2019/10/31 by Robert McRae · 1 citation
Mathematics · Physics and Astronomy · #math.QA #math-ph #math.MP #math.RT #msc:17B69 #msc:18D10 #msc:81R10
paper · pdf · doi:10.1007/s00220-020-03882-2
published as Comm. Math. Phys., Vol. 383 (2021), no. 3, 1939-2019 · 56 pages, updated contact information and minor edits in this version
arxiv created 2020/02/19 · arxiv updated 2021/04/20
A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories C of modules for the fixed-point vertex operator subalgebra VG of a vertex operator (super)algebra V with finite automorphism group G. The main results are that every VG-module in C with a unital and associative V-action is a direct sum of g-twisted V-modules for possibly several g∈ G, that the category of all such twisted V-modules is a braided G-crossed (super)category, and that the G-equivariantization of this braided G-crossed (super)category is braided tensor equivalent to the original category C of VG-modules. This generalizes results of Kirillov and Müger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether VG is strongly rational if V is strongly rational. We show that VG is indeed strongly rational if V is strongly rational, G is any finite automorphism group, and VG is C2-cofinite.