Factorial Fock free fermions
Daniel Bump, Andrew Hardt, Travis Scrimshaw
TL;DR
The paper develops a two-parameter deformation of free fermions using shifted powers $(z; \boldsymbol{\alpha})^k$ and $(z|\boldsymbol{\beta})^k$, yielding deformed fermion fields, currents, and shift operators that preserve the vacuum correlators. This enables a deformed boson–fermion correspondence in which the natural basis maps to double factorial Schur functions $s_{\lambda}(\mathbf{p}||\boldsymbol{\alpha};\boldsymbol{\beta})$, with Jacobi–Trudi, Murnaghan–Nakayama, Giambelli, and skew Cauchy identities proven via Wick’s theorem. The double factorial Schur functions form a basis of the completed symmetric function algebra and appear as tau functions of the KP hierarchy and 2D Toda lattice, providing a bridge to integrable systems. Additionally, the authors connect the deformed vertex operators to Naprienko’s solvable lattice models through row transfer matrices, establishing a concrete lattice-model realization of the algebraic framework and highlighting the role of column and row parameters in the factorial deformation. An alternative deformation via Miwa parameters and a second boson–fermion correspondence broadens the landscape, linking to generalized factorials and dualities in a unified fermionic/bosonic setting.
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.