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

TL;DR

The paper addresses non-perturbative dualities of local $N=2$ Yang–Mills theories by embedding the rigid Seiberg–Witten solution into a gravitational setting via dynamical Calabi–Yau threefolds. It develops a framework in which the rigid monodromy and $R$-symmetry are realized as substructures of the local symplectic duality group $\Gamma_D$, with explicit constructions for $SU(r+1)$ using a genus-$r$ hyperelliptic Riemann surface and its Picard–Fuchs system. It then extends to local theories by identifying a corresponding $r+1$ parameter family of Calabi–Yau threefolds ${\cal M}_3[r]$ that embed the rigid dualities, and derives central charges and BPS spectra from three-form cohomology, linking CY geometry, mirror symmetry, and heterotic/type II dualities. The work provides a concrete route to realize and classify non-perturbative dualities in gravity-coupled $N=2$ theories and highlights the role of CY geometry and string dualities in organizing the BPS spectrum and moduli-space structure.

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 $Γ_D$ associated with the moduli space of the dynamical Calabi--Yau threefold.