Equivariant sl(n)-link homology

TL;DR

This paper introduces a new bigraded homology theory for links that generalizes existing $sl_n$-homology, connecting link invariants with equivariant cohomology and extending Khovanov-Rozansky theories.

Contribution

It constructs a novel $sl_n$-link homology theory that unifies and extends previous homologies, incorporating equivariant cohomology structures.

Findings

Homology theory closely related to $U(n)$-equivariant cohomology of $ ext{CP}^{n-1}$

Specializes to Khovanov-Rozansky $sl_n$-homology

Provides a universal framework inspired by Frobenius extensions

Abstract

For every positive integer $n$ we construct a bigraded homology theory for links, such that the corresponding invariant of the unknot is closely related to the U(n)-equivariant cohomology ring of $\mathbb{CP}^{n-1}$; our construction specializes to the Khovanov-Rozansky $sl_n$-homology. We are motivated by the "universal" rank two Frobenius extension studied by M. Khovanov in \cite{Kh3} for $sl_2$-homology.