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Projective objects and the modified trace in factorisable finite tensor categories

Published online by Cambridge University Press:  26 March 2020

Azat M. Gainutdinov
Institut Denis Poisson, CNRS, Université de Tours, Université d’Orléans, Parc de Grandmont, 37200Tours, France email Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146Hamburg, Germany
Ingo Runkel
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146Hamburg, Germany email


For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:

  1. (1) ${\mathcal{C}}$ always contains a simple projective object;

  2. (2) if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;

  3. (3) the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;

  4. (4) the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.

Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

Research Article
© The Authors 2020

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