We give an account of the theory of factorization spaces, categories, functors, and algebras, following the approach of [Ras1]. We apply these results to give geometric constructions of factorization $\mathbb{ E}_n$ algebras describing mixed holomorphic-topological twists of supersymmetric gauge theories in low dimensions. We formulate and prove several recent predictions from the physics literature in this language: We recall the Coulomb branch construction of [BFN1] from this perspective. We prove a conjecture from [CosG] that the Coulomb branch factorization $\mathbb{ E}_1$ algebra $\mathcal{ A}(G,N)$ acts on the factorization algebra of chiral differential operators $\mathcal{ D}^{ ch}(Y)$ on the quotient stack $Y=N/G$. We identify the latter with the semi-infinite cohomology of $\mathcal{ D}^{ ch}(N)$ with respect to $\hat{ \mathfrak{ g}}$, following the results of [Ras3]. Both these results require the hypothesis that $Y$ admits a Tate structure, or equivalently that $\mathcal{ D}^{ ch}(N)$ admits an action of $\hat{ \mathfrak{ g}}$ at level $\kappa=-\text{ Tate}$. We construct an analogous factorization $\mathbb{ E}_2$ algebra $\mathcal{ F}(Y)$ describing the local observables of the mixed holomorphic-B twist of four dimensional $\mathcal{ N}=2$ gauge theory. We apply the theory of equivariant factorization algebras of the prequel [Bu1] in this example: we identify $S^1$ equivariant structures on $\mathcal{ F}(Y)$ with Tate structures on $Y=N/G$, and prove that the corresponding filtered quantization of $\iota^!\mathcal{ F}(Y)$ is given by the two-periodic Rees algebra of chiral differential operators on $Y$. This gives a mathematical account of the results of [Beem4]. Finally, we apply the equivariant cigar reduction principle of [Bu1] to explain the relationship between these results and our account of the results of [CosG] described above.
Categories: math.RT hep-th math-ph math.AG math.AT math.MP math.QA