Higgs and Coulomb branch descriptions of the volume of the vortex moduli space
Higgs and Coulomb branch descriptions of the volume of the vortex moduli space
复制标题
涡旋模空间体积的希格斯和库仑分支描述
DOI：
10.1093/ptep/ptz016
复制
发表时间：
2019
影响因子：
3.5
通讯作者：
Kazutoshi Ohta and Norisuke Sakai
中科院分区：
文献类型：
--
作者：
Kazutoshi Ohta and Norisuke Sakai
将任意亏格的闭Riemann曲面上的BPS涡系嵌入到二维超对称Yang-Mills理论中。我们打开背景规范场，以在弯曲空间上保持一半的刚性超对称（拓扑A扭曲）。我们考虑两个互补的描述：希格斯和库仑分支。通过在Higgs分支中的局域化，路径积分降为零模积分。玻色子零模上的积分直接给出模空间的体积形式上的积分，而费米子零模则通过适当的算子插入来补偿。在库仑分支描述中，同样的操作插入，路径积分减少到一个有限维的剩余积分。算子插入自动确定积分轮廓的选择，从而得到Jeffrey-Kirwan留数公式。这一结果保证了BPS涡方程解的存在性，并解释了BPS涡的Bradlow界。我们还讨论了生成函数的体积的涡模空间，并显示减少的模空间从半本地到本地的旋涡。
BPS vortex systems on closed Riemann surfaces with arbitrary genus are embedded into 2D supersymmetric Yang–Mills theory with matters. We turn on background-gauge fields to keep half of the rigid supersymmetry (topological A-twist) on the curved space. We consider two complementary descriptions: Higgs and Coulomb branches. The path integral reduces to the zero mode integral by the localization in the Higgs branch. The integral over the bosonic zero modes directly gives an integral over the volume form of the moduli space, whereas the fermionic zero modes are compensated by an appropriate operator insertion. In the Coulomb branch description with the same operator insertion, the path integral reduces to a finite-dimensional residue integral. The operator insertion automatically determines a choice of integral contours, leading to the Jeffrey–Kirwan residue formula. This result ensures the existence of the solution to the BPS vortex equation and explains the Bradlow bounds of the BPS vortex. We also discuss a generating function of the volume of the vortex moduli space and show a reduction of the moduli space from semi-local to local vortices.