A Physical Origin for Singular Support Conditions in Geometric Langlands Theory

TL;DR

This paper links the nilpotent singular support condition in geometric Langlands to N=4 supersymmetric gauge theory, revealing its natural origin and proposing a broader categorical framework involving vacua and symmetry breaking.

Contribution

It introduces a physical interpretation of the singular support condition in geometric Langlands via topological quantum field theory and explores its implications for gauge symmetry breaking and factorization.

Findings

Nilpotent singular support arises from vacuum restriction in gauge theory

Categories for larger strata relate to gauge symmetry breaking to Levi subgroups

Proposes a hidden factorization structure in geometric Langlands

Abstract

We explain how the nilpotent singular support condition introduced into the geometric Langlands conjecture by Arinkin and Gaitsgory arises naturally from the point of view of N = 4 supersymmetric gauge theory. We define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum, both in functorial field theory and in the language of factorization algebras. For B-twisted N = 4 gauge theory with gauge group G, the moduli space of vacua is equivalent to h*/W , and the nilpotent singular support condition arises by restricting to the vacuum 0 in h*/W. We then investigate the categories obtained by restricting to points in larger strata, and conjecture that these categories are equivalent to the geometric Langlands categories with gauge symmetry broken to a Levi subgroup, and furthermore that by assembling such for the groups GL_n for all positive integers n one finds a hidden factorization structure for the geometric Langlands theory.