Mirror Symmetry, Hitchin’s Equations, and Langlands Duality

Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold C. But understanding these statements is extremely difficult without picking a complex structure on C and using Hitchin’s equations. We sketch the essential statements both for the “unramified” case that C is a compact oriented two-manifold without boundary, and the “ramified” case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory. 1. The A-Model And The B-Model Let G be a compact Lie group and let GC be its complexification. And let C be a compact oriented two-manifold without boundary. We write Y(G,C) (or simply Y(G) or Y if the context is clear) for the moduli space of flat GC bundles E → C, modulo gauge transformations. Equivalently, Y(G,C) parametrizes homomorphisms of the fundamental group of C to GC. Y(G,C) is in a natural way a complex symplectic manifold, that is a complex manifold with a nondegenerate holomorphic two-form. The complex structure comes simply from the complex structure of GC, and the symplectic form, which we call Ω, comes from the intersection pairing on H(C, ad(E)), where ad(E) is the adjoint bundle associated to a flat bundle E. Since Y(G,C) is a complex symplectic manifold, in particular it follows that its canonical line bundle is naturally trivial. Geometric Langlands duality is concerned with certain topological field theories associated with Y(G,C). The most basic of these are the B-model that is defined by viewing Y(G,C) as a complex manifold with trivial canonical bundle, and the A-model that is defined by viewing it as a real symplectic manifold with symplectic form ω = ImΩ. Date: February, 2007. Supported in part by NSF Grant Phy-0503584. 1Actually, it is best to define Y(G, C) as a geometric invariant theory quotient that parametrizes stable homomorphisms plus equivalence classes of semi-stable ones. This refinement will not concern us here. See section 6.1. 2The definition of this intersection pairing depends on the choice of an invariant quadratic form on the Lie algebra of G. It can be shown using Hitchin’s C action on the moduli space of Higgs bundles that the A-model that we define shortly is independent of this choice, up to a natural isomorphism. The geometric Langlands duality that one ultimately defines likewise does not depend on this choice. 3The usual definition of Ω is such that ImΩ is cohomologically trivial, while ReΩ is not. The fact that ω = ImΩ is cohomologically trivial is a partial explanation of the fact, mentioned in the last footnote, that the A-model of Y is invariant under scaling of ω.