Twisted modules and \(G\)-equivariantization in logarithmic conformal field theory. (English) Zbl 1480.18014
From a mathematical standpoint, a two-dimensional chiral conformal field theory is to be viewed as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are appropriate in study of logarithmic conformal field theory, in which correlation functions are of logarithmic singularities arising from non-semisimple modules for the chiral algebra, beholden to the logarithmic tensor category theory [Y.-Z. Huang et al., in: Conformal field theories and tensor categories. Proceedings of a workshop held at Beijing International Center for Mathematical Research, Beijing, China, June 13–17, 2011. Heidelberg: Springer. 169–248 (2014; Zbl 1345.81112)].
This paper is concerned with not-necessarily-semisimple or rigid braided tensor categories \(\mathcal{C}\)of modules for the fixed-point vertex operator subalgebra \(V^{G}\) of a vertex operator (super)algebra \(V\) with finite automorphism group \(G\). The principal results in the paper, which are generalizations of the corresponding results of [A. Kirillov jun., Commun. Math. Phys. 229, No. 2, 309–335 (2002; Zbl 1073.17011); “Modular categories and orbifold models. II”, Preprint, arXiv:math/0110221; “On \(G\)-equivariant modular categories”, Preprint, arXiv:math/0401119; M. Müger, J. Algebra 277, No. 1, 256–281 (2004; Zbl 1052.18004); Commun. Math. Phys. 260, No. 3, 727–762 (2005; Zbl 1160.81454)] established by exploiting rigidity and semisimplicity, are as follows.
The author applies these results to the orbifold rationality problem asking whether \(V^{G}\)is strongly rational if \(V\)is strongly rational. It is shown that \(V^{G}\) is indeed strongly rational if \(V\) is strongly rational, \(G\) is any finite automorphism group, and \(V^{G}\) is \(C_{2}\)-cofinite.
This paper is concerned with not-necessarily-semisimple or rigid braided tensor categories \(\mathcal{C}\)of modules for the fixed-point vertex operator subalgebra \(V^{G}\) of a vertex operator (super)algebra \(V\) with finite automorphism group \(G\). The principal results in the paper, which are generalizations of the corresponding results of [A. Kirillov jun., Commun. Math. Phys. 229, No. 2, 309–335 (2002; Zbl 1073.17011); “Modular categories and orbifold models. II”, Preprint, arXiv:math/0110221; “On \(G\)-equivariant modular categories”, Preprint, arXiv:math/0401119; M. Müger, J. Algebra 277, No. 1, 256–281 (2004; Zbl 1052.18004); Commun. Math. Phys. 260, No. 3, 727–762 (2005; Zbl 1160.81454)] established by exploiting rigidity and semisimplicity, are as follows.
- 1.
- Every \(V^{G}\)-module in \(\mathcal{C}\)with a unital and associative \(V\)-action is a direct sum of \(g\)-twisted \(V\)-modules for possibly several \(g\in G\);
- 2.
- The category of all such twisted \(V\)-modules is a braided \(G\)-crossed (super)category;
- 3.
- The \(G\)-equivariantization of this braided \(G\)-crossed (super)category is braided tensor equivalent to the original category \(\mathcal{C}\)of \(V^{G}\)-modules.
The author applies these results to the orbifold rationality problem asking whether \(V^{G}\)is strongly rational if \(V\)is strongly rational. It is shown that \(V^{G}\) is indeed strongly rational if \(V\) is strongly rational, \(G\) is any finite automorphism group, and \(V^{G}\) is \(C_{2}\)-cofinite.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18M20 | Fusion categories, modular tensor categories, modular functors |
| 16B70 | Applications of logic in associative algebras |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 18G40 | Spectral sequences, hypercohomology |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
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