M2-brane

In theoretical physics, an M2-brane, is a spatially extended mathematical object (brane) that appears in string theory and in related theories (e.g. M-theory, F-theory). In particular, it is a solution of eleven-dimensional supergravity which possesses a three-dimensional world volume.

Description
The M2-brane solution can be found by requiring $$(Poincare)_{3}\times SO(8)$$ symmetry of the solution and solving the supergravity equations of motion with the p-brane ansatz. The solution is given by a metric and three-form gauge field which, in isotropic coordinates, can be written as
 * $$ \begin{align} ds^{2}_{M2} &= \left(1+\frac{q}{r^{6}}\right)^{-\frac{2}{3}}dx^{\mu} dx^{\nu}\eta_{\mu\nu} + \left(1+\frac{q}{r^{6}}\right)^{\frac{1}{3}}dx^{i}dx^{j}\delta_{ij} \\

F_{i\mu_{1}\mu_{2}\mu_{3}} &= \epsilon_{\mu_{1}\mu_{2}\mu_{3}} \partial_{i}\left(1+\frac{q}{r^6}\right)^{-1}, \quad \mu=1,\ldots ,3 \quad i=4,\ldots, 11,\end{align} $$ where $$\eta$$ is the flat-space metric and the distinction has been made between world volume $$x^\mu$$ and transverse $$x^i$$ coordinates. The constant $$q$$ is proportional to the charge of the brane which is given by the integral of $$F$$ over the boundary of the transverse space of the brane.