Seiberg-Witten Geometries Revisited

TL;DR

This paper presents a unified geometric solution for 4d N=2 gauge theories with gauge group G=A,D,E, using non-compact Calabi-Yau geometries constructed from known polynomials, supported by multiple lines of evidence.

Contribution

It introduces a novel geometric framework for solving 4d N=2 gauge theories with specific beta function conditions, connecting gauge theory and Calabi-Yau geometries.

Findings

Explicit construction of Calabi-Yau geometries from known polynomials

Validation through analysis of 6d N=(2,0) theory compactifications

Unified solution applicable to multiple gauge groups G=A,D,E

Abstract

We provide a uniform solution to 4d N=2 gauge theory with a single gauge group G=A,D,E when the one-loop contribution to the beta function from any irreducible component R of the hypermultiplets is less than or equal to half of that of the adjoint representation. The solution is given by a non-compact Calabi-Yau geometry, whose defining equation is built from explicitly known polynomials W_G and X_R, associated respectively to the gauge group G and each irreducible component R. We provide many pieces of supporting evidence, for example by analyzing the system from the point of view of the 6d N=(2,0) theory compactified on a sphere.