A Physical Origin for Singular Support Conditions in Geometric Langlands Theory

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.Comment: 55 pages, 5 figures, more improvements to the expositio

Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry

We develop techniques for describing the derived moduli spaces of solutions to the equations of motion in twists of supersymmetric gauge theories as derived algebraic stacks. We introduce a holomorphic twist of N=4 supersymmetric gauge theory and compute the derived moduli space. We then compute the moduli spaces for the Kapustin-Witten topological twists as its further twists. The resulting spaces for the A- and B-twist are closely related to the de Rham stack of the moduli space of algebraic bundles and the de Rham moduli space of flat bundles, respectively. In particular, we find the unexpected result that the moduli spaces following a topological twist need not be entirely topological, but can continue to capture subtle algebraic structures of interest for the geometric Langlands program.Comment: 55 pages; minor correction

Asymptotic Freedom in the BV formalism

We define the beta-function of a perturbative quantum field theory in the mathematical framework introduced by Costello -- combining perturbative renormalization and the BV formalism -- as the cohomology class of a certain element in the obstruction-deformation complex. We show that the one-loop beta-function is a well-defined element of the local deformation complex for translation-invariant and classically scale-invariant theories, and furthermore that it is locally constant as a function on the space of classical interactions and computable as a rescaling anomaly, or as the logarithmic one-loop counterterm. We compute the one-loop beta-function in first-order Yang--Mills theory, recovering the famous asymptotic freedom for Yang--Mills in a mathematical context.Comment: Minor edits mad