Geometry of Langlands Duality

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

This research project aims to relate and unify recent developments in mathematics and theoretical particle physics. The mathematical origin of these developments is the so-called Langlands program, which is the modern view on the classical mathematical discipline of number theory. The Langlands program functions today as a bridge between central disciplines of mathematics, such as representation theory, algebraic geometry, and number theory, and central disciplines in physics, such as gauge theory and string theory. This project will use this connection in both directions by applying ideas from physics to mathematics, through a new use of locality, and applying mathematical constructions to physics, by constructing the "theory X," the conjectural six-dimensional conformal field theory that should unify all important quantum field theories in lower dimensions. The central theme of this project is to extend the geometric Langlands program from objects of dimension one (in algebraic geometry) to arbitrary dimensions. The goal of the project is to develop conceptual breakthroughs into a full mathematical machinery. These breakthroughs include an extension of the basic case of the geometric Langlands program (the so called geometric Class Field Theory) to arbitrary dimension. This development is expected to be influential in mathematics, in particular through the introduction of the relative motivic cohomology. The general geometric Langlands program will also be investigated, based on recent results in constructing and generalizing the loop Grassmannian from the point of view of statistical mechanics.