The bilinear fermionic form for KP and BKP hierarchies

TL;DR

This paper introduces a new method using lifting operators to connect fermionic functions in KP and BKP hierarchies, enabling efficient computation of two-point functions for various models.

Contribution

It develops a novel approach linking fermionic two-point and one-point functions via lifting operators for KP and BKP tau-functions.

Findings

Derived concise formulas for fermionic two-point functions in specific models

Provided an effective method to compute fermionic two-point functions

Applied the approach to models like the r-spin and Brézin–Gross–Witten models

Abstract

For a tau-function of the KP or BKP hierarchy, we introduce the notion of lifting operator and derive an equation connecting the corresponding fermionic two-point function and fermionic one-point function through the lifting operator. This provides an effective approach to determine the fermionic two-point function of the tau-function from the lifting operator and the fermionic one-point function. As practical applications, we derive concise formulas for the fermionic two-point functions of several models, like the $r$-spin model and the Br{\' e}zin--Gross--Witten model, which respectively serve as examples for KP and BKP tau-functions.