# Vector bundles, Higgs bundles and the Langlands program

This question is somewhat vaguely structured. But, I hope someone can make it more precise (or) it is indeed possible to answer it in the form that I am stating it.

Background : I recently chanced upon the following lecture note of Langlands : http://publications.ias.edu/rpl/paper/2578 . In this, he has various comments/proposals on geometrical approaches to aspects of the Langlands program.  I don't understand several things about it but the following question came to my mind.

Qn : If you are Beilinson-Drinfeld in the early 90s looking to formulate a version of Geometric Langlands. Why are moduli spaces of stable Higgs bundles (arising in Hitchin systems) on Riemann surfaces the right place to look at and not just moduli spaces of stable vector bundles on Riemann surfaces ?

(the connection to the Lecture note linked above is that Langlands discusses the theory of stable vector bundles, especially the work of Harder-Narasimhan, Atiyah-Bott etc and provides certain conjectures in the language of this theory. )

PS : There is actually a 'physics answer' that I can think of.  As is well known, Atiyah-Bott were studying two dimensional Yang-Mills (gauge) theories. Such theories do not have an electric magnetic duality relating the theory for $G$ and the theory for $G^\vee$. However, such dualities can occur in four dimensional gauge theory. The most robust such duality is for the maximally symmetric $\mathcal{N}=4$ Supersymmetric Yang Mills theory in four dimensions. And in the gauge theory approach of Kapustin-Witten to the Geometric Langlands Program, this electric magnetic duality is indeed the starting point for obtaining statements about geometric Langlands. When you compactify the four dimensional theory on a Riemann surface, one naturally obtains Hitchin system(s) from the physics (I am being telegraphic here, omitting details about twists, defects etc). Plenty of MO threads do a better job of this. But, the point is that if a 4d field theory (more accurately, a TQFT built from it) is the starting point, then the Hitchin system is what one gets naturally. But, I suspect there might be another answer since Beilinson-Drinfeld's original approach was not motivated by dualities in four dimensional field theories. Hence the question.

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I don't know enough about geom. Langlands to comment on the main question, but if one is interested in representations of $\pi_1$ into $GL_n$ rather than $U(n)$, then stable Higgs bundles rather than vector bundles are the right things to consider. – Donu Arapura Jul 14 '14 at 13:50

I think that understanding what can be a "motivation" for Beilinson-Drinfeld work, at least several things should be kept in mind.

1) V. Drinfeld made principal contributions for functional fields case of Geometric Langlands - he established it for GL(2), as well as, creating basic constructions which lead Lafforgue to establish GL(n) case - he introduced "schutakas", and a way to use them in order to get Langlands correspondence. ( Somewhat curious, that his contribution to this field dated back to 1979, and since that time he seems to moved his principal interests to quantum groups and "math. physics". And it seems that it is kind of astonishing that work on geometric Langlands over complex numbers unified both of his interestes - "math. phys" and Langlands.)

2) A. Beilinson (with J. Bernstein) introduced so-called "Beilison-Bernstein localization" in their way to prove Kazhdan-Lusztig conjectures on Verma modules and intersection cohomology. This construction plays an important role for "understanding-motivating" Beilinson-Drinfeld work. It defines a correspondence between D-modules on flag variety and certain representations of the semisimple Lie algebra. ( In Beilinson-Drinfeld story: similar D-modules apppear as "Hitchin eigensheaves", flag manifold is substituted by moduli space of vector bundles, semi-simple Lie algebra substituted by affine Lie algebra).

3) Hitchin's paper on integrable system appeared in 1987, and pay attention that it appeared in special issued of Duke journal dedicated to 50-th anniversary of Yu.I. Manin, who was the teacher of both V.Drinfeld and A.Beilinson. So it was of course known to them.

From the position of our nowdays knowledge it is easy to understand the relevance of Beilionson-Bernstein (BB) localization to Hitchin's paper: roughly speaking the center of universal enveloping of affine Lie algebra on the critical level which play crucial role in BB-localization gives exactly the quantum version of the Hitchin integrable system. However, I think in late 1980-ies, probably no one except Beilinson and Drinfeld were able to see through the clouds. (Since the center itself was not actually proved to exist in 1987, and many many other things were not yet known).

Concerning the more detailed version of the question.

Why are moduli spaces of stable Higgs bundles (arising in Hitchin systems) on Riemann surfaces the right place to look at and not just moduli spaces of stable vector bundles on Riemann surfaces ?

I am not sure I fully understand the point to make stress on the moduli space of Higgs bundles vs. moduli space of vector bundles. In my undestanding one should stress on the following things.

1) Moduli space of Higgs bundles is (modula details) COTANGENT BUNDLE to moduli space of vector bundles. Contangent bundles to manifold appears naturally when you speak about D-modules or "quantization" - it is "classical phase space" in quantization story or the space where characteristic manifold of D-modules lives.

2) What is surpsising that in functional field case developped by V.Drinfeld in late 70-ies, he needs to work with moduli space of "schtukas" - which are vector bundles plus additional structures related to Frobenius, while Beilinson-Drinfeld story is more simple in that respect - you do not need "schtukas" and you can actually work with vector bundles.

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@Chervov Thats a very useful summary of the motivations behind Beilinson-Drinfeld. I am not familiar with 'schtukas', but I understand that to make a statement with D-modules, the cotangent bundle $T^*Bun_G$ is the place to look. My question though is can some part/version of the Langlands philosophy/program (not necessarily the version of BD) still fit into the story about stable vector bundles ? Again, I understand this is vague. I am curious about this since Langlands seems to think (if I understand atleast the broad drift in his lecture note) that this could be possible. – Aswin Jul 14 '14 at 14:46
@aswin do we agree that Hitching d-module lives on modli space of bundles NOT cotangent? – Alexander Chervov Jul 14 '14 at 15:53
@Aswin to speak physically : BUN is configuration space of Hitchin system. Cotangent is phase space. These two are twin brotherss – Alexander Chervov Jul 14 '14 at 16:02
@Chervov Partially, since I find the sense in which the words quantum/classical limit used here to be confusing. But, let me actually set that aside. There is certainly a restriction of the space of solutions to Hitchin equations for say the pair (A,ϕ) that reduces to moduli space of vector bundles. This would correspond to just setting ϕ=0. Is there a version/limit of BD (say their identification of the Hecke operator in this setting) that knows only about $Bun_G$ and $Bun_{G^\vee}$ and not about $T^*Bun_G,T^*Bun_{G^\vee}$. Maybe there is one and this is what is called 'classical limit' ? – Aswin Jul 14 '14 at 21:13
@Chervov The previous comment was getting very long. Let me just add this : my PS in the question was to suggest a physical reason why one can expect $(Bun_G,Bun_{G^\vee})$ to be less of an interesting playground than $(T^∗Bun_G,T^∗Bun_{G^\vee})$. – Aswin Jul 14 '14 at 21:21