Gauge Theory and the Analytic Form of the Geometric Langlands Program

Abstract

We present a gauge-theoretic interpretation of the "analytic" version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction -such as electric-magnetic duality between Wilson and 't Hooft line operators in four-dimensional gauge theory -are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.

Automorphic functions on moduli spaces of bundles on curves over local fields: a survey

“…But among these we should only choose those that extend to C/τ (and then the multiplicity of eigenvalue may be related to the number of such extensions). This agrees with the picture [19], 6.2 coming from 4-dimensional supersymmetric gauge theory. 13 More precisely, recall that by a result of Beilinson and Drinfeld [3], opers for adjoint groups have no nontrivial automorphisms.…”

“…In this section we would like to describe the conjectural picture of the analytic Langlands correspondence in the case F = R. This picture has been developed by P. Etingof, E. Frenkel, D. Gaiotto, D. Kazhdan and E. Witten, and is discussed in [19], Section 6.…”

“…Let ζ be a local system on the non-orientable surface C/τ with structure group L G. Let us say that ζ is an L-system if it attaches to every orientation-reversing path in C/τ a conjugacy class in L G that maps to the nontrivial element in Z/2. The following conjecture is formulated in [19], Section 6 (in the case of the compact inner class). Namely, in this case the spectral local systems are ζ which are isomorphic to ζ τ and such that ζ is an oper (hence also an anti-oper), so ζ is a real oper "with real coefficients".…”

“…If C(R) = ∅, the story gets more complicated, and we will not discuss the details here. Let us just indicate that, as explained in [19], Section 6, to define the appropriate moduli space and the spectral problem on it, we need to fix a real form G i of G in the inner class s for each component (oval) C i of C(R), and the eigenvalues of Hecke operators are conjecturally parametrized by a certain kind of "real" opers corresponding to this data, i.e., opers with real coefficients satisfying appropriate reality conditions on the monodromy of the corresponding G ∨ -connection. Furthermore, in the tamely ramified case, when we also have a collection of marked points D on C defined over R, to define the most general version of our spectral problem, we need to fix a unitary representation π i of the real group G i for every marked point c ∈ D on C i and a unitary representation of the complex group G C for every pair of complex conjugate marked points c, c ∈ D not belonging to C(R).…”

“…Relation of the archimedian case to geometric Langlands correspondence and conformal field theory. In the case when the field F is archimedian our program is related to the quantum Gauge Theory (see [19]).…”