Abstract
We study N = 2 superconformal theories with gauge group SU(N) and 2N fundamental flavours in a locus of the Coulomb branch with a Z_N symmetry. In this special vacuum, we calculate the prepotential, the dual periods and the period matrix using equivariant localization. When the flavours are massless, we find that the period matrix is completely specified by [N/2] effective couplings. On each of these, we show that the S-duality group acts as a generalized triangle group and that its hauptmodul can be used to write a non-perturbatively exact relation between each effective coupling and the bare one. For N = 2, 3, 4 and 6, the generalized triangle group is an arithmetic Hecke group which contains a subgroup that is also a congruence subgroup of the modular group PSL(2,Z). For these cases, we introduce mass deformations that respect the symmetries of the special vacuum and show that the constraints arising from S-duality make it possible to resum the instanton contributions to the period matrix in terms of meromorphic modular forms which solve modular anomaly equations.
Highlights
One of the most interesting properties of supersymmetric gauge theories is the existence of non-perturbative S-dualities that relate their weak- and strong-coupling behaviour.1 Recently, there has been much progress in understanding these dualities in conformally invariant N = 2 supersymmetric gauge theories in four dimensions, especially following the seminal work of Gaiotto [2]
We introduce mass deformations that respect the symmetries of the special vacuum and show that the constraints arising from S-duality make it possible to resum the instanton contributions to the period matrix in terms of meromorphic modular forms which solve modular anomaly equations
For linear quiver gauge theories in the weak coupling limit, the Riemann surface degenerates into a collection of three-punctured spheres connected by long thin tubes, and the sewing parameters are identified with the bare coupling constants of the superconformal gauge theory
Summary
For N = 2 theories (i.e. mass deformed N = 4 theories) with unitary gauge groups it has been shown [10,11,12,13] that the constraints coming from S-duality take the form of a modular anomaly equation whose solution allows one to reconstruct the prepotential on the Coulomb branch order by order in the mass of the adjoint hypermultiplet to all orders in the gauge coupling To achieve this result one has to organize the low-energy effective prepotential as a semi-classical expansion in inverse powers of the vacuum expectation values of the scalar fields in the gauge vector multiplet and realize that the coefficients of this expansion satisfy a recursion relation whose solution can be written in terms of quasi-modular forms of PSL(2,Z) acting on the bare gauge coupling. The paper ends with a discussion of the results and some future directions for work, and with two technical appendices
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