On a Selberg-Schur integral

TL;DR

This paper generalizes the Selberg integral using Schur polynomials related to partitions with entries up to 2, providing explicit formulas and complex extensions with applications in conformal field theory.

Contribution

It introduces a new explicit computation of a generalized Selberg-Schur integral and its complex extension, connecting these to conformal blocks in physics.

Findings

Explicit formulas for the generalized Selberg-Schur integral

Complex extension of the integral with analytical continuation

Applications to conformal field theory and WZNW models

Abstract

A generalization of Selberg's beta integral involving Schur polynomials associated with partitions with entries not greater than 2 is explicitly computed. The complex version of this integral is given after proving a general statement concerning the complex extensions of Selberg-Schur integrals. All these results have interesting applications in both mathematics and physics, particularly in conformal field theory, since the conformal blocks for the $SL(2,\mathbb{R})$ Wess-Zumino-Novikov-Witten model can be obtained by analytical continuation of these integrals.