Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations

TL;DR

This paper constructs a large set of eigenstates for a three-dimensional quantum q-oscillator model, linking the evolution operator's action to quantized discrete BKP equations, revealing new insights into quantum discrete integrable systems.

Contribution

It introduces a novel construction of eigenstates for the 3D quantum q-oscillator evolution operator, connecting it to quantum discrete BKP equations through boundary states of specific R-matrices.

Findings

Eigenstates form a subspace where evolution acts as quantized discrete BKP equations

Construction relies on boundary states of 3D R-matrices associated with quantum groups

Provides a framework for understanding 3D quantum integrable models

Abstract

In this paper we consider three-dimensional quantum q-oscillator field theory without spectral parameters. We construct an essentially big set of eigenstates of evolution with unity eigenvalue of discrete time evolution operator. All these eigenstates belong to a subspace of total Hilbert space where an action of evolution operator can be identified with quantized discrete BKP equations (synonym Miwa equations). The key ingredients of our construction are specific eigenstates of a single three-dimensional R-matrix. These eigenstates are boundary states for hidden three-dimensional structures of U_q(B_n^1) and U_q(D_n^1)$.