TY - JOUR

T1 - The Quantum Group Dual of the First-Row Subcategory for the Generic Virasoro VOA

AU - Koshida, Shinji

AU - Kytölä, Kalle

N1 - Funding Information:
SK is supported by the Grant-in-Aid for JSPS Fellows (No. 19J01279).
Publisher Copyright:
© 2021, The Author(s).

PY - 2022/1

Y1 - 2022/1

N2 - 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 Uq(sl2).

AB - 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 Uq(sl2).

UR - http://www.scopus.com/inward/record.url?scp=85120343519&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04266-w

DO - 10.1007/s00220-021-04266-w

M3 - Article

AN - SCOPUS:85120343519

SN - 0010-3616

VL - 389

SP - 1135

EP - 1213

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -