The B-Model Approach to Topological String Theory on Calabi-Yau n-Folds

The main goal of these lectures is to explain, to both mathematicians and physicists, some of the new ideas and techniques that have been developed in the course of considering compactifications of string theories from their critical dimension on special holonomy manifolds M (such as Calabi–Yau manifolds) to four dimensions. The physical motivation in this subject is simply to describe the four-dimensional physical theories which arise in this construction, while for mathematicians, the interest is that the physical perspective was effective in discovering unexpected mathematical structures of these manifolds. Physicists and mathematicians alike find interesting structures and simplifications in supersymmetric theories, in particular exactly solvable subsectors, which might shed light on more general mathematical properties of quantum theories of fields and strings. We give an in-depth overview from a physical perspective, covering such topics as the monodromy group and Picard-Fuchs equations, tt* structures, the holomorphic anomaly equation, and modularity properties.