Vortex counting from field theory

TL;DR

The paper addresses counting vortices in $2d$ $\mathcal{N}=(2,2)$ $U(N)$ gauge theories by deriving vortex partition functions through a field-theoretical moduli-matrix approach combined with localization. It develops fixed-point data and tangent-space characters directly from $H_0(z)$, and extends the analysis beyond the fundamental sector to include anti-fundamental and adjoint matter, highlighting the crucial role of fermionic zero modes. The authors also formulate the orbifold vortex partition function on $\mathbb{C}/\mathbb{Z}_n$ by imposing orbifold projections on the moduli data and torus action. These results are shown to be consistent with the Kähler-quotient method, and the framework is poised for broad applications, including semi-local vortices, other gauge groups, quiver theories, and connections to 4d/2d correspondences and AGT-like structures.

Abstract

The vortex partition function in 2d N = (2,2) U(N) gauge theory is derived from the field theoretical point of view by using the moduli matrix approach. The character for the tangent space at each moduli space fixed point is written in terms of the moduli matrix, and then the vortex partition function is obtained by applying the localization formula. We find that dealing with the fermionic zero modes is crucial to obtain the vortex partition function with the anti-fundamental and adjoint matters in addition to the fundamental chiral multiplets. The orbifold vortex partition function is also investigated from the field theoretical point of view.