Theory HEP Seminar

Superconformal algebra and Jack superpolynomials.

Pierre Mathieu

Université Laval

Via the free-field representation, symmetric polynomials provide representations of the states in Virasoro highest-weight modules. Among all basis of such polynomials, the Jack-polynomial basis is distinguished in that the singular vector in a Kac module can be represented by a single Jack polynomial. This remarkable relation is reviewed and is extended to the supersymmetric case where the role of the Jacks, eigenfunctions of the Calogero-Sutherland model, is played by the eigenfunctions of its supersymmetric counterpart, the Jack superpolynomials (sJacks). Specifically, the singular vector of a super-Virasoro Kac module is shown to be represented by a simple linear combination of sJacks. 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.