# The Hitchin connection as a twisted D-module?

The famous Hitchin connection is a flat projective connection on the (projectivization of) the vector bundle of non-abelian theta functions, over the moduli space of curves $\mathcal{M}_g$. There are a number of approaches to this, but the original work of Hitchin (following earlier work of Welters on the projective connection on the bundle of abelian theta functions given by the heat equation) specifies an intrinsically projective connection, determined by a projective heat operator.

It is also well know, by work of Laszlo, that this connection is isomorphic to the (projectivization of the) TUY/WZW connection on the bundle of conformal blocks $V_{k,\mathfrak{g}}\rightarrow\mathcal{M}_g$. This connection is defined not so much as a projective connection, but as a twisted $D$-module, or rather as a vector bundle equipped with the action of an Atiyah algebra. This Atiyah algebra formally is the $\frac{c}{2}$-th power of the algebra of first order differential operators on the Hodge determinant line bundle $\lambda$ (here $c$ is the central charge of the conformal field theory).

Now, in general any flat projective connection on (the projectivization of) a vector bundle $E$ can be understood to arise in this way, where one takes the Atiyah algebra that is the $\frac{1}{\text{rk}(E)}$-th power of the determinant line bundle of $E$.

However, my question is:

Is it possible to construct the Hitchin connection ab initio as a twisted $D$-module?

In particular, it is natural to introduce the notion of a twisted heat operator, which hypothetically would do this job as follows:

First recall that the action of the Atiyah algebra $\mathcal{A}$ on a vector bundle $V\rightarrow \mathcal{M}_g$ is given by a commutative diagram (of sheaves of Lie algebras)

$$\require{AMScd}\begin{CD} 0@>>>\mathcal{O}_{\mathcal{M}_{g}} @>>>\mathcal{A}@>>>\mathcal{T}_{\mathcal{M}_g}@>>>0\\ \ @V.\text{id}_{V}VV @VVV @V\text{id}VV\\ 0@>>>\text{End}(V)@>>>\text{sb}^{-1}(\mathcal{T}_{\mathcal{M}_g})@>>>\mathcal{T}_{\mathcal{M}_g}@>>>0 \end{CD}$$ where $\text{sb}^{-1}(\mathcal{T}_{\mathcal{M}_g})$ is the preimage of $\mathcal{T}_{\mathcal{M}_g}\otimes \text{id}$ under the symbol map $$\text{sb}:\mathcal{D}^{\leq 1}(V)\rightarrow \mathcal{T}_{\mathcal{M}_g}\otimes \text{End}(V)$$ for first order differential operators on $V$. In this commutative diagram the bottom row is the Atiyah algebroid of the frame bundle of $V$; an ordinary connection would be a splitting of this short exact sequence.

Now let $\pi:\mathcal{SU_r}\rightarrow \mathcal{M}_g$ be the morphism from the moduli space of (semi-stable) bundles with trivial determinant on curves to $\mathcal{M}_g$, and let $\mathcal{L}\rightarrow \mathcal{SU_r}$ be the line bundle whose sections are the non-abelian theta functions. Let $$\mathcal{W}_{\mathcal{SU_r}/\mathcal{M}_g}= \mathcal{D}^{\leq 1}_{\mathcal{SU,r}}(\mathcal{L})+\mathcal{D}^{\leq 2}_{\mathcal{SU_r}/\mathcal{M}_g}(\mathcal{L})\subset \mathcal{D}^{\leq 2}_{\mathcal{SU_r}}(\mathcal{L})$$ be the sheaf of differential operators on $\mathcal{L}$ that are at most first order in the horizontal direction, and at most second order in the vertical direction. This fits into a short exact sequence (a double symbol) $$0\rightarrow \mathcal{D}^{\leq 1}_{\mathcal{SU_r}/\mathcal{M}_g}(\mathcal{L})\rightarrow \mathcal{W}_{\mathcal{SU_r}/\mathcal{M}_g}\overset{\rho_1+ \rho_2}{\longrightarrow} \pi^*\mathcal{T}_{M_g} \oplus \text{Sym}^2(\mathcal{T}_{\mathcal{SU_r}/\mathcal{M}_g})\rightarrow 0$$

A heat operator is now a morphism $\mathcal{T_{\mathcal{M}_g}}\rightarrow \pi_*(\mathcal{W}_{\mathcal{SU_r}/\mathcal{M}_g})$ which composes with $\rho_1$ to the identity. This gives rise to a connection on $\pi_*(\mathcal{L})$. Similarly a projective heat operator is a morphism $\mathcal{T}_{\mathcal{SU_r}}\rightarrow \pi_*(\mathcal{W}_{\mathcal{SU_r}/\mathcal{M}_g})/\mathcal{O}_{\mathcal{M}_g}$, which gives rise to a projective connection.

A twisted heat operator would now be a commutative diagram $$\require{AMScd}\begin{CD} 0@>>>\pi^*\mathcal{O}_{\mathcal{M}_{g}} @>>>\pi^*\mathcal{A}@>>>\pi^*\mathcal{T}_{\mathcal{M}_g}@>>>0\\ \ @VVV @VVV @V\text{id}+\dots VV\\ 0@>>> \mathcal{D}^{\leq 1}_{\mathcal{SU_r}/\mathcal{M}_g}(\mathcal{L})@>>> \mathcal{W}_{\mathcal{SU_r}/\mathcal{M}_g}@>\rho_1+ \rho_2>>\pi^*\mathcal{T}_{M_g} \oplus \text{Sym}^2(\mathcal{T}_{\mathcal{SU_r}/\mathcal{M}_g})@>>> 0 \end{CD}$$ and likewise this would give rise to a twisted $D$-module. My main aim is that I want to have control over $\mathcal{A}$ in advance, to avoid having to determine afterwards what the determinant of the bundle of non-abelian theta functions is. I am unaware however of a construction in the literature (perhaps implied) of such a twisted heat operator.