No abstract

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. We show that this dg algebra is formal, and calculate its homology explicitly, for all Coxeter groups. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and identify the derived horizontal trace of Soergel bimodules in type A with the homotopy category of perfect dg modules of an explicit algebra. As an application we obtain a derived annular Khovanov-Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus. CONTENTS 1. Introduction Acknowledgements Funding 2. Facts from homological algebra 3. Standard dg categorical constructions 4. Homotopy idempotents and the Karoubi envelope 5. The bar complex for dg categories 6. The dg monoidal center and trace 7. Traces of the Soergel category 8. The Soergel category in type A References