Minimal D = 7 supergravity and the supersymmetry of Arnold-Beltrami flux branes

In this paper we study some properties of the newly found Arnold-Beltrami flux-brane solutions to the minimal D = 7 supergravity. To this end we first single out the appropriate Free Differential Algebra containing both a gauge 3-form B[3] and a gauge 2-form B[2]: then we present the complete rheonomic parametrization of all the generalized curvatures. This allows us to identify two-brane configurations with Arnold-Beltrami fluxes in the transverse space with exact solutions of supergravity and to analyze the Killing spinor equation in their background. We find that there is no preserved supersymmetry if there are no additional translational Killing vectors. Guided by this principle we explicitly construct Arnold-Beltrami flux two-branes that preserve 0, 1818 and 1414 of the original supersymmetry. Two-branes without fluxes are instead BPS states and preserve 1212 supersymmetry. For each two-brane solution we carefully study its discrete symmetry that is always given by some appropriate crystallographic group Γ. Such symmetry groups Γ are transmitted to the D = 3 gauge theories on the brane world-volume that would occur in the gauge/gravity correspondence. Furthermore we illustrate the intriguing relation between gauge fluxes in two-brane solutions and hyperinstantons in D = 4 topological sigma-models.