Non-perturbative Gauge Groups and Local Mirror Symmetry

TL;DR

This paper investigates how non-perturbative gauge symmetries of type A_{n-1} emerge in local mirror symmetry of resolved C^2/Z_n singularities, revealing the quantum geometric structure and roots of the gauge algebra.

Contribution

It provides an explicit solution of the GKZ system showing the emergence of all positive roots of A_{n-1} in quantum geometry, extending classical geometric understanding.

Findings

All positive roots of A_{n-1} appear in quantum geometry.

The point where discriminant branches coincide signals non-perturbative gauge symmetry.

The analysis connects local mirror symmetry with McKay correspondence.

Abstract

We analyze D-brane states and their central charges on the resolution of C^2/Z_n by using local mirror symmetry. There is a point in the moduli space where all n(n-1)/2 branches of the principal component of the discriminant locus coincide. We argue that this is the point where compactifications of Type IIA theory on a K3 manifold containing such a local geometry acquire a non-perturbative gauge symmetry of the type A_{n-1}. This analysis, which involves an explicit solution of the GKZ system of the local geometry, explains how the quantum geometry exhibits all positive roots of A_{n-1} and not just the simple roots that manifest themselves as the exceptional curves of the classical geometry. We also make some remarks related to McKay correspondence.