Free field realization of the BMS Ising model

TL;DR

This paper constructs a free field realization of the BMS Ising model using inhomogeneous BMS free fermion theory, revealing enhanced symmetries and an underlying W(2,2,1) algebra structure.

Contribution

It introduces a novel BMS-Kac-Moody algebra extension and connects the BMS Ising model to a W-algebra, expanding the understanding of BMS symmetries in fermionic models.

Findings

Discovery of an anisotropic scaling symmetry in BMS free fermion theory.

Identification of an enhanced BMS-Kac-Moody algebra with non-zero level.

Establishment of a fermion-boson duality leading to the BMS Ising model.

Abstract

In this work, we study the inhomogeneous BMS free fermion theory, and show that it gives a free field realization of the BMS Ising model. We find that besides the BMS symmetry there exists an anisotropic scaling symmetry in BMS free fermion theory. As a result, the symmetry of the theory gets enhanced to an infinite dimensional symmetry generated by a new type of BMS-Kac-Moody algebra, different from the one found in the BMS free scalar model. Besides the different coupling of the $u(1)$ Kac-Moody current to the BMS algebra, the Kac-Moody level is nonvanishing now such that the corresponding modules are further enlarged to BMS-Kac-Moody staggered modules. We show that there exists an underlying $W(2,2,1)$ structure in the operator product expansion of the currents, and the BMS-Kac-Moody staggered modules can be viewed as highest-weight modules of this $W$-algebra. Moreover we obtain the BMS Ising model by a fermion-boson duality. This BMS Ising model is not a minimal model with respect to BMS$_3$, since the minimal model construction based on BMS Kac determinant always leads to chiral Virasoro minimal models. Instead, the underlying algebra of the BMS Ising model is the $W(2,2,1)$-algebra, which can be understood as a quantum conformal BMS$_3$ algebra.