A∞-Structures on an Elliptic Curve
By Alexander Polishchuk. 2000
68 citations recorded by the source corpus.
Abstract:The main result of this paper is the proof of the ‘‘transversal part’’ of the homological mirror symmetry conjecture for an elliptic curve that states an equivalence of two A∞-categories: one is built using holomorphic vector bundles on an elliptic curve and another is a subcategory in the Fukaya A∞-category of a torus. The proof is based on the study of A∞-structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particular, m1 should be zero, m2 should coincide with the usual composition). The key observation is that such a structure is uniquely determined up to equivalence by certain triple products.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e77eb11cdde07281dfc13dd2f483dc2552ae10c9
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In Geometry and Topology
- P. Aspinwall
- T. Bridgeland
- Yukinobu Toda
- Alexander Polishchuk
- G. Moore
- A. Kapustin
- Balázs Szendrői
- G. Segal
In Mathematical Physics
- G. Moore
- P. Aspinwall
- Alexander Polishchuk
- Yukinobu Toda
- T. Bridgeland
- A. Kapustin
- Michael R Douglas
- G. Segal
Citations in the corpus
- D-Branes on Calabi-Yau Manifolds (2004, background): “ of technical details that have made this subject the object of a good deal of attention for the past ten years. We refer to [56,62,66–68], for example, for more of the gory details. We also refer to [69,70] where the Fukaya category (complete with its A∞ structure) is determined explicitly for the ”
- Classical Yang–Baxter Equation and the A∞-Constraint (2002, background): “Loosely speaking Massey products in A ∞ -categories are expressions in m n 's which are invariant under arbitrary homotopy of A ∞ -structure (see [18] for the definition).”
- Homological mirror symmetry for the 4-torus (2009, unknown): “Polishchuk later refined this by checking equivalence of Massey products [Pol00], and verified that the A∞-structure is in a sense determined by a triple product [ Pol04 ].”
- Dirichlet Branes and Mirror Symmetry (2009, methodology): “The case of the elliptic curve has already been studied in detail in [397, 396]; we will take a somewhat different approach here inspired by Abouzaid in [2] and developed in [1] which uses the Čech approach to defining the A∞ structure on the derived category.”