Holomorphic anomaly equations in topological string theory

Abstract

In this thesis we discuss various aspects of topological string theories. In particular we provide a derivation of the holomorphic anomaly equation for open strings and study aspects of the Ooguri, Strominger, and Vafa conjecture. Topological string theory is a computable theory. The amplitudes of the closed topological string satisfy a holomorphic anomaly equation, which is a recursive differential equation. Recently this equation has been extended to the open topological string. We discuss the derivation of this open holomorphic anomaly equation. We find that open topological string amplitudes have new anomalies that spoil the recursive structure of the equation and introduce dependence on wrong moduli (such as complex structure moduli in the A-model), unless the disk one-point functions vanish. We also show that a general solution to the extended holomorphic anomaly equation for the open topological string on D-branes in a Calabi-Yau manifold, is obtained from the general solution to the holomorphic anomaly equations for the closed topological string on the same manifold, by shifting the closed string moduli by amounts proportional to the 't Hooft coupling. An important application of closed topological string theory is the Ooguri, Strominger, and Vafa conjecture, which states that a certain black hole partition function is a product of topological and anti-topological string partition functions. However when the black hole has finite size, the relation becomes complicated. In a specific example, we find a new factorization rule in terms of a pair of functions which we interpret as the non-perturbative' completion of the topological string partition functions.