Topological string theory from the worldsheet: a Batalin-Vilkovisky perspective

The Batalin-Vilkovisky (BV) formalism is a powerful mathematical framework for studying field theories. Originally developed as a homological method for quantizing gauge theories, it has been reinterpreted in recent years from the perspective of derived deformation theory. In this thesis, we first explore these mathematical structures and their role in the study of perturbative field theories. We then apply these techniques to a concrete model: topological string theory. From the worldsheet perspective, topological string theory is a two-dimensional sigma model coupled to topological gravity. However, an alternative approach was initiated by Bershadsky, Cecotti, Ooguri, and Vafa, who proposed a formulation as a quantum field theory living on the target space. More recently, Costello and Li, using the BV formalism and techniques similar to those used in this thesis, provided a precise formulation of the target-space BCOV model. Our goal is to connect these two perspectives -- worldsheet and target space -- by constructing a model for topological string theory from the worldsheet and comparing it with the BCOV theory as formulated by Costello and Li.