Convoluted solutions in supergravity

Abstract

Abstract. Inspired by the convoluted solutions for two intersecting M2 branes in elevendimensional supergravity, in which one brane in the system is completely localized along the overall and relative transverse coordinates while the other brane in the system is localized only along the overall transverse coordinates, we construct two classes of exact solutions to Einstein-Maxwell theory in six and higher dimensions. We show that the membrane configuration preserves four supersymmetries and upon dimensional reduction, the solutions provide intersecting configurations of three D-branes in type IIA supergravity. Moreover, we show that the metric functions in six and higher dimensions can be written as convolution-like integrals of two special functions. The solutions are regular everywhere and show a bolt structure on a single point in any dimensionality. Also, we find the exact nonstationary solutions to the Einstein-Maxwell theory with positive cosmological constant. We show that the cosmological solutions are expanding patches in asymptotically de Sitter spacetime. IntroductionFinding the supergravity solutions for the intersecting D-branes is relatively easy as long as the D-branes are sufficiently smeared [1]- [4]. However, it is far harder to find exact solutions for the fully localized D-branes that are valid everywhere; near, as well as far from the core of D-branes. To overcome this problem, in an interesting paper [5] the authors constructed the exact solutions for the fully localized type IIA D2 branes intersecting D6 branes. By lifting a D6 brane to a four-dimensional Taub-NUT geometry embedded in M-theory and then placing M2 branes in the Taub-NUT background geometry, the authors found convolution-like solutions for the M2 branes. The solutions for the system of intersecting D-branes can be obtained by compactifying the corresponding convolution-like M2 brane solutions over a circle of transverse self-dual Taub-NUT geometry. The solutions are not restricted to near core of D6 branes and moreover the solutions are supersymmetric. Inspired by other related works in [6]-[41] to find exact solutions for M-branes and intersecting D-branes as well as solutions to Einstein-Maxwell theory with different matter fields, we can find some exact convoluted solutions to six and higher dimensional Einstein-Maxwell theory [42]. The purpose of this talk (article) is to summarize the progress in finding the convoluted solutions in supergravity and Einstein-Maxwell theory presented in references [14], [21], [42]. In section 2, we present the different M2 brane solutions that upon compactification lead to different intersecting brane systems. In section 3, we present the solutions for two M2 branes and then in section 4 we explicitly calculate the number of preserved supersymmetries. In section 5, we present the exact convoluted cosmological solutions to Einstein-Maxwell theory in D-dimensions and discuss the properties of the solutions. The concluding remarks are in section 6.