Remarks on A-branes, mirror symmetry, and the Fukaya category

Creators: Kapustin, Anton; Orlov, Dmitri

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Abstract

We discuss D-branes of the topological A-model (A-branes), which are believed to be closely related to the Fukaya category. We give string theory arguments which show that A-branes are not necessarily Lagrangian submanifolds in the Calabi–Yau: more general coisotropic branes are also allowed, if the line bundle on the brane is not flat. We show that a coisotropic A-brane has a natural structure of a foliated manifold with a transverse holomorphic structure. We argue that the Fukaya category must be enlarged with such objects for the Homological Mirror Symmetry Conjecture to be true.

Additional Information

© 2003 Elsevier Science B.V. Received 31 May 2002, Accepted 28 January 2003, Available online 2 April 2003. We are grateful to Dan Freed for useful suggestions and to Ezra Getzler for pointing out the relevance of bihamiltonian geometry. Some preliminary results have been presented by AK at the Duality Workshop, ITP, Santa Barbara, 18 June–13 July, 2001. AK would like to thank Ron Donagi, Dan Freed, Ezra Getzler, Tony Pantev, and other participants for stimulating discussions, and the organizers for making this workshop possible. AK was supported in part by DOE grants DE-FG02-90ER40542 and DE-FG03-92-ER40701. DO was supported in part by RFFI grant 99-01-01144 and a grant for support of leading scientific groups N 00-15-96085. The research described in this publication was made possible in part by Award No RMl-2089 of the US. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).

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