Abhijit Gadde

Gauge Theories and Macdonald Polynomials

We study the

2d index and surface operators

{\mathcal{N}=2}

The \mathcal{N} = 1 superconformal index for class \mathcal{S} fixed points

four-dimensional superconformal index in various interesting limits, such that only states annihilated by more than one supercharge contribute. Extrapolating from the SU(2) generalized quivers, which have a Lagrangian description, we conjecture explicit formulae for all A-type quivers of class

(0, 2) trialities

{\mathcal S}

On the reduction of 4d \( \mathcal{N}=1 \) theories on \( {\mathbb{S}}^2 \)

, which in general do not have one. We test our proposals against several expected dualities. The index can always be interpreted as a correlator in a two-dimensional topological theory, which we identify in each limit as a certain deformation of two-dimensional Yang-Mills theory. The structure constants of the topological algebra are diagonal in the basis of Macdonald polynomials of the holonomies.

A "Lagrangian" for a non-Lagrangian theory

In this letter we describe a new powerful duality, relating physics in four and in two dimensions. We will argue that for a large class of four-dimensional superconformal gauge theories, non-trivial information about the operator spectrum is captured by correlators of a twodimensional non-supersymmetric gauge theory. The 4d side of the duality is generically strongly-coupled, and difficult to analyze directly; on the other hand calculations on the 2d side will be explicit and algorithmic. Thus our conjecture gives new information about stronglycoupled 4d field theories. Our proposal is in the same spirit as the AldayGaiotto-Tachikawa (AGT) relation between the partition function of a 4d N = 2 gauge theory on S and a correlator in 2d Liouville/Toda theory [1, 2]. In our case, the 4d observable is a (twisted) supersymmetric partition function of an N = 2 superconformal field theory on S × S, also known as the superconformal index. We will focus on a “reduced” index that depends on a single fugacity q. On the 2d side, instead of Liouville/Toda we have the zero-area limit of q-deformed Yang-Mills theory. The topological nature of this 2d theory dovetails with the independence of the 4d index on the gauge theory moduli. We begin by reviewing the 4d side of the duality. The full N = 2 superconformal index is defined as [3] I = Tr(−1) pE−R 2 1q E−R 2 −j1u−(r+R) , (1)