Modular tensor category

A modular tensor category (also called a modular fusion category) is a type of tensor category that plays a role in the areas of topological quantum field theory, conformal field theory, and quantum algebra. Modular tensor categories were introduced in 1989 by the physicists Greg Moore and Nathan Seiberg in the context of rational conformal field theory.^[1] In the context of quantum field theory, modular tensor categories are used to store algebraic data for rational conformal field theories in 2 dimensional spacetime,^[1] and topological quantum field theories in 3 dimensional spacetime.^[2] In the context of condensed matter physics, modular tensor categories play a role in the algebraic theory of topological quantum information, as they are used to store the algebraic data describing anyons in topological quantum phases of matter.^[3]

Mathematically, a modular tensor category is a rigid, semisimple,^[4] braided fusion category with a non-degenerate braiding, ensuring a well-defined notion of topological invariance. These categories naturally arise in quantum groups, representation theory, and low-dimensional topology, where they are used to construct knot and three-manifold invariants.

^ ^a ^b Moore, G; Seiberg, N (1989-09-01). Lectures on RCFT (Rational Conformal Field Theory) (Report). doi:10.2172/7038633. OSTI 7038633.
^ Bartlett, Bruce; Douglas, Christopher L.; Schommer-Pries, Christopher J.; Vicary, Jamie (2015-09-22). "Modular categories as representations of the 3-dimensional bordism 2-category". arXiv:1509.06811 [math.AT].
^ Kitaev, Alexei (2006). "Anyons in an exactly solved model and beyond". Annals of Physics. 321 (1). Elsevier BV: 2–111. arXiv:cond-mat/0506438. Bibcode:2006AnPhy.321....2K. doi:10.1016/j.aop.2005.10.005. ISSN 0003-4916.
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