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

A Physical Origin for Singular Support Conditions in Geometric Langlands Theory