[Paper Review] Mirror symmetry and actions of braid groups on derived categories

Richard Thomas|ArXiv.org|Jan 7, 2000

Homotopy and Cohomology in Algebraic Topology14 references16 citations

TL;DR

This paper establishes that braid group actions arise naturally on derived categories of coherent sheaves on Calabi-Yau manifolds via spherical objects, providing a derived category counterpart to symplectic braid group actions on Lagrangian spheres in mirror duals. The key contribution is a proof of faithfulness of these actions in dimensions ≥2, linking mirror symmetry to mutations of exceptional collections and Fourier-Mukai transforms.

ABSTRACT

Talk given at Harvard, January 1999, published in the Proceedings of the Harvard Winter School on mirror symmetry, vector bundles and lagrangian cycles, 1999, International Press. Surveys the joint work [ST, KS] with Paul Seidel and Mikhail Khovanov.

Motivation & Objective

• To establish a derived category analogue of mirror symmetry by linking braid group actions on coherent sheaves to symplectic automorphisms on the mirror.
• To prove that braid group actions induced by spherical objects in derived categories are faithful in dimensions ≥2.
• To clarify the relationship between exceptional collections on Fano manifolds and spherical objects on Calabi-Yau threefolds via simple maps.
• To provide a derived category framework for understanding mutations of bundles and their mirror duals in the context of singularities and resolutions.

Proposed method

• Construct autoequivalences of derived categories using Fourier-Mukai transforms associated with spherical objects.
• Define spherical objects as those satisfying Ext*(F,F) ≅ ℂ[0] ⊕ ℂ[N] in the derived category.
• Use the twist functor T_F associated with a spherical object F to generate braid group actions on D^b(X).
• Apply the theory of differential graded algebras and their cohomology to analyze the faithfulness of braid group actions.
• Prove intrinsic formality of the graded algebra Ext*(F_i, F_j) for A_n-chains of N-spherical objects with N ≥ 2.
• Relate symplectic Dehn twists on Lagrangian spheres to derived autoequivalences via mirror symmetry conjectures.

Experimental results

Research questions

• RQ1 How do braid group actions on derived categories of coherent sheaves arise from spherical objects in Calabi-Yau threefolds?
• RQ2 What is the mirror dual of a braid group action generated by spherical twists in the derived category?
• RQ3 Under what conditions does a map from a Calabi-Yau to a Fano manifold induce spherical objects from exceptional ones?
• RQ4 Is the braid group action on derived categories faithful in dimensions ≥2, and what conditions ensure this?
• RQ5 How do mutations of exceptional collections on Fano manifolds relate to spherical twists on the mirror Calabi-Yau?

Key findings

• Braid group actions on D^b(X) are constructed via spherical objects, with each spherical object giving rise to a twist functor T_F.
• The action of the braid group B_{n+1} on D^b(X) is faithful when the spherical objects form an A_n-chain and the dimension N ≥ 2.
• The graded algebra of Ext groups between spherical objects in an A_n-chain is intrinsically formal, ensuring faithfulness of the action.
• For a Fano divisor Y in a Calabi-Yau X with normal bundle ω_Y, the pushforward of an exceptional sheaf on Y yields a spherical object on X.
• In dimension 1, the braid group action may fail to be faithful; for example, if L is a degree-zero line bundle of order 2 on an elliptic curve, then (T_L^{-1}T_O)^2 ≅ id.
• The mirror dual of symplectic Dehn twists on Lagrangian spheres is realized as spherical twist functors on the derived category of coherent sheaves.