We study N=1 theories on Hermitian manifolds of the form M^4=S^1xM^3 with M^3 a U(1) fibration over S^2, and their 3d N=2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D^2xT^2 and D^2xS^1. We prove that when the 4d and 3d anomalies are cancelled the matrix integrands in the Coulomb branch partition functions can be factorised in terms of 1-loop factors on D^2xT^2 and D^2xS^1 respectively. By evaluating the Coulomb branch matrix integrals we show that the 4d and 3d partition functions can be expressed as sums of products of 4d and 3d holomorphic blocks.
Factorisation of N = 2 theories on the squashed 3-sphere
最近已证明，N=2理论的配分函数在压缩3-球面上局部化为矩阵积分。通过显式地评估矩阵积分，我们表明，阿贝尔配分函数可以表示为两个块的产品之和。我们用涡旋理论的配分函数来确定第一块，其中等变参数hbar=2 Pi i b^2，定义在R^2 x S_1上，对应于椭球的b->0退化。第二块给出了在对偶R^2 x S_1上对应于1/b ->0退化的具有等变参数hbar^L=2 Pi i/b^2的涡旋理论的配分函数。椭球分区似乎提供了涡旋理论的hbar -> hbar^L模不变非微扰完备性。
We continue the exploration of multipoint scalar comb channel blocks for conformal field theories in 3D and 4D. The central goal here is to construct novel comb channel cross ratios that are well adapted to perform projections onto all intermediate primary fields. More concretely, our new set of cross ratios includes three for each intermediate mixed symmetry tensor exchange. These variables are designed such that the associated power series expansion coincides with the sum over descendants. The leading term of this expansion is argued to factorise into a product of lower point blocks. We esta…
We give an explicit realization of the 4d local operator / 2d conformal block correspondence of Costello and Paquette in the case of gauge theories. This is accomplished by lifting the 4d local operators to non-local operators in twistor space using a holomorphic generalization of the Wilson line. This procedure automatically constructs the 2d conformal blocks corresponding to the local operator. We interpret this lifting as effectively integrating out the 2d degrees of freedom living on the defect. We present some 2d chiral CFT representations of the defect algebra whose correlators reproduce…
We propose a novel approach of uncovering Stokes phenomenon exhibited by the holomorphic blocks of $\mathbb{CP}^1$ model by considering it as a specific decoupling limit of SQED$_2$ model. This approach involves using a $\mathbb{Z}_3$ symmetry that leaves the supersymmetric parameter space of SQED$_2$ model invariant to transform a pair of SQED$_2$ holomorphic blocks to get two new pairs of blocks. The original pair obtained by solving the line operator identities of the SQED$_2$ model and the two new transformed pairs turn out to be related by Stokes-like matrices. These three pairs of holomo…
Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the $\{3nj\}$-symbols of the recoupling of the $\SU(2)$ representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's gener…
Connecting Mirror Symmetry in 3d and 2d via Localization
我们明确地应用定域化结果来研究具有四个超荷的阿贝尔规范理论的三维和二维镜像对称性之间的内插。首先用椭球S_b^3配分函数验证了一对一般的三维N=2阿贝尔Chern-Simons规范规范理论之间的镜像对称性。这些表达式很容易分解成全纯块和它们的反全纯副本，因此我们也可以通过融合过程获得S^1 x S^2上的配分函数。然后我们证明了三维阿贝尔规范理论的S^1 × S^2配分函数可以在维数上约化为相应的二维镜像对的N=（2，2）GLSM和Landau-Ginzburg模型的S^2配分函数，正如先前在\cite{Aganagic:2001uw}中所预期的那样。我们还讨论了非阿贝尔规范理论的类似插值，并计算了一个简单极限的K理论涡旋配分函数，以验证全纯块的预言。
Conformal blocks and braiding matrices for RCFTs I: Virasoro minimal models
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Centre for Operator Algebras, Geometry, Matter and Spacetime（算子代数、几何、物质与时空中心）；Asia Pacific Center for Theoretical Physics (APCTP)（亚太理论物理中心）
5D partition functions, q-Virasoro systems and integrable spin-chains
我们分析了S5和S4 x S1上的N = 1理论，展示了它们的配分函数如何可以用一组基本的5d全纯块来表示。我们证明，当5d质量参数解析地继续到合适的值时，S5和S4 x S1配分函数退化到S3和S2 x S1的配分函数。我们解释了这一机制，通过最近提出的对应关系5d配分函数和潜在的q-Virasoro对称性的电子。从q-Virasoro 3-点函数，我们公理化地导出了一组相关的反射系数，并表明它们可以在量子对称空间的Harish-Chandra c-函数的几何解释。然后，我们将这些特定的c-函数连接到Jost函数中出现的类型，Jost函数编码可积自旋链中散射的渐近性，将XYZ模型的不同极限转换为XXZ型。
Refined Painlevé/gauge theory correspondence and quantum tau functions
International School for Advanced Studies (SISSA)（高级研究国际学院）；National Institute for Nuclear Physics (INFN), Section of Trieste（意大利国家核物理研究所的里雅斯特分部）；Institute for Geometry and Physics (IGAP)（几何与物理研究所）
