Super-Whittaker vector at c=3/2

TL;DR

This paper derives a closed-form expression for the norm of the degenerate super-Whittaker vector at central charge c=3/2, linking it to supersymmetric gauge theory partition functions and resolving orthogonality issues in the representation.

Contribution

It introduces a method to obtain a closed-form norm of super-Whittaker vectors at c=3/2, addressing orthogonality problems in Jack superpolynomial representations.

Findings

Closed-form expression for super-Whittaker vector norm at c=3/2

Connection to Z_2-symmetric SU(2) instanton partition functions

Resolution of orthogonality issues in superconformal algebra representations

Abstract

The degenerate Whittaker vector of the superconformal algebra can be represented in terms of Jack superpolynomials. However, in this representation the norm of the Whittaker vector involves a scalar product with respect to which the Jack superpolynomials are not orthogonal. In this note, we point out that this defect can be cured at c=3/2 by means of a trick specific to the supersymmetric case. At c=3/2, we thus end up with a closed-form expression for the norm of the degenerate super-Whittaker vector. Granting the super-version of the AGT conjecture, this closed-form expression should be equal to the Z_2-symmetric SU(2) pure-gauge instanton partition function -- the corresponding equality taking the form of a rather nontrivial combinatorial identity.