$(l,q)$-Deformed Grassmann Field and the Two-dimensional Ising Model

Abstract

In this paper we construct the exact representation of the Ising partition function in the form of the $ SL_q(2,R)$-invariant functional integral for the lattice free $(l,q)$-fermion field theory ($l=q=-1$). It is shown that the $(l,q)$-fermionization allows one to re-express the partition function of the eight-vertex model in external field through functional integral with four-fermion interaction. To construct these representations, we define a lattice $(l,q,s)$-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At $l=q=-1, s=1$ we obtain the lattice $(l,q)$-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over $(q,s)$-Grassmann variables is expressed through the $(q,s)$-deformed Pfaffian which is equal to square root of the determinant of some matrix at $q=\pm 1, s=\pm 1$.