Modular group representation

In mathematics, the modular group representation (or simply modular representation) of a modular tensor category ${\displaystyle {\mathcal {C}}}$ is a representation of the modular group ${\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}$ associated to ${\displaystyle {\mathcal {C}}}$. It is from the existence of the modular representation that modular tensor categories get their name.^[1]

From the perspective of topological quantum field theory, the modular representation of ${\displaystyle {\mathcal {C}}}$ arrises naturally as the representation of the mapping class group of the torus associated to the Reshetikhin–Turaev topological quantum field theory associated to ${\displaystyle {\mathcal {C}}}$.^[2] As such, modular tensor categories can be used to define projective representations of the mapping class groups of all closed surfaces.