Electrified Fuzzy Spheres and Funnels in Curved Backgrounds

TL;DR

This paper investigates the dynamics of multiple D-branes in curved backgrounds with electric fields, deriving static and time-dependent fuzzy funnel solutions, and exploring dualities and energy minimization in various geometries.

Contribution

It extends the analysis of fuzzy sphere and funnel solutions to curved backgrounds with electric fields, including dualities, automorphisms, and general static solutions, with consistency checks in Abelian theories.

Findings

Static fuzzy funnel solutions for arbitrary metrics.

Time-dependent fuzzy funnels in NS5-brane backgrounds.

Double Wick rotation invariance of static equations.

Abstract

We use the non-Abelian DBI action to study the dynamics of $N$ coincident $Dp$-branes in an arbitrary curved background, with the presence of a homogenous world-volume electric field. The solutions are natural extensions of those without electric fields, and imply that the spheres will collapse toward zero size. We then go on to consider the $D1-D3$ intersection in a curved background and find various dualities and automorphisms of the general equations of motion. It is possible to map the dynamical equation of motion to the static one via Wick rotation, however the additional spatial dependence of the metric prevents this mapping from being invertible. Instead we find that a double Wick rotation leaves the static equation invariant. This is very different from the behaviour in Minkowski space. We go on to construct the most general static fuzzy funnel solutions for an arbitrary metric either by solving the static equations of motion, or by finding configurations which minimise the energy. As a consistency check we construct the Abelian $D3$-brane world-volume theory in the same generic background and find solutions consistent with energy minimisation. In the $NS$5-brane background we find time dependent solutions to the equations of motion, representing a time dependent fuzzy funnel. These solutions match those obtained from the $D$-string picture to leading order suggesting that the action in the large $N$ limit does not need corrections. We conclude by generalising our solutions to higher dimensional fuzzy funnels.