Abstract
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 $\mathbf{B}^{[3]}$ and a gauge $2$-form $\mathbf{B}^{[2]}$: 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 $0$, $1/8$ and $1/4$ of the original supersymmetry. Two-branes without fluxes are instead BPS states and preserve $1/2$ supersymmetry. For each two-brane solution we carefully study its discrete symmetry that is always given by some appropriate crystallographic group $\Gamma$. Such symmetry groups $\Gamma$ are transmitted to the $D=3$ gauge theories on the brane world--volume that 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.
Highlights
B) Secondly, comparing the supersymmetry transformation rules derived in [1] with those that follow from our rheonomic solutions of the Bianchi identities, we work out the rescalings that connect our normalizations of the supergravity fields with those of [1] and of the standard flux 2-brane form of eq (2.10)
The main result of the present paper is the analysis the supersymmetry properties of the Arnold-Beltrami flux 2-branes suitably embedded in supergravity
This required the study of the Killing spinor equation on the corresponding background and of its solutions
Summary
The Beltrami vector fields live on three-dimensional tori and in mathematical hydrodynamics are interpreted as velocity fields of some fluid They can be used as compactification fluxes in the transverse space to the world volume of 2-brane solutions of D = 7 supergravity theory. We shall consider a transverse space has the topology of R+ × T 3, which is suitable for the introduction of the Arnold-Beltrami fluxes In this case the solution can still be interpreted as a 2-brane since, in the absence of these extra fluxes, it has the form given above, i.e. of an extended two-dimensional object electrically coupled to the 3-form, H(y) is a harmonic function on R+ × T 3. We shall just briefly comment on it at the end of subsection 2.2.1
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