BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions

TL;DR

This paper derives a new formula for connected n-point functions of BKP hierarchy tau-functions using affine coordinates, extending KP results, and introduces algorithms for computing free energies of specific tau-functions.

Contribution

It provides a BKP-analogue of Zhou's KP formula, relates KP and BKP affine coordinates, and offers new computational methods for free energies of Witten-Kontsevich and Brézin-Gross-Witten tau-functions.

Findings

Derived a formula for connected n-point functions in BKP hierarchy.

Established a relation between KP and BKP affine coordinates.

Developed algorithms for computing free energies of specific tau-functions.

Abstract

We derive a formula for the connected $n$-point functions of a tau-function of the BKP hierarchy in terms of its affine coordinates. This is a BKP-analogue of a formula for KP tau-functions proved by Zhou in [arXiv:1507.01679]. Moreover, we prove a simple relation between the KP-affine coordinates of a tau-function $\tau(\mathbf{t})$ of the KdV hierarchy and the BKP-affine coordinates of $\tau(\mathbf{t}/2)$. As applications, we present a new algorithm to compute the free energies of the Witten-Kontsevich tau-function and the Br\'ezin-Gross-Witten tau-function.