Operator lift of the Reshetikhin-Turaev formalism to Khovanov-Rozansky topological quantum field theory

Topological quantum field theory (TQFT) is a powerful tool to describe homologies, which normally involve complexes and a variety of maps/morphisms, what makes a functional integration approach with a sum over a single kind of map seemingly problematic. In TQFT this problem is overcome by exploiting the rich set of zero modes of BRST operators, which appear sufficient to describe complexes. We explain what this approach looks like for the important class of Khovanov-Rozansky (KR) cohomologies, which categorify the observables (Wilson lines or knot polynomials) in 3D Chern-Simons theory. We develop a construction of odd differential operators, associated with all link diagrams, including tangles with open ends. These operators become nilpotent only for a diagram with no external legs, but even for open tangles one can develop a factorization formalism, which preserves Reidemeister/topological invariance—the symmetry of the problem. This technique seems much more “physical” than the conventional language of homological algebra and should have many applications to various problems beyond Chern-Simons theory. We also hope that this language will provide efficient algorithms, and finally allow one to computerize the calculation of KR cohomologies—for closed diagrams and for open tangles.