Brane

Builds on Lorentzian manifold Submanifold

Definition

Let $(M, g)$ be a $D$ -dimensional Lorentzian manifold and let $0 \le p \le D - 1$ . A $p$ -brane in $M$ is an extended object whose history is a $(p+1)$ -dimensional submanifold $W \subset M$ , its worldvolume, on which the induced metric $h = g|_W$ is Lorentzian. In the simplest case its dynamics is given by the Nambu–Goto–Dirac action $S[W] = -T_p \int_W \sqrt{-\det h_{\alpha\beta}} \; d^{p+1}\sigma.$ Here $\sigma^\alpha$ are coordinates on $W$ , $h_{\alpha\beta}$ are the components of $h$ , and $T_p > 0$ is a constant, the tension of the brane.

Remarks

The action is $-T_p$ times the $(p+1)$ -volume of $W$ , so the classical worldvolumes are those of stationary volume. Example: for $p = 0$ , $W$ is a timelike curve and $S$ is $-T_0$ times its proper time. This is the action of a free relativistic particle of mass $T_0$ , in units $c = 1$ . For $p = 1$ one gets the Nambu–Goto action of a string. Non-example: a spacelike surface in $M$ is not the worldvolume of a brane, since its induced metric is not Lorentzian. A D $p$ -brane in string theory is a $p$ -brane on which open strings can end, with Dirichlet boundary conditions in the normal directions. Its action, the Dirac–Born–Infeld action, also contains a gauge field on $W$ .

Reading path

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Lorentzian manifold concept
Submanifold concept
Brane definition

Tag	`WD5EM0V75H`
URI	`m2://WD5EM0V75H`
Kind	definition
Writing	W0 stub
MSC	81T30, 83E30
Defines	Brane
Uses	Lorentzian manifold, Submanifold
Changed	3 November 2026

Brane

Definition

Remarks

Reading path

Further reading

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