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 \({\mathfrak{gl}(1|1)}\) 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 \({\mathfrak{sp}_\infty}\).
Similar content being viewed by others
References
Rohsiepe, F.: On reducible but indecomposable representations of the Virasoro algebra. arXiv:hep-th/9611160
Gaberdiel M.: Fusion in conformal field theory as the tensor product of the symmetry algebra. Int. J. Mod. Phys. A 9, 4619 (1994)
Gaberdiel M., Kausch H.: Indecomposable fusion products. Nucl. Phys. B 477, 293–318 (1996)
Mathieu P., Ridout D.: From percolation to logarithmic conformal field theory. Phys. Lett. B 657, 120 (2007)
Eberle H., Flohr M.: Virasoro representations and fusion for general augmented minimal models. J. Phys. A 39, 15245–15286 (2006)
Pearce P., Rasmussen J., Zuber J.B.: Logarithmic minimal models. J. Stat. Mech. 0611, 017 (2006)
Read N., Saleur H.: Associative-algebraic approach to logarithmic conformal field theories. Nucl. Phys. B 777, 316 (2007)
Pasquier V., Saleur H.: Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B 330, 523 (1990)
Martin P.P.: Potts Models and Related Problems in Statistical Mechanics. World Scientific, Singapore (1991)
Read N., Saleur H.: Enlarged symmetry algebras of spin chains, loop models, and S-matrices. Nucl. Phys. B 777, 263 (2007)
Rasmussen J., Pearce P.: Fusion algebras of logarithmic minimal models. J. Phys. A 40, 13711–13734 (2007)
Kytölä K., Ridout D.: On staggered indecomposable Virasoro modules. J. Math. Phys. 50, 123503 (2009)
Feigin B.L., Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu.: Kazhdan–Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT. Theor. Math. Phys. 148, 1210–1235 (2006)
Feigin B.L., Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu.: Kazhdan–Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models. J. Math. Phys. 48, 032303 (2007)
Bushlanov P.V., Feigin B.L., Gainutdinov A.M., Tipunin I.Yu.: Lusztig limit of quantum \({s\ell(2)}\) at root of unity and fusion of (1, p) Virasoro logarithmic minimal models. Nucl. Phys. B 818 [FS], 179–195 (2009)
Gaberdiel M., Runkel I.: From boundary to bulk in logarithmic CFT. J. Phys. A 41, 075402 (2008)
Gaberdiel M., Runkel I., Wood S.: A modular invariant bulk theory for the \({c=0}\) triplet model. J. Phys. A 44, 015204 (2011)
Kausch H.G.: Extended conformal algebras generated by a multiplet of primary fields. Phys. Lett. B 259, 448 (1991)
Feigin B.L., Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu.: Logarithmic extensions of minimal models: characters and modular transformations. Nucl. Phys. B 757, 303–343 (2006)
Saleur H., Schomerus V.: The \({GL(1|1)}\) WZW model: from supergeometry to logarithmic CFT. Nucl. Phys. B 734, 221–245 (2006)
Saleur H., Schomerus V.: On the \({SU(2|1)}\) WZW model and its statistical mechanics applications. Nucl. Phys. B 775, 312 (2007)
Gainutdinov A.M., Read N., Saleur H.: Continuum limit and symmetries of the periodic \({\mathfrak{gl}(1|1)}\) spin chain. Nucl. Phys. B 871 [FS], 245–288 (2013)
Gainutdinov A.M., Read N., Saleur H.: Bimodule structure in the periodic \({\mathfrak{gl}(1|1)}\) spin chain. Nucl. Phys. B 871 [FS], 289–329 (2013)
Howe, R.: Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons. In: Flato, M., Sally, P., Zuckerman, G. (eds.) Applications of Group Theory in Physics and Mathematical Physics. Lectures in Applied Math., vol. 21, pp. 179–207. American Mathematical Society, Providence (1985)
Kausch H.: Symplectic fermions. Nucl.Phys. B 583, 513–541 (2000)
Fjelstad J., Fuchs J., Hwang S., Semikhatov A.M., Tipunin I.Yu.: Logarithmic conformal field theories via logarithmic deformations. Nucl. Phys. B 633, 379 (2002)
Read N., Saleur H.: Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions. Nucl. Phys. B 613, 409 (2001)
Dubail J., Jacobsen J., Saleur H.: Conformal field theory at central charge \({c=0}\) : a measure of the indecomposability (b) parameters. Nucl. Phys. B 834, 399 (2010)
Vasseur R., Jacobsen J., Saleur H.: Indecomposability parameters in chiral logarithmic conformal field theory. Nucl. Phys. B 851, 314–345 (2011)
Gainutdinov A.M., Vasseur R.: Lattice fusion rules and logarithmic operator product expansions. Nucl. Phys. B 868, 223–270 (2013)
Bushlanov P.V., Gainutdinov A.M., Tipunin I.Yu.: Kazhdan–Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models. Nucl. Phys. B 862, 232–269 (2012)
Graham J.J., Lehrer G.I.: The representation theory of affine Temperley–Lieb algebras. L’Ens. Math. 44, 173 (1998)
Graham J.J., Lehrer G.I.: The two-step nilpotent representations of the extended Affine Hecke algebra of type A. Compos. Math. 133, 173 (2002)
Martin P.P., Saleur H.: On an algebraic approach to higher-dimensional statistical mechanics. Commun. Math. Phys. 158, 155 (1993)
Martin P.P., Saleur H.: The blob algebra and the periodic Temperley–Lieb algebra. Lett. Math. Phys. 30, 189 (1994)
Gainutdinov A.M., Read N., Saleur H., Vasseur R.: The periodic \({s\ell(2|1)}\) alternating spin chain and its continuum limit as a bulk logarithmic conformal field theory at \({c=0}\). JHEP 1565, 114 (2015)
Graham J.J., Lehrer G.I.: Cellular algebras. Invent. Math. 123, 1–34 (1996)
Goodman, R., Wallach, N.R.: Symmetry, representations, and invariants. In: Graduate Texts in Mathematics (Book 255). Springer, New York (2009)
Martin, P.P., McAnally, D.: On commutants, dual pairs and non-semisimple algebras from statistical mechanics. Int. J. Mod. Phys. A 7(Supp. 1B), 675 (1992)
Martin, P.P.: On Schur–Weyl duality, A n Hecke algebras and quantum sl(N) on \({\otimes^{n+1}{\mathbb C}^N}\). Int. J. Mod. Phys. A 7(Supp. 1B), 645 (1992)
Brocker, T., Dieck, T.T.: Representations of Compact Lie Groups, pp. 271–272. Springer, New York (2013)
Koo W.M., Saleur H.: Representations of the Virasoro algebra from lattice models. Int. J. Mod. Phys. A 8, 5165 (1993)
Batchelor M.T., Cardy J.: Extraordinary transition in the two-dimensional O(n) model. Nucl. Phys. B 506, 553 (1997)
Gainutdinov, A.M., Saleur, H., Tipunin, I.Y.: Lattice W-algebras and logarithmic CFTs. J. Phys. A Math. Theor. 47, 495401 (2014)
Ottesen, J.T.: Infinite dimensional groups and algebras in quantum physics. In: Lecture Notes in Physics. Springer, New York (1995)
Kac, V.: Infinite-Dimensional Lie Algebras. Cambridge University Press, Cambridge (1994)
Gainutdinov, A.M.: TheVirasoro bimodule structure of the bulk symplectic fermions LCFT (unpublished)
Gaberdiel M.R., Kausch H.G.: A local logarithmic conformal field theory. Nucl. Phys. B 538, 631–658 (1999)
Donkin, S.: The q-Schur algebra. In: London Math. Soc. Lecture Note Series, vol. 253. Cambridge University Press, Cambridge (1999)
Cardy J.: Operator content of two-dimensional conformally invariant theories. Nucl. Phys. 270, 186 (1986)
Baake, M., Christe, Ph., Rittenberg, V.: Higher spin conserved currents in c = 1 conformally invariant systems. Nucl. Phys. B 300, 637 (1988)
Grimm, U., Rittenberg,V.: Themodified XXZ Heisenberg chain: conformal invariance, surface exponents of c < 1 systems and hidden symmetries of finite chains. Int. J. Mod. Phys. B 4, 969 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Gainutdinov, A.M., Read, N. & Saleur, H. 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. Commun. Math. Phys. 341, 35–103 (2016). https://doi.org/10.1007/s00220-015-2483-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-015-2483-9