K-theoretic boson–fermion correspondence and melting crystals

Abstract

We study non-Hermitian integrable fermion and boson systems from the perspectives of Grothendieck polynomials. The models considered in this article are the five-vertex model as a fermion system and the non-Hermitian phase model as a boson system. Both models are characterized by different solutions satisfying the same Yang–Baxter relation. From our previous works on the identification between the wavefunctions of the five-vertex model and Grothendieck polynomials, we introduce skew Grothendieck polynomials and derive the addition theorem among them. Using these relations, we derive the wavefunctions of the non-Hermitian phase model as a determinant form, which can also be expressed as Grothendieck polynomials. Namely, we establish a K-theoretic boson–fermion correspondence at the level of wavefunctions. As a by-product, the partition function of the statistical mechanical model of a three-dimensional (3D) melting crystal is exactly calculated by use of the scalar products of the wavefunctions of the phase model. The resultant expression can be regarded as a K-theoretic generalization of the MacMahon function describing the generating function of the plane partitions, which interpolates the generating functions of two-dimensional (2D) and (3D) Young diagrams.