Generating functions of dual $K$-theoretic $P$- and $Q$-functions and boson-fermion correspondence

TL;DR

This paper develops a new algebraic framework using deformed fermionic operators to represent $K$-theoretic $P$- and $Q$-functions and their duals, providing explicit generating functions and a boson-fermion correspondence.

Contribution

It introduces $eta$-deformed neutral fermion operators and vertex operators, extending the algebraic description of $K$-theoretic functions and deriving their generating functions.

Findings

Realization of $K$-theoretic functions as vacuum expectation values

Derivation of $K$-theoretic Cauchy kernel from operator relations

Conjectured generating functions confirmed using fermionic operators

Abstract

In this paper, we present a new algebraic description of Ikeda-Naruse's $K$-theoretic Schur $P$- and $Q$-functions and their dual functions in terms of neutral fermion operators. We introduce four families of ``$\beta$-deformed neutral-fermion operators'' depending on a parameter $\beta$, which reduce to the usual neutral-fermion operators when $\beta$ is zero. Using these operators, we introduce two families of $\beta$-deformed vertex operators, power sums, and boson-fermion correspondences. From commutation relations among these operators, we naturally derive the $K$-theoretic Cauchy kernel of Nakagawa-Naruse. Exploiting this fact, we show that the four $K$-theoretic functions can be realized as vacuum expectation values of certain $\beta$-deformed fermionic operators. This presentation also allows us to derive generating functions for the dual $K$-theoretic $P$-and $Q$-functions, as conjectured by Nakagawa-Naruse.