[Paper Review] Soergel bimodules and matrix factorizations

Alexei Oblomkov, Lev Rozansky|arXiv (Cornell University)|Oct 27, 2020

Geometric and Algebraic Topology32 references4 citations

TL;DR

This paper establishes a canonical isomorphism between the algebraically defined Khovanov-Rozansky triply graded link homology (HHH_alg) and the geometrically constructed homology via Soergel bimodules and matrix factorizations (HHH_geo), proving a geometric realization of HHH as derived sections of a T_{qt}-equivariant sheaf on the Hilbert scheme. The key result confirms the Gorsky-Negut-Rasmussen conjecture and implies the q→t/q symmetry for knot homology, with explicit computation for torus links via cohomology of the punctual Hilbert scheme.

ABSTRACT

We establish an isomorphism between the Khovanov-Rozansky triply graded link homology and the geometric triply graded homology due to the authors. Hence we provide an interpretation of the Khovanov-Rozansky homology of the closure of a braid $β$ as the space of derived sections of a $\mathbb{C}^* imes \mathbb{C}^*$- equivariant sheaf $Tr(β)$ on the Hilbert scheme $Hilb_n(\mathbb{C}^2)$, thus proving a version of Gorsky-Negut-Rasmussen conjecture \cite{GorskyNegutRasmussen16}. As a consequence we prove that Khovanov-Rozansky homology of knots satisfies the $q o t/q$ symmetry conjectured by Dunfield-Gukov-Rasmussen \cite{DunfieldGukovRasmussen06}. We also apply our main result to compute the Khovanov-Rozansky homology of torus links.

Motivation & Objective

• Establish a canonical isomorphism between algebraically defined Khovanov-Rozansky homology (HHH_alg) and geometrically constructed homology (HHH_geo) for braid closures.
• Provide a geometric interpretation of HHH as derived sections of a T_{qt}-equivariant sheaf Tr(β) on the Hilbert scheme Hilb_n(C^2), thus proving a version of the Gorsky-Negut-Rasmussen conjecture.
• Prove the q→t/q symmetry for knot homology, confirming a long-standing conjecture by Dunfield-Gukov-Rasmussen.
• Compute the Khovanov-Rozansky homology of torus links using cohomology of the punctual Hilbert scheme and determinant bundles.
• Construct a fully faithful monoidal functor from stable matrix factorizations to Soergel bimodules, paving the way for categorical equivalences.

Proposed method

• Define a trace functor Tr: Br_n → D^per_{T_{qt}}(Hilb_n(C^2)) that assigns to each braid β a two-periodic complex of T_{qt}-equivariant coherent sheaves.
• Construct a monoidal and fully-faithful functor B: MF_n^flat → SBim_n from a subcategory of stable matrix factorizations to the category of Soergel bimodules.
• Use the derived RHom construction RHom(O⊗Λ^•B, Tr(β)) to define the geometric link homology HHH_geo(β).
• Prove that HHH_geo(β) is invariant under isotopy of the braid closure L(β), establishing it as a link invariant.
• Apply the relation Tr(β·FT) = Tr(β)⊗det(B) for the full twist braid FT to compute HHH for torus links.
• Utilize Haiman's results on cohomology vanishing and localization to compute H^0(Hilb_n(C^2,0), det(B)^k ⊗ Λ(B)) for torus knots T_{n,nk+1}.

Experimental results

Research questions

• RQ1 Is the algebraic Khovanov-Rozansky homology HHH_alg(β) canonically isomorphic to the geometric HHH_geo(β) constructed via Soergel bimodules and matrix factorizations?
• RQ2 Can the HHH homology of a braid closure be realized as the derived sections of a T_{qt}-equivariant sheaf on the Hilbert scheme Hilb_n(C^2)?
• RQ3 Does the HHH homology of knots satisfy the q→t/q symmetry conjectured by Dunfield-Gukov-Rasmussen?
• RQ4 What is the geometric formula for the HHH homology of torus links T_{n,nk+1}?
• RQ5 Is the functor B: MF_n^flat → SBim_n an equivalence, or can it be extended to an equivalence between MF_n^st and Ho(SBim_n)?

Key findings

• The algebraic and geometric triply graded link homologies are canonically isomorphic: HHH_alg(β) ≅ HHH_geo(β) for any braid β ∈ Br_n.
• The HHH homology of a knot is invariant under the transformation q → t/q, confirming the Dunfield-Gukov-Rasmussen conjecture.
• The HHH homology of the torus link T_{n,nk+1} is given by (1−q^2)·HHH_alg(T_{n,nk+1}) = H^0(Hilb_n(C^2,0), det(B)^k ⊗ Λ(B)).
• Higher cohomology groups of the sheaf det(B)^k ⊗ Λ(B) on the punctual Hilbert scheme vanish, as established by Haiman.
• The geometric trace functor satisfies Tr(β·FT) = Tr(β)⊗det(B), enabling the computation of HHH for torus links via the full twist.
• The functor B: MF_n^flat → SBim_n is monoidal and fully faithful, suggesting a deep categorical equivalence between matrix factorizations and Soergel bimodules.