A Physical Origin for Singular Support Conditions in Geometric Langlands Theory

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.

55 pages, 5 figures, more improvements to the exposition

References in corpus (8)

• Higher and derived stacks: a global overview
• SU(5)-invariant decomposition of ten-dimensional Yang-Mills supersymmetry
• Derived Algebraic Geometry and Deformation Quantization
• Braces and Poisson additivity
• Generators in formal deformations of categories
• Spectral action in Betti Geometric Langlands
• Derived smooth stacks and prequantum categories
• Sheaves of categories with local actions of Hochschild cochains