Stringy corrections to spacetime superpotentials

Over the last few years, string compactifications have come back into fashion as an approach to phenomenology (KKLT, landscapes, ...), and nonperturbative corrections to superpotentials play an important role. Most of the work done recently revolves around type II string compactifications with fluxes, but a smaller community has also been interested in using heterotic strings to the same ends. However, nonperturbative corrections to heterotic spacetime superpotentials have historically been more difficult to compute. In this talk, we will describe some work on computing stringy nonperturbative corrections to heterotic spacetime superpotentials. Technically, this amounts to understanding (0,2) generalizations of the A and B model topological field theories, meaning (0,2) generalizations of curve-counting, which also has applications to understanding (0,2) mirror symmetry. Curiously, the (0,2) generalizations of the A and B models are equivalent -- the A/B distinction is lost off the (2,2) locus. We will describe formally how one defines these (0,2) generalizations, and will discuss how gauged linear sigma models solve technical issues, such as extensions of induced bundles of zero modes over compactifications of bosonic moduli spaces. Heterotic quantum cohomology rings will be discussed and computed. Finally, as a minor offshoot of this work, we will show how the topological B model, on the (2,2) locus, can be consistently defined on more complex manifolds than just Calabi-Yau's -- one only needs the tensor square of the canonical bundle to be trivializable, not the canonical bundle itself.