Polynomial KP and BKP $\tau$-functions and correlators

TL;DR

This paper develops explicit formulas for polynomial KP and BKP tau-functions and their correlators using fermionic representations, generalizing classical identities like Jacobi's bialternant and Nimmo's Pfaffian ratio.

Contribution

It introduces a novel fermionic framework to express polynomial KP and BKP tau-functions and correlators, extending classical symmetric function identities to these integrable hierarchies.

Findings

Expressed tau-functions via generalized Jacobi and Nimmo formulas.

Derived multipair KP and multipoint BKP correlation functions.

Connected fermionic vacuum expectation values with classical symmetric function identities.

Abstract

Lattices of polynomial KP and BKP $\tau$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions are expressed via generalizations of Jacobi's bialternant formula for Schur functions and Nimmo's Pfaffian ratio formula for Schur $Q$-functions. These are obtained by applying Wick's theorem to fermionic vacuum expectation value representations in which the infinite group element acting on the lattice of basis states stabilizes the vacuum.