To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. I also want to say that this is the type of question that I would ask at a coffee break during a conference about this subject. If my question is too disorganized for mathoverflow, I apologize, and there will be no hard feelings if this question is closed because it is too speculative.

Given a manifold $M$, one has a canonical symplectic structure on the cotangent bundle $T^{*}M$. Given a Poisson sub-algebra of functions $A$ on $T^{*}M$, one aim of quantization asks if there exists an algebra of differential operators $A^{'}$ on $M,$ (generally they may be differential operators on sections of line bundles over $M$), for which the principal symbol map $A^{'}\rightarrow A$ is a morphism, where commutator of differential operators corresponds to Poisson bracket of functions on $T^{*}M.$

This question, which has its roots in the passage from classical to quantum mechanics, was employed by Beilinson-Bernstein in their localization procedure for producing modules over the universal enveloping algebra of a semisimple complex Lie algebra $\mathfrak{g}$ from $D$-modules on the flag variety $G/B$ where $G$ integrates $\mathfrak{g}$ and $B<G$ is a Borel sub-group.

Beilinson and Drinfeld, in a stroke of genius, realized that for $G$ a connected, complex reductive algebraic group over $\mathbb{C},$ this process can be applied to $T^{*}{Bun}_{G}(X)$ where $X$ is a compact Riemann surface, and ${Bun}_{G}(X)$ is the moduli stack of principal $G$-bundles over $X.$ Here, the sub-algebra of functions they consider is the Hitchin integral system. Morally speaking, a cotangent vector $\xi$ to a principal $G$ bundle $E_{G}$ is a global section $\xi\in H^{0}(X, K\otimes E_{G}(\mathfrak{g})),$ and a $G$-invariant symmetric homogeneous polynomial on the Lie algebra $\mathfrak{g}$ (almost) produces an function on $T^{*}{Bun}_{G}(X)$ via evaluation: this quickly leads to a completely integrable system on $T^{*}{Bun}_{G}(X)$ called the Hitchin fibration.

What Beilinson and Drinfeld do is produce a commuting algebra of (twisted) differential operators on $Bun_{G}(X)$ whose principal symbols map to functions appearing in the Hitchin integrable system.

While certainly an attractive story on face, as a pedestrian differential geometer, the theory of stacks and local algebra through which this construction passes hides, for me, what these differential operators actually are.

This is where my question begins. Instead of the stack $Bun_{G}(X),$ consider instead the projective variety (analytic space) parameterizing stable $G$-bundles on $X.$ I'm being very loose here, so maybe I want to switch to $GL(n, \mathbb{C})$, and also add extra hypotheses to make the following question well formed.

Question: Is there an avatar of the Beilinson-Drinfeld commuting algebra of differential operators on the moduli space of stable $G$-bundles on $X?$ For $GL(1, \mathbb{C})$, I think this question has a nice answer, and is basically part of the passage from abelian class field theory to geometric class field theory as espoused by many important figures in the study of the geometric Langlands conjecture.

Moving to non-abelian groups like $GL(2,\mathbb{C}),$ I already have no precise idea what might be going on. In this case, line bundles over the moduli space of stable $G$-bundles are a very important object, and subsequent objects like generalized Theta functions play an important role, for example in the Verlinde formula. It's possible that hidden inside these ideas I should find the differential operators I am seeking, but this is the point of my question.

Refined question: Is there a commuting family of differential operators on the space of stable $G$-bundles which corresponds to the Beilinson-Drinfeld construction, which has a formulation in terms of generalized theta functions etc.

In an attempt to not be a total idiot, I understand that in the Beilinson-Drinfeld application, they're focusing on $G$-bundles which are very unstable, those corresponding to $G$-opers, and therefore the stack point of view is essential. In this vein, I'm not asking for an explanation of their work which can be understood in the language of stable bundles. I'm just asking if their construction produces something interesting, perhaps already studied before, when restricted to the space of stable bundles.

I apologize if this is a series of paragraphs, each compounding the next, resulting in a question that makes no sense. If this is not the case, I appreciate any responses, and thank you for reading to the end.


1 Answer 1


Answer is Yes for all. These commuting differential operators can be defined, on the moduli space of stable bundles and written in theta functions terms (but it might not be illuminating or suggestive). In GL(1) case it is just "free system" just $H_i = \partial_i^2$ on the Jacobian. I'll give refrences below.

Sometimes in Russia we say: "Всё гениальное просто", meaning "All genius things are simple", so let me try to comment how one probably should think on that in the spirit of that saying.

Step 0. Everybody knows that $Tr(M^k)$ or $det(M-x)$ are invariant with respect to GL(n) action on matrices by conjugation.

So can guess:

Step 1. Universal envelopping algebra of $U(gl(n))$ has a center generated by these "slightly quantized" $Tr(M^k)$ or $det(M-x)$ and respectively they give commuting operators on $GL(n)$, (which are toy-model for Beilinson-Drinfeld-Hitchin operators). That is general for any Lie group: Lie algebra $g$ are vector fields on group, $U(g)$ are differential operators, (and S(g) functions on $T^*G$ ).

And we are here:

Step 2: consider loop group/algebra $GL((z))$, universal envelopping algebra of $U(gl((z))$ has a center generated by these "slightly quantized" $Tr(M(z)^k)$ or $det(M(z)-x)$, (here "slightly" is some 20 years long story, which I took part [CT06], but let us concentrate on the spirit). Again center of universal enveloping is the same as bi-invartiant differential operators on the group GL((z)). And these are true "universal" Beilinson-Drinfeld-Hitchin operators.

Finally, to G-bundles (which actually not the essence of the story).

Step 3. Nowadays "standard" trick is to represent $Bun_G = G_{out}\backslash GL((z))/G_{in}$ where $G_{out}$ are $G$-valued functions holomorphic outside point some point $p$, and $G_{in}$ are holomophic in neighbourhood of $p$. (It is specific for curves that every bundle is trivial on curve minus point (since curve minus point is affine manifold) and so we have such representations).

So since one has huge set commuting and BI-variant (both left and right) operators on GL((z)), they factorizes to commuting operators on $Bun_G = G_{out}\backslash GL((z))/G_{in}$.

That is it - Beilinon-Drinfeld-Hitchin operators on $Bun(G)$.

Concerning the problem writting these operators "explicitly". The actual issue that there are not convenient coordinates on BunG, so not having "nice" coordinates, no hope for "nice" explicit formulas. However "nice" is subjective, and for example for SL(2) one may do the following:

Hecke-Tyurin parametrization of the Hitchin and KZB systems B. Enriquez, V. Rubtsov https://arxiv.org/abs/math/9911087

The very specific case is genus 2 and SL(2), so $BunG = CP^3$, and here are nice coordinates and nice formulas: Hitchin Systems at Low Genera Krzysztof Gawedzki, Pascal Tran-Ngoc-Bich https://arxiv.org/abs/hep-th/9803101

It is amazing that Hitchin came to his part completely from different side, and Beilinson and Drinfeld understood that picture above gives his differential operators, with the main insight lying further - Hitchin D-module will be Hecke eig-module, with the "eigen"-value (G-oper) given by the quantization of the Hitchin's spectral curve (geometric Langlands).

  • $\begingroup$ Thank you for this wonderful answer Alexander! The papers, in particular, address exactly the kind of thing I was thinking about. You are my hero for the day! $\endgroup$ Dec 4, 2018 at 14:44
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    $\begingroup$ You are welcome ! By the way , many years ago when we start thinking about that, we decided to consider rational curves with node/cusp/ other singularity - then there is very simple and explicit description of vector bundles and so we can describe Hitchin system: arxiv.org/abs/hep-th/0303069, I would say it much more simple rather then theta-functions... $\endgroup$ Dec 11, 2018 at 19:09
  • $\begingroup$ Thank you for this beautiful answer! $\endgroup$ Jan 28, 2019 at 18:02
  • $\begingroup$ Just some comments: 1) $CP^3$ is the coarse moduli of semi-stable SL2-bundles on a genus two curve, not the stack Bun_G. 2) Is it true that Hitchin came up with the differential operators when trying to define a (flat projective) connection on the bundle of non-abelian theta functions on the moduli space of curves? 3) Since the rest of your answer was so nice, can you please say a little more about the eigen G-oper given by quantization of Hitchin's spectral curve? i.e. what is the quantization of the spectral curve and how it gives the sought after Oper? Thanks! $\endgroup$ Jan 28, 2019 at 18:10
  • $\begingroup$ @supersymmetric thank you for your remarks. 1 Agree 2. Should be! It is natural to think so and At least Atiyah book 'geometry and physics of knots' says like that, but I cannot say for sure. Hitchin wrote two papers around that time and topic: one is about integrable system another Self duality equation on a Rieman surface - it might be the second contains more precise statements on projective connection, i do not remember. Self duality paper has been later generalized quite famously by Carlos Simpson to higher dimensions. 3) may be you ask a separate question and i will write an answer.. $\endgroup$ Jan 28, 2019 at 20:55

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