The bilinear fermionic form for KP and BKP hierarchies

TL;DR

This work establishes a unified lifting-operator framework to compute fermionic two-point functions for tau-functions of the KP and BKP hierarchies from their one-point data. The core relation, $(l_u^*-l_v)\Psi(u,v)=\Psi^*(u)\Psi(v)$ for KP and its BKP analogue, enables compact closed-form expressions of $\Psi(u,v)$ in terms of a lifting operator and the corresponding one-point functions. By applying the method to several enumerative models—Generalized Kontsevich, $r$-spin, and BGW among others—the authors derive explicit two-point formulas, often of the form $\Psi(u,v)=\dfrac{\text{polynomial in }l_u,l_v}{\text{polynomial in }x(u)-x(v)}\Psi^*(u)\Psi(v)$ (KP) or its BKP counterpart. This framework provides a powerful and versatile tool to connect geometric enumerations to KP/BKP integrable structures, with broad applicability across Hurwitz-type problems, Gromov–Witten theory, and topological string-related 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.

Theorems & Definitions (35)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 2.1: boson-fermionic correspondence
• Proposition 2.2
• Lemma 3.1
• proof
• Example 3.2
• Lemma 3.3
• Definition 3.4