Five-Dimensional Gauge Theories and Local Mirror Symmetry

TL;DR

This work shows that five-dimensional ${SU}(2)$ gauge theories on ${M_4\times S^1}$ can be exactly captured by compactifications of Type II/M-theory on local Calabi–Yau geometries, with the prepotential matching the string theoretic prepotential on the local ${\bf F}_2$ surface. Local mirror symmetry provides a precise vehicle to compute the prepotential via GKZ/Picard–Fuchs analyses, reproducing the Seiberg–Witten results in the $R\to0$ limit and the intrinsic 5D behavior in the $R\to\infty$ limit, including the infinite bare coupling characteristic of the 5D fixed point. The construction extends to $N_f$ flavors up to 4 by blowing up ${\bf F}_2$ at $N_f$ points, yielding mass parameters $M_i$ and corresponding moduli that reproduce both 4D SW beta-functions and 5D Kaluza–Klein enhanced structure, suggesting an M-theoretic lift of certain 4D quantum field theories to higher dimensions. The analysis highlights a unique role for the local ${\bf F}_2$ geometry in encoding 5D physics and points to an avenue toward understanding higher-dimensional lifts and possible $E_n$ symmetries for $N_f\ge5$.

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.