Topological String, Supersymmetric Gauge Theory and

In this thesis we study the Donaldson-Thomas theory on the local curve geometry, which arises in the context of geometric engineering of supersymmetric gauge theory from type IIA string compactification. The topological A-model amplitude gives the F-term interaction of the com-pactified theory. In particular, it is related to the instanton partition function via Nekrasov conjecture. We will introduce ADHM sheaves on curve, as an alternative description of local Donaldson-Thomas theory. We derive the wallcrossing of ADHM invariants and their refinements. We show that it is equivalent to the semi-primitive wallcrossing from supergravity, and the Kontsevich-Soibelman wallcrossing formula. As an application, we discuss the connection between ADHM moduli space with Hitchin system. In particular we give a recursive formula for the Poincare polynomial of Hitchin system in terms of instanton partition function, from refined wallcrossing. We also introduce higher rank generalization of Donaldson-Thomas invariant in the context of ADHM sheaves. We study their wallcrossing and discuss their physical interpretation via string duality. ii Acknowledgements First I would like to express my deepest gratitude to my advisor, Prof. Emanuel Diaconescu, for all the guidance, help and encouragement, for teaching me string theory and algebraic geometry, for sharing his ideas and discussing physics with me. He is always patient and willing to answer my questions. I cannot imagine having a better advisor. I also thank Wu-yen Chuang and Greg Moore for past collaborations. I learned a lot from our discussions, which gave rise to many interesting ideas and fruitful results. I would like to thank the faculties in NHETC, Tom Banks, Dan Friedan, Sergei Lukyanov