2D CFT blocks for the 4D class $\mathcal{S}_k$ theories

We study instanton partition functions for N=2 superconformal Sp(1) and SO(4) gauge theories. We find that they agree with the corresponding U(2) instanton partitions functions only after a non-trivial mapping of the microscopic gauge couplings, since the instanton counting involves different renormalization schemes. Geometrically, this mapping relates the Gaiotto curves of the different realizations as double coverings. We then formulate an AGT-type correspondence between Sp...

A summary of results is presented, which provide exact description of the low-energy $4d$ $N=2$ and $N=4$ SUSY gauge theories in terms of $1d$ integrable systems.

Some time ago, Seiberg and Witten solved for moduli spaces of vacua parameterized by scalar vacuum expectation values in $\mathcal{N}=2$ gauge theories. More recently, new vacua associated to soft theorems and asymptotic symmetries have been found. This paper takes some first steps towards a complete picture of the infrared geometry of $\mathcal{N}=2$ gauge theory incorporating both of these infrared structures.

In this work we launch a systematic theory of superconformal blocks for four-point functions of arbitrary supermultiplets. Our results apply to a large class of superconformal field theories including 4-dimensional models with any number $\mathcal{N}$ of supersymmetries. The central new ingredient is a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick ...

Analytic expressions for the two dimensional $\mathcal{N}=1$ SLFT blocks in the light semi-classical limit are found for both Neveu-Schwarz and Ramond sectors. The calculations are done by using the duality between $SU(2)$ $\mathcal{N}=2$ super-symmetric gauge theories living on $R^4/Z_2$ space and two dimensional $\mathcal{N}=1$ super Liouville field theory. It is shown that in the light asymptotic limit only a restricted set of Young diagrams contribute to the partition fun...

In this paper we begin mapping out the space of rank-2 $\mathcal{N}=2$ superconformal field theories (SCFTs) in four dimensions. This represents an ideal set of theories which can be potentially classified using purely quantum field-theoretic tools, thus providing a precious case study to probe the completeness of the current understanding of SCFTs, primarily derived from string theory constructions. Here, we collect and systematize a large amount of field theoretic data char...

This is the writeup of the lectures given at the Winter School "YRISW 2018" to appear in a special issue of JPhysA. In the first part of these lecture notes we review some important facts about 4D $\mathcal{N}=2$ SCFTs. We begin with basic textbook material, the supersymmetry algebra and its massless representations and the construction of Lagrangians using superspace. Then we turn to more modern topics, the study of the $\mathcal{N}=2$ SCA and its representation theory. Our ...

This is the third article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J. Teschner. It is explained how to compute the instanton partition functions. The results can be written as sums over bases for the equivariant cohomology of instanton moduli spaces. The known results relating the symmetries of these spaces to the symmetries of conformal field theory are reviewed.

Applying the Casimir operator to four-point functions in CFTs allows us to find the conformal blocks for any external operators. In this work, we initiate the program to find the superconformal blocks, using the super Casimir operator, for $4D$ ${\mathcal N}=1$ SCFTs. We begin by finding the most general four-point function with zero $U(1)_R$-charge, including all the possible nilpotent structures allowed by the superconformal algebra. We then study particular cases where som...

This is the second in a series of three papers on systematic analysis of rank 1 Coulomb branch geometries of four dimensional $\mathcal{N}$=2 SCFTs. In the first paper we developed a strategy for classifying physical rank-1 CB geometries of $\mathcal{N}$=2 SCFTs. Here we show how to carry out this strategy computationally to construct the Seiberg-Witten curves and one-forms for all the rank-1 SCFTs. Explicit expressions are given for all cases, with the exception of the $N_f$...