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[1] 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{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 η {\displaystyle \eta } \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.[2]

String theory
Membrane (M-theory)
M-theory

References

K. Stelle, "BPS Branes in Supergravity"

A. Miemiec, I. Schnakenburg "Basics of M-theory"

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String theory
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