Five-Dimensional Gauge Theories and Local Mirror Symmetry

TL;DR

This paper explores 5D gauge theories via local mirror symmetry, revealing exact prepotentials related to string compactifications on local F2 surfaces, connecting 4D Seiberg-Witten and 5D theories.

Contribution

It demonstrates the exact computation of 5D gauge theory prepotentials using local mirror symmetry and relates these to string theory on local F2 geometries, including matter effects.

Findings

Prepotential matches type II string theory on local F2.

Reproduces Seiberg-Witten results in 4D limit.

Describes theories with matter via blown-up F2 geometries.

Abstract

We study the dynamics of 5-dimensional gauge theory on $M_4\times S^1$ by compactifying type II/M theory on degenerate Calabi-Yau manifolds. We use the local mirror symmetry and shall show that the prepotential of the 5-dimensional SU(2) gauge theory without matter is given exactly by that of the type II string theory compactified on the local ${\bf F}_2$, i.e. Hirzebruch surface ${\bf F}_2$ lying inside a non-compact Calabi-Yau manifold. It is shown that our result reproduces the Seiberg-Witten theory at the 4-dimensional limit $R\to 0$ ($R$ denotes the radius of $S^1$) and also the result of the uncompactified 5-dimensional theory at $R\to \infty$. We also discuss SU(2) gauge theory with $1\le N_f\le 4$ matter in vector representations and show that they are described by the geometry of the local ${\bf F}_2$ blown up at $N_f$ points.