Projective objects and the modified trace in factorisable finite tensor categories

Azat M. Gainutdinov (1, 2) , Ingo Runkel (2)

1 IDP - Institut Denis Poisson
2 Fachbereich Mathematik [Hamburg]

For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff 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 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.

Mots clés: category: tensor; fermion: symplectic; S-matrix; modular; algebra; Hopf

Domaines: Physique mathématique [math-ph]; Physique des Hautes Energies - Théorie [hep-th]

Azat M. Gainutdinov, Ingo Runkel. Projective objects and the modified trace in factorisable finite tensor categories. Compos.Math., 2020, 156, pp.770-821. ⟨10.1112/S0010437X20007034⟩. ⟨hal-02536321⟩