Superconformal field theory and Jack superpolynomials

TL;DR

This paper reveals a deep link between 2D superconformal field theory and Jack superpolynomials, providing explicit formulas for singular vectors and Whittaker vectors, and reformulating the supersymmetric AGT conjecture in combinatorial terms.

Contribution

It explicitly connects superconformal field theory singular vectors with Jack superpolynomials and reformulates the supersymmetric AGT conjecture using their combinatorics.

Findings

Singular vectors are expressed as sums of Jack superpolynomials within a rectangular diagram.

Degenerate Whittaker vectors are simple linear combinations of Jack superpolynomials.

Formulation of the supersymmetric AGT conjecture in terms of Jack superpolynomials' combinatorics.

Abstract

We uncover a deep connection between the $\mathcal{N}=1$ superconformal field theory in 2D and eigenfunctions of the supersymmetric Sutherland model known as Jack superpolynomials (sJacks). Specifically, the singular vector at level $rs/2$ of the Kac module labeled by the two integers $r$ and $s$ are given explicitly as a sum of sJacks whose indexing diagrams are contained in a rectangle with $r$ columns and $s$ rows As a second compelling evidence for the distinguished status of the sJack-basis in SCFT, we find that the degenerate Whittaker vectors (Gaiotto states) can be expressed as a remarkably simple linear combination of sJacks. As a consequence, we are able to reformulate the supersymmetric version of the (degenerate) AGT conjecture in terms of the combinatorics of sJacks. Note that the closed-form formulas for the singular vectors and the degenerate Whittaker vectors, although only conjectured in general, have been heavily tested (in some cases, up to level 33/2). Note also that both the Neveu-Schwarz and Ramond sectors are treated.