The Seiberg-Witten Kahler Potential as a Two-Sphere Partition Function

TL;DR

This work shows that the Seiberg–Witten Kahler potential for 4D ${\cal N}=2$ $SU(2)$ SYM can be obtained from the two-sphere partition function of a GLSM describing the non-compact Calabi–Yau $\mathcal{O}(-2,-2)\rightarrow \mathbb{P}^1\times\mathbb{P}^1$, by taking a field-theory limit of the Kahler parameters. A small positive R-charge $\mathfrak{q}$ is introduced to regulate the non-compact direction, and the leading behavior of $Z_{S^2}$ as $\epsilon\to 0$ and $\mathfrak{q}\to 0^+$ yields $-\pi \partial\bar{\partial} K_{SW}(u,\Lambda)$ up to the prescribed identifications $q_b=\epsilon^4\Lambda^4$ and $q_f=\tfrac{1}{4}-\epsilon^2 u$. The method leverages the known relation between $S^2$ partition functions and Kahler potentials for Calabi–Yau geometries, and interprets the result in terms of the mirror coordinate $u$ on the Seiberg–Witten curve. The authors outline a generalization to other theories engineered by toric Calabi–Yau threefolds, providing a concrete program for obtaining gauge-theory Kahler potentials from $S^2$ partition functions, while noting subtleties in field-theory limits and R-charge choices for more complex cases.

Abstract

Recently it has been shown that the two-sphere partition function of a gauged linear sigma model of a Calabi-Yau manifold yields the exact quantum Kahler potential of the Kahler moduli space of that manifold. Since four-dimensional N=2 gauge theories can be engineered by non-compact Calabi-Yau threefolds, this implies that it is possible to obtain exact gauge theory Kahler potentials from two-sphere partition functions. In this paper, we demonstrate that the Seiberg-Witten Kahler potential can indeed be obtained as a two-sphere partition function. To be precise, we extract the quantum Kahler metric of 4D N=2 SU(2) Super-Yang-Mills theory by taking the field theory limit of the Kahler parameters of the O(-2,-2) bundle over P1 x P1. We expect this method of computing the Kahler potential to generalize to other four-dimensional N=2 gauge theories that can be geometrically engineered by toric Calabi-Yau threefolds.