Flux-branes and the Dielectric Effect in String Theory

TL;DR

This work develops a gravity-side description of flux-branes in string/M-theory and demonstrates how dielectric effects stabilize flux-brane configurations. By constructing an exact M-theory reduction to type IIA, it yields a dielectric D4-brane expanding into a spherical D6-brane in a flux 7-brane background, with a calculable gravitational energy and scalar VEV in the decoupling limit. The analysis connects to gauge/gravity duality by showing that the deformation due to the flux-brane is UV-irrelevant in the decoupled geometry and corresponds to a Scherk–Schwarz reduction of the (2,0) theory, including a Myers-like cubic coupling. Through D6-brane probes, the authors identify dual field theory structures, including inner and outer dielectric shells and a reducible SU(2) representation for the dielectric branes, illuminating the IR dynamics of the associated deformed theories. Overall, the paper provides the first exact gravity solution for the dielectric effect in this context and clarifies the role of flux-branes in the gauge/gravity correspondence, with implications for extending dielectric constructions to other brane configurations.

Abstract

We consider the generalization to String and M-theory of the Melvin solution. These are flux p-branes which have (p+1)-dimensional Poincare invariance and are associated to an electric (p+1)-form field strength along their worldvolume. When a stack of Dp-branes is placed along the worldvolume of a flux (p+3)-brane it will expand to a spherical D(p+2)-brane due to the dielectric effect. This provides a new setup to consider the gauge theory/gravity duality. Compactifying M-theory on a circle we find the exact gravity solution of the type IIA theory describing the dielectric expansion of N D4-branes into a spherical bound state of D4-D6-branes, due to the presence of a flux 7-brane. In the decoupling limit, the deformation of the dual field theory associated with the presence of the flux brane is irrelevant in the UV. We calculate the gravitational radius and energy of the dielectric brane which give, respectively, a prediction for the VEV of scalars and vacuum energy of the dual field theory. Consideration of a spherical D6-brane probe with n units of D4-brane charge in the dielectric brane geometry suggests that the dual theory arises as the Scherk-Schwarz reduction of the M5-branes (2,0) conformal field theory. The probe potential has one minimum placed at the locus of the bulk dielectric brane and another associated to an inner dielectric brane shell.

Figures (4)

• Figure 1: Left: The $(\tilde{r},\tau)$ Euclidean plan with $\tilde{r}^2=r^2-r_H^{ 2}$. At $\tilde{r}=0$ there is a 4--sphere of radius $r_H$; Right: The parameterization of the 4--sphere; the subscript on the $S^2$ denotes the radius.
• Figure 2: Geometry of the hypersurface ${\mathit M}$, defined as the restriction to the transverse space of the intersection of $r=r_H$ with a partial Cauchy surface. For $B=0$, ${\mathit M}$ is a point in the transverse space and can be thought of as the physical singularity associated to the locus of the D4--branes. As $B$ is turned on, ${\mathit M}$ expands into a regular three-space bounded by a singular $S^2$, which can be thought of as the locus of the D4--branes which have expanded into a D4--D6--bound state. In the next section we will be able to smoothly interpolate between these two configurations.
• Figure 3: The region of validity of the gravitational approximation, in terms of the variables $v=U\cos{\theta}/(2\pi)$ and $\rho=U\sin{\theta}/(2\pi)$. There are curvature corrections around $U=U_0$ and $\theta=0$ (point P), where the dielectric brane is placed. Inside the dielectric sphere, i.e. for $U=U_0$ and $0<\theta\le\pi/2$, there is a large region surrounding the center of the sphere (point Q at $\theta=\pi/2$) where the curvature is small. Far away the appropriate description is eleven--dimensional.
• Figure 4: The two minima of the potential for $g^2_{YM}B=0.01$ and $n/N=0.3$. In the top graph the dimensionless radial coordinate $x$ is defined, for $0\le x<1$, by $U=U_0$, $x=\cos{\theta}$ and it is the radial coordinate inside the dielectric sphere. For $x>1$, we have $x=U/U_0$ and $\theta=0$ as appropriate for the radial coordinate outside the dielectric sphere on the $\theta=0$ three--plane. The other two plots zoom into the minima whose interpretation is given in the main text.