Catalan polynomials for the Wishart-Laguerre $\beta$-ensemble

Abstract

By means of $W$-representation, we investigate the Wishart-Laguerre $\beta$-ensemble and present a compact expression for Catalan polynomials. For a special Nekrasov partition function associated with the Wishart-Laguerre $\beta$-ensemble, we show that it can be expressed as a exponential form of the special Catalan polynomials. Furthermore, we make a rule to calculate the Catalan polynomials by means of Feynman diagrams and present another expression for Catalan polynomials. We also establish the connection between certain Catalan polynomials and knot theory.

15 pages, accepted for publication in Physics Letters B