Factorial Fock free fermions

TL;DR

This paper introduces deformed free fermion operators linked to factorial Schur functions, connecting integrable models, symmetric functions, and tau functions of the 2D Toda lattice.

Contribution

It develops a double deformation of free fermion fields and establishes their relation to factorial Schur functions and integrable systems.

Findings

Deformed half vertex operators correspond to row transfer matrices of a solvable six vertex model.

Specialization yields factorial Schur functions from the model.

Natural basis maps to double factorial Schur functions and tau functions of 2D Toda.

Abstract

We use a double shifted power analog of free fermion fields to introduce current operators, Hamiltonians, and vertex operators which are deformed by two families of parameters and satisfy analogous formulas to the classical case. We show that the deformed half vertex operators correspond to the row transfer matrices of a solvable six vertex model recently given by Naprienko [arXiv:2301.12110], which under a specialization yields the factorial Schur functions (up to a reindexing of parameters). As a consequence, we show that under the boson-fermion correspondence using our deformed half vertex operators, the natural basis (under this specialization) maps to the double factorial Schur functions. Furthermore, the image of the natural basis vectors are tau function solutions to the 2D Toda lattice.