论文标题
分数化同源性和数学物理应用中的应用中的均等定位II：量规理论应用
Equivariant localization in factorization homology and applications in mathematical physics II: Gauge theory applications
论文作者
论文摘要
我们遵循[RAS1]的方法来说明分解空间，类别，函子和代数的理论。我们将这些结果应用于分解的几何结构$ \ mathbb {e} _n $代数，描述了在低维度中超对称仪表理论的混合全体形态曲折。我们用这种语言制定并证明了物理学文献的一些最新预测： 我们从这个角度回想起[BFN1]的库仑分支结构。我们证明了[cosg]的一个猜想，即库仑分支分解$ \ mathbb {e} _1 $ algebra $ \ mathcal {a}（g，n）$ cos to ragral差异操作员的因数代数$ \ mathcal $ \ mathcal {d}^d}^{ch ch}（y）$ y = y =我们以$ \ mathcal {d}^{ch}（n）$相对于$ \ hat {\ mathfrak {\ mathfrak {g}} $识别后者的$ \ mathcal {d}^{ch}（n）$的半无限共同体。这两个结果都要求$ y $接受泰特结构的假设，或等效地说$ \ mathcal {d}^{ch}^{ch}（n）$允许$ \ hat {\ mathfrak {g Mathfrak {g}} $ at Level $κ= - \ text {tate} $。 我们构建一个类似的分解$ \ mathbb {e} _2 $ algebra $ \ mathcal {f}（y）$，描述了四维$ \ MATHCAL {n} = 2 $ gauge理论的四维$ \ MATHCAL {n}的混合全态B扭曲的局部可观察结果。在此示例中，我们应用了前传代数[BU1]的模棱两可的理论：我们在$ \ Mathcal {f}（y）上确定了$ y = n/g $的泰特结构上的$ s^1 $ epivariant结构，并通过$ y = n/g $，并通过$ y^！\ natercal iS $ re^！\ res fraira（y）（y）（y）（y）（y）（y）（y）（y）（y）（y） $ y $的手性差异操作员。这给出了[BEEM4]结果的数学说明。最后，我们应用[bu1]的雪茄还原原理来解释这些结果与我们对上述[COSG]结果的描述之间的关系。
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 $κ=-\text{Tate}$. 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 $ι^!\mathcal{F}(Y)$ 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.
