We find a four-dimensional N = 1 gauge theory which flows to the minimal interacting N = 2 superconformal field theory, the Argyres-Douglas theory, in the infrared up to the extra free chiral multiplets. The gauge theory is obtained from a certain N = 1 preserving deformation of the N = 2 SU (2) gauge theory with four fundamental hypermultiplets. From this description, we compute the full superconformal index and find agreements with the known results in special limits.

We study N = 2 supersymmetric SU(2) gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional A 1 (2,0) theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Ω-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.

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We continue to investigate the N = 1 deformations of four-dimensional N = 2 superconformal field theories (SCFTs) labeled by a nilpotent element of the flavor symmetry [1]. This triggers a renormalization group (RG) flow to an N = 1 SCFT. We systematically analyze all possible deformations of this type for certain classes of N = 2 SCFTs: conformal SQCDs, generalized Argyres-Douglas theories and the E 6 SCFT. We find a number of examples where the amount of supersymmetry gets enhanced to N = 2 at the end point of the RG flow. Most notably, we find that the SU(N ) and Sp(N ) conformal SQCDs can be deformed to flow to the Argyres-Douglas (AD) theories of type (A 1 , D 2N −1 ) and (A 1 , D 2N ) respectively. This RG flow therefore allows us to compute the full superconformal index of the (A 1 , D N ) class of AD theories. Moreover, we find an infrared duality between N = 1 theories where the fixed point is described by an N = 2 AD theory. We observe that the classes of examples that exhibit supersymmetry enhancement saturate certain bounds for the central charges implied by the associated two-dimensional chiral algebra.

We study the dual descriptions recently discovered for the Seiberg-Witten theory in the presence of surface operators. The Nekrasov partition function for a four-dimensional N = 2 gauge theory with a surface operator is believed equal to the wave-function of the corresponding integrable system, or the Hitchin system, and is identified with the conformal block with a degenerate field via the AGT relation. We verify the conjecture by showing that the null state condition leads to the Schrödinger equations of the integrable systems. Furthermore, we show that the deformed prepotential emerging from the period integrals of the principal function corresponds to monodromy operation of the conformal block. We also give the instanton partition functions for the asymptotically free SU(2) gauge theories in the presence of the surface operator via the AGT relation. We find that these partition functions involve the counting of two-and four-dimensional instantons. *

Abstract:We elucidate the relation between Painlevé equations and four-dimensional rank one N = 2 theories by identifying the connection associated to Painlevé isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated to gauge theories and by studying the corresponding renormalisation group flow. Based on this correspondence we provide long-distance expansions at various canonical rays for all Painlevé τ -functions in terms of magnetic and dyonic Nekrasov partition functions for N = 2 SQCD and Argyres-Douglas theories at self-dual Omega background ǫ 1 + ǫ 2 = 0, or equivalently in terms of c = 1 irregular conformal blocks.

Abstract:We study certain N = 1 preserving deformations of four-dimensional N = 2 superconformal field theories (SCFTs) with non-abelian flavor symmetry. The deformation is described by adding an N = 1 chiral multiplet transforming in the adjoint representation of the flavor symmetry with a superpotential coupling, and giving a nilpotent vacuum expectation value to the chiral multiplet which breaks the flavor symmetry. This triggers a renormalization group flow to an infrared SCFT. Remarkably, we find classes of theories flow to enhanced N = 2 supersymmetric fixed points in the infrared under the deformation. They include generalized Argyres-Douglas theories and rank-one SCFTs with non-abelian flavor symmetries. Most notably, we find renormalization group flows from the deformed conformal SQCDs to the (A 1 , A n ) Argyres-Douglas theories. From these "Lagrangian descriptions," we compute the full superconformal indices of the (A 1 , A n ) theories and find agreements with the previous results. Furthermore, we study the cases, including the T N and R 0,N theories of class S and some of rank-one SCFTs, where the deformation gives genuine N = 1 fixed points.

We show that the N =1 supersymmetric SU (N ) gauge theory with 2N flavors without superpotential has not only the standard Seiberg dual description but also another dual description involving two copies of the so-called T N theory. This is a natural generalization to N > 2 of a dual description of SU (2) gauge theory with 4 flavors found by Csaki, Schmaltz, Skiba and Terning. We also study dualities of other N =1 SCFTs involving copies of T N theories. Our duality is the basic operation from which a recently-found web of N =1 dualities obtained by compactifying M5-branes on Riemann surfaces can be derived field-theoretically.