on the highest weights. This is checked via a direct
calculation generalizing the non-parabolic counterparts in the works on Laszlo
[39] and Tsuchiya-Ueno-Yamada [66]. The counterpart of this statement on the
moduli of parabolic bundles side for the parabolic Hitchin symbol is therefore
the crux of the argument. This is carried out in Proposition 5.6 using
Corollary 5.5. These are the key new features/differences in the proof of
Theorem 1.1 to that in non-parabolic case considered by Laszlo [39].

##### Application

Now we discuss an application of the parabolic generalization of Theorem 1.1
by giving a geometric reconstruction of the Knizhnik-Zamolodchikov (KZ)
equation. Let us now focus on the genus zero case. Since $\mathbb{P}^{1}$ has
a global coordinate and a global meromorphic two form on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ with second order poles along the
diagonal, the WZW/TUY connection gives a flat (honest) connection on the
bundle of conformal blocks. The equations for the flat sections are known as
_Knizhnik-Zamolodchikov equations_ [34]. Thus, the KZ equations constitute a
system of first order differential equations, arising from the conformal Ward
identities, that determines $n$-point correlation functions in the Wess-
Zumino-Witten-Novikov model of two dimensional conformal field theory. The KZ
equations have remarkable realizations in many other areas. For example higher
dimensional generalizations of hypergeometric functions are known to be
solutions of these equations [54]. The KZ equations can also be regarded as
quantizations of the isomonodromy problem for differential equations of
Fuchsian type [53]. The Kohno-Drinfeld [23, 35] theorem relates the monodromy
representation of the braid group induced by the KZ connection with solutions
of the Yang-Baxter equation.

Here, we consider the KZ equations as equations for flat sections of the
trivial vector bundle $\mathbb{A}_{\vec{\lambda}}$ over the configuration
space $X_{n}$ of $n$-points in $\mathbb{C}$ with fibers

$A_{\vec{\lambda}}\,:=\,\operatorname{Hom}_{\mathfrak{g}}(V_{\lambda_{1}}\otimes\cdots\otimes
V_{\lambda_{n}},\,\mathbb{C})\,.$

We restrict the projective heat operator constructed in [19] to the open
substack $\mathcal{P}ar^{c}_{G}(\mathbb{P}^{1},\bm{p},\vec{\lambda})$ of
quasiparabolic bundles in case of genus zero, where the underlying principal
$G$-bundle is trivial. This turns out to be the quotient stack

$\mathcal{P}ar^{c}_{G}(\mathbb{P}^{1},\bm{p},\vec{\lambda})\,=\,[\left(G/P_{\lambda_{1}}\times\cdots\times
G/P_{\lambda_{n}}\right)/G],$

where $P_{\lambda_{i}}$ are parabolic subgroup determined by $\lambda_{i}$ and
the global sections of the homogeneous line bundle
$\mathscr{L}_{\vec{\lambda}}$ are just the invariants $A_{\vec{\lambda}}$.
Thus, we obtain a flat connection on the vector bundle
$\mathbb{A}_{\vec{\lambda}}$ over $X_{n}$. Finally using Theorem 1.1, we
identify this connection with the KZ connection. This gives an alternative
geometric construction of the KZ equations. We refer the reader to Section 7
and Corollary 7.1 for more details and precise statements.

The outline of the paper is as follows. In Sections 2 and 3 we review in some
detail the construction of the WZW/TUY connection in the parabolic setting. In
Section 4, we review the construction from [19] of the Hitchin connection in
the parabolic setting. This involves the metaplectic correction in a central
way. An important step in this section is the re-expression of one of the
“controlling equations” of van Geemen-de Jong for the existence of a
projective heat operator on elements of the rational Picard group (see Section
4.2). Finally, in Section 5.2 we state the fundamental result, Theorem 5.3,
which provides a simplification of the expression for parabolic Hitchin
symbol. This is the geometric reflection of the aforementioned fact that the
Sugawara operators do not depend on the highest weights.

Finally, in Section 6, using this result we directly relate the symbols of the
Sugawara tensor and the parabolic Hitchin connection, thus proving Theorem 1.1
(see Section 6.4). In the last Section 7 we elaborate on the special case of
genus zero curves. In Appendix A, we define the line bundles over moduli
spaces of parabolic $G$-bundles whose sections give rise to the Friedan-
Shenker bundles. We also relate these line bundles to the determinant of
cohomology in the relative setting.

#####  Acknowledgments.

We thank Prakash Belkale for suggesting the question of constructing the
Knizhnik-Zamolodchikov connection geometrically. S. M. would like to thank
Arvind Nair for useful discussions.

##  2\. Conformal Blocks

In this section we recall the basic notions of conformal blocks, following
Tsuchiya-Ueno-Yamada [66]. Let $\mathfrak{g}$ be a complex simple Lie algebra
and $\mathfrak{h}\,\subset\,\mathfrak{g}$ a Cartan subalgebra. Let $\Delta$ be
a system of roots and $\kappa_{\mathfrak{g}}$ the Cartan-Killing form,
normalized so that
$\kappa_{\mathfrak{g}}(\theta_{\mathfrak{g}},\,\theta_{\mathfrak{g}})\,=\,2$
for the longest root $\theta_{\mathfrak{g}}$. We let
$\nu_{\mathfrak{g}}:\mathfrak{h}^{\ast}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\mathfrak{h}$
denote the isomorphism induced by $\kappa_{\mathfrak{g}}$.

###  2.1. Affine Lie algebras and integrable modules

Let $\xi$ be a formal parameter. The affine Lie algebra

$\widehat{\mathfrak{g}}\,:=\,\mathfrak{g}\otimes\mathbb{C}((\xi))\oplus\mathbb{C}\cdot
c$

is a central extension of the loop algebra $\mathbb{C}((\xi))$ by $c$. The Lie
bracket operation on $\widehat{\mathfrak{g}}$ is given by the formula

(2.1) $\displaystyle[X\otimes f,\,Y\otimes g]$ $\displaystyle:=[X,\,Y]\otimes
fg+\kappa_{\mathfrak{g}}(X,\,Y)\operatorname{Res}_{t=0}(gdf)\cdot c,$

where $X,\ Y\in\mathfrak{g}$ and $f,\ g$ are elements of $\mathbb{C}((\xi))$.

We now briefly recall the basic objects in the representation theory of
$\widehat{\mathfrak{g}}$. It is well-known that the finite dimensional
$\mathfrak{g}$ modules are parametrized by the subset
$P_{+}(\mathfrak{g})\,\subset\,\mathfrak{h}^{*}$ consisting of dominant
integral weights. The representation corresponding to a weight $\lambda$ will
be denoted by $V_{\lambda}$. Let $\ell\,>\,0$ be a positive integer, and
consider the set
$P_{\ell}(\mathfrak{g})\,:=\,\\{\lambda\,\in\,P_{+}(\mathfrak{g})\,\,\mid\,\,\kappa_{\mathfrak{g}}(\lambda,\theta_{\mathfrak{g}})\,\leq\,\ell\\}.$
The highest weight irreducible integrable representations of
$\widehat{\mathfrak{g}}$ at level $\ell$ are classified by the set
$P_{\ell}(\mathfrak{g})$ defined above. The $\widehat{\mathfrak{g}}$-module
corresponding to $\lambda$ will be denoted by $\mathcal{H}_{\lambda}$.

###  2.2. Sheaf of Conformal Blocks

Integrable representations of affine Lie algebras were used by Tsuchiya-Ueno-
Yamada [66] and Tsuchiya-Kanie [65] to define conformal blocks. In this paper,
we will restrict ourselves to conformal blocks associated to smooth curves.
Let $\pi\,:\,\mathcal{C}\,\to\,S$ be a family of smooth projective curves, and
let ${\bm{p}}\,=\,(p_{1},\,\cdots,\,p_{n})$ be $n$ non-intersecting sections
of the map $\pi$ such that the complement
$\mathcal{C}\backslash\cup_{i=1}^{n}p_{i}(S)$ is an affine scheme.

Consider formal coordinates $\xi_{1},\,\cdots,\,\xi_{n}$ around the sections
$\bm{p}\,=\,(p_{1},\,\cdots,\,p_{n})$ giving isomorphisms
$\lim_{m}\mathcal{O}_{\mathcal{C}}/\mathcal{I}^{m}_{p_{i}}\,\cong\,\mathcal{O}_{S}[[\xi_{i}]]$,
where $\mathcal{I}_{p_{i}}$ is the ideal sheaf of the section $p_{i}$. Let
$\mathfrak{g}$ be a simple Lie algebra. As before, we get a sheaf of
$\mathcal{O}_{S}$ Lie algebras defined by

(2.2)
$\widehat{\mathfrak{g}}_{n}(S)\,:=\,\mathfrak{g}\otimes\left(\bigoplus_{i=1}^{n}\mathcal{O}_{S}((\xi_{i}))\right)\oplus\mathcal{O}_{S}\cdot
c$

The Lie algebra $\widehat{\mathfrak{g}}_{n}(S)$ in (2.2) contains a natural
subsheaf of Lie algebras
$\mathfrak{g}\otimes_{\mathbb{C}}\pi_{*}\mathcal{O}_{\mathcal{C}}(*D)$, where
$D\,=\,\sum_{i=1}^{n}p_{i}(S)$. That it satisfies the Lie algebra condition is
actually guaranteed by the residue theorem.

For any choice of $n$-tuple of weights
$\vec{\lambda}\,=\,(\lambda_{1},\,\cdots,\,\lambda_{n})$ in
$P_{\ell}(\mathfrak{g})$, consider the $\widehat{\mathfrak{g}}_{n}(S)$-module
(hence it is also a
$\mathfrak{g}\otimes_{\mathbb{C}}\pi_{*}\mathcal{O}_{\mathcal{C}}(*D)$-module)

$\mathcal{H}_{\vec{\lambda}}(S)\,:=\,\mathcal{H}_{\lambda_{1}}\otimes\mathcal{H}_{\lambda_{2}}\otimes\dots\otimes\mathcal{H}_{\lambda_{n}}\otimes_{\mathbb{C}}\mathcal{O}_{S}.$

######  Definition 2.1.

The sheaf of covacua
$\mathcal{V}_{\vec{\lambda}}(\mathfrak{g},\mathcal{C}/S,\,\bm{p},\,\ell)$ at
level $\ell$ is defined to be the largest quotient of
$\mathcal{H}_{\vec{\lambda}}(S)$ on which
$\mathfrak{g}\otimes_{\mathbb{C}}\pi_{*}\mathcal{O}_{\mathcal{C}}(*D)$ acts
trivially. The sheaf of conformal blocks
$\mathcal{V}^{\dagger}_{\vec{\lambda}}(\mathfrak{g},\mathcal{C}/S,\bm{p},\ell)$
is defined to be the $\mathcal{O}_{S}$ dual of the sheaf of covacua.

Since the above definition uses only the fact that the formal coordinates
identify the completed local ring with the Laurent series ring, the definition
of sheaf of covacua and the sheaf of conformal blocks can be extended in a
straightforward manner to families of nodal curves with Deligne-Mumford
stability property.

###  2.3. Coordinate free construction

The sheaf $\widehat{\mathfrak{g}}_{n}$ and its integrable modules can be
defined without the choice of formal coordinates $\bm{\xi}$; we recall this
from [64]. Let $\pi\,:\,\mathcal{C}\,\to\,S$ be as above. Consider the sheaf
of formal meromorphic functions on $\mathcal{C}$ with poles along the marked
sections

$\mathscr{K}_{\mathcal{C}/p_{i}}\,\,:=\,\,\lim_{a\to+\infty}\lim_{m\to+\infty}\mathcal{O}(ap_{i}(S))/\mathcal{I}^{m}_{p_{i}}.$

Using this, the coordinate free affine Lie algebra is defined as follows:

(2.3)
$\widehat{\mathfrak{g}}(\mathcal{C}/p_{i})\,:=\,\mathfrak{g}\otimes\mathscr{K}_{\mathcal{C}/p_{i}}\oplus\mathcal{O}_{S}\cdot
c.$

Its $n$-pointed analog is defined to be

$\widehat{\mathfrak{g}}_{n}(\mathcal{C}/S)\,:=\,\left(\mathfrak{g}\otimes\bigoplus_{i=1}^{n}\mathscr{K}_{\mathcal{C}/p_{i}}\right)\oplus\mathcal{O}_{S}\cdot
c.$

The Lie bracket operation is defined as in the previous section.

The Verma module $\mathcal{M}_{\lambda}(\mathcal{C}/S)$ (resp.the coordinate
free highest weight integrable module $\mathbb{H}_{\lambda}(\mathcal{C}/S)$)
can be defined similarly by inducing representations (and taking quotients)
using a parabolic subalgebra

$\widehat{\mathfrak{p}}_{p_{i}}\,:=\,\mathfrak{g}\otimes\widehat{\mathcal{O}}_{\mathcal{C}/p_{i}}(S)\oplus\mathcal{O}_{S}\cdot
c$

on the finite dimensional module $V_{\lambda}$ via evaluation. More precisely
$\mathcal{M}_{\lambda_{i}}(\mathcal{C}/S)=\operatorname{Ind}_{\widehat{\mathfrak{p}}_{p_{i}}}^{\widehat{\mathfrak{g}}(\mathcal{C}/p_{i})}V_{\lambda_{i}}$
refer the reader to [64] for more details.

######  Definition 2.2.

The coordinate free sheaf of covacua
$\mathbb{V}_{\vec{\lambda}}(\mathfrak{g},\,\mathcal{C}/S,\,\ell)$ is defined
to be the sheaf of coinvariants
$\mathbb{H}_{\vec{\lambda}}(\mathcal{C}/S)/\left(\mathfrak{g}\otimes_{\mathbb{C}}\pi_{*}\mathcal{O}_{\mathcal{C}}(*D)\mathbb{H}_{\vec{\lambda}}(\mathcal{C}/S)\right)$.
As before, the sheaf of coordinate free conformal blocks
$\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\mathcal{C}/S,\,\ell)$ is
defined to be the $\mathcal{O}_{S}$-module dual of
$\mathbb{V}_{\vec{\lambda}}(\mathfrak{g},\,\mathcal{C}/S,\,\ell)$.

Observe that a choice of formal coordinates around the marked points actually
induces isomorphisms between $\widehat{\mathfrak{g}}(\mathcal{C}/p_{i})$ and
the sheaf
$\widehat{\mathfrak{g}}_{\xi_{i}}\,:=\,\mathfrak{g}\otimes\mathcal{O}_{S}((\xi_{i}))\oplus\mathcal{O}_{S}\cdot
c$. This identifies the coordinate free conformal blocks and the sheaf of
covacua with those obtained by a choice of coordinates.

We now recall some important properties of the sheaf of conformal blocks. The
reader is referred to [66], [64], [25], and [58] for proofs and further
exposition.

######  Theorem 2.3.

Let $\mathfrak{g}$ be a simple Lie algebra and $\ell\,>\,0$ a positive
integer. Then the following statements hold:

  1. (1)

The sheaf of conformal blocks
$\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\mathcal{C}/S,\ell)$
carries a flat projective connection and hence it is locally free [64].The
Verlinde formula, [27, 62], computes the rank of the vector bundle
$\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\ell)$.

  2. (2)

Let $\overline{\mathcal{M}^{\prime}}_{g,n}$ be the moduli stack of stable
curves along with a choice of formal coordinates around the marked points.
Then the sheaf
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\mathcal{C}/S,\,\ell)$
gives a vector bundle with a flat projective connection with logarithmic
singularities along the boundary.

  3. (3)

The sheaf of conformal blocks descends to a vector bundle
$\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$ with a flat
projective connection on the moduli stack $\mathcal{M}_{g,n}$. Moreover, the
vector bundle $\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$
extends to a vector bundle over the Deligne-Mumford compactification
$\overline{\mathcal{M}}_{g,n}$, and the projective connection extends to a
projective connection with logarithmic singularities along the boundary.

  4. (4)

The natural forgetful map form
$F\,:\,\overline{\mathcal{M}}^{\prime}_{g,n}\,\to\,\overline{\mathcal{M}}_{g,n}$
identifies the vector bundle
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\ell)$ with the pull-
back $F^{*}\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\ell)$.

We will refer to the connections in Theorem 2.3 as the TUY/WZW connections;
their construction is recalled in Section 3.

######  Remark 2.4.

There are several other important properties — e.g. “propagation of vacua” and
“factorization theorems” — exhibited by conformal blocks. We refer the reader
to [66] for more details.

We also point out that
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\ell)$ and
$\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\ell)$ do not agree as
twisted $\mathcal{D}$-modules. This issue does not appear in [39], as the
weights there are zero. We refer the reader to [64] for the computation of the
Atiyah algebra of $\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\ell)$.

##  3\. Energy momentum tensor and the Sugawara construction

In this section, following the discussion in [66] we recall the definition of
the Sugawara tensor, which will be used in defining the connections on the
sheaf of covacua and conformal blocks.

For any $X\,\in\,\mathfrak{g}$, the element
$X\otimes\xi^{m}\,\in\,\widehat{\mathfrak{g}}$ will be denoted by $X(m)$. The
energy momentum tensor $T(z)$ at level $\ell$ is defined by the formula

(3.1)
$T(z)\,\,=\,\,\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{a=1}^{\dim\mathfrak{g}}\vcentcolon\mathrel{J^{a}(z)J^{a}(z)}\vcentcolon\,,$

where $\vcentcolon\mathrel{\ }\vcentcolon$ is the normal ordering (cf. [66, p.
467]), $h^{\vee}(\mathfrak{g})$ is the dual Coxeter number of $\mathfrak{g}$,
and $\\{J^{1},\,\cdots,\,J^{\dim\mathfrak{g}}\\}$ is an orthonormal basis of
$\mathfrak{g}$ with respect to the normalized Cartan-Killing form. Also,
define

$X(z)\,\,:=\,\,\sum_{n\in\mathbb{Z}}X(n)z^{-n-1}$

for any element $X\,\in\,\mathfrak{g}$. The $n$-th Virasoro operator $L_{n}$
is defined by the formula (see [33]):

(3.2)
$L_{n}\,\,:=\,\,\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{m\in\mathbb{Z}}\sum_{a=1}^{\dim\mathfrak{g}}\vcentcolon\mathrel{J^{a}(m)J^{a}(n-m)}\vcentcolon\,.$

The operators $L_{n}$ act on the module $\mathcal{H}_{\lambda}$ defined in
Section 2.1.

For $X\,\in\,\mathfrak{g}$, $f(z)\,\in\,\mathbb{C}((z))$ and
$\underline{\ell}\,=\,\ell(z)\frac{d}{dz}$, define (as in [66])

$\displaystyle X[f]\,:=\,\operatorname{Res}_{z=0}\left(X(z)f(z)\right)dz,\ \ \
T[\underline{\ell}]\,:=\,\operatorname{Res}_{z=0}\left(T(z)\ell(z)\right)dz.$

###  3.1. Construction of the WZW/TUY connection

Let $C$ be an irreducible smooth projective curve with $n$-marked points
${\bm{p}}\,=\,(p_{1},\,\cdots,\,p_{n})$. For every $1\,\leq\,i\,\leq\,n$, we
choose a formal coordinate $\xi_{i}$ around the marked point $p_{i}$ on the
curve $C$. Following [66], let us briefly recall the construction of a flat
projective connection.

Let $\pi\,:\,\mathcal{C}\,\to\,S$ be a versal family of $n$-pointed stable
curves equipped with $n$ non-intersecting sections
$p_{i}\,:\,S\,\to\,\mathcal{C}$. Let $D\,=\,\sum_{i=1}^{n}p_{i}(S)$ be the
corresponding divisor on $\mathcal{C}$. We have a short exact sequence of
sheaves

(3.3)
$0\,\longrightarrow\,\pi_{*}\mathcal{T}_{\mathcal{C}/S}(*D)\,\longrightarrow\,\pi_{*}\mathcal{T}_{\mathcal{C},\pi}(*D)\,\longrightarrow\,\mathcal{T}_{S}\,\longrightarrow\,0$

on $S$. On the other hand, we also have the short exact sequence

(3.4)
$0\,\longrightarrow\,\pi_{*}\mathcal{T}_{\mathcal{C}/S}(*D)\,\longrightarrow\,\bigoplus_{i=1}^{n}\mathcal{O}_{S}[\xi_{i}^{-1}]\frac{d}{d\xi_{i}}\,\longrightarrow\,R^{1}\pi_{*}\mathcal{T}_{\mathcal{C}/S}(-D)\,\longrightarrow\,0.$

obtained from pushing forward the following short exact sequence

(3.5)
$0\,\longrightarrow\,\mathcal{T}_{\mathcal{C}/S}(-D)\longrightarrow\mathcal{T}_{\mathcal{C}/S}(*D)\longrightarrow\bigoplus_{i=1}^{n}\mathcal{O}_{S}[\xi_{i}^{-1}]\frac{d}{d\xi_{i}}\longrightarrow
0.$

Since the family of $n$-pointed curves $\mathcal{C}$ is versal, we have the
Kodaira-Spencer isomorphism
$\mathcal{T}_{S}\,\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\,R^{1}\pi_{*}\mathcal{T}_{\mathcal{C}/S}(-D)$.
Combining (3.3) and (3.4), the following commutative diagram of homomorphisms
is obtained:

(3.6)
${0}$${\pi_{*}\mathcal{T}_{\mathcal{C}/S}(*D)}$${\pi_{*}\mathcal{T}_{\mathcal{C},\pi}(*D)}$${\mathcal{T}_{S}}$${0}$${0}$${\pi_{*}\mathcal{T}_{\mathcal{C}/S}(*D)}$${\bigoplus_{i=1}^{n}\mathcal{O}_{S}[\xi_{i}^{-1}]\frac{d}{d\xi_{i}}}$${R^{1}\pi_{*}\mathcal{T}_{\mathcal{C}/S}(-D)}$${0,}$$\scriptstyle{\bf
t}$$\scriptstyle{KS}$$\scriptstyle{\theta}$

where ${\bf t}$ is the projection to the polar part of the Laurent expansion
of sections in terms of the given coordinates $\xi_{i}$ around the divisors
$p_{i}(S)$. This map ${\bf t}$ is an isomorphism because the family
$\mathcal{C}$ is versal.

Let $\vec{\ell}\,=\,(l_{1},\,\cdots,\,l_{n})$ and
$\vec{m}\,=\,(m_{1},\,\cdots,\,m_{n})$ be two formal vector fields; both
$l_{i}$ and $m_{i}$ are defined on a formal neighborhood of $p_{i}(S)$. The
Lie bracket $[\vec{\ell},\,\vec{m}]_{d}$ is given by the formula

(3.7)
$[\vec{\ell},\,\vec{m}]_{d}\,:=\,[\vec{\ell},\,\vec{m}]_{0}+\theta(\vec{\ell})(\vec{m})-\theta(\vec{m})(\vec{\ell}),$

where $[\ ,\ ]_{0}$ is the usual Lie bracket of formal vector fields and
$\theta(\vec{\ell})$ acts componentwise using the formal parameters $\xi_{i}$.
Now for any formal vector field $\vec{\ell}$, define the operator
$\mathscr{D}(\vec{\ell})$ on $\mathcal{H}_{\vec{\lambda}}$ by the formula

(3.8)
$\mathscr{D}(\vec{\ell})(F\otimes|\Phi\rangle)\,\,:=\,\,\theta(\vec{\ell})(F)\otimes|\Phi\rangle-F\cdot\left(\sum_{j=1}^{n}\rho_{j}(T[\underline{l}_{j}])\right)|\Phi\rangle,$

where $\rho_{j}$ is the action on $\mathcal{H}_{\lambda_{j}}$ defined on [66,
p. 475].

###  3.2. WZW/TUY connection

After possibly shrinking $S$, we can find a symmetric bidifferential $\omega$
on $\mathcal{C}\times_{S}\mathcal{C}$ with a pole of order two on the diagonal
such that the biresidue is $1$. For any formal vector field ${\vec{\ell}}$
define

$a_{\omega}(\vec{\ell})\,:=\,-\frac{c_{v}}{12}\sum_{i=1}^{n}\operatorname{Res}_{\xi_{i}=0}\left(\ell_{i}(\xi_{i})S_{\omega}(\xi_{i})d\xi_{i}\right),$

where $S_{\omega}$ is the projective connection associated to $\omega$, and
$\displaystyle
c_{v}\,=\,\frac{\ell\dim\mathfrak{g}}{\ell+h^{\vee}(\mathfrak{g})}$ is the
central charge.

Now let $\tau$ be a vector field on $S$. Take a lift of $\tau$ to
$\pi_{*}\mathcal{T}_{\mathcal{C}/S}(*D)$ and denote it by $\vec{\ell}$. With
the choice of a bidifferential $\omega$ as above, we define the following
operator on the sheaf of conformal blocks:

(3.9)
$\nabla_{\tau}^{(\omega)}(\langle\Psi|)\,:=\,\mathscr{D}(\vec{\ell})(\langle\Psi|)+a_{\omega}(\vec{\ell})(\langle\Psi|).$

We recall the following result (see [66, Thm. 5.3.3]).

######  Proposition 3.1.

The operator $\nabla^{(\omega)}_{\tau}$ in (3.9) defines a flat projective
connection on the sheaf of conformal blocks
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\,\ell)$.

######  Remark 3.2.

Consider the natural forgetful map
$F\,:\,\overline{\mathcal{M}}_{g,n}^{\prime}\,\to\,\overline{\mathcal{M}}_{g,n}$
constructed by forgetting the choice of formal parameters at the $n$-marked
points. Then the natural identification between
$F^{*}\mathbb{V}_{\vec{\lambda}}(\mathfrak{g},\,\ell)$ and
$\mathcal{V}_{\vec{\lambda}}(\mathfrak{g},\,\ell)$ as locally free sheaves
intertwines, up to a first order operator, the pull-back of the coordinate
free TUY connection on $F^{*}\mathbb{V}_{\vec{\lambda}}(\mathfrak{g},\ell)$
and the TUY connection on $\mathcal{V}_{\vec{\lambda}}(\mathfrak{g},\,\ell)$.

##  4\. Projective connections via heat operators

###  4.1. Heat operators and the Hitchin-van Geemen-de Jong equation

Let $\pi\,:\,M\,\to\,S$ be a smooth map of smooth varieties, and let
$\mathcal{L}$ be a line bundle on $M$. The Kodaira-Spencer infinitesimal
deformation map is given by:

$KS_{M/S}:\mathcal{T}_{S}\longrightarrow R^{1}\pi_{*}\mathcal{T}_{M/S}.$

On the other hand, we have the coboundary map

$\mu_{\mathcal{L}}:\pi_{*}\operatorname{Sym}^{2}\mathcal{T}_{M/S}\longrightarrow
R^{1}\pi_{*}\mathcal{T}_{M/S},$

occurring in the long exact sequence obtained from the push forward $\pi_{*}$
of the fundamental short exact sequence of differential operators

$0\longrightarrow\mathcal{T}_{M/S}\cong\mathcal{D}_{M/S}^{\leq
1}(\mathcal{L})/\mathcal{O}_{M}\longrightarrow\mathcal{D}_{M/S}^{\leq
2}(\mathcal{L})/\mathcal{O}_{M}\stackrel{{\scriptstyle
s_{2}}}{{\longrightarrow}}\operatorname{Sym}^{2}\mathcal{T}_{M/S}\longrightarrow
0\ ,$

where $s_{2}$ is the symbol map and $\mathcal{D}^{\leq i}_{M/S}(\mathcal{L})$
is the sheaf of relative differential operators of order atmost $i$. Following
[67], consider the sheaf $\mathcal{W}(\mathcal{L})$ defined by

(4.1) $\mathcal{W}(\mathcal{L})\,:=\,\mathcal{D}_{M}^{\leq
1}(\mathcal{L})+\mathcal{D}_{M/S}^{\leq 2}(\mathcal{L}).$

There is a natural short exact sequence

(4.2) $0\,\longrightarrow\,\mathcal{D}_{M/S}^{\leq
1}(\mathcal{L})\,\longrightarrow\,\mathcal{W}(\mathcal{L})\,\stackrel{{\scriptstyle
q_{0}}}{{\longrightarrow}}\,\pi^{*}\mathcal{T}_{S}\oplus\operatorname{Sym}^{2}\mathcal{T}_{M/S}\,\longrightarrow\,0.$

######  Definition 4.1.

A heat operator
$\mathbb{D}\,:\,\pi^{*}\mathcal{T}_{S}\,\to\,\mathcal{W}({\mathcal{L}})$ is a
section of the natural projection map
$\mathcal{W}(\mathcal{L})\,\to\,\pi^{*}\mathcal{T}_{S}$. A projective heat
operator is a section of
$\mathcal{W}(\mathcal{L})/\mathcal{O}_{M}\,\to\,\pi^{*}\mathcal{T}_{S}$.

A projective heat operator evidently lifts, locally, to a heat operator. Given
a homomorphism
$\rho\,:\,\pi^{*}\mathcal{T}_{S}\,\to\,\operatorname{Sym}^{2}\mathcal{T}_{M/S}$,
one can ask whether there is a canonical projective heat operator
$\mathbb{D}\,:\,\pi^{*}\mathcal{T}_{S}\,\to\,\mathcal{W}(\mathcal{L})/\mathcal{O}_{M}$
such that $q_{1}\circ\mathbb{D}\,=\,\rho$. The following theorems of Hitchin
[32] and van Geemen-de Jong [67], answer a fundamental question on existence
of projective heat operators.

######  Theorem 4.2 ([32, 67]).

Assume that the following conditions hold:

  * •

$KS_{M/S}+\mu_{\mathcal{L}}\circ\rho\,=\,0$;

  * •

$\cup[\mathcal{L}]\,:\,\pi_{*}\mathcal{T}_{M/S}\,\to\,R^{1}\pi_{*}\mathcal{O}_{M}$
is an isomorphism;

  * •

$\pi_{*}\mathcal{O}_{M}\,=\,\mathcal{O}_{S}$.

Then there exists a unique projective heat operator $\overline{\mathbb{D}}$
lifting any candidate symbol
$\rho\,:\,\pi^{*}\mathcal{T}_{S}\,\to\,\operatorname{Sym}^{2}\mathcal{T}_{M/S}$.
Moreover the coherent sheaf $\pi_{*}\mathcal{L}$ carries a projective
connection.

######  Remark 4.3.

We can take $\mathcal{L}$ to be an object in the rational Picard group
$\operatorname{Pic}(M)\otimes\mathbb{Q}$. All the sheaves that appear in the
statement of Theorem 4.2 are well-defined, and the proof in [67] does not
require the assumption that $\mathcal{L}$ be a line bundle.

###  4.2. Heat operators and metaplectic quantization

We are interested in the case where the Kodaira-Spencer map $KS_{M/S}$, the
candidate symbol
$\rho\,:\,\mathcal{T}_{S}\,\to\,\pi_{*}\operatorname{Sym}^{2}\mathcal{T}_{M/S}$
and the class of $\mathcal{L}$ are intertwined by the equation:

(4.3) $KS_{M/S}+\cup[\mathcal{L}]\circ\rho\,=\,0.$

We refer to (4.3) as the equation controlling the deformations. Recall that
the connecting homomorphism

$\mu_{\mathcal{L}^{{\otimes
k}}}\,:\,\pi_{*}\operatorname{Sym}^{2}\mathcal{T}_{M/S}\,\longrightarrow\,R^{1}\pi_{*}\mathcal{T}_{M/S}$

is given by the formula (see [4])

(4.4) $\mu_{\mathcal{L}^{\otimes
k}}\,=\,\cup\left([k\cdot\mathcal{L}]-\frac{1}{2}[K_{M/S}]\right).$

####  4.2.1. Rewriting the deformation equation

We will now rewrite (4.3) to produce a projective heat operator on
$\mathcal{L}$. Throughout this subsection we assume that for any positive
$k\,\in\,\mathbb{Q}$, the connecting homomorphism $\mu_{\mathcal{L}^{\otimes
k}}$ in (4.4) is an isomorphism. This condition holds, for example, in the
case of moduli spaces of parabolic bundles.

Now

$\displaystyle KS_{M/S}+\cup[\mathcal{L}]\circ\rho$
$\displaystyle=\,KS_{M/S}+\frac{1}{k}\left(\cup\left(k[\mathcal{L}]-\frac{1}{2}K_{M/S}\right)+\cup\frac{1}{2}[K_{M/S}]\right)\circ\rho$
$\displaystyle=\,KS_{M/S}+\mu_{\mathcal{L}^{\otimes
k}}\circ\left(1+\mu_{\mathcal{L}^{\otimes
k}}^{-1}\circ(\cup\frac{1}{2}[K_{M/S}])\right)\circ\frac{1}{k}\rho$
$\displaystyle=\,KS_{M/S}+\mu_{\mathcal{L}^{\otimes
k}}\circ\widetilde{\rho}_{k},$

where $\widetilde{\rho}_{k}\,=\,\left(1+\mu_{\mathcal{L}^{\otimes
k}}^{-1}\circ\left(\cup\frac{1}{2}[K_{M/S}]\right)\right)\circ\frac{1}{k}{\rho}$
is the symbol map. Again assuming that the conditions of Theorem 4.2 are
satisfied, we get a projective heat operator $\overline{\mathscr{D}}$ on
$\mathcal{L}$ with symbol $\widetilde{\rho}_{k}$ such that the following
diagram commutes

(4.5) ${\pi_{*}\mathcal{D}^{\leq 2}_{M}(\mathcal{L}^{\otimes
k})}$${\mathcal{T}_{S}}$${\pi_{*}\operatorname{Sym}^{2}\mathcal{T}_{M/S}}$$\scriptstyle{symb}$$\scriptstyle{\rho_{k}}$

This induces a projective connection on $\pi_{*}\mathcal{L}^{{\otimes k}}$ for
any positive $k\,\in\,\mathbb{Z}$ with symbol $\widetilde{\rho}_{k}$

####  4.2.2. Metaplectic correction d’après Scheinost-Schottenloher

We can rewrite the left-hand side of (4.3) as follows:

$\displaystyle KS_{M/S}+\cup[\mathcal{L}]\circ\rho$
$\displaystyle=\,KS_{M/S}+\frac{1}{k}\left(\cup\left([k\cdot\mathcal{L}]+\frac{1}{2}[K_{M/S}]\right)-\cup\frac{1}{2}[K_{M/S}]\right)\circ\rho$
$\displaystyle=\,KS_{M/S}+\mu_{\mathcal{L}^{{\otimes k}}\otimes
K^{\frac{1}{2}}_{M/S}}\circ\rho_{k},$

where $\rho_{k}\,:=\,\frac{1}{k}\rho$ and $\mathcal{L}^{\otimes k}\otimes
K_{M/S}^{1/2}$ is considered as an element of the rational Picard group. Thus,
from (4.3) the following equation is obtained:

(4.6) $KS_{M/S}+\mu_{\mathcal{L}^{{\otimes k}}\otimes
K_{M/S}^{\frac{1}{2}}}\circ\rho_{k}\,=\,0.$

Assume that the other conditions of the Hitchin-van Geemen-de Jong existence
theorem are satisfied. Then Theorem 4.2 tells us that there exists a unique
projective heat operator $\widehat{\mathbb{D}}_{k}$ with symbol $\rho_{k}$ and
a connection on $\pi_{*}(\mathcal{L}^{k}\otimes K_{M/S}^{\frac{1}{2}})$. As
pointed out in Remark 4.3, the projective heat operator makes sense even if
the square-root of $K_{M/S}$ does not exist.

It is easy to see that for any candidate symbol, there exists a second order
projective operator $\widehat{\mathbb{D}}$ on $K^{\frac{1}{2}}$ with the same
given symbol. However, this operator is not a projective heat operator, since
the natural projection of it to $\pi^{*}\mathcal{T}_{S}$ is zero. On the other
hand, we have a projective heat operator $\overline{\mathbb{D}}$ on
$\mathcal{L}^{\otimes k}$ with the same symbol $\widetilde{\rho}_{k}$. The
following is then a natural question.

######  Question 4.4.

Using the projective heat operator $\overline{\mathbb{D}}$ and the projective
operator $\widehat{\mathbb{D}}$ with symbol ${\rho}_{k}$, can we construct a
projective heat operator $\widetilde{\mathbb{D}}$ on $\mathcal{L}^{k}\otimes
K_{M/S}^{\frac{1}{2}}$?

######  Remark 4.5.

Observe that the equations in Theorem 4.2 imply that there exists at most one
heat operator provided

$\mu_{\mathcal{L}^{\otimes k}\otimes
K_{M/S}^{\frac{1}{2}}}\,:\,\pi_{*}\operatorname{Sym}^{2}\mathcal{T}_{M/S}\,\longrightarrow\,R^{1}\pi_{*}\mathcal{T}_{M/S}$

is an isomorphism. A positive answer to Question 4.4 would immediately imply
that the symbol of $\widetilde{\mathbb{D}}$ satisfies the equation for
$\rho_{k}$ given in (4.6). This will provide a necessary relation that the
linear maps $\cup[{\mathcal{L}}]$ and $\cup[{K_{M/S}}]$ are scalar multiples
of each other. This would give an alternative, more conceptual understanding
of Theorem 4.1 in [19]

##  5\. Parabolic Hitchin symbol as in Biswas-Mukhopadhyay-Wentworth

In this section we first briefly recall the construction of the Hitchin
connection for the moduli space of parabolic bundles
$M^{par,rs}_{G,\bm{\tau}}$ obtained in [19]. We refer the reader to Appendices
A.1 and A.3 for a brief review of the moduli stack of parabolic bundles and
the parabolic determinant of cohomology line bundles. We will freely use the
correspondence between parabolic $G$ bundles on a curve $C$ and equivariant
($\Gamma$-$G$)-bundles on a Galois cover $\widehat{C}$ of the curve $C$ with
Galois group $\Gamma$. This is recalled in Appendix A.2.

In particular, we focus on the parabolic Hitchin symbol defined in the paper
[19]. Using restriction to fibers of the Hitchin map, we give a simplification
of the expression for the symbol that enables us to compare the parabolic
Hitchin symbol with the symbol of the Sugawara operators as constructed in
[66]. The main result of this section is Proposition 5.6. This is a key new
feature and one of the main technical difficulties in the parabolic set-up
that does not appear in [39].

###  5.1. The Hitchin symbol

In [19] (recalled in Appendix A.2), we identified the moduli space
$M^{par}_{G,\bm{\tau}}$ of parabolic bundles with the moduli space
$M^{\bm{\tau},ss}_{G}$ of $(\Gamma,\,G)$-bundles on a Galois cover
$\widehat{C}$ of the curve $C$ via the invariant pushforward functor [5, 6].

This includes the following identifications: Let $\mathcal{P}$ be a regularly
stable parabolic bundle and $\widehat{\mathcal{P}}$ the corresponding
$(\Gamma,\,G)$ bundle (see Appendix A.2). Let
$\operatorname{Par}(\mathcal{P})$ (resp. $\operatorname{Spar}(\mathcal{P})$)
denote the bundle of parabolic (resp. strictly parabolic) automorphisms of
$\mathcal{P}$. Then

(5.1) $H^{0}(C,\,\operatorname{Spar}(\mathcal{P})\otimes
K_{C}(D))\,\cong\,H^{0}(\widehat{C},\,\operatorname{ad}\mathcal{P}\otimes
K_{\widehat{C}})^{\Gamma}$

and

(5.2)
$H^{1}(C,\,\operatorname{Par}(\mathcal{P}))\,\cong\,H^{1}(\widehat{C},\,\operatorname{ad}\mathcal{P})^{\Gamma}.$

The Hitchin symbol was defined using the natural map

(5.3) $H^{0}(\widehat{C},\,\operatorname{ad}\mathcal{P}\otimes
K_{\widehat{C}})^{\Gamma}\otimes
H^{1}(\widehat{C},\,\operatorname{ad}\mathcal{P})^{\Gamma}\xrightarrow{\
\kappa_{g}\
}H^{1}(\widehat{C},\,K_{\widehat{C}})^{\Gamma}\,\cong\,\mathbb{C};$

where the last isomorphism is given by the Serre duality on $\widehat{C}$. As
in [32, Prop. 2.16], this induces a natural map

(5.4)
$\rho_{sym}\,:\,R^{1}\pi_{s*}\mathcal{T}_{\mathcal{C}/S}(-D)\,\longrightarrow\,\pi_{e*}\operatorname{Sym}^{2}\mathcal{T}_{M^{\bm{\tau},rs}_{G}/S}\,,$

where $\pi_{e}\,:\,M_{G}^{\bm{\tau},rs}\,\to\,S$ and
$\pi_{s}\,:\,\mathcal{C}\,\to\,S$ are the projections. On the other hand we
also have the pairing

(5.5) ${H^{0}(C,\,\operatorname{Spar}(\mathcal{P})\otimes K_{C}(D))\otimes
H^{1}(C,\,\operatorname{Par}(\mathcal{P}))}$${H^{1}(C,\,\operatorname{Spar}(\mathcal{P})(D)\otimes\operatorname{Par}(\mathcal{P})\otimes
K_{C})}$${H^{1}(C,\,K_{C})\,\cong\,\mathbb{C}.}$$\scriptstyle{\kappa_{\mathfrak{g}}}$

This also induces a map

(5.6)
$\widetilde{\rho}_{sym}\,:\,R^{1}\pi_{s*}\mathcal{T}_{\mathcal{C}/S}(-D)\,\longrightarrow\,\pi_{e*}\operatorname{Sym}^{2}\mathcal{T}_{M_{G,\bm{\tau}}^{par,rs}/S}.$

The isomorphisms $H^{1}(C,\,K_{C})\,\cong\,\mathbb{C}$ in (5.5) and
$H^{1}(\widehat{C},\,K_{\widehat{C}})\cong\,\mathbb{C}$ are both given by the
residue theorem and Serre duality, but for different curves. Hence, the map on
Hitchin symbols $\rho_{sym}$ and $\widetilde{\rho}_{sym}$ do not commute under
the identifications given by (5.1) and (5.2). However they are related as
follows:

######  Lemma 5.1.

Let $\rho_{sym}$ and $\widetilde{\rho}_{sym}$ be as above. Then
$|\Gamma|\cdot\rho_{sym}\,=\,\widetilde{\rho}_{sym}.$

###### Proof.

This is immediate from the commutativity of

${H^{1}(C,K_{C})}$${H^{1}(\widehat{C},\,K_{\widehat{C}})^{\Gamma}}$${\mathbb{C}}$${\mathbb{C}}$$\scriptstyle{\sim}$$\scriptstyle{\times|\Gamma|}$

∎

###  5.2. Parabolic Hitchin connection via heat operators

Let $\phi\,:\,G\,\to\,\mathsf{SL}(V)$ be a linear representation satisfying
the hypothesis of Section A.3. Let $\mathcal{L}_{\phi}$ be the pullback of the
determinant of cohomology line bundle to $M^{\bm{\tau},ss}_{G}$. Via the
identification of parabolic $G$-bundles as equivariant bundles and Proposition
A.7, we have identified it with the parabolic determinant of cohomology
$\operatorname{Det}_{par}(\nu_{\mathfrak{sl}(V)}(\phi(\bm{\tau})))$. For
notational convenience, we will drop $\phi$ when denoting the line bundle
$\operatorname{Det}_{par,\phi}(\mathcal{P},\bm{\tau})$ and simply write
$\operatorname{Det}_{par}(\mathcal{P},\bm{\tau})$.

In [19], the authors produced a projective heat operator on the line bundle
$\mathcal{L}^{\otimes k}_{\phi}$ whose symbol satisfies the following Hitchin-
van Geemen-de Jong equation:

(5.7)
$KS_{M^{par,rs}_{G,\bm{\tau}}/S}+\mu_{\mathcal{L}^{k}_{\phi}}\circ\left(\frac{1}{m_{\phi}k}\operatorname{Id}+\mu^{-1}_{\mathcal{L}^{k}_{\phi}}\circ\left(\cup\frac{1}{2m_{\phi}k}[K_{M^{par,rs}_{G,\bm{\tau}}/S}]\right)\right)\circ\rho_{sym}\circ
KS_{\mathcal{C}/S}=0.$

Setting $\rho=\rho_{sym}\circ KS_{\mathcal{C}/S}$ gives

(5.8)
$KS_{M^{par,rs}_{G,\bm{\tau}}/S}+\mu_{\mathcal{L}^{k}_{\phi}}\circ\left(\operatorname{Id}+\mu^{-1}_{\mathcal{L}^{k}_{\phi}}\circ\left(\cup\frac{1}{2}[K_{M^{par,rs}_{G,\bm{\tau}}/S}]\right)\right)\circ\frac{1}{m_{\phi}\cdot
k}\rho=0,$

where $k$ is a rational number.

Let $k\,=\,\ell/|\Gamma|$. Using the identification
$\mathcal{L}_{\phi}\cong\left(\operatorname{Det}_{par}(\mathcal{P},\bm{\tau})\right)^{\frac{|\Gamma|}{\ell}}$
in Proposition A.7, from (5.8) we get that

$KS_{M^{par,rs}_{G,\bm{\tau}}/S}+\mu_{\operatorname{Det}_{par}(\mathcal{P},\bm{\tau})\otimes
K^{1/2}_{M^{par,ss}_{G,\bm{\tau}}/S}}\circ\frac{|\Gamma|}{m_{\phi}\cdot\ell}\rho\,=\,0.$

######  Definition 5.2.

For any rational number $a$, we will denote the projective heat operator on
$\operatorname{Det}^{\otimes a}_{par,\phi}(\mathcal{P},\bm{\tau})$ obtained in
[19] by $\mathbb{D}(\mathfrak{g},a\cdot m_{\phi}\cdot\ell)$.

The following is one of the main results of [19, Theorem 4.1].

######  Theorem 5.3.

Let $\mathbb{L}$ be an element of
$\operatorname{Pic}(M^{par,rs}_{G,\bm{\tau}})\otimes\mathbb{Q}$ of level $a$.
Then there is an equality $\cup[\mathbb{L}]\,=\,\cup a[\operatorname{Det}]$ as
linear maps
$\pi_{e*}\operatorname{Sym}^{2}\mathcal{T}_{M^{par,rs}_{G,\bm{\tau}}/S}\,\to\,R^{1}\pi_{e*}\mathcal{T}_{M^{par,rs}_{G,\bm{\tau}}/S}$,
where $\operatorname{Det}$ is the determinant of cohomology (non-parabolic)
line bundle.

######  Remark 5.4.

The above result should be put in the more general context of deformation
theory of the moduli space of the parabolic bundles as studied in Boden-
Yokogawa [22], and the birational variation of these moduli spaces as the
weights vary.

Since line bundles on $M_{G,\bm{\tau}}^{par,rs}$ are pull-backs of rational
powers of line bundles from $M_{\mathsf{SL}(V),\bm{\alpha}}^{par,s}$ for an
appropriate choice of representation $(\phi,\,V)$ of $G$, the following is an
immediate consequence of Theorem 5.3.

######  Corollary 5.5.

Let $\widehat{M}_{G}$ and $M_{G,\bm{\tau}}^{par,rs}$ be as in Section A.3. Let
$(\phi,\,V)$ be a representation of $G$. Then the line bundle
$K_{\widehat{M}_{G}/S}$ restricted to $M_{G,\bm{\tau}}^{par,rs}$ is of level
$-\frac{2h^{\vee}(\mathfrak{g}).|\Gamma|}{m_{\phi}}$ with respect to
$\mathsf{SL}(V)$, and hence

$\cup[K_{M_{G,\bm{\tau}}^{par,rs}/S}]\,=\,\cup\frac{[{K_{\widehat{M}_{G}/S}}]}{|\Gamma|}$

as linear maps from
$\pi_{*}\operatorname{Sym}^{2}\mathcal{T}_{M_{G,\bm{\tau}}^{par,rs}/S}$ to
$R^{1}\pi_{*}\mathcal{T}_{M_{G,\bm{\tau}}^{par,rs}/S}$.

###### Proof.

On the one hand, we have the fact from [37] that $K_{\widehat{M}_{G}/S}$ is
$-2h^{\vee}(\mathfrak{g})\mathcal{L}_{\phi}$, where $\mathcal{L}_{\phi}$ is
the ample generator of the Picard group of the moduli stack of
$\mathsf{SL}(V)$ bundles on $\widehat{C}$, where $\widehat{C}\to C$ is a
$\Gamma$-cover. Moreover, $\mathcal{L}_{\phi}$ restricted to
$M_{G,\bm{\tau}}^{par,rs}/S$ is of level $|\Gamma|$ (see Appendix A.3). On the
other hand, the canonical bundle $K_{M_{G,\bm{\tau}}^{par,rs}/S}$ has a
component that is $-2h^{\vee}(\mathfrak{g})$ times the ample generator of the
Picard group of the moduli stack of $G$ bundles on $C$, which in turn is of
level $\frac{1}{m_{\phi}}$ with respect to the $\mathfrak{sl}(V)$ embedding of
$\mathfrak{g}$. This proves the result. ∎

Corollary 5.5 further simplifies the parabolic Hitchin symbol, as shown by the
following.

######  Proposition 5.6.

Assume that $m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})\,\not=\,0$. Then

(5.9)
$\frac{1}{m_{\phi}\ell}\left(1+\mu_{\operatorname{Det}_{par}(\bm{\tau})}^{-1}\circ\left(\frac{1}{2}\cup[K_{M^{par,rs}_{\bm{\tau},G}/S}]\right)\right)\circ{\widetilde{\rho}_{sym}}\,=\,\frac{\widetilde{\rho}_{sym}}{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}.$

###### Proof.

Since $\widetilde{\rho}_{sym}$ is invertible, we need to show that

$\frac{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}{m_{\phi}\cdot\ell}\left(1+\mu_{\operatorname{Det}_{par}(\bm{\tau})}^{-1}\left(\frac{1}{2}\cup[K_{M^{par,rs}_{\vec{\alpha},\mathsf{SL}_{r}}/S}]\right)\right)\,=\,\operatorname{Id}.$

So it suffices to prove that

$\mu_{\operatorname{Det}_{par}}^{-1}(\bm{\tau})\circ\left(\frac{1}{2}\cup[K_{M^{par,rs}_{G,\bm{\tau}}/S}]\right)\,=\,\left(-1+\frac{m_{\phi}\cdot\ell}{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}\right)\operatorname{Id}.$

By multiplying with $\mu_{\operatorname{Det}_{par}(\bm{\tau})}$, it suffices
to show that

(5.10)
$\cup[K_{M^{par,rs}_{G,\bm{\tau}}/S}]\,=\,\left(\frac{-2h^{\vee}(\mathfrak{g})}{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}\right)\mu_{\operatorname{Det}_{par}(\bm{\tau})}.$

Now by [37, 38], applied to the moduli space $\widehat{M}_{G}$ of principal
$G$ bundle on $\widehat{C}$, we get that

(5.11)
$[K_{\widehat{M}_{G}}]\,=\,-2h^{\vee}(\mathfrak{g}).m_{\phi}.[\mathcal{L}_{\phi}].$

By Corollary 5.5,

$\cup[K_{M^{par,rs}_{G,\bm{\tau}}/S}]=\cup\frac{-2h^{\vee}(\mathfrak{g})[\mathcal{L}_{\phi}]}{m_{\phi}\cdot|\Gamma|}=\cup\frac{-2h^{\vee}(\mathfrak{g})}{m_{\phi}\cdot|\Gamma|}\left(\frac{|\Gamma|[\operatorname{Det}_{par}(\bm{\tau})]}{\ell}\right).$

Rewriting the above equation, we find

$\displaystyle\cup m_{\phi}\cdot\ell[K_{M^{par,rs}_{G,\bm{\tau}}/S}]$
$\displaystyle\,=\,$
$\displaystyle\cup\left(\frac{-2h^{\vee}(\mathfrak{g})\cdot
m_{\phi}\cdot\ell}{|\Gamma|\cdot
m_{\phi}}\right)\frac{|\Gamma|[\operatorname{Det}_{par}(\bm{\tau})]}{\ell},$
$\displaystyle\cup\frac{m_{\phi}\cdot\ell}{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}[K_{M^{par}_{G,\bm{\tau}}/S}]$
$\displaystyle\,=\,$
$\displaystyle\left(\cup\frac{1}{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}\left(-2h^{\vee}(\mathfrak{g})[\operatorname{Det}_{par}]\right)\right),$
$\displaystyle\cup[K_{M^{par,rs}_{G,\bm{\tau}}/S}]$ $\displaystyle\,=\,$
$\displaystyle\left(\cup\frac{-2h^{\vee}(\mathfrak{g})}{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}\left([\operatorname{Det}_{par}(\bm{\tau})]-\frac{1}{2}[K_{M^{par,rs}_{\bm{\tau},G}/S}]\right)\right),$
$\displaystyle\cup\frac{1}{2}[K_{M^{par}_{\bm{\tau},G}/S}]$
$\displaystyle\,=\,$
$\displaystyle\left(\frac{-h^{\vee}(\mathfrak{g})}{m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g})}\right)\cdot\mu_{\operatorname{Det}_{par}(\bm{\tau})}\qquad\qquad\text{(from
\eqref{ch})}\ .$

This completes the proof. ∎

##  6\. Proof of Theorem 1.1

In this section, we give a proof of the main theorem in this article by
comparing the Sugawara tensor and the parabolic analog of the heat operator
with a given symbol constructed by the authors in [19].

Let $\mathcal{P}ar^{rs}_{G}(C,\bm{p},\bm{\tau})$ be the open substack
parametrizing regularly stable parabolic bundles of parabolic type
$\bm{\tau}$. Then the natural map
$\mathcal{P}ar_{G}^{rs}(C,\bm{p},\bm{\tau})\,\to\,M_{G,\bm{\tau}}^{par,rs}(C,\bm{p})$
is a gerbe banded by the center $Z(G)$ of the group $G$.

Similarly, let ${\mathcal{Q}}_{\bm{\tau}}^{rs}$ be the open ind-subscheme of
${\mathcal{Q}}_{\bm{\tau}}$ parametrizing the regularly stable bundles. The
natural map $\pi^{reg}$ given by the composition

$\pi^{reg}\,:\,{\mathcal{Q}}_{\bm{\tau}}^{rs}\,\longrightarrow\,\mathcal{P}ar_{G}^{rs}(\bm{\tau})\,\longrightarrow\,M_{G,\bm{\tau}}^{par,rs}$

is a $L_{\mathcal{C}^{\prime},G}/Z(G)$ torsor which is étale locally trivial.
Here, $L_{\mathcal{C}^{\prime},G}$ is the loop group associated to a punctured
curve.

###  6.1. Twisted $\mathcal{D}$-modules via quasi-section of Drinfeld-Simpson

Let $\pi_{s}\,:\,\mathcal{C}\,\to\,S$ be a versal family of $n$-pointed smooth
curves of genus $g$. We choose formal coordinates
$\bm{\xi}\,=\,(\xi_{1},\,\cdots,\,\xi_{n})$ along the sections $\bm{p}$.

Let $\bm{\tau}\,=\,(\tau_{1},\,\cdots,\,\tau_{n})$ be as in Section B.1, and
let $\mathcal{Q}_{\tau_{i}}$ be the affine flag variety associated to
$\tau_{i}$. The above choice of coordinates gives an identification of
$\mathcal{Q}_{\bm{\tau}}$ with $\prod_{i=1}^{n}L_{G}/\mathcal{P}_{\tau_{i}}$.
By the discussion in [9, Secs. 5.2.9-5.2.12], the infinitesimal action, of the
central extension $\widehat{L}_{G}$ of the loop group, on
$\mathcal{Q}_{\tau_{i}}$ gives a map

$\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}\,\longrightarrow\,H^{0}(\mathcal{Q}_{\tau_{i}},\,\mathcal{D}_{\mathcal{Q}_{\tau_{i}}/S}(\mathscr{L}_{\vec{\lambda}})).$

Here
$\mathcal{D}_{\mathcal{Q}_{\tau_{i}}/S}(\mathscr{\mathscr{L}}_{\lambda_{i}})$
is the ring of relative $\mathscr{L}_{{\lambda}_{i}}$-twisted differential
operators on $\mathcal{Q}_{\tau_{i}}$, and
$\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})$ is a suitable completion of
the universal enveloping algebra of $\widehat{\mathfrak{g}}_{\xi_{i}}$, and
$\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}$ is the opposite
algebra. Summing over all the coordinates, we get a map

(6.1)
$\bigoplus_{i=1}^{n}\left(\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}\right)\,\longrightarrow\,H^{0}(\mathcal{Q}_{\bm{\tau}},\,\mathcal{D}_{\mathcal{Q}_{\bm{\tau}}/S}(\mathscr{L}_{\vec{\lambda}}))$

which via further restriction gives a map
$\bigoplus_{i=1}^{n}\left(\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}\right)\,\to\,H^{0}(\mathcal{Q}^{rs}_{\bm{\tau}},\,\mathcal{D}_{\mathcal{Q}_{\bm{\tau}}/S}(\mathscr{L}_{\vec{\lambda}}))$.
Both sides of (6.1) carry natural filtrations and the map in (6.1) is a map of
filtered sheaves of algebras.

As in [39], we consider a quasi-section of $\pi^{rs}$. The result of [24]
implies that the natural étale locally trivial torsor
$\pi^{rs}\,:\,\mathcal{Q}_{\bm{\tau}}^{rs}\,\to\,M_{G,\bm{\tau}}^{rs}$ has a
quasi-section $N^{par,rs}_{G,\bm{\tau}}\,\xrightarrow{\,\ r\,\
}\,M_{G,\bm{\tau}}^{par,rs}$ such that $r$ is an étale epimorphism, and there
is a map $\sigma\,:\,N_{G,\tau}^{par,rs}\,\to\,\mathcal{Q}_{\bm{\tau}}^{rs}$
such that the following diagram commutes

(6.2)
${\mathcal{Q}^{rs}_{\bm{\tau}}}$${N_{G,\bm{\tau}}^{par,rs}}$${M_{G,\bm{\tau}}^{par,rs}}$${S}$$\scriptstyle{\pi^{rs}}$$\scriptstyle{r}$$\scriptstyle{\sigma}$$\scriptstyle{\pi_{e}}$

Now since the map $r$ is étale, we get an isomorphism

$H^{0}(N_{G,\bm{\tau}}^{par,rs},\,r^{*}\mathcal{T}_{M_{G,\bm{\tau}}^{par,rs}/S})\,=\,H^{0}(N_{G,\bm{\tau}}^{par,rs},\,\mathcal{T}_{N_{G,\bm{\tau}}^{par,rs}/S}).$

Given any relative differential operator $\mathfrak{D}$ on the line bundle
$\mathscr{L}_{\vec{\lambda}}$, we can pull it back via $\sigma$ (see Section
8.1 and 8.7 in [39]) to a differential operator on the line bundle
$\sigma^{*}\mathscr{L}_{\vec{\lambda}}$ which, by an abuse of notation, is
again denote by $\mathscr{L}_{\vec{\lambda}}$. Thus, (6.1) gives the following
map of filtered sheaves of algebras

(6.3)
$\hbar_{\mathscr{L}_{\vec{\lambda}}}\,:\,\bigoplus_{i=1}^{n}\left(\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}\right)\,\longrightarrow\,H^{0}(N_{G,\bm{\tau}}^{par,rs},\,\mathcal{D}_{N_{G,\bm{\tau}}^{par,rs}/S}(\mathscr{L}_{\vec{\lambda}})).$

The sheaf of Lie algebras
$\bigoplus_{i=1}^{n}\left(\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}\right)$
carries a natural PBW filtration and we let
$\left(\bigoplus_{i=1}^{n}\left(\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}\right)\right)^{\leq
m}$ be the $m$-th part of the filtration. Then the following diagram is
commutative

(6.4)
${H^{0}(N_{G,\bm{\tau}}^{par,rs},\mathcal{D}_{N_{G,\bm{\tau}}^{par,rs}/S}^{\leq
m}(\mathscr{L}_{\vec{\lambda}}))}$${\left(\bigoplus_{i=1}^{n}\left(\overline{U}(\widehat{\mathfrak{g}}_{\xi_{i}})^{opp}\right)\right)^{\leq
m}}$${H^{0}(N_{G,\bm{\tau}}^{par,rs},\operatorname{Sym}^{m}\mathcal{T}_{N_{G,\bm{\tau}}^{par,rs}/S})}$${H^{0}(N_{G,\bm{\tau}}^{par,rs},\mathcal{D}_{N_{G,\bm{\tau}}^{par,rs}/S}^{\leq
m}(\mathcal{O}_{N_{G,\bm{\tau}}^{par,rs}}))}$$\scriptstyle{symb^{\leq
m}}$$\scriptstyle{\hbar_{\mathscr{L}_{\vec{\lambda}}}^{\leq
m}}$$\scriptstyle{\hbar^{\leq m}_{\mathcal{O}}}$$\scriptstyle{symb^{\leq m}}$

where $symb^{\leq m}$ denotes the principal $m$-th order symbol map of a
differential operator.

###  6.2. Projective heat operator from Sugawara

We now give a local description of the map
$\hbar_{\mathscr{L}_{\vec{\lambda}}}$. Let $\mathcal{P}$ be a regularly stable
parabolic $G$-bundle in the moduli space of parabolic bundles of parabolic
weights $\vec{\lambda}$ on a curve $C$ with parabolic structure over $\bm{p}$.
We consider it as a point in $N_{G,\bm{\tau}}^{par,rs}$. The tangent space at
$\mathcal{P}$ is given by $H^{1}(C,\,\operatorname{Par}(\mathcal{P}))$, where
$\operatorname{Par}(\mathcal{P})$ is the sheaf of Lie algebras given by
parabolic endomorphisms of the bundle $\mathcal{P}$.

Let $P_{i}\,\subset\,{G}$ be the parabolic subgroup determined by the weight
$\lambda_{i}$ attached to the point $p_{i}\,\in\,\bm{p}$, and let
$\mathfrak{p}_{i}$ be the corresponding Lie algebra. We denote by
$\mathfrak{p}_{i}^{-}$ the opposite parabolic and by $\mathfrak{n}_{i}^{-}$
the nilpotent radical of $\mathfrak{p}_{i}^{-}$. We have a short exact
sequence of sheaves

(6.5)
$0\,\longrightarrow\,\operatorname{Par}(\mathcal{P})\,\longrightarrow\,\operatorname{Par}(\mathcal{P})\left(\sum_{i=1}^{n}m_{i}p_{i}\right)\,\longrightarrow\,\bigoplus_{i=1}^{n}\left(\mathfrak{n}_{i}^{-}\oplus\bigoplus_{j=1}^{m_{i}}\mathfrak{g}\otimes\xi_{i}^{j}\right)\,\longrightarrow\,0,$

where $m_{1},\,\cdots,\,m_{n}$ are nonnegative integers. Taking the long exact
sequence of cohomologies associated to (6.5), we get a homomorphism

(6.6)
$\bigoplus_{i=1}^{n}\left(\mathfrak{n}_{i}^{-}\oplus\mathfrak{g}\otimes\mathbb{C}[\xi_{i}^{-1}]\xi_{i}^{-1}\right)\,\longrightarrow\,H^{1}(C,\,\operatorname{Par}(\mathcal{P})).$

Combining this with the natural projection
$\mathfrak{g}\otimes\mathbb{C}((\xi_{i}))\,\to\,\mathfrak{n}_{i}^{-}\oplus\mathfrak{g}\otimes\mathbb{C}[\xi_{i}^{-1}]\xi_{i}^{-1}$
for each $1\,\leq\,i\,\leq\,n$, we get a homomorphism

(6.7)
$\rho_{i}\,:\,\mathfrak{g}\otimes\mathbb{C}((\xi_{i}))\,\longrightarrow\,\mathfrak{n}_{i}^{-}\oplus\mathfrak{g}\otimes\mathbb{C}[\xi_{i}^{-1}]\xi_{i}^{-1}\,\longrightarrow\,H^{1}(C,\,\operatorname{Par}(\mathcal{P})).$

The composition of maps $\rho_{i}$ in (6.7) is the local description of
$\hbar_{\mathcal{O}}$ (defined in (6.3))

(6.8) $\left(\overline{U}(\widehat{\mathfrak{g}}_{p_{i}})^{opp}\right)^{\oplus
n}\,\longrightarrow\,H^{0}(N_{G,\bm{\tau}}^{par,rs},\,\mathcal{D}_{N_{G,\bm{\tau}}^{par,rs}/S}(\mathcal{O})).$

The operator $\mathscr{D}(\vec{\ell})$ defined in (3.8) gives a relative
second order differential operator $\mathfrak{D}$ on
$N_{G,\bm{\tau}}^{par,rs}$ which acts on the $i$-th factor by
$(T[\underline{l}_{i}])$ (see (3) and (6.7)). Thus we have the following
diagram

(6.9)
${\bigoplus_{i=1}^{n}\mathcal{O}_{S}((\xi_{i}))\frac{d}{d\xi_{i}}}$${H^{0}(N_{G,\bm{\tau}}^{par,rs},\,\mathcal{D}^{\leq
2}_{N_{G,\bm{\tau}}^{par,rs}/S}(\mathscr{L}_{\vec{\lambda}}))}$${H^{0}(S,\,T_{S}).}$$\scriptstyle{\mathfrak{D}}$$\scriptstyle{\theta}$

We can realize $\mathfrak{D}$ as a projective heat operator by taking a lift
of a vector field on $S$ to an element of
$\bigoplus_{i=1}^{n}\mathcal{O}_{S}((\xi_{i}))\frac{d}{d\xi_{i}}$. Now as
described in the previous section, the difference between two lifts can be
understood as a $\mathcal{O}_{S}$-module homomorphism $a_{\omega}$. Thus the
map $\mathfrak{D}$ descends to a projective heat operator, and we will also
denote the descended operator by ${\mathfrak{D}}$. In the rest of this section
we show that the symbol of $\mathfrak{D}$ is the Hitchin symbol
$\widetilde{\rho}_{sym}$, which will complete the proof of Theorem 1.1.

###  6.3. The parabolic duality pairing and the Hitchin symbol

Recall that the Cartan-Killing form induces a nondegenerate bilinear form
between the sheaves

$\kappa_{\mathfrak{g}}\,:\,\operatorname{Spar}(\mathcal{P})(D)\otimes\operatorname{Par}(\mathcal{P})\,\longrightarrow\,\mathcal{O}_{C}.$

Let $D_{p_{i}}$ be a formal disc around each marked point $p_{i}$ in $C$, and
let $C^{*}\,=\,C\backslash\\{p_{1},\,\cdots,\,p_{n}\\}$ be the complement.
Consider the following open covering:

$C\,=\,C^{*}\cup\left(\sqcup_{i=1}^{n}D_{p_{i}}\right).$

A section of $\operatorname{Spar}(\mathcal{P})$ restricted to $D_{p_{i}}$
consists of an element of $\mathfrak{g}\otimes\mathbb{C}[[\xi_{i}]]$ whose
image under the natural evaluation map

$\operatorname{ev}_{p_{i}}\,:\,\mathfrak{g}\otimes\mathbb{C}[[\xi_{i}]]\,\longrightarrow\,\mathfrak{g}$

is contained in the nilradical $\mathfrak{n}_{i}$ of the parabolic subalgebra
$\mathfrak{p}_{i}$. Similarly, $\operatorname{Par}(\mathcal{P})$ consists of
sections whose restriction to any formal disc $D_{p_{i}}$ has the property
that the image of the evaluation map is in $\mathfrak{p}_{i}$.

Let $\\{\overline{P}_{i}\\}$ be a Čech cocycle representative in
$\prod_{i=1}^{n}\left(\operatorname{Par}(\mathcal{P})(D_{p_{i}}^{*})\right)$
of a cohomology class of $H^{1}(C,\,\operatorname{Par}({\mathcal{P}}))$ with
respect to the covering $C\,=\,C^{*}\cup\left(\sqcup D_{p_{i}}\right)$. Here
we have $\overline{P}_{i}\,\in\,\mathfrak{g}\otimes\mathbb{C}((\xi_{i}))$,
under a trivialization of $\mathcal{P}$ restricted to $D_{p_{i}}$. Similarly
we let
$\\{\phi_{i}d\xi_{i}\\}\,\in\,\mathfrak{g}\otimes\mathbb{C}((\xi_{i}))d\xi_{i}$
denote the restriction of an element of
$H^{0}(C,\,\operatorname{Spar}(\mathcal{P})\otimes K_{C}(D))$ to $\sqcup
D_{p_{i}}^{*}$.

The natural pairing in (5.5) takes the form

(6.10) $\displaystyle H^{0}(C,\,\operatorname{Spar}(\mathcal{P})\otimes
K_{C}(D))\otimes
H^{1}(C,\,\operatorname{Par}(\mathcal{P}))\,\longrightarrow\,\mathbb{C}$
(6.11)
$\displaystyle\\{\phi_{i}d_{\xi_{i}}\\}\times\\{\overline{P}_{i}\\}\,\longmapsto\,\sum_{i=1}^{n}\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(\phi_{i},\overline{P}_{i})d\xi_{i}.$

Now consider a Čech representative
$\vec{\ell}\,=\,\\{\underline{l}_{i}\\}\,\in\,\oplus_{i=1}^{n}\mathbb{C}((\xi_{i}))\frac{d}{d\xi_{i}}$
of a cohomology class in $H^{1}(C,\,T_{C}(-D))$. Let $\phi$ be a global
section of the sheaf $\operatorname{Spar}(\mathcal{P})\otimes K_{C}(D)$. For
each $i$, we have

$\underline{l}_{i}\,=\,\sum_{m=-m_{i}}^{\infty}l_{i,m}\xi_{i}^{m+1}\frac{d}{d\xi_{i}},$

and $\phi$ restricted to $D_{p_{i}}^{*}$ is of the form

$\phi_{i}d\xi_{i}\,=\,\sum_{m\in\mathbb{Z}}X_{i,m}\xi_{i}^{-m-1}d\xi_{i}\in\mathfrak{g}\otimes\mathbb{C}((\xi_{i}))d\xi_{i}.$

Since the diagram in (6.4) commutes, we can evaluate the symbol of
$\mathfrak{D}$ by computing the following:

$\displaystyle\langle\phi\otimes\phi,\,\sum_{i=1}^{n}T[\underline{l}_{i}]\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,\sum_{m=-m_{i}}^{\infty}l_{i,m}L_{m}\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\sum_{m=-m_{i}}^{\infty}l_{i,m}\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,L_{m}\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\sum_{m=-m_{i}}^{\infty}l_{i,m}\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\sum_{a=1}^{\dim\mathfrak{g}}\vcentcolon\mathrel{J^{a}(k)J^{a}(m-k)}\vcentcolon\rangle.$

Now if $\underline{l}_{i}\,=\,\xi_{i}^{n_{i}+1}\frac{d}{d\xi_{i}}$, we get
that

$\displaystyle\langle\phi\otimes\phi,\,\sum_{i=1}^{n}T[\xi_{i}^{n_{i}+1}\frac{d}{d\xi_{i}}]\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,L_{n_{i}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{i=1}^{n}\sum_{k\in\mathbb{Z}}\sum_{a=1}^{\dim\mathfrak{g}}\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,\vcentcolon\mathrel{J^{a}(k)J^{a}(n_{i}-k)}\vcentcolon\rangle.$

If $n_{i}\,\neq\,0$, then we get that

$\displaystyle\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,L_{n_{i}}\rangle$
$\displaystyle=\,\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\sum_{a=1}^{\dim\mathfrak{g}}\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,J^{a}(k)J^{a}(n_{i}-k)\rangle$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\sum_{a=1}^{\dim\mathfrak{g}}\operatorname{Res}_{\xi_{i}=0}\langle{\phi}_{i},\,J^{a}(k)\rangle
d\xi_{i}.\operatorname{Res}_{\xi_{i}=0}\langle{\phi}_{i},\,J^{a}(n_{i}-k)\rangle
d\xi_{i}$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\sum_{a=1}^{\dim\mathfrak{g}}\left(\sum_{m\in\mathbb{Z}}\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(X_{i,m},\,J^{a})\xi^{k-m-1}d\xi_{i}\right)$
$\displaystyle\qquad\qquad\times\left(\sum_{m\in\mathbb{Z}}\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(X_{i,m},\,J^{a})\xi^{n_{i}-k-m-1}d\xi_{i}\right)$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\sum_{a=1}^{\dim\mathfrak{g}}\kappa_{\mathfrak{g}}(X_{i,k},\,J^{a})\kappa_{\mathfrak{g}}(X_{i,n_{i}-k},\,J^{a})$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\kappa_{\mathfrak{g}}(X_{i,k},\,X_{i,n_{i}-k}).$

The zero-th Virasoro operator $L_{0}$ can be rewritten without normal ordering
as follows:

$L_{0}\,=\,\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{a=1}^{\dim\mathfrak{g}}J^{a}J^{a}+\frac{1}{(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k=1}^{\infty}J^{a}(-k)J^{a}(k).$

Thus we get the following:

$\displaystyle\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,L_{0}\rangle$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{a=1}^{\dim\mathfrak{g}}\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,J^{a}J^{a}\rangle+\frac{1}{(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k=1}^{\infty}\langle\phi_{i}\phi_{i}d\xi_{i}^{2},\,J^{a}(-k)J^{a}(k)\rangle$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{a=1}^{\dim\mathfrak{g}}\left(\sum_{m\in\mathbb{Z}}\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(X_{i,m},\,J^{a})\xi_{i}^{-m-1}d\xi_{i}\right)^{2}$
$\displaystyle\quad+\frac{1}{(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k=1}^{\infty}\sum_{a=1}^{\dim\mathfrak{g}}\Bigg{(}\left(\sum_{m\in\mathbb{Z}}\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(X_{i,m},\,J^{a})\xi_{i}^{-k-m-1}d\xi_{i}\right)$
$\displaystyle\quad\quad\times\left(\sum_{m\in\mathbb{Z}}\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(X_{i,m},\,J^{a})\xi_{i}^{k-m-1}d\xi_{i}\right)\Bigg{)}$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{a=1}^{\dim\mathfrak{g}}\kappa_{\mathfrak{g}}(X_{i,0},\,J^{a})\kappa_{\mathfrak{g}}(X_{i,0},\,J^{a}).$
$\displaystyle\quad+\frac{1}{(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k=1}^{\infty}\sum_{a=1}^{\dim\mathfrak{g}}\kappa_{\mathfrak{g}}(X_{i,-k},\,J^{a})\kappa_{\mathfrak{g}}(X_{i,k},\,J^{a})$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\kappa_{\mathfrak{g}}(X_{i,0},\,X_{i,0})+\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}\backslash\\{0\\}}\kappa_{\mathfrak{g}}(X_{i,-k},\,X_{i,k})$
$\displaystyle=\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\kappa_{\mathfrak{g}}(X_{i,k},\,X_{i,-k}).$

We summarize the above calculations in the following proposition.

######  Proposition 6.1.

For any $1\,\leq\,i\,\neq\,n$, and for any $m_{i}\,\in\,\mathbb{Z}$,

$\langle\phi_{i}\otimes\phi_{i}d\xi_{i}^{2},\,L_{m_{i}}\rangle\,=\,\frac{1}{2(\ell+h^{\vee}(\mathfrak{g}))}\sum_{k\in\mathbb{Z}}\kappa_{\mathfrak{g}}(X_{i,k},\,X_{i,m_{i}-k}).$

###  6.4. Proof of the Main theorem (Theorem 1.1)

Recall that the product

$H^{1}(C,\,T_{C}(-D))\otimes H^{0}(C,\,\operatorname{SPar}(\mathcal{P})\otimes
K_{C}(D))\,\longrightarrow\,H^{1}(C,\,\operatorname{Par}\mathcal{P})$

induces a homomorphism

(6.12)
$\widetilde{\rho}_{sym}\,:\,R^{1}\pi_{n*}\mathcal{T}_{\mathfrak{X}_{G}^{par}/M_{G,\bm{\tau}}^{par,rs}}(-D)\,\longrightarrow\,\pi_{n*}\operatorname{Sym}^{2}\mathcal{T}_{M_{G,\bm{\tau}}^{par,rs}/S}.$

Consider the Čech cover of $C$ given by $C\,=\
C^{*}\cup\left(\sqcup_{i=1}^{n}D_{p_{i}}\right)$. In particular, given any
Čech cohomology class $\\{\xi_{i}^{n_{i}+1}\frac{d}{d\xi_{i}}\\}$ in
$H^{1}(C,\,T_{C}(-D))$, using Serre duality and the identification of
$\operatorname{SPar}(\mathcal{P})(D)$ with
$\operatorname{Par}(\mathcal{P})^{\vee}$, we get a symmetric bilinear form on
$H^{0}(C,\,\operatorname{SPar}(\mathcal{P})\otimes K_{C}(D))$.

As in the previous section, consider a section $\phi\in
H^{0}(C,\,\operatorname{SPar}(\mathcal{P})\otimes K_{C}(D)$. For each $i$, the
section $\phi$ restricted to $D_{p_{i}}^{*}$ is of the form

$\phi_{i}d\xi_{i}\,=\,\sum_{m\in\mathbb{Z}}X_{i,m}\xi_{i}^{-m-1}d\xi_{i}\in\mathfrak{g}\otimes\mathbb{C}((\xi_{i}))d\xi_{i}.$

Thus evaluating a cocycle class $\\{\xi_{i}^{n_{i}+1}\frac{d}{d\xi_{i}}\\}$
against a section $\phi$ written in the Čech cover as
$\\{\phi_{i}d\xi_{i}\\}$, we get that

$\displaystyle\\{\xi_{i}^{n_{i}+1}\frac{d}{d\xi_{i}}\\}(\phi)$
$\displaystyle:=$
$\displaystyle\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(\phi_{i}d\xi_{i}\otimes\langle\xi_{i}^{n_{i}+1}\frac{d}{d\xi_{i}},\,\phi_{i}d\xi_{i}\rangle)$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(\phi_{i}d\xi_{i},\,\sum_{m\in\mathbb{Z}}X_{i,m}\xi_{i}^{n_{i}-m})$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{\xi_{i}=0}\kappa_{\mathfrak{g}}(\sum_{k\in\mathbb{Z}}X_{i,k}\xi_{i}^{-k-1},\,\sum_{m\in\mathbb{Z}}X_{i,m}\xi_{i}^{n_{i}-m})d\xi_{i}$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathbb{Z}}\kappa_{\mathfrak{g}}(X_{i,k},\,X_{i,n_{i}-k}).$

We summarize the discussion in this subsection in the following proposition
which completes the proof the Theorem 1.1.

######  Proposition 6.2.

Let $a$ be any rational number and $\phi$ be a faithful representation with
Dynkin index $m_{\phi}$. Then the symbol of the projective heat operator
${\mathfrak{D}}$ acting on $\mathscr{L}_{\vec{\lambda}}^{\otimes
a.{m_{\phi}}}\cong\operatorname{Det}^{\otimes a}_{par,\phi}(\bm{\tau})$
constructed from the Sugawara tensor and uniformization (see Section 6.2)
coincides with

$\frac{1}{2(a\cdot
m_{\phi}\cdot\ell+h^{\vee}(\mathfrak{g}))}\widetilde{\rho}_{sym}.$

Hence, the projective heat operator $\mathfrak{D}$ and the projective heat
operator constructed in [19] via (5.7) coincide.

######  Remark 6.3.

If a square-root of $K_{M^{par,s}_{\mathsf{SL}_{r},\bm{\alpha}}/S}$ exists,
then it follows that the push-forward of the line bundle
$\operatorname{Det}_{par}(\bm{\alpha})\otimes
K^{\frac{1}{2}}_{M^{par,s}_{\mathsf{SL}_{r},\bm{\alpha}}/S}$ produces
conformal blocks of level $\ell-h^{\vee}(\mathfrak{sl}(r))$, where
$h^{\vee}(\mathfrak{g})$ is the dual Coxeter number of a Lie algebra
$\mathfrak{g}$. From the calculations in this section, and the fact that the
tangent and cotangent spaces of the moduli space
$M^{par,s}_{\mathsf{SL}_{r},\bm{\alpha}}$ are only dependent on the flag type
of $\bm{\alpha}$, it follows that the symbol ${\rho_{sym}/\ell}$ equals the
symbol of the differential operator that induces the TUY connection. However,
it should be mentioned that even if
$K_{M^{par,s}_{\mathsf{SL}_{r},\bm{\alpha}}/S}$ has a square-root, the
pushforward $\pi_{*}\left(\operatorname{Det}_{par}(\bm{\alpha})\otimes
K^{\frac{1}{2}}_{M^{par,s}_{\mathsf{SL}_{r},\bm{\alpha}}/S}\right)$ may not
have any sections.

##  7\. Geometrization of the KZ equation on invariants

In this section, we show a geometric construction of the Knizhnik-
Zamolodchikov connection (KZ). This question was suggested to us by Professor
P. Belkale. Let us first recall the classical construction of the KZ
connection [34].

###  7.1. KZ connection

Let $\mathfrak{g}$ be a fixed semisimple Lie algebra, and let
$\vec{\lambda}\,=\,(\lambda_{1},\,\cdots,\,\lambda_{n})$ be an $n$-tuple of
highest weights. Consider the vector space of invariants of tensor product of
representations

$A_{\vec{\lambda}}(\mathfrak{g})\,:=\,\operatorname{Hom}_{\mathfrak{g}}(V_{\lambda_{1}}\otimes\dots\otimes
V_{\lambda_{n}},\mathbb{C}).$

The space of invariants sits inside the zero weight space
$(V_{\lambda_{1}}\otimes\dots\otimes V_{\lambda_{n}})^{*}_{0}$ of the dual of
the tensor product of representations.

Let
$X_{n}\,=\,\\{\bm{z}\,=\,(z_{1},\,\cdots,\,z_{n})\,\in\,\mathbb{C}^{n}\,\,\mid\,\,z_{i}\,\neq\,z_{j}\\}$,
and consider the trivial vector bundle $\mathbb{A}_{\vec{\lambda}}$ on the
configuration space of points $X_{n}$ whose fiber is
$A_{\vec{\lambda}}(\mathfrak{g})$. It is well known [29, 28, 66] that the
space of conformal blocks
$\mathcal{V}^{\dagger}_{\vec{\lambda}}(C,\mathfrak{g},\ell,\bm{z})$ on
$\mathbb{P}^{1}\,=\,\mathbb{C}\sqcup\\{\infty\\}$ with $n$ marked points
$(z_{1},\,\cdots,\,z_{n})$ for $\mathfrak{g}$ at level $\ell$ and weights
$\vec{\lambda}$ injects into $A_{\vec{\lambda}}(\mathfrak{g})$:

(7.1)
$\iota\,:\,\mathcal{V}^{\dagger}_{\vec{\lambda}}(\mathbb{P}^{1},\mathfrak{g},\ell,\bm{z})\,\hookrightarrow\,A_{\vec{\lambda}}(\mathfrak{g}).$

This map is actually an isomorphism for $\ell\,\gg\,0$. Specific bounds for
$\ell$ are given in Belkale-Gibney-Mukhopadhyay ([14, 15]).

As in Section 3, consider an orthonormal basis
$J^{1},\,\cdots,\,J^{\dim\mathfrak{g}}$ of the Lie algebra $\mathfrak{g}$ for
the normalized Cartan-Killing form. Define the Casimir operator
$\Omega\,=\,\sum_{a=1}^{\dim\mathfrak{g}}J^{a}J^{a}$. For pairs of integers
$1\,\leq\,i\,\neq\,j\,\leq\,n$, and vectors $v_{1}\otimes\cdots\otimes
v_{n}\,\in\,V_{\lambda_{1}}\otimes\cdots\otimes V_{\lambda_{n}}$, let

$\Omega_{i,j}(v_{1}\otimes\cdots\otimes
v_{n})\,:=\,\sum_{a=1}^{\dim\mathfrak{g}}v_{1}\otimes\cdots\otimes
J^{a}v_{i}\otimes\cdots\otimes J^{a}v_{j}\otimes\cdots\otimes v_{n}.$

For any complex number $\kappa\,\neq\,0$, the formula

(7.2) $\left(\nabla^{(\kappa)}_{\frac{\partial}{\partial
z_{i}}}(f\otimes\langle\Psi|\right)(|\Phi\rangle)\,:=\,\frac{\partial
f}{\partial z_{i}}\langle\Psi|\Phi\rangle-\frac{f}{\kappa}\sum_{j\neq
i}\frac{\langle\Psi|\Omega_{i,j}(\phi)\rangle}{z_{i}-z_{j}},$

defines a flat connection on $(V_{\lambda_{1}}\otimes\cdots\otimes
V_{\lambda_{n}})^{*}_{0}\otimes\mathcal{O}_{X_{n}}$ over $X_{n}$ that
preserves the subbundle $\mathbb{A}_{\vec{\lambda}}$. Hence, its monodromy
gives a representation of the pure braid group $\pi_{1}(X_{n},\,\bm{z})$.

In this discussion, we restrict ourselves to the case where
$\kappa\,=\,\ell+h^{\vee}(\mathfrak{g})$, and
$\kappa_{\mathfrak{g}}(\lambda_{i},\,\theta_{\mathfrak{g}})\,<\,1$ for all
$i$. In this case it is known that the connection
$\nabla^{(\ell+h^{\vee}(\mathfrak{g}))}$ preserves the bundle
$\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$ of conformal blocks
and it is equal to the TUY/WZW connection [28, 29, 66].

###  7.2. Invariants as global sections

As in Section A.1 consider the moduli stack of quasi-parabolic bundles
$\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ of local type $\bm{\tau}$
on $\mathbb{P}^{1}$, where $\bm{\tau}$ and $\vec{\lambda}$ are related by the
usual exponential map as before. Consider the open substack
$\mathcal{P}ar^{c}_{G}(\mathbb{P}^{1},\bm{z},\tau)$ of
$\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ parametrizing quasi-
parabolic bundle on $\mathbb{P}^{1}$ whose underlying bundle is trivial. By
construction, we have an isomorphism of
$\mathcal{P}ar_{G}^{c}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ with the quotient
stack

(7.3) $[\left(G/P_{1}\times\cdots\times G/P_{n}\right)/G],$

where $P_{1},\,\cdots,\,P_{n}$ are the parabolics determined by
$\tau_{1},\,\cdots,\,\tau_{n}$ and $G$ acts diagonally on the product of
partial flag varieties.

Let $\mathscr{L}_{\vec{\lambda}}$ be the Borel-Weil-Bott line bundle on
$\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$, and consider the
restriction of $\mathscr{L}_{\vec{\lambda}}$ to
$\mathcal{P}ar_{G}^{c}(\mathbb{P}^{1},\bm{z},\bm{\tau})$. We get a natural map

(7.4)
$H^{0}(\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau}),\,\mathscr{L}_{\vec{\lambda}})\,\longrightarrow\,H^{0}(\mathcal{P}ar_{G}^{c}(\mathbb{P}^{1},\bm{z},\bm{\tau}),\,\mathscr{L}_{\vec{\lambda}}).$

Now the restriction of $\mathscr{L}_{\vec{\lambda}}$ to
$[\left(G/P_{1}\times\cdots\times G/P_{n}\right)/G]$ is
$L_{\lambda_{1}}\boxtimes\dots\boxtimes L_{\lambda_{n}}$, where the
$L_{\lambda_{i}}$ are the natural homogeneous line bundles on $G/P_{i}$
determined by the weights $\lambda_{i}$. Moreover, by the Borel-Weil theorem,
we have $H^{0}(G/P_{i},\,L_{\lambda_{i}})\,=\,V_{\lambda_{i}}^{*}$. Thus, from
the restriction we get the natural commutative diagram

(7.5)
${H^{0}(\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau}),\mathscr{L}_{\vec{\lambda}})}$${H^{0}(\mathcal{P}ar_{G}^{c}(\mathbb{P}^{1},\bm{z},\bm{\tau}),\mathscr{L}_{\vec{\lambda}})}$${\left(\bigotimes_{i=1}^{n}H^{0}(G/P_{i},L_{\lambda_{i}})\right)^{\mathfrak{g}}}$${\mathcal{V}_{\vec{\lambda}}^{\dagger}(\mathbb{P}^{1},\mathfrak{g},\ell,\bm{z})}$${\operatorname{Hom}_{\mathfrak{g}}(V_{\lambda_{1}}\otimes\cdots\otimes
V_{\lambda_{n}},\mathbb{C})}$${\big{(}V^{*}_{\lambda_{1}}\otimes\cdots\otimes
V^{*}_{\lambda_{n}}\big{)}^{\mathfrak{g}}.}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{res}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\iota}$

Here the left vertical isomorphism is due to Laszlo-Sorger [40]; the diagram
was used in [14]. Now it follows that the complement of
$\mathcal{P}ar_{G}^{c}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ in
$\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ is just the ordinary
theta divisor.

###  7.3. Differential operators

Recall the notion of a good stack from Beilinson-Drinfeld [9]: an
equidimensional algebraic stack $\mathcal{Y}$ over complex numbers is good if
the dimension of $\mathcal{Y}$ is half the dimension of the cotangent stack
$\mathcal{T}^{\vee}_{\mathcal{Y}}$. Let $\mathcal{Y}_{sm}$ be the smooth
topology of $\mathcal{Y}$. For any object $S\,\in\,\mathcal{Y}_{sm}$ and a
smooth $1$-morphism $\pi_{S}\,\in\,\mathcal{Y}$, we have the exact sequence

$\mathcal{T}_{S/\mathcal{Y}}\,\longrightarrow\,\mathcal{T}_{S}\,\longrightarrow\,\pi_{S}^{*}\mathcal{T}_{Y}\,\longrightarrow\,0.$

Consider the sheaf of differential operators $\mathcal{D}_{S}$ on $S$ and the
left ideal
$I\,=\,\mathcal{D}_{S}\mathcal{T}_{S/\mathcal{Y}}\,\subset\,\mathcal{D}_{S}$.
Set $\mathcal{D}_{\mathcal{Y}}(S)\,:=\,\mathcal{D}_{S}/I,$. This
$\mathcal{D}_{\mathcal{Y}}$ is an $\mathcal{O}_{\mathcal{Y}}$ module along
with a natural filtration such that
$\operatorname{Sym}\mathcal{T}_{\mathcal{Y}}\,\cong\,\operatorname{gr}\mathcal{D}_{\mathcal{Y}}$.
The above also works for differential operators twisted by a line bundle.

Since the nilpotent cone of the moduli space of parabolic Higgs bundles is
isotropic of exactly half the dimension [7, 26, 41], it follows that the stack
$\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ is good. Moreover, since
both $\mathcal{P}ar^{c}_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ and
$\mathcal{P}ar^{rs}_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ are quotients of a
smooth scheme by a reductive group, they are also good.

Now we know that the line bundle $\mathscr{L}_{\vec{\lambda}}$ descends to a
line bundle on $\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$. The
construction of the projective heat operator (cf. Definition 5.2) with symbol
(5.7) gives a second order differential operator $\mathbb{D}$ on
$\mathscr{L}_{\vec{\lambda}}$ over the moduli stack
$\mathcal{P}ar^{rs}_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$. Since the sheaf
$\mathcal{D}^{\leq 2}(\mathscr{L}_{\vec{\lambda}})$ is coherent and
$\mathcal{P}ar^{rs}_{G}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ has complement of
dimension at least two (provided $\bm{\tau}$ satisfies the conditions in the
statement of Theorem B.1) applying Hartogs theorem, we get a differential
operator on $\mathscr{L}_{\vec{\lambda}}$ over the entire stack
$\mathcal{P}ar_{G}(\mathbb{P}^{1},\bm{z},\tau)$, which we will still denote by
$\mathbb{D}$.

Recall that via the uniformization theorem and the Sugawara construction, we
have a degree two differential operator $\mathscr{D}$ on
$\mathscr{L}_{\vec{\lambda}}$, which by Theorem 1.1 agrees with $\mathbb{D}$.
Since the Sugawara construction restricted to the open substack
$\mathcal{P}ar_{G}^{c}(\mathbb{P}^{1},\bm{z},\bm{\tau})$ induces the KZ
connection, we have the following corollary obtained by restricting
$\mathbb{D}$ to $\mathcal{P}ar_{G}^{c}(\mathbb{P}^{1},\bm{z},\bm{\tau})$.

######  Corollary 7.1.

Let $\pi^{c}\,:\,\mathcal{P}ar_{G}^{c}(\bm{\tau})\,\to\,X_{n}$ be the relative
open substack of quasi-parabolic bundles whose underlying bundle is trivial.
Then the heat operator $\mathbb{D}$ induces a flat connection on the vector
bundle $\pi^{c}_{*}\mathscr{L}_{\vec{\lambda}}$ over $X_{n}$ whose fiber at a
point $\bm{z}$ is
$H^{0}(\mathcal{P}ar^{c}_{G}(C,\bm{z},\bm{\tau}),\mathscr{L}_{\vec{\lambda}})$.
Moreover, the natural identification of
$\pi_{*}^{c}\mathscr{L}_{\vec{\lambda}}$ with $\mathbb{A}_{\vec{\lambda}}$ is
flat for the geometric connection on $\pi_{*}^{c}\mathscr{L}_{\vec{\lambda}}$
and the Knizhnik-Zamolodchikov connection on $\mathbb{A}_{\vec{\lambda}}$.

##  Appendix A Moduli spaces of parabolic bundles

In this section, we briefly recall the basic notion of parabolic bundles and
the natural line bundles on their moduli spaces.

Let $C,\bm{p}$ be as in Section 3.1. Let $E$ be a vector bundles on $C$. A
quasi-parabolic structure on $E$ at a point $p\,\in\,\bm{p}$ is a strictly
decreasing flag

$E_{p}\,=\,F^{1}E_{p}\,\supset\,F^{2}E_{p}\,\cdots\,\supset\,\cdots\,\supset\,F^{k_{p}}E_{p}\,\supset\,F^{k_{p}+1}E_{x}\,=\,0.$

of linear subspaces in $E_{p}$. The above integer $k_{p}$ is the length of the
flag at $p$, and the tuple

$(r_{1}(F^{\bullet}E_{p}),\,\cdots,\,r_{k_{p}}(F^{\bullet}E_{p}))$

records the jumps in the dimension of the subspaces and is defined by

$r_{j}(F^{\bullet}E_{p})\,:=\,\dim F^{j}E_{p}-\dim F^{j+1}E_{p}.$

A parabolic structure on $E$ at $p$ is a quasi-parabolic structure as above
together with a sequence of rational numbers

$0\,\leq\,\alpha_{1}\,<\,\alpha_{2}\,<\,\cdots\,<\,\alpha_{k_{p}}<1$

known as the weights. A parabolic bundle $(E,\,{\bm{\alpha}},\,\bm{r})$ on $C$
with parabolic divisor $\bm{p}$ is a vector bundle $E$ on $C$ along with
parabolic structure over the points in $\bm{p}$. Using the weights
$\bm{\alpha}$, the parabolic degree of $E$ is defined to be

${\rm
pdeg}(E)\,:=\,\deg(E)+\sum_{i=1}^{n}\sum_{j=1}^{k_{p_{i}}}r_{j}(F^{\bullet}(E_{p_{i}}))\alpha_{j}(F^{\bullet}(E_{p_{i}})).$

Stable and semistable parabolic bundles are defined using the parabolic degree
(see [43]). Mehta and Seshadri constructed the moduli space
$M_{\bm{\alpha}}^{par}$ of semistable parabolic bundles [43].

We now discuss some natural ample line bundles on $M_{\bm{\alpha}}^{par}$,
following [20]. Let ${\bm{\alpha}}$ be a fixed set of weights for fixed flag
type $\bm{r}$, and let $(E,\,\bm{\alpha},\,\bm{r})$ be a parabolic bundle on
$(C,\,\bm{p})$. Define the parabolic Euler characteristic

$\chi_{\bm{p}}(E)\,:=\,\chi(E)-\sum_{i=1}^{n}\sum_{j=1}^{k_{p_{i}}}r_{j}(F^{\bullet}(E_{p_{i}}))\alpha_{j}(F^{\bullet}(E_{p_{i}})).$

Let $\mathcal{E}$ be a family of parabolic bundles on $C$, parametrized by a
scheme $T$, of rank $r$, weight ${\bm{\alpha}}$ and flag type ${\bm{r}}$. For
each point $p_{i}$, we have a string of rational numbers

$\bm{\alpha}_{p_{i}}\,=\,(0\,\leq\,\alpha_{1}(p_{i})\,<\,\cdots\,<\,\alpha_{j}(p_{i})\,<\,\cdots\,<\,\alpha_{k_{p_{i}}}(p_{i})\,<\,1)$

which are the parabolic weights. Observe that the parabolic Euler
characteristic $\chi_{\bm{p}}$ remains constant in a connected family.

Let $\ell$ be the least common multiple of all denominators of all the
rational numbers appearing in $\bm{\alpha}$.

######  Definition A.1.

The parabolic determinant bundle of level $\ell$ on $M_{\bm{\alpha}}^{par}$ is
the element of the rational Picard group $\operatorname{Pic}(T)_{\mathbb{Q}}$
given by

(A.1)
$\operatorname{ParDet}(\mathcal{E},\bm{\alpha})\,:=\,\operatorname{Det}(\mathcal{E})^{\ell}\bigotimes\left(\bigotimes_{i=1}^{n}\left(\bigotimes_{j=1}^{k_{p_{i}}}\det\operatorname{Gr}^{j}\mathscr{F}_{\bullet,p_{i}}(\mathcal{E}_{|T\times
p_{i}})\right)^{\ell.\alpha_{j}(p_{i})}\right)\otimes\left(\det\mathcal{E}_{|T\times
p_{0}}\right)^{\frac{\ell.\chi_{\bm{p}}(\mathcal{E})}{r}},$

where $p_{0}$ is a fixed point of $C$, and ${\rm Gr}^{j}$ denotes the $j$-th
graded piece of the filtration $\mathscr{F}_{\bullet,p_{i}}$ on
$\mathcal{E}_{|T\times p_{i}}$ (cf. [20, Prop. 4.5]).

Let $M_{\mathsf{SL}_{r},\bm{\alpha}}^{par,ss}$ be the moduli space of
semistable parabolic $\mathsf{SL}_{r}$ bundles or equivalently parabolic
bundles with trivialized determinant. Then
$\operatorname{ParDet}(\mathcal{E},\bm{\alpha})$ descends to a line bundle on
$M_{\mathsf{SL}_{r},\bm{\alpha}}^{par,ss}$, which will be denoted by
$\operatorname{Det}_{par}(\bm{\alpha})$.

###  A.1. Parabolic $G$ bundles

We shall follow the notation in [19, App. A] and refer the reader there for
more details. Consider the fundamental alcove $\Phi_{0}$, and let
$\bm{\tau}\,=\,(\tau_{1},\,\cdots,\,\tau_{n})$ be a choice of $n$-tuple of
weights in $\Phi_{0}$ which will be referred to as parabolic weights.

######  Definition A.2.

Let $G$ be a connected complex reductive group. A parabolic structure on a
principal $G$–bundle $E\,\to\,C$ with parabolic structures at the points
$\bm{p}\,=\,(p_{1},\,\cdots,\,p_{n})$ is a choice of parabolic weights
$\bm{\tau}$ along with a section $\sigma_{i}$ of the homogeneous space
$E_{p_{i}}/P(\tau_{i})$, for each $1\,\leq\,i\,\leq\,n$, where $P(\tau_{i})$
is the standard parabolic associated to $\tau_{i}$. Throughout this paper we
will assume that $\theta_{\mathfrak{g}}(\tau_{i})\,<\,1$ for all
$1\,\leq\,i\,\leq\,n$.

We observe that when $G\,=\,\mathsf{GL}_{r}$, the associated bundle
constructed via the standard representation of $\mathsf{GL}_{r}$ recovers the
notion of parabolic bundles and parabolic weights as in the beginning of the
present section. The notions of stability and semistability for parabolic
$G$-bundles appear in the work of Bhosle-Ramanathan [16]; for
$G\,=\,\mathsf{GL}_{r}$ they coincide with the notions of stable and
semistable parabolic vector bundles.

Let $\bm{\tau}$ be an $n$-tuple of parabolic weights in the interior of the
Weyl alcove of $G$. The corresponding moduli space $M^{par,ss}_{G,\bm{\tau}}$
(respectively, $M^{par,s}_{G,\bm{\tau}}$) of semistable (respectively, stable)
parabolic $G$-bundles was constructed in [5]. These moduli spaces are normal
irreducible quasi-projective varieties. The smooth locus of
$M^{par,ss}_{G,\bm{\tau}}$ is denoted by $M^{par,rs}_{G,\bm{\tau}}$ and it
parametrizes regularly stable parabolic bundles [18] or equivalently stable
parabolic bundles with minimal automorphisms.

Let $\iota\,:\,G\,\to\,G^{\prime}$ be an embedding of connected semisimple
groups. This homomorphism $\iota$ produces a map
$M^{par,ss}_{G,\bm{\tau}}\,\to\,M^{par,ss}_{G^{\prime},\bm{\tau}^{\prime}}$
which is a finite morphism. The weights $\bm{\tau}^{\prime}$ and $\bm{\tau}$
are related by $\iota$. This plays a key role in construction of the moduli
spaces. In fact, choosing an appropriate representation of the group $G$, one
can reduce the question of construction to the corresponding question on
parabolic vector bundles.

######  Remark A.3.

Let $\mathcal{C}\,\to\,S$ be a family of smooth curves with $n$ disjoint
sections. We will denote the corresponding semistable and regularly stable
moduli spaces also by $M^{par,ss}_{G,\bm{\tau}}$ and
$M^{par,rs}_{G,\bm{\tau}}$ respectively. When there is a scope of confusion,
for any $n$-pointed smooth curve $(C,\,\bm{p})$, we will use the notation
$M^{par,ss}_{G,\bm{\tau}}(C,\,\bm{p})$ and
$M^{par,rs}_{G,\bm{\tau}}(C,\,\bm{p})$ respectively.

###  A.2. Parabolic bundles as equivariant bundles

We now discuss parabolic bundles from the point of view of equivariant
bundles. We refer the reader to [6], [17], [57], and [19, App. B] for more
details. This was used in [19] to construct a Hitchin type connection for
parabolic bundles and it will be crucial here as well.

######  Definition A.4.

Let $p\,:\,\widehat{C}\,\to\,C$ be a Galois cover of curves with Galois group
$\Gamma$. A $(\Gamma,\,G)$-bundle is a principal $G$-bundle $\widehat{E}$ on
$\widehat{C}$ together with a lift of the action of $\Gamma$ on $\widehat{E}$
as bundle automorphism that commutes with the action of $G$ on $\widehat{E}$.

Assume that the map $p\,:\,\widehat{C}\,\to\,C$ is ramified over
$p_{i}\,\in\,C$, $1\,\leq\,i\,\leq\,n$. Let
$\Gamma_{q_{i}}\,\subset\,\Gamma\,=\,{\rm Gal}(p)$ be the isotropy subgroup
for some $q_{i}$ over $p_{i}$. A $(\Gamma,\,G)$–bundle on a formal disc around
$q_{i}$ is uniquely determined by the conjugacy class of a homomorphism
$\rho_{i}\,:\,\Gamma_{q_{i}}\,\to\,G$ given by the action of $\Gamma_{q_{i}}$
on the fiber of the principal $G$–bundle over the point $q_{i}$ (see [6, 63]).
Fix a generator $\gamma_{i}$ of the cyclic group $\Gamma_{q_{i}}$. Now
consider a string of parabolic weights
${\bm{\tau}}\,=\,(\tau_{1},\,\cdots,\,\tau_{n})$ such that
$\rho_{i}(\gamma_{i})$ is conjugate to $\tau_{i}$ for each
$1\,\leq\,i\,\leq\,n$. We will refer to this $\bm{\tau}$ as the local type of
a $(\Gamma,\,G)$–bundle.

The notions of stability and semistability for $(\Gamma,\,G)$–bundles are
similar to those for the usual principal $G$-bundles; more precisely, the
inequality is checked only for the $\Gamma$ equivariant reductions of the
structure group to a parabolic subgroup of $G$ ([5, 51]). Let
$M^{\bm{\tau},ss}_{G}$ (respectively, $M^{\bm{\tau},s}_{G}$) denote the moduli
spaces of semistable (respectively, stable) $(\Gamma,\,G)$ bundles of local
type $\bm{\tau}$.

Recall the isomorphism
$\nu_{\mathfrak{g}}:\mathfrak{g}^{\vee}\xrightarrow{\,\smash{\raisebox{-2.15277pt}{$\sim$}}\,}\mathfrak{g}$
from the Killing form. Given a string of parabolic weights
$\bm{\tau}\,=\,(\tau_{1},\,\cdots,\,\tau_{n})$, choose a minimal integer
$\ell$ such that
$\exp\left(2\pi\sqrt{-1}\left(\ell\cdot\nu_{\mathfrak{g}}(\tau_{i})\right)\right)\,=\,1$.
Then by [46, 56], we can find a ramified Galois cover
$p\,:\,\widehat{C}\,\to\,C$ with ramification exactly over $n$-points
$\\{p_{i}\\}_{i=1}^{n}$ whose isotropy at any ramification point is a cyclic
group of order $\ell$. From now on we will restrict ourselves only to such
Galois covers. The following theorem is due to [5, 6, 17].

######  Theorem A.5.

Consider the moduli stack $\mathcal{B}un^{\bm{\tau}}_{\Gamma,G}(\widehat{C})$
of $(\Gamma,\,G)$–bundles of fixed local type $\bm{\tau}$. The invariant
direct image functor identifies the stack
$\mathcal{B}un^{\bm{\tau}}_{\Gamma,G}(\widehat{C})$ with the moduli stack
$\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})$ of quasi-parabolic bundles of flag
type $\bm{\tau}$. Moreover, the invariant push-forward functor also induces an
isomorphism between the moduli spaces $M^{par,ss}_{G,\bm{\tau}}$
(respectively, $M^{par,s}_{G,\bm{\tau}}$) and $M^{\bm{\tau},ss}_{G}$
(respectively, $M^{\bm{\tau},s}_{G}$).

###  A.3. Parabolic determinants as equivariant determinants

Consider the moduli space $M_{G}^{\bm{\tau},ss}$ of $(\Gamma,\,G)$ bundles
associated to a Galois cover $p\,:\,\widehat{C}\,\to\,C$ with Galois group
$\Gamma$. Let $\widehat{M}_{G}$ be the moduli space of semistable principal
$G$-bundles on the curve $\widehat{C}$. There is a natural forgetful map
$M^{par,ss}_{G,\bm{\tau}}\,\to\,\widehat{M}_{G}$ that simply forgets the
action of $\Gamma$.

Given a representation $\phi\,:\,G\,\to\,\mathsf{SL}_{r}$, consider the
associated morphism
$\overline{\phi}\,:\,\widehat{M}_{G}\,\to\,\widehat{M}_{\mathsf{SL}_{r}}$
between the corresponding moduli spaces. Let $\mathcal{L}$ be the determinant
of cohomology line bundle on $\widehat{M}_{\mathsf{SL}_{r}}$. Let

$\mathcal{L}_{\phi}\,:=\,\overline{\phi}^{*}\mathcal{L}$

be its pullback to $\widehat{M}_{G}$. If $G\,=\,\mathsf{SL}_{r}$, then $\phi$
can be taken to be the standard representation. Now Theorem A.5 realizes the
moduli space $M^{par,ss}_{G,\bm{\tau}}$ of parabolic bundles as a moduli space
$M^{\bm{\tau},ss}_{G}$ of $(\Gamma,\,G)$–bundles on $\widehat{C}$, which maps
further into $\widehat{M}_{G}$ by forgetting the action of $\Gamma$. Thus
using the identification between $M_{G}^{\bm{\tau},ss}$ and
$M^{par,ss}_{G,\bm{\tau}}$, we get a natural line bundle $\mathcal{L}_{\phi}$
on $M^{par,ss}_{G,\bm{\tau}}$.

On the other hand, using the parabolic determinant of cohomology, one can
construct natural line bundles on $M^{par,ss}_{G,\bm{\tau}}$ as follows:

Let $\bm{\tau}\,=\,(\tau_{1},\,\cdots,\,\tau_{n})$ be a string of parabolic
weights such that $\theta_{\mathfrak{g}}(\tau_{i})\,<\,1$ for all
$1\,\leq\,i\,\leq\,n$. Take a faithful representation $(\phi,\,V)$ of the
group $G$ satisfying the following condition:

  * •

The local type
$\phi(\bm{\tau})\,=\,(\phi(\tau_{1}),\,\cdots,\,\phi(\tau_{n}))$ is rational,
and $\theta_{\mathfrak{sl}(V)}(\phi(\tau_{i}))\,<\,1$.

Here, $\theta_{\mathfrak{g}}$ and $\theta_{\mathfrak{sl}(V)}$ are the highest
roots of the Lie algebras $\mathfrak{g}$ and $\mathfrak{sl}(V)$, respectively.
We now recall the definition of the parabolic determinant of cohomology for
$\operatorname{G}$–bundles.

######  Definition A.6.

Let $\mathcal{E}$ be a family of parabolic $G$–bundles on a curve $C$ with
$n$-marked points, and let $\phi\,:\,\operatorname{G}\,\to\,\mathsf{SL}(V)$ be
a faithful representation. Then the parabolic $G$-determinant bundle
$\operatorname{Det}_{par,\phi}(\bm{\tau})$ with weight $\bm{\tau}$ is defined
to be the line bundle
$\operatorname{Det}_{par}(\nu_{\mathfrak{sl}(V)}(\phi(\bm{\tau})))$.

The following is recalled from [20].

######  Proposition A.7.

Let $\ell$ be the order of the stabilizer at each ramification point of the
Galois cover $p\,:\,\widehat{C}\,\to\,C$ with Galois group $\Gamma$, then
under the isomorphism in Theorem A.5, the parabolic determinant of cohomology
is related to $\mathcal{L}_{\phi}$ by the formula

$\mathcal{L}_{\phi}\,\,\cong\,\,\left(\operatorname{Det}_{par,\phi}(\bm{\tau})\right)^{\frac{|\Gamma|}{\ell}},$

where the $\Gamma$ cover $\widehat{C}$ is determined by the parabolic weight
data $\nu_{\mathfrak{sl}(V)}(\phi\bm{(}\tau))$.

##  Appendix B Uniformization of moduli spaces and conformal blocks

In this section, following the work of Belkale-Fakhruddin [12], Laszlo [39],
and Laszlo-Sorger [40], we discuss the universal isomorphism between the
sections of the parabolic determinant of cohomology bundle and the spaces of
conformal blocks. If $(C,\,\vec{p})$ is a fixed smooth $n$-pointed curve, this
identification is due to Beauville-Laszlo [8] ($G\,=\,\mathsf{SL}_{r}$ and
$n\,=\,0$), Faltings [27], Kumar-Narasimhan-Ramanathan [37] (for $n\,=\,0$),
Pauly [49] (for $G\,=\,\mathsf{SL}_{r}$) and Laszlo-Sorger [40]. The result
has been extended to nodal curves by Belkale-Fakhruddin [12]. All of the
results use a key uniformization theorem of Harder [31] and Drinfeld-Simpson
[24] in the smooth case and its generalization in [12, 13] for the nodal case.
We mostly follow the discussion in [12, Sec. 6].

###  B.1. The line bundle on the universal moduli stack

Consider the moduli stack $\mathcal{M}_{g,n}$ parametrizing smooth $n$-pointed
curves of genus $g$. Recall from Section A.1 that given a tuple
$\bm{\tau}\,=\,(\tau_{1},\,\cdots,\,\tau_{n})$ in the fundamental Weyl alcove
$\Phi$ of a simple Lie algebra $\mathfrak{g}$, we have the moduli stack
$\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})$ of quasi-parabolic $G$ bundles of type
$\bm{\tau}$ on a smooth curve $C$. This construction for families of smooth
$n$-pointed curves gives relative moduli stacks
$\pi_{e}\,:\,\mathcal{P}ar_{G}(\bm{\tau})\,\to\,\mathcal{M}_{g,n}$ such that
for any smooth curve $(C,\,\vec{p})$ we have
$\pi_{e}^{-1}(C,\bm{p})\,=\,\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})$. Throughout
this discussion, it is assumed that $\theta_{\mathfrak{g}}(\tau_{i})\,<\,1$
for all $1\,\leq\,i\,\leq\,n$.

Following [12] and [39], we construct a line bundle
$\mathscr{L}_{\vec{\lambda}}\to\mathcal{P}ar_{G}(\bm{\tau})$, such that
$\pi_{*}\mathscr{L}_{\vec{\lambda}}\,=\,\mathbb{V}^{*}_{\vec{\lambda}}(\mathfrak{g},\ell),$
where $\vec{\lambda}$ and $\ell$ are related to $\bm{\tau}$ by the exponential
map. The construction in [12] extends to the stable nodal curves.

####  B.1.1. The relative affine flag varieties

Let $\mathcal{C}\,\to\,S$ be a family of smooth $n$-pointed curves, and let
$S\,=\,\operatorname{Spec}{R}$. Consider the affine curve
$\mathcal{C}^{\prime}\,=\,\mathcal{C}-\sqcup_{i=1}^{n}p_{i}(S)$. Let
$\mathcal{C}_{A}\,=\,\mathcal{C}\times_{R}\operatorname{Spec}(A)$ for an $R$
algebra $A$ and similarly define $\mathcal{C}_{A}^{\prime}$. Let
$\widehat{\mathcal{C}}_{A}$ denote the completion of $\mathcal{C}_{A}$ along
the sections $\bm{p}$. The sections $\bm{p}$ induce sections of
$\widehat{\mathcal{C}}_{A}$, and $\widehat{\mathcal{C}}_{A}^{\prime}$ denotes
its complement.

Consider the following:

  1. (1)

$L_{\mathcal{C}^{\prime},G}(A)\,=\,\operatorname{Mor}_{k}(\mathcal{C}^{\prime}_{A},\,G)$.

  2. (2)

$\mathcal{LG}(A)\,=\,G(\Gamma(\widehat{\mathcal{C}}_{A}^{\prime},\,\mathcal{O}))$.

Each $\tau_{i}$ determines a parabolic subgroup $P(\tau_{i})\,\subset\,G$, and
we consider the standard parahoric subgroup $\mathcal{P}_{\bm{\tau}}$ given by
the inverse image of $\prod_{i=1}^{n}P(\tau_{i})$ under the natural evaluation
map $L_{G}(A)\,\longrightarrow\,G^{n}$. Proposition 6.3 of [12] shows that the
$R$ group $L_{\mathcal{C}^{\prime},G}$ is relatively ind-affine and formally
smooth with connected integral geometric fibers over $\operatorname{Spec}(A)$.
Observe that if $n\,=\,1$ and $\lambda\,=\,0$, and if $t$ is a formal
coordinate at the marked point $p_{1}$, then $\mathcal{LG}$ gets identified
with the loop group $L_{G}$ and $\mathcal{P}_{\bm{\tau}}$ is the group of
positive loops $L^{+}_{G}$.

####  B.1.2. The central extension

Faltings ([27], and also [8, Lemma 8.3 ], [40]) constructed a projective
representation of $\mathcal{LG}$ on
$\mathbb{H}_{\vec{\lambda}}\,=\,R\otimes\left(\bigotimes\mathcal{H}_{\lambda_{i}}(\mathfrak{g},\ell)\right)$
whose derivative coincides with the natural projective action of the Lie
algebra of $\mathcal{LG}$. This gives us a central extension

(B.1)
$1\,\longrightarrow\,\mathbb{G}_{m}\,\longrightarrow\,\widehat{\mathcal{LG}}\,\longrightarrow\,\mathcal{LG}\,\longrightarrow\,1.$

The extension $\widehat{\mathcal{LG}}$ splits over $\mathcal{P}_{\bm{\tau}}$
(see [60, Lemma 7.3.5]), and the central extension $\widehat{\mathcal{LG}}$ is
independent of the chosen representations $\vec{\lambda}$. Moreover the
extension (B.1) splits over $L_{\mathcal{C}^{\prime},G}$ ([59], [12, Lemma
6.5]).

###  B.2. The relative uniformization and parabolic theta functions

Let
$\widehat{\mathcal{P}}_{\bm{\tau}}\,:=\,\mathcal{P}_{\bm{\tau}}\times\mathbb{G}_{m}$.
The weight vectors $\vec{\lambda}$ give natural characters on
$\widehat{\mathcal{P}}_{\bm{\tau}}$ and the product of characters induces a
line bundle

$\mathscr{L}_{\vec{\lambda}}\,\longrightarrow\,{\mathcal{Q}}_{\bm{\tau}}\,:=\,\widehat{\mathcal{LG}}/\widehat{\mathcal{P}}_{\bm{\tau}}.$

Moreover, from the uniformization theorems [12, 8, 24, 31], it follows that
the quotient of ${\mathcal{Q}}_{\bm{\tau}}$ by $L_{\mathcal{C}^{\prime},G}$ is
isomorphic to the pullback $\mathcal{P}ar_{G}(\bm{\tau})_{S}$ of the stack
$\mathcal{P}ar_{G}(\bm{\tau})$ to $S$. Now since the extension in (B.1) splits
over $L_{\mathcal{C}^{\prime},G}$, the line bundle
$\mathscr{L}_{\vec{\lambda}}$ descends to a line bundle over the stack
$\mathcal{P}ar_{G}(\bm{\tau})$ which we will also denote by
$\mathscr{L}_{\vec{\lambda}}$. Observe that the line bundle
$\mathscr{L}_{\vec{\lambda}}$ is trivialized along the trivial section of
$\mathcal{P}ar_{G}(\bm{\tau})$ over $S$, and such data determine the line
bundle up to canonical isomorphism. We will refer to the line bundle
$\mathscr{L}_{\vec{\lambda}}$ as the Borel-Weil-Bott line bundle.

####  B.2.1. Parabolic determinant as the Borel-Weil-Bott line bundle

We now compare the parabolic determinant of cohomology of the universal bundle
with the line bundle $\mathscr{L}_{\vec{\lambda}}$.

Recall from Definition A.6 the notion of the parabolic determinant
$\operatorname{Det}_{par,\phi}(\bm{\tau})$ of cohomology associated to a
family of parabolic $G$ bundles on $\mathcal{C}\,\to\,S$ and a suitable
representation $\phi\,:\,G\,\to\,\mathsf{SL}(V)$. Now for the fixed
$n$-pointed curve $(C,\,\bm{p})$, it is known that the line bundles
$\mathscr{L}^{\otimes m_{\phi}}_{\vec{\lambda}}$ and
$\operatorname{Det}_{par,\phi}(\bm{\tau})$ on
$\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})$ are isomorphic, where $m_{\phi}$ is
the Dynkin index of the embedding $\phi$. Since these line bundles are
determined up to a normalizing factor, it follows that the corresponding
projective bundles are identified as

(B.2)
$\mathbb{P}{\pi_{e}}_{*}\left(\operatorname{Det}_{par,\phi}(\bm{\tau})\right)\,\cong\,\mathbb{P}{\pi_{e}}_{*}\left(\mathscr{L}_{\vec{\lambda}}^{\otimes
m_{\phi}}\right),$

where $\pi_{e}:\mathcal{P}ar_{G}(\bm{\tau})\,\to\,\mathcal{M}_{g,n}$ is the
natural projection.

####  B.2.2. Parabolic theta functions and conformal blocks

For any choice of formal parameters, the ind-scheme
${\mathcal{Q}}_{\bm{\tau}}$ can be identified with the product of affine flag
varieties $\prod_{i=1}^{n}L_{G}/\mathcal{P}_{\tau_{i}}$ and the line bundle
$\mathscr{L}_{\vec{\lambda}}$ pulls back to the corresponding line bundle on
$L_{G}/\mathcal{P}_{\tau_{i}}$ given by the character $\lambda_{i}$. Now by
Kumar [36] and Mathieu [42], we get that

(B.3)
$H^{0}(\mathcal{Q}_{\bm{\tau}},\,\,\mathscr{L}_{\vec{\lambda}})\,=\,\mathbb{H}_{\vec{\lambda}}^{*}.$

We end this discussion with the following theorem (see [12, Theorem 1.7] and
[39, Sec. 5.7]) which we will refer to as the universal identification of the
parabolic theta functions and the conformal blocks. In the case when $S$ is a
point, the result can be found in [8, 27, 37, 40].

######  Theorem B.1.

The push-forward of $\mathscr{L}_{\vec{\lambda}}$ along the map
$\pi_{e}\,:\,\mathcal{P}ar_{G}(\bm{\tau})\,\to\,\mathcal{M}_{g,n}$ can be
identified canonically with the bundle of coordinate free conformal blocks
$\mathbb{V}^{\dagger}_{\vec{\lambda}}(\mathfrak{g},\ell)$. Moreover,
$\mathscr{L}_{\vec{\lambda}}$ descends to a line bundle on
$M_{G,\bm{\tau}}^{par,rs}$, and
$({\pi_{e}}_{|M^{par,rs}_{G,\bm{\tau}}})_{*}\mathscr{L}_{\vec{\lambda}}$ is
isomorphic to $\mathbb{V}_{\vec{\lambda}}^{\dagger}(\mathfrak{g},\ell)$
provided the following conditions hold:

  * •

The genus of the orbifold curve determined by $\bm{\tau}$ is at least $2$, if
$G$ is not $\mathsf{SL}_{2}$.

  * •

The genus of the orbifold curve is at least $3$, if $G=\mathsf{SL}_{2}$.

######  Remark B.2.

The last conditions ensure that for any smooth pointed curve $(C,\,\bm{p})$,
the codimension of the moduli space $M_{G,\bm{\tau}}^{par,rs}(C,\bm{p})$ in
the moduli stack $\mathcal{P}ar_{G}(C,\bm{p},\bm{\tau})$ is at least two. We
refer the reader to [19, App. C].

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