[Paper Review] Braid cobordisms, triangulated categories, and flag varieties

Mikhail Khovanov, Richard Thomas|ArXiv.org|Sep 12, 2006

Algebraic structures and combinatorial models35 references4 citations

TL;DR

This paper extends braid group actions on triangulated categories to projective actions of the category of braid cobordisms, constructing a faithful action of the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety. The key contribution is a geometric categorification of braid group representations via Fourier-Mukai transforms and sheaf-theoretic functors, realizing the Yang-Baxter relation and braid movie moves in a coherent sheaf setting.

ABSTRACT

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

Motivation & Objective

• To extend weak braid group actions on triangulated categories to genuine actions of the category of braid cobordisms.
• To construct a projective action of the affine braid group on the derived category of coherent sheaves on the cotangent bundle of the full flag variety.
• To realize braid group representations geometrically through sheaf-theoretic functors and Fourier-Mukai transforms.
• To verify that the constructed functors satisfy the Yang-Baxter equation and braid movie moves, ensuring consistency with topological invariance.

Proposed method

• Utilizes the category of braid cobordisms as a monoidal category with braid isotopy classes as objects and natural transformations as morphisms.
• Constructs functors on the derived category $D^b(T^*Fl)$ using Fourier-Mukai transforms associated with specific coherent sheaves on flag varieties and their cotangent bundles.
• Applies sheaf-theoretic functors $T_i$, $U_i$, and their adjoints to model braid group generators and their relations.
• Employs relative cohomology and Chern classes to analyze the behavior of functors under composition and isotopy.
• Verifies the Yang-Baxter relation by showing isomorphisms between compositions of functors via explicit isomorphisms in the derived category.
• Proves invariance under braid movie moves by checking commutativity of diagrams involving evaluation and coevaluation maps, using properties of $l_{X_i}$ and $r_{X_i}$.

Experimental results

Research questions

• RQ1 Can braid group actions on triangulated categories be lifted to actions of the category of braid cobordisms?
• RQ2 How can the affine braid group act projectively on the derived category of coherent sheaves on $T^*Fl$?
• RQ3 Do the constructed functors satisfy the Yang-Baxter relation and braid movie moves in the derived category?
• RQ4 What is the geometric realization of braid group representations via coherent sheaves and Fourier-Mukai transforms?
• RQ5 How do the functors $T_i$ and $U_i$ interact under composition and isotopy in the context of flag varieties?

Key findings

• The affine braid group acts projectively on $D^b(T^*Fl)$ via functors induced by coherent sheaves and Fourier-Mukai transforms.
• The functors $T_i$ and $U_i$ satisfy the Yang-Baxter relation, with $T_i oxtimes T_j oxtimes T_i o T_j oxtimes T_i oxtimes T_j$ realized as an isomorphism in the derived category.
• Braid movie moves, including those with negative branch points, are verified to commute via isomorphisms involving $l_{X_i}$ and $r_{X_i}$, ensuring topological invariance.
• The isomorphism $U_i oxtimes T_j oxtimes T_i o T_j oxtimes T_i oxtimes U_j$ intertwines evaluation maps, confirming consistency of the construction.
• The compositions $ (l_{X_{i+1}} - l_{X_{i-1}})oxtimes ext{id} $ and $ ext{id} oxtimes (r_{X_{j+1}} - r_{X_{j-1}}) $ are equal, confirming commutativity in the negative move 13.
• The construction generalizes to surfaces with chains of rational curves and provides a geometric categorification of braid group actions in the context of flag varieties.