Some aspects of topological string theory

Abstract

In the thesis we study topological aspects of string and M-theory. We derive a large N holomorphic string expansion for the Macdonald-deformed U(N) Yang-Mills theory on a closed Riemann surface. Macdonald deformation of two-dimensional Yang-Mills theory computes entropies of BPS black holes and it is also dual to refined topological string theory. In the classical limit, the expansion defines a new β-deformation for Hurwitz theory of branched covers wherein the refined partition function is a generating function for certain parameterized Euler characters. We also apply the large N expansion to observables corresponding to open surfaces and Wilson loops. We study AKSZ constructions for the A and B sigma-models of topological string theory within a double field theory formulation that incorporates backgrounds with geometric and non-geometric fluxes. AKSZ formulations provide natural geometric methods for constructing BV quantized sigma-models. After a section condition, we relate the Aand B-model to a three-dimensional Courant sigma-model, corresponding to a generalized complex structure, which reduces to the Aor B-models on the boundary. We introduce S-duality at the level of the three-dimensional sigma-model based on the generalized complex structure, which exchanges the related AKSZ field theories, and interpret it as topological S-duality of the Aand B-models. We also study AKSZ constructions for closed topological membranes on G2-manifolds. These membranes were originally introduced to be the worldvolume formulation for topological M-theory, which is intended to capture a topological sector of physical Mtheory. We propose two inequivalent AKSZ membrane theories, in each of which the two existing topological membranes appear as different gauge fixed versions, and their dimensional reductions give new AKSZ constructions for the topological A-model. We show that the two AKSZ membrane models originate through worldvolume dimensional reduction of a single AKSZ three-brane theory, which gives the higher Courant bracket of exceptional generalized geometry of M-theory as the underlying derived bracket. The thesis is based on three papers [1–3].