M2-brane

The M2-brane solution can be found[1] by requiring ${\displaystyle (Poincare)_{3}\times SO(8)}$ 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

${\displaystyle {\begin{aligned}ds_{M2}^{2}&=\left(1+{\frac {q}{r^{6}}}\right)^{-{\frac {2}{3}}}dx^{\mu }dx^{\nu }\eta _{\mu \nu }+\left(1+{\frac {q}{r^{6}}}\right)^{\frac {1}{3}}dx^{i}dx^{j}\delta _{ij}\\F_{i\mu _{1}\mu _{2}\mu _{3}}&=\epsilon _{\mu _{1}\mu _{2}\mu _{3}}\partial _{i}\left(1+{\frac {q}{r^{6}}}\right)^{-1},\quad \mu =1,\ldots ,3\quad i=4,\ldots ,11,\end{aligned}}}$

where ${\displaystyle \eta }$ is the flat-space metric and the distinction has been made between world volume ${\displaystyle x^{\mu }}$ and transverse ${\displaystyle x^{i}}$ coordinates. The constant ${\displaystyle q}$ is proportional to the charge of the brane which is given by the integral of ${\displaystyle F}$ over the boundary of the transverse space of the brane.[2]

• String theory
• Membrane (M-theory)
• M-theory