Bimodule structure in the periodic gl(1|1) spin chain

Abstract

This paper is second in a series devoted to the study of periodic super-spin chains. In our first paper at 2011, we have studied the symmetry algebra of the periodic gl(1|1) spin chain. In technical terms, this spin chain is built out of the alternating product of the gl(1|1) fundamental representation and its dual. The local energy densities - the nearest neighbor Heisenberg-like couplings - provide a representation of the Jones Temperley Lieb (JTL) algebra. The symmetry algebra is then the centralizer of JTL, and turns out to be smaller than for the open chain, since it is now only a subalgebra of U_q sl(2) at q=i, dubbed U_q^{odd} sl(2). A crucial step in our associative algebraic approach to bulk logarithmic conformal field theory (LCFT) is then the analysis of the spin chain as a bimodule over U_q^{odd} sl(2) and JTL. While our ultimate goal is to use this bimodule to deduce properties of the LCFT in the continuum limit, its derivation is sufficiently involved to be the sole subject of this paper. We describe representation theory of the centralizer and then use it to find a decomposition of the periodic gl(1|1) spin chain over JTL for any even number N of tensorands and ultimately a corresponding bimodule structure. Applications of our results to the analysis of the bulk LCFT will then be discussed in the third part of this series.