Fusion Rules for the Virasoro Algebra of Central Charge 25
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Fusion Rules for the Virasoro Algebra of Central Charge 25 Florencia Orosz Hunziker1 Received: 17 April 2019 / Accepted: 2 September 2019 / © Springer Nature B.V. 2019
Abstract Let F25 be the family of irreducible lowest weight modules for the Virasoro algebra of central charge 25 which are not isomorphic to Verma modules. Let L(25, 0) be the Virasoro vertex operator algebra of central charge 25. We prove that the fusion rules for the L(25, 0)-modules in F25 are in correspondence with the tensor rules for the irreducible finite dimensional representations of sl(2, C), extending the known correspondence between modules for the Virasoro algebras of dual central charges 1 and 25. Keywords Vertex algebras · Virasoro algebra · Virasoro vertex algebras · Fusion rules Mathematics Subject Classification (2010) 17B69
1 introduction In 1990, in their important paper [3], Feigin and Fuchs described the structure of Verma modules for the Virasoro algebra and the homomorphisms between them. They stated the projection formulas for singular vectors on the density modules and described the duality between the category of Verma modules with central charge c and the category of Verma modules with central charge 26 − c for c ∈ C. In particular, they established an antiequivalence of additive categories between the category of Verma modules for V irc=1 and the Verma modules for V irc=25 which assigns the Verma module of central charge 1 and lowest weight h, M(1, h), to the Verma module of central charge 25 and lowest weight 1 − h, M(25, 1 − h) and reverses morphisms. On the other hand, Segal on his 1981 paper [18] noted a correspondence between the finite dimensional irreducible representations of sl(2, C) and certain representations of the Virasoro algebra of central charge 1 by a dual pair type of argument. Later, Frenkel and Zhu, proved in [6] that for c ∈ C, L(c, 0), the irreducible quotient of the Verma module M(c, 0), has a vertex algebra operator algebra Presented by: Vyjayanthi Chari Florencia Orosz Hunziker
[email protected] 1
Department of Mathematics, Yale University, 442 Dunham Lab, 10 Hillhouse Ave, New Haven, CT 06511, USA
F.O. Hunziker
structure and that the irreducible quotient of M(c, h), L(c, h), is an L(c, 0)-module for h ∈ C. The vertex algebra version of the decomposition noted in [18] was proved by Dong and Griess in [2]. In his doctorate thesis [19], Styrkas used the dual pair decomposition of Segal to state an antiequivalence of tensor categories between the category of irreducible finite dimensional modules for sl(2, C) and the semisimple tensor category generated by the irreducible L(1, 0)-modules which are not Verma modules, i.e the modules of the from n2 L 1, 4 for n ≥ 0. Milas, independently proved in [14], that the fusion rules for the irreducible non verma L(1, 0) modules coincide with the tensor rules of the irreducible finite dimensional representations of sl(2, C). More recently, McRae proved in [17] using the fusion rules from [14] that the semisimple category generated by the L(1
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