Free field realization of the BMS Ising model

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 BMS3, 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 BMS3 algebra.