Projective objects and the modified trace in factorisable finite tensor categories

<jats:p>For <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline1.png" /><jats:tex-math>${\mathcal{C}}$</jats:tex-math></jats:alternatives></jats:inline-formula> a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:<jats:list list-type="number"><jats:list-item><jats:label>(1)</jats:label><jats:p><jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline2.png" /><jats:tex-math>${\mathcal{C}}$</jats:tex-math></jats:alternatives></jats:inline-formula> always contains a simple projective object;</jats:p></jats:list-item><jats:list-item><jats:label>(2)</jats:label><jats:p>if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline3.png" /><jats:tex-math>${\mathcal{C}}$</jats:tex-math></jats:alternatives></jats:inline-formula> is in addition ribbon, the internal characters of projective modules span a submodule for the projective <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline4.png" /><jats:tex-math>$\text{SL}(2,\mathbb{Z})$</jats:tex-math></jats:alternatives></jats:inline-formula>-action;</jats:p></jats:list-item><jats:list-item><jats:label>(3)</jats:label><jats:p>the action of the Grothendieck ring of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline5.png" /><jats:tex-math>${\mathcal{C}}$</jats:tex-math></jats:alternatives></jats:inline-formula> on the span of internal characters of projective objects can be diagonalised;</jats:p></jats:list-item><jats:list-item><jats:label>(4)</jats:label><jats:p>the linearised Grothendieck ring of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline6.png" /><jats:tex-math>${\mathcal{C}}$</jats:tex-math></jats:alternatives></jats:inline-formula> is semisimple if and only if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline7.png" /><jats:tex-math>${\mathcal{C}}$</jats:tex-math></jats:alternatives></jats:inline-formula> is semisimple.</jats:p></jats:list-item></jats:list></jats:p><jats:p>Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline8.png" /><jats:tex-math>${\mathcal{C}}$</jats:tex-math></jats:alternatives></jats:inline-formula> carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline9.png" /><jats:tex-math>$S$</jats:tex-math></jats:alternatives></jats:inline-formula>-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007034_inline10.png" /><jats:tex-math>$S$</jats:tex-math></jats:alternatives></jats:inline-formula>-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.</jats:p>