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

Abstract

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $gl(1|1)$ spin-chain and its continuum limit - the $c=-2$ symplectic fermions theory - and rely on two technical companion papers, "Continuum limit and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over in the continuum limit to a bigger algebra than the product of the left and right Virasoro algebras. This algebra, S - which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field $S(z,\bar{z})=S_{ab}{\psi }^a(z){\bar{\psi }}^b(\bar{z})$, with a symmetric form $S_{ab}$ and conformal weights (1,1). We discuss in details how the Hilbert space of the LCFT decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL_N in the $gl(1|1)$ spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of $sp(N-2)$. The semi-simple part of JTL_N is represented by $Usp(N-2)$, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL image represented in the spin-chain. On the continuum side, simple modules over the interchiral algebra S are identified with "fundamental" representations of $sp(\infty )$.

69 pp., 10 figs, v2: the paper has been substantially modified - new proofs, new refs, new App C with inductive limits construction, etc

References in corpus (14)

• Logarithmic extensions of minimal models: characters and modular transformations
• Associative-algebraic approach to logarithmic conformal field theories
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• From Percolation to Logarithmic Conformal Field Theory
• Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
• Virasoro representations and fusion for general augmented minimal models
• On the SU(2|1) WZW model and its statistical mechanics applications
• From boundary to bulk in logarithmic CFT
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Fusion Algebras of Logarithmic Minimal Models
• A modular invariant bulk theory for the c=0 triplet model
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Lattice W-algebras and logarithmic CFTs

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• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• Spaces of states of the two-dimensional O(n) and Potts models
• Electrical varieties as vertex integrable statistical models
• On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• Critical loop models are exactly solvable
• Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra
• Spin chains as modules over the affine Temperley-Lieb algebra
• Logarithmic operators in $c=0$ bulk CFTs