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Home > Progress of Theoretical and Experimental Physics (OUP/JPS) > Higgs and Coulomb branch descriptions of the volume of the vortex moduli space |

Ohta, Kazutoshi (Institute of Physics, Meiji Gakuin University, Yokohama, Kanagawa 244-8539, Japan) ; Sakai, Norisuke (Department of Physics, and Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan, and iTHEMS, RIKEN 2-1 Hirasawa, Wako, Saitama 351-0198, Japan)

16 April 2019

**Abstract: ** 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.

**Published in: ****PTEP 2019 (2019) 043B01**
**DOI: **10.1093/ptep/ptz016

**License: **CC-BY-3.0