Gluing Branes, I

Abstract

We consider several aspects of holomorphic brane configurations. We recently showed that an important part of the defining data of such a configuration is the gluing morphism, which specifies how the constituents of a configuration are glued together, but is usually assumed to be vanishing. Here we explain the rules for computing spectra and interactions for configurations with non-vanishing gluing VEVs. We further give a detailed discussion of the D-terms for Higgs bundles, spectral covers and ALE fibrations. We highlight a stability criterion that applies to degenerate configurations of the spectral data, and address an apparent discrepancy between the field theory and ALE descriptions. This allows us to show that one gets walls of marginal stability in F-theory even though they are absent in the 11d supergravity description. We also propose a numerical approach for approximating the hermitian-Einstein metric of the Higgs bundle using balanced metrics.

Figures (1)

• Figure 1: Picture of the branch structure. The cone represents the $3d$ Coulomb branch, where one resolves the singularities of the Calabi-Yau four-fold. The plane represents a 'non-abelian' $F$-theory branch where wrapped $M2$-branes have condensed, eg. a branch with a non-zero Fayet-Iliopoulos parameter. This branch is visible in the Higgs bundle description but not in the Calabi-Yau four-fold description. The picture is schematic in several respects, for example it is not guaranteed that every $F$-theory branch is connected to an $M$-theory branch.