Higgs and Coulomb branch descriptions of the volume of the vortex moduli space

Abstract

BPS vortex systems on closed Riemann surfaces with arbitrary genus are embedded into two-dimensional supersymmetric Yang-Mills theory with matters. We turn on background R-gauge fields to keep half of 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.

Vortex counting and the quantum Hall effect

“…In [5,6,33], the volume of the vortex moduli space was computed using methods similar to those that we use here for nonAbelian vortices. In the Abelian case, we use a more direct method and show that our results in this case specialise to reproduce those of [5,31,32].…”

“…This can be thought of as the space of holomorphic line bundles with normalised holomorphic section, which facilitates the following construction of the vortex moduli space as a complex manifold. 6 Write P k (Σ) for the moduli space of holomorphic line bundles on Σ of degree k. This is a torus of complex dimension g. Let U → P k (Σ) × Σ be the universal degree k line bundle. This is well-defined only after choosing a point x ∈ Σ and asking that U| P k (Σ)×{x} be trivial.…”

“…We do not give a complete account of this technique here, instead giving a fairly high-level exposition of the computation in the context relevant to us. We refer the reader to the original works [1,2,80] and to [5,6], which give a clear exposition of the technique in the context of computing the volumes of vortex moduli spaces. Our computation of χ(L λ ) generalises the results of these latter works.…”

“…The volumes of the vortex moduli spaces describe the statistical mechanics of vortex gases [31] and so represent interesting quantities. They have been computed for standard Abelian and nonAbelian vortices in [5,6,32,33,87], although there remains some confusion in the case of nonAbelian vortices on surfaces of genus greater than zero, and for systems of vortices and antivortices in [88].…”

“…The vortex moduli space is a symplectic quotient (of an infinite-dimensional space by an infinite-dimensional group), so this is available to us (at least formally). This technique has been used in [5,6] to compute the volumes of vortex moduli spaces.…”