Quantization of Hitchin’s integrable system and the geometric Langlands conjecture

Quantization of Hitchin’s Integrable System and Hecke Eigensheaves

To Robert MacPherson on the occasion of his 60th birthday GEOMETRIC LANGLANDS CORRESPONDENCE FOR D-MODULES IN PRIME CHARACTERISTIC: THE GL(n) CASE

Let X be a smooth projective algebraic curve of genus > 1 over and algebraically closed field k of characteristic p > 0. Denote by Bunn (resp. Locn) the moduli stack of vector bundles of rank n on X (resp. the moduli stack of vector bundles of rank n endowed with a connection). Let also DBunn denote the sheaf of crystalline differential operators on Bunn (cf. e.g. [3]). In this paper we constru...

To Robert MacPherson on the occasion of his 60ht birthday GEOMETRIC LANGLANDS CORRESPONDENCE FOR D-MODULES IN PRIME CHARACTERISTIC: THE GL(n) CASE

Let X be a smooth projective algebraic curve of genus > 1 over and algebraically closed field k of characteristic p > 0. Denote by Bunn (resp. Locn) the moduli stack of vector bundles of rank n on X (resp. the moduli stack of vector bundles of rank n endowed with a connection). Let also DBunn denote the sheaf of crystalline differential operators on Bunn (cf. e.g. [3]). In this paper we constru...

Selfduality and Chern–Simons Theory

We propose a relation between the operator of S-duality (of N = 4 super Yang–Mills theory in 3+1D) and a topological theory in one dimension lower. We construct the topological theory by compactifying N = 4 super Yang–Mills on S with an S-duality and R-symmetry twist. The S-duality twist requires a selfdual coupling constant. We argue that for a sufficiently low rank of the gauge group the thre...

Opers versus Nonabelian Hodge

For a complex simple simply connected Lie group G, and a compact Riemann surface C, we consider two sorts of families of flat G-connections over C. Each family is determined by a point u of the base of Hitchin’s integrable system for (G, C). One family ∇h̄,u consists of G-opers, and depends on h̄ ∈ C×. The other family ∇R,ζ,u is built from solutions of Hitchin’s equations, and depends on ζ ∈ C×, ...