Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs

TL;DR

The paper introduces an operator-based reformulation of KR homology, replacing matrix-factorization language with odd differential operators ${\mathscr{O}}_L$ attached to link diagrams, including open tangles. By organizing vertical and horizontal morphisms into a KR double complex and defining a boundary-enhanced total operator ${\mathscr{O}}_L$, the authors derive invariants as zero modes $H^*({\mathscr{O}}_L)$, preserving Reidemeister invariance and enabling algorithmic computation. They develop localization techniques, MOY-move invariances, and explicit constructions for blinking and splitting 4-valent vertices, paving the way to practical KR cohomology calculations and connections to RT $R$-matrix formalism. The framework connects KR categorification to Morse/Floer-type perspectives and suggests efficient avenues for computing superpolynomials for knots and tangles, with Hopf-link and other examples illustrating the approach. Overall, this operator lift provides a physically intuitive, locality-driven, and potentially automatable route to KR invariants with broad applicability beyond Chern–Simons theory.

Abstract

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 maps seemingly problematic. In TQFT this problem is overcame 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 diagram with no external legs, but even for open tangles one can develop a factorization formalism, which preserve Reidemeister/topological invariance -- the symmetry of the problem. This technique seems much more ``physical'' than 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 to computerize the calculation of KR cohomologies -- for closed diagrams and for open tangles.

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