Abstract
In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules, we prove the analyticity of compositions of intertwining operators, and we show that the conformal blocks are fully determined by the quantum group method. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the 6j-symbols of the quantum group. Our results constitute a concrete duality between a VOA and a quantum group, and they will serve as the key tools to establish the equivalence of the first-row subcategory of modules of the generic Virasoro VOA and the category of (type-1) finite-dimensional representations of {mathcal {U}}_q (mathfrak {sl}_2).
Highlights
Conformal field theories, vertex operator algebras, and quantum groups
Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules, we prove the analyticity of compositions of intertwining operators, and we show that the conformal blocks are fully determined by the quantum group method
Our results constitute a concrete duality between a vertex operator algebra (VOA) and a quantum group, and they will serve as the key tools to establish the equivalence of the first-row subcategory of modules of the generic Virasoro VOA and the category of finite-dimensional representations of Uq
Summary
Vertex operator algebras, and quantum groups. Twodimensional conformal field theories (CFT) are an outstanding example of extremely fruitful interaction of physics and mathematics [DFMS97,Gaw99,Hua12,Nah00]. It is the semisimple category whose infinitely many simple modules are the irreducible Virasoro highest weight modules “in the first row of the Kac table”, i.e., with highest weights h = h1,s, s ∈ Z>0, when hr,s, r, s ∈ Z>0, denote the usual Kac labeled highest weights [Kac79] These correspond to a certain infinite set of (chiral) primary fields in a CFT, which has been found to be relevant in particular to questions in conformally invariant random geometry— the two simplest of these primary fields after the identity, with Kac labeled conformal weights h1,2 and h1,3, correspond to SLE-type curves’ starting points [BB03b,Dub07] and boundary visit points [BB03a,JJK16,Dub15], respectively. Our main results are that the first row subcategory of modules of the generic Virasoro VOA is stable under fusion, and detailed calculations of the fusions with the quantum group method
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