Ground states of the Heisenberg evolution operator in discrete three-dimensional spacetime and quantum discrete BKP equations

In this paper we consider three-dimensional quantum q-oscillator field theory without spectral parameters. We construct an essentially large set of eigenstates of evolution with unity eigenvalue of discrete time-evolution operator. All these eigenstates belong to a subspace of a total Hilbert space where an action of the 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 \mathscr{U}_q\big(B_n^{(1)}\big) and \mathscr{U}_q\big(D_n^{(1)}\big)