Continuum limit and symmetries of the periodic gl(1|1) spin chain

Abstract

This paper is the first in a series devoted to the study of logarithmic conformal field theories (LCFT) in the bulk. Building on earlier work in the boundary case, our general strategy consists in analyzing the algebraic properties of lattice regularizations (quantum spin chains) of these theories. In the boundary case, a crucial step was the identification of the space of states as a bimodule over the Temperley Lieb (TL) algebra and the quantum group U_q sl(2). The extension of this analysis in the bulk case involves considerable difficulties, since the U_q sl(2) symmetry is partly lost, while the TL algebra is replaced by a much richer version (the Jones Temperley Lieb - JTL - algebra). Even the simplest case of the gl(1|1) spin chain - corresponding to the c=-2 symplectic fermions theory in the continuum limit - presents very rich aspects, which we will discuss in several papers. In this first work, we focus on the symmetries of the spin chain, that is, the centralizer of the JTL algebra in the alternating tensor product of the gl(1|1) fundamental representation and its dual. We prove that this centralizer is only a subalgebra of U_q sl(2) at q=i that we dub U_q^{odd} sl(2). We then begin the analysis of the continuum limit of the JTL algebra: using general arguments about the regularization of the stress energy-tensor, we identify families of JTL elements going over to the Virasoro generators L_n, \bar{L}_n in the continuum limit. We then discuss the SU(2) symmetry of the (continuum limit) symplectic fermions theory from the lattice and JTL point of view. The analysis of the spin chain as a bimodule over U_q^{odd} sl(2) and JTL is discussed in the second paper of this series.