论文标题
3D $ \ MATHCAL {N} = 4 $扭曲指标和涡流模量空间的几何形状的分解
Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space
论文作者
论文摘要
我们研究了$ \ Mathcal {n} = 4 $ supersymmortric量规的扭曲索引，并在空间$ s^{2} $上进行三个维度，并具有角动量精致。我们证明了在存在通用通量和散发性的情况下，该指数将索引分解为$ t [su（n）] $理论的全体形态块。我们还研究了扭曲的指数，希尔伯特系列和涡流的模量空间之间的关系。特别是，我们表明，每个全体形状块与“ local”涡流的Moduli空间的$χ_{t} $属的生成功能相吻合。扭曲的索引本身与“全局”涡流的$χ_{t} $属的相应生成函数一致，这与Bullimore等人的建议一致。 al。我们将这种扭曲指数的几何解释推广到包括通量和Chern-Simons水平。对于$ t [su（n）] $理论，相关的模量空间分别是劳蒙空间的本地和全球版本，我们使用数学文献的结果明确地证明了所提出的协议。最后，我们在库仑分支希尔伯特系列和相应涡流模量空间的庞加莱多项式之间表现出了精确的关系。
We study the twisted indices of $\mathcal{N}=4$ supersymmetric gauge theories in three dimensions on spatial $S^{2}$ with an angular momentum refinement. We demonstrate factorisation of the index into holomorphic blocks for the $T[SU(N)]$ theory in the presence of generic fluxes and fugacities. We also investigate the relation between the twisted index, Hilbert series and the moduli space of vortices. In particular, we show that each holomorphic block coincides with a generating function for the $χ_{t}$ genera of the moduli spaces of "local" vortices. The twisted index itself coincides with a corresponding generating function for the $χ_{t}$ genera of moduli spaces of "global" vortices in agreement with a proposal of Bullimore et. al. We generalise this geometric interpretation of the twisted index to include fluxes and Chern-Simons levels. For the $T[SU(N)]$ theory, the relevant moduli spaces are the local and global versions of Laumon space respectively and we demonstrate the proposed agreements explicitly using results from the mathematical literature. Finally, we exhibit a precise relation between the Coulomb branch Hilbert series and the Poincaré polynomials of the corresponding vortex moduli spaces.
