Superpotentials for Vector Bundle Moduli

TL;DR

The authors develop a concrete method to compute non-perturbative superpotentials in heterotic and M-theory by evaluating the Pfaffian ${\rm Pfaff}({\cal D}_{-})$ of the Dirac operator on a holomorphic curve. They reduce the vanishing of the Pfaffian to the existence of holomorphic sections of the restricted bundle $V|_{C}(-1)$ and implement this through a spectral-cover construction, computing an explicit example where $W \propto {\cal P}^{4}$ with ${\cal P}$ a quintic in seven transition moduli and independent of other moduli. The zeros of $W$ thus occur on the hypersurface ${\cal P}=0$ inside ${\mathbb P}^{12}$, offering a precise, holomorphic expression for the vector-bundle moduli contribution to the superpotential. These results have potential implications for vacuum stability, small instanton transitions, and cosmology in heterotic string and M-theory frameworks, and the authors outline broader applicability in follow-up work (BDOnew).

Abstract

We present a method for explicitly computing the non-perturbative superpotentials associated with the vector bundle moduli in heterotic superstrings and M-theory. This method is applicable to any stable, holomorphic vector bundle over an elliptically fibered Calabi-Yau threefold. For specificity, the vector bundle moduli superpotential, for a vector bundle with structure group G=SU(3), generated by a heterotic superstring wrapped once over an isolated curve in a Calabi-Yau threefold with base B=F1, is explicitly calculated. Its locus of critical points is discussed. Superpotentials of vector bundle moduli potentially have important implications for small instanton phase transitions and the vacuum stability and cosmology of superstrings and M-theory.