Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds
Thomas W. Grimm, Albrecht Klemm, Denis Klevers
TL;DR
The paper develops a unifying framework for space-time filling five-branes on Calabi–Yau threefolds by blowing up brane curves to exceptional divisors, thereby converting open brane deformations into complex-structure deformations of a non-Calabi–Yau space ŜZ3. It derives a complete open–closed Picard–Fuchs system via residue methods and GKZ technology, enabling explicit brane superpotentials and disk-instanton counts in toric settings (notably the quintic and Z3(1,1,1,6,9)). It then promotes the blow-up construction to a dynamical setting with an SU(3) structure, proposing a non-Kähler twist that reconciles brane fluxes with four-dimensional N=1 supersymmetry, and sketches a path to fully back-reacted heterotic backgrounds. The work provides concrete computational tools for open–closed moduli, links brane obstructions to flux-induced superpotentials, and offers a platform for connecting brane physics with non-Kähler geometry and heterotic/F-theory duality. Overall, it advances a geometrical unification of brane and bulk dynamics and delivers actionable predictions for disk invariants and flux vacua in toric Calabi–Yau compactifications.
Abstract
We investigate the dynamics of space-time filling five-branes wrapped on curves in heterotic and orientifold Calabi-Yau compactifications. We first study the leading N=1 scalar potential on the infinite deformation space of the brane-curve around a supersymmetric configuration. The higher order potential is also determined by a brane superpotential which we compute for a subset of light deformations. We argue that these deformations map to new complex structure deformations of a non-Calabi-Yau manifold which is obtained by blowing up the brane-curve into a four-cycle and by replacing the brane by background fluxes. This translates the original brane-bulk system into a unifying geometrical formulation. Using this blow-up geometry we compute the complete set of open-closed Picard-Fuchs differential equations and identify the brane superpotential at special points in the field space for five-branes in toric Calabi-Yau hypersurfaces. This has an interpretation in open mirror symmetry and enables us to list compact disk instanton invariants. As a first step towards promoting the blow-up geometry to a supersymmetric heterotic background we propose a non-Kaehler SU(3) structure and an identification of the three-form flux.