F-Theory, T-Duality on K3 Surfaces and N=2 Supersymmetric Gauge Theories in Four Dimensions

TL;DR

The paper develops a generalized T-duality on K3 surfaces via a Fourier–Mukai (Mukai) transform to relate D-brane moduli spaces describing Higgs branches of four-dimensional N=2 gauge theories arising in F-theory compactifications. The duality maps 0- and 4-brane charges through a precise Mukai-vector transformation, enabling an exchange between SU(N_c) with N_f flavors and SU(N_f−N_c) with N_f flavors, while preserving the K3 lattice and inverting the K3 volume; it identifies the baryonic branches and clarifies their geometric realization as determinants of Grassmannian bundles, though Coulomb branches and N=1 cases require additional twists or duality inputs. The analysis connects to heterotic–IIA dualities and Hilbert-scheme descriptions, and highlights where the N=1 duality on rational surfaces needs further modification beyond the straightforward T-duality picture.

Abstract

We construct T-duality on K3 surfaces. The T-duality exchanges a 4-brane R-R charge and a 0-brane R-R charge. We study the action of the T-duality on the moduli space of 0-branes located at points of K3 and 4-branes wrapping it. We apply the construction to F-theory compactified on a Calabi-Yau 4-fold and study the duality of N=2 SU(N_c) gauge theories in four dimensions. We discuss the generalization to the N=1 duality scenario.