Projective objects and the modified trace in factorisable finite tensor categories

Abstract

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.

61 pages, v2: minor changes + references updated

References in corpus (1)

• SL(2,Z)-action for ribbon quasi-Hopf algebras

Cited by in corpus (8)

• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• Non-Semisimple Extended Topological Quantum Field Theories
• Mapping Class Group Representations From Non-Semisimple TQFTs
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
• The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre
• Modified traces for quasi-Hopf algebras
• Modified traces and the Nakayama functor
• Non-semisimple link and manifold invariants for symplectic fermions