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.
