Langlands Reciprocity for Algebraic Surfaces

TL;DR

This paper explores extending the geometric Langlands conjecture from algebraic curves to surfaces, introducing Hecke operators related to quantum toroidal algebras, and providing new geometric constructions of affine quantum groups.

Contribution

It introduces Hecke-type operators on algebraic surfaces and links their algebra to quantum toroidal algebras, extending Langlands reciprocity to higher dimensions.

Findings

Hecke operators form a homomorphic image of quantum toroidal algebra.

Provides a geometric construction of affine quantum groups of types A, D, E.

Extends geometric Langlands conjecture to algebraic surfaces.

Abstract

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra generated by the Hecke operators turns out to be a homomorphic image of the {\it quantum toroidal algebra}. The latter is a quantization, in the spirit of Drinfeld-Jimbo, of the universal enveloping algebra of the universal central extension of a "double-loop" Lie algebra. This yields, in particular, a new geometric construction of affine quantum groups of types A, D E in terms of Hecke operators for an elliptic surface.