We prove in this paper that the genus zero data of a modular functor determines the modular functor. We do this by establishing that the S-matrix in genus one with one point labeled arbitrarily can be expressed in terms of the genus zero information and we give an explicit formula. We do not assume the modular functor in question has duality or is unitary, in order to establish this.
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Jørgen Ellegaard Andersen, Kenji Ueno, Modular functors are determined by their genus zero data. Quantum Topol. 3 (2012), no. 3/4, pp. 255–291DOI 10.4171/QT/29