A-model and generalized Chern-Simons theory

The relation between open topological strings and Chern-Simons theory was discovered by E. Witten. He proved that A-model on T ∗M where M is a three-dimensional manifold is equivalent to Chern-Simons theory on M and that A-model on arbitrary Calabi-Yau 3-fold is related to Chern-Simons theory with instanton corrections. In present paper we discuss multidimensional generalization of these results. 0. Introduction In present paper we analyze the relation between multidimensional A-model of open topological strings and generalized Chern-Simons theory. Such a relation was discovered by E. Witten [19] in three-dimensional case; we generalize his results. Our approach is based on rigorous mathematical results of [3], [4], [5], [7], [13]; in three-dimensional case it gives mathematical justification of some of Witten’s statements. In modern language Witten considers A-model in presence of a stack of N coinciding D-branes wrapping a Lagrangian submanifold M . In the neighborhood of Lagrangian submanifold a symplectic manifold V looks like T M . In the case V = T M , dimM = 3 Witten shows that A-model is equivalent to ChernSimons theory on M . He considers also the case when V is a CalabiYau 3-fold and shows that in this case Chern-Simons action functional on M acquires instanton corrections. We remark that one can analyze instanton corrections to Chern-Simons functional combining results by Fukaya [5] and Cattaneo-Froehlich-Pedrini [3] and that this approach works also in multidimensional case. To study the origin of Chern-Simons functional and its generalizations one can replace the stack of N coinciding D-branes by N Lagrangian submanifolds depending on ε and tending to the same limit as ε → 0. This situation was studied by Fukaya-Oh [7] and Kontsevich-Soibelman [13]; we will show that the appearance of Chern-Simons functional follows from their results. 1. Generalized Chern-Simons theory Partly supported by NSF grant No. DMS 0204927.