Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the ${\mathfrak{gl}(1|1)}$ periodic spin chain, Howe duality and the interchiral algebra
Résumé
We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed ${\mathfrak{gl}(1|1)}$ spin-chain and its continuum limit—the ${c=-2}$ symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). 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 ${\boldsymbol{\mathcal{V}}}$ , the product of the left and right Virasoro algebras. This algebra, ${\mathcal{S}}$ —which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field ${S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}$ , with a symmetric form ${S_{\alpha\beta}}$ and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) 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 ${\mathfrak{gl}(1|1)}$ spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of ${\mathfrak{sp}_{N-2}}$ . The semi-simple part of JTL$_{N}$ is represented by ${U \mathfrak{sp}_{N-2}}$ , providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL$_{N}$ image represented in the spin-chain. On the continuum side, simple modules over ${\mathcal{S}}$ are identified with “fundamental” representations of ${\mathfrak{sp}_\infty}$ .