Equivariant localization in factorization homology and applications in mathematical physics I: Foundations

Abstract

We develop a theory of equivariant factorization algebras on varieties with an action of a connected algebraic group $G$, extending the definitions of Francis-Gaitsgory [FG] and Beilinson-Drinfeld [BD1] to the equivariant setting. We define an equivariant analogue of factorization homology, valued in modules over ${\text{H}}_G^{\bullet }(\text{pt})$, and in the case $G=({\mathbb{C}}^{\times }{)}^n$ we prove an equivariant localization theorem for factorization homology, analogous to the classical localization theorem [AtB]. We establish a relationship between ${\mathbb{C}}^{\times }$ equivariant factorization algebras and filtered quantizations of their restrictions to the fixed point subvariety. These results provide a model for predictions from the physics literature about the $\Omega$-background construction introduced in [Nek1], interpreting factorization ${\mathbb{E}}_n$ algebras as observables in mixed holomorphic-topological quantum field theories. In the companion paper [Bu2], we develop tools to give geometric constructions of factorization ${\mathbb{E}}_n$ algebras, and apply them to define those corresponding to holomorphic-topological twists of supersymmetric gauge theories in low dimensions. Further, we apply our above results in these examples to give an account of the predictions of [CosG] as well as [Beem4], and explain the relation between these constructions from this perspective.

95 pages, 5 figures, Part I of a series

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• Equivariant localization in factorization homology and applications in mathematical physics II: Gauge theory applications