A Search for Non-Perturbative Dualities of Local $N=2$ Yang--Mills Theories from Calabi--Yau Threefolds

TL;DR

This paper explores non-perturbative dualities in local N=2 Yang--Mills theories using Calabi--Yau threefolds, extending duality group understanding from rigid to supergravity contexts.

Contribution

It generalizes the special geometry framework to supergravity via dynamical Calabi--Yau threefolds and determines the full duality group for arbitrary rigid SU(r+1) gauge theories.

Findings

Full duality group for rigid SU(r+1) theories derived

Duality symmetries linked to dynamical Calabi--Yau threefolds

Embedding of R-symmetry and monodromy groups into symplectic duality group

Abstract

The generalisation of the rigid special geometry of the vector multiplet quantum moduli space to the case of supergravity is discussed through the notion of a dynamical Calabi--Yau threefold. Duality symmetries of this manifold are connected with the analogous dualities associated with the dynamical Riemann surface of the rigid theory. N=2 rigid gauge theories are reviewed in a framework ready for comparison with the local case. As a byproduct we give in general the full duality group (quantum monodromy) for an arbitrary rigid $SU(r+1)$ gauge theory, extending previous explicit constructions for the $r=1,2$ cases. In the coupling to gravity, R--symmetry and monodromy groups of the dynamical Riemann surface, whose structure we discuss in detail, are embedded into the symplectic duality group $\Gamma_D$ associated with the moduli space of the dynamical Calabi--Yau threefold.