# Projective objects and the modified trace in factorisable finite tensor categories

## TL;DR

This paper explores the structure of projective objects in factorisable finite tensor categories, establishing their properties, actions, and traces, with applications to symplectic fermion categories and a Verlinde-like formula.

## Contribution

It introduces new results on the existence and properties of projective objects, their internal characters, and the modified trace in such categories, including a Verlinde-like formula.

## Key findings

- Existence of a simple projective object in C
- Internal characters of projectives span a submodule for SL(2,Z) action
- Modified trace of open Hopf links expressed via S-matrix elements

## 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.