Bimodule structure in the periodic [formula omitted] spin chain

Abstract

This paper is the second in a series devoted to the study of periodic super-spin chains. In our first paper (Gainutdinov et al., 2013 [3]), 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 neighbour Heisenberg-like couplings — provide a representation of the Jones–Temperley–Lieb (JTL) algebra JTLN. The symmetry algebra is then the centralizer of JTLN, and turns out to be smaller than for the open chain, since it is now only a subalgebra of Uqsℓ(2) at q=i — dubbed Uqoddsℓ(2) in Gainutdinov et al. (2013) [3]. 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 Uqoddsℓ(2) and JTLN. 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 JTLN for any even N 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.