A cobordism category attached to Khovanov-Rozansky link homologies based on operads

TL;DR

This paper constructs a cobordism category with operad actions to categorify quantum invariants of links, extending Khovanov-Rozansky homologies using operads and matrix factorizations.

Contribution

It introduces a new cobordism category with operad actions that categorifies $sl_n$ quantum invariants and connects to matrix factorizations.

Findings

Constructed a cobordism category with operad actions from planar arc diagrams.

Provides a categorification of $sl_n$ quantum invariants inspired by Bar-Natan.

Proposes a conjecture on the functorial relationship to matrix factorizations.

Abstract

We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams with values in the homotopy category attached to the cobordism category. Motivated by Bar-Natan's categorification of the Jones polynomial, it categorifies the quantum $sl_n$ quantum invariants and is adapted to the categorification of the $sl_n$ quantum invariants by Khovanov and Rozansky using matrix factorizations. We conjecture to exist the consistency of the cobordism category and to have an explicit functor from the cobordism category to a category of matrix factorizations.