BKP hierarchy and Pfaffian point process

Zhi-Lan Wang,Shi-Hao Li

doi:10.1016/j.nuclphysb.2018.12.028

Abstract

Inspired by Okounkov's work (2001) [20] which relates KP hierarchy to determinant point process, we establish a relationship between BKP hierarchy and Pfaffian point process. We prove that the correlation function of the shifted Schur measures on strict partitions can be expressed as a Pfaffian of skew symmetric matrix kernel, whose elements are certain vacuum expectations of neutral fermions. We further show that the matrix integrals solution of BKP hierarchy can also induce a certain Pfaffian point process.

Highlights

There is a connection revealed by Okounkov [20] between random partitions and KP hierarchy of type A∞
The correlation function can be realized as a determinant point process via the Fock space formalism, which satisfies the KP hierarchy
The tau function can be viewed as an element in the Bosonic Fock space, and it can be expressed in terms of Schur funcions

Summary

Introduction

There is a connection revealed by Okounkov [20] between random partitions and KP hierarchy of type A∞. As the BKP tau functions can be described respectively in the Bosonic picture and the Fermionic picture [31], the Boson-Fermion correspondence of type B∞ allows us to relate projective Schur functions to vacuum expectations of neutral fermions. Inspired by these facts, we generalize Okounkov’s results to BKP hierarchy in this paper.

A brief introduction to neutral fermions

Schur Q-functions and Pfaffian point process