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.

latex, 42 pp., 13 figures + 5 figures in color, many comments added

References in corpus (6)

• Associative-algebraic approach to logarithmic conformal field theories
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Continuum limit and symmetries of the periodic gl(1|1) spin chain
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• On Representations of Affine Temperley--Lieb Algebras

Cited by in corpus (19)

• Logarithmic Conformal Field Theory: a Lattice Approach
• Continuum limit and symmetries of the periodic gl(1|1) spin chain
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Boundary algebras and Kac modules for logarithmic minimal models
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• Modular invariant partition function of critical dense polymers
• Lattice W-algebras and logarithmic CFTs
• Jordan cells of periodic loop models
• On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories
• On the reality of spectra of ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-invariant XXZ Hamiltonians
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• Integrable Floquet systems related to logarithmic conformal field theory
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Conformal Field Theories as Scaling Limit of Anyonic Chains
• Spin chains as modules over the affine Temperley-Lieb algebra