Abstract:We study the superconformal index for the class of N = 2 4d superconformal field theories recently introduced by Gaiotto [1]. These theories are defined by compactifying the (2, 0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A 1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult [2].

In this article we use 5-brane junctions to study the 5D T N SCFTs corresponding to the 5D N = 1 uplift of the 4D N = 2 strongly coupled gauge theories, which are obtained by compactifying N M5 branes on a sphere with three full punctures. Even though these theories have no Lagrangian description, by using the 5-brane junctions proposed by Benini, Benvenuti and Tachikawa, we are able to derive their Seiberg-Witten curves and Nekrasov partition functions. We cross-check our results with the 5D superconformal index proposed by Kim, Kim and Lee. Through the AGTW correspondence, we discuss the relations between 5D superconformal indices and n-point functions of the q-deformed W N Toda theories.

In this article we explore the duality between the low energy effective theory of five-dimensional N=1 SU(N)^{M-1} and SU(M)^{N-1} linear quiver gauge theories compactified on S^1. The theories we study are the five-dimensional uplifts of four-dimensional superconformal linear quivers. We study this duality by comparing the Seiberg-Witten curves and the Nekrasov partition functions of the two dual theories. The Seiberg-Witten curves are obtained by minimizing the worldvolume of an M5-brane with nontrivial geometry. Nekrasov partition functions are computed using topological string theory. The result of our study is a map between the gauge theory parameters, i.e., Coulomb moduli, masses and UV coupling constants, of the two dual theories. Apart from the obvious physical interest, this duality also leads to compelling mathematical identities. Through the AGTW conjecture these five-dimentional gauge theories are related to q-deformed Liouville and Toda SCFTs in two-dimensions. The duality we study implies the relations between Liouville and Toda correlation functions through the map we derive.Comment: 58 pages, 17 figures; v2: minor corrections, references adde

In this paper we find preliminary evidence that N = 2 superconformal QCD, the SU (N c ) SYM theory with N f = 2N c fundamental hypermultiplets, might be integrable in the large N Veneziano limit. We evaluate the one-loop dilation operator in the scalar sector of the N = 2 superconformal quiver with SU (N c )×SU (Nč) gauge group, for N c ≡ Nč. Both gauge couplings g andǧ are exactly marginal. This theory interpolates between the Z 2 orbifold of N = 4 SYM, which corresponds toǧ = g, and N = 2 superconformal QCD, which is obtained foř g → 0. The planar one-loop dilation operator takes the form of a nearest-neighbor spinchain Hamiltonian. For superconformal QCD the spin chain is of novel form: besides the color-adjoint fields φ a b , which occupy individual sites of the chain, there are "dimers" Q a iQ i b of flavor-contracted fundamental fields, which occupy two neighboring sites. We solve the two-body scattering problem of magnon excitations and study the spectrum of bound states, for generalǧ/g. The dimeric excitations of superconformal QCD are seen to arise smoothly forǧ → 0 as the limit of bound wavefunctions of the interpolating theory. Finally we check the Yang-Baxter equation for the two-magnon S-matrix. It holds as expected at the orbifold pointǧ = g. While violated for generalǧ = g, it holds again in the limitǧ → 0, hinting at one-loop integrability of planar N = 2 superconformal QCD.

In non-supersymmetric orbifolds of N = 4 super Yang-Mills, conformal invariance is broken by the logarithmic running of double-trace operators -a leading effect at large N . A tachyonic instability in AdS 5 has been proposed as the bulk dual of doubletrace running. In this paper we make this correspondence more precise. By standard field theory methods, we show that the double-trace beta function is quadratic in the coupling, to all orders in planar perturbation theory. Tuning the double-trace coupling to its (complex) fixed point, we find conformal dimensions of the form 2 ± i b(λ), as formally expected for operators dual to bulk scalars that violate the stability bound. We also show that conformal invariance is broken in perturbation theory if and only if dynamical symmetry breaking occurs. Our analysis is applicable to a general large N field theory with vanishing single-trace beta functions.