Derived traces of Soergel categories

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

Cyclotomic expansions for $\mathfrak{gl}_N$ knot invariants via interpolation Macdonald polynomials

“…, so c ζ,V indeed defines a braiding. Even more concretely, we get the braiding as the composition (11)…”

“…Finally, we would like to comment on potential ideas for categorification of these results. The ring of symmetric polynomials in N variables is naturally categorified by the category of annular gl N -webs, with morphisms given by annular foams [6,32,33,13,11]. By the work of the second author and Wedrich [13], one can interpret it as a symmetric monoidal Karoubian category generated by one object E corresponding to a single essential circle.…”

“…On the other hand, the conjectures of the second author, Negut , and Rasmussen ( [12], see [10,11] for further discussions) relate a version of the annular category to the derived category of the Hilbert scheme of points on the plane. The interpolation Macdonald polynomials appear in that context as well [7].…”