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

TL;DR

This paper analyzes the bimodule structure of the periodic gl(1|1) spin chain, focusing on the representation theory of its symmetry algebra and its relation to logarithmic conformal field theory, providing foundational results for future continuum limit analysis.

Contribution

It provides a detailed description of the bimodule decomposition of the periodic gl(1|1) spin chain over the JTL algebra and its symmetry algebra, advancing the algebraic understanding of this model.

Findings

Decomposition of the spin chain over JTL algebra for any even N.

Representation theory of the centralizer algebra U_q^{odd} sl(2).

Bimodule structure elucidated for 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.