Modular Structures in Superconformal field theories[HBNI Th144]

Abstract

We study the modular structures underlying N = 2 superconformal gauge theories in four dimensions. Using constraints from a nonperturbative duality symmetry called S-duality, we show that observables of interest may be resummed into quasimodular forms of the S-duality group. We study these gauge theories coupled to different kinds of matter. Adjoint Matter : We study chiral observables in U(N) gauge theories and show that they may be resummed into quasimodular forms of the nonperturbative S-duality group. We do this using a number of complimentary approaches: explicitly evaluating the period integrals; invoking a correspondence between these gauge theories and an integrable model called the elliptic Calogero-Moser system; and microscopically evaluating nonperturbative contributions to chiral observables using the machinery of equivariant localization. Fundamental Matter : We study SQCD theories with gauge group SU(N) with Nf = 2N fundamental hypermultiplets. When the flavours are massless, we focus on the period matrix of these theories in a ZN -symmetric locus on the Coulomb moduli space: the special vacuum. We clarify the underlying modular structure, in particular to understand the manner in which the S-duality group acts on the renormalized couplings, and show that this action is consistent with more recent studies of S-duality that focus on the bare couplings of the gauge theory. We also study massive hypermultiplet configurations that respect the ZN symmetry of the special vacuum. Here, we find that the modular structure of the massless theory will is deformed; more specifically, we find that the renormalized couplings admit semiclassical expansions with mass-dependent coefficients. We use constraints from S-duality to derive modular anomaly equations, which are then used to solve for the mass-dependent coefficients order-by-order in the mass expansion.