On finite symmetries and their gauging in two dimensions
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It is well-known that if we gauge a ℤ n symmetry in two dimensions, a dual ℤ n symmetry appears, such that re-gauging this dual ℤ n symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.
KeywordsAnyons Discrete Symmetries Global Symmetries Topological Field Theories
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- P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, AMS, Providence, RI (2015).Google Scholar
- G.W. Moore and N. Seiberg, Lectures on RCFT, in Strings ’89, Proceedings of the Trieste Spring School on Superstrings, World Scientific (1990) [http://www.physics.rutgers.edu/~gmoore/LecturesRCFT.pdf] [INSPIRE].
- P. Etingof and S. Gelaki, Isocategorical groups, Int. Math. Res. Not. (2001) 59, [math/0007196].
- V. Ostrik, Fusion categories of rank 2, Math. Res. Lett. 10 (2003) 177 [math/0203255].
- J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Defect lines, dualities and generalised orbifolds, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, August 3-8, 2009 [DOI: https://doi.org/10.1142/9789814304634_0056] [arXiv:0909.5013] [INSPIRE].
- V. Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003) 177 [math/0111139].
- D. Naidu, Categorical Morita equivalence for group-theoretical categories, Comm. Algebra 35 (2007) 3544 [math/0605530].
- B. Uribe, On the classification of pointed fusion categories up to weak morita equivalence, arXiv:1511.05522.
- P. Etingof, S. Gelaki and V. Ostrik, Classification of fusion categories of dimension pq, Int. Math. Res. Not. 57 (2004) 3041 [math/0304194].
- G.I. Kac and V.G. Paljutkin, Finite ring groups, Trans. Moscow Math. Soc. (1966) 251 [Tr. Mosk. Mat. Obs. 15 (1966) 224].Google Scholar
- D. Handel, On products in the cohomology of the dihedral groups, Tohoku Math. J. (2) 45 (1993) 13.Google Scholar