Superconformal field theory and Jack superpolynomials

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.

37 pages. v3: correction of some equations (not corrected in the published version), most importantly, (6.9), (6.10), and (B.27)