Branes and Calibrated Geometries

TL;DR

This work shows that supersymmetric configurations of the M5-brane in eleven dimensions, including intersecting fivebranes and their worldvolume solitons, are governed by Bogomol'nyi equations that enforce calibrated embeddings. By deriving explicit spinor variation conditions from intersecting-brane projections, the authors connect the resulting BPS equations to Harvey–Lawson calibrated geometries, including Kähler, Special Lagrangian, Cayley, associative, and coassociative cases. The framework yields concrete differential equations for the transverse scalars that ensure minimal worldvolume volume in various ambient spaces, and it suggests generalizations to curved targets and nonzero self-dual three-forms. Overall, the paper provides a unifying view of how M5-brane worldvolumes realize calibrated geometries via supersymmetry constraints, with potential applications to low-energy gauge dynamics and MQCD constructions.

Abstract

The fivebrane worldvolume theory in eleven dimensions is known to contain BPS threebrane solitons which can also be interpreted as a fivebrane whose worldvolume is wrapped around a Riemann surface. By considering configurations of intersecting fivebranes and hence intersecting threebrane solitons, we determine the Bogomol'nyi equations for more general BPS configurations. We obtain differential equations, generalising Cauchy-Riemann equations, which imply that the worldvolume of the fivebrane is wrapped around a calibrated geometry.