[Paper Review] Clasp technology to knot homology via the affine Grassmannian

Sabin Cautis|arXiv (Cornell University)|Jul 9, 2012

Algebraic structures and combinatorial models41 references17 citations

TL;DR

This paper presents a categorification of Reshetikhin-Turaev tangle invariants for $υ\text{-}\mathfrak{sl}_m$ using infinite twists in the affine Grassmannian to realize generalized Jones-Wenzl projectors (clasps) as limits of full twists. The key contribution is a uniform construction of homological knot invariants via the $U_q(\mathfrak{sl}_\infty)$-module $\Lambda_q^{m\infty}(\mathbb{C}^m \otimes \mathbb{C}^{2\infty})$, realized through convolution algebras on the affine Grassmannian or Nakajima quiver varieties, yielding equivalent invariants via skew Howe duality.

ABSTRACT

We categorify all the Reshetikhin-Turaev tangle invariants of type A. Our main tool is a categorification of the generalized Jones-Wenzl projectors (a.k.a. clasps) as infinite twists. Applying this to certain convolution product varieties on the affine Grassmannian we extend our earlier work with Kamnitzer from standard to arbitrary representations.

Motivation & Objective

• To provide a uniform categorification of all Reshetikhin-Turaev tangle invariants for $\mathfrak{sl}_m$ using higher representation theory.
• To construct generalized Jones-Wenzl projectors (clasps) as limits of full twists in the affine Grassmannian.
• To establish equivalence between invariants arising from the affine Grassmannian and Nakajima quiver varieties via skew Howe duality.
• To extend earlier results from standard to arbitrary representations using the $U_q(\mathfrak{sl}_\infty)$-module $\Lambda_q^{m\infty}(\mathbb{C}^m \otimes \mathbb{C}^{2\infty})$.
• To unify categorified tangle invariants through infinite twist limits and 2-categorical structures on convolution varieties.

Proposed method

• Categorify clasps as the limit $\lim_{\ell \to \infty} T_\omega^{2\ell} \mathbf{1}_{\underline{i}}$, where $T_\omega$ is the full twist on $n$ strands.
• Use the action of the braid group on $\Lambda_q^{N}(\mathbb{C}^m \otimes \mathbb{C}^{2N})$ via skew Howe duality to define tangle invariants for finite $N$.
• Pass to the $\infty$-limit $N \to \infty$ to obtain the $U_q(\mathfrak{sl}_\infty)$-module $\Lambda_q^{m\infty}(\mathbb{C}^m \otimes \mathbb{C}^{2\infty})$ for a uniform categorification.
• Realize the categorified module via convolution algebras on the affine Grassmannian of $\mathrm{PGL}_m$, giving a geometric categorification.
• Construct the same invariants using Nakajima quiver varieties, showing equivalence via the 2-category $\mathcal{K}_{\mathrm{Gr},m}$ and $\mathcal{K}_{\mathrm{Q},m}$.
• Use Rickard complexes and 2-representations of $U_q(\mathfrak{sl}_\infty)$ to define the categorified tangle functors and verify relations.

Experimental results

Research questions

• RQ1 How can the generalized Jones-Wenzl projectors (clasps) for $\mathfrak{sl}_m$ be categorified using geometric and categorical constructions?
• RQ2 Can the Reshetikhin-Turaev tangle invariants for arbitrary representations of $\mathfrak{sl}_m$ be uniformly categorified via a limit construction?
• RQ3 What is the relationship between categorifications via the affine Grassmannian and via Nakajima quiver varieties for the same $U_q(\mathfrak{sl}_\infty)$-module?
• RQ4 How does the infinite twist limit $\lim_{\ell \to \infty} T_\omega^{2\ell}$ recover the clasp projector in the categorified setting?
• RQ5 To what extent do the 2-categorical structures on the affine Grassmannian and quiver varieties yield equivalent homological knot invariants?

Key findings

• The clasp projector $P\mathbf{1}_{\underline{i}}$ is realized as the limit $\lim_{\ell \to \infty} T_\omega^{2\ell} \mathbf{1}_{\underline{i}}$, providing a geometric categorification of the Jones-Wenzl projector in type $A$.
• The $U_q(\mathfrak{sl}_\infty)$-module $\Lambda_q^{m\infty}(\mathbb{C}^m \otimes \mathbb{C}^{2\infty})$ supports a uniform categorification of all Reshetikhin-Turaev tangle invariants for $\mathfrak{sl}_m$.
• Categorified invariants constructed via the affine Grassmannian of $\mathrm{PGL}_m$ are equivalent to those from Nakajima quiver varieties, via the 2-category $\mathcal{K}_{\mathrm{Gr},m}$ and $\mathcal{K}_{\mathrm{Q},m}$.
• The braid group action on $\Lambda_q^N(\mathbb{C}^m \otimes \mathbb{C}^{2N})$ via skew Howe duality matches the Reshetikhin-Turaev R-matrix construction for $\mathfrak{sl}_m$.
• The categorified clasps are realized as derived tensor products of bimodules over $A_{1,2} = \mathbb{C}[t]/t^2$, with Koszul duality relating them to earlier constructions in type $A_1$.
• The construction extends previous work with Kamnitzer to arbitrary representations, not just fundamental ones, by working in the $\infty$-limit.