Brane ( $p -brane$ )

A brane is an extended object with

p

spatial dimensions. Its history is a

(p + 1)

-dimensional worldvolume embedded in target spacetime. A point particle has

p = 0

, a string has

p = 1

, and a membrane has

p = 2

. Geometric brane actions depend on the induced worldvolume metric; D-branes also carry open-string endpoint and gauge-field data.

Table of contents
- Brane tension Brane
- Worldvolume Brane
  - Auxiliary worldvolume metric Worldvolume
    - Auxiliary-metric brane action Auxiliary worldvolume metric
      - Weyl-invariance exception for the string among branes Auxiliary-metric brane action
  - Induced worldvolume metric Worldvolume

Brane tension ( $T_{p}$ )

The brane tension is the energy per unit spatial brane volume and multiplies its geometric action. In units

ℏ = c = 1

, a

p

-brane tension has mass dimension

p + 1

. The

p = 1

case is the string tension.

Worldvolume ( $Σ^{p + 1}$ )

The worldvolume is the history of a brane. An embedding

X : Σ \to M

maps its local coordinates into target spacetime. A string worldvolume is a worldsheet.

Auxiliary worldvolume metric ( $γ_{μν}$ )

An auxiliary worldvolume metric is an independent metric introduced in a brane action, distinct from the induced worldvolume metric. Its algebraic equation can eliminate it. For the Polyakov action of a string, it is determined only up to a Weyl transformation.

Auxiliary-metric brane action ( $\int - γ (γ^{μν} h_{μν} - (p - 1))$ )

The auxiliary-metric brane action is a quadratic embedding action with an independent worldvolume metric. Its embedding equation is

\partial_{μ} (- γ γ^{μν} \partial_{ν} X^{a}) = 0

. Its metric equation is

h_{μν} - γ_{μν} (γ^{ρ σ} h_{ρ σ} - (p - 1)) /2 = 0

. For

p \neq = 1

, tracing determines

γ^{ρ σ} h_{ρ σ} = p + 1

and hence

γ = h

; eliminating the auxiliary field gives twice the invariant worldvolume volume, before including the overall brane tension. The Weyl-invariance exception for the string among branes prevents unique elimination at

p = 1

.

Weyl-invariance exception for the string among branes ( $p = 1$ )

Under

γ_{μν} \mapsto e^{2 ω} γ_{μν}

, the quadratic kinetic density of a

p

-brane scales as

e^{(p - 1) ω}

and its constant density scales as

e^{(p + 1) ω}

. At

p = 1

the first factor is one and the constant term in the auxiliary-metric brane action vanishes. The traced metric equation is then an identity, leaving a Weyl transformation freedom. For

p \neq = 1

, the metric equation instead identifies the auxiliary and induced worldvolume metrics.

Induced worldvolume metric ( $h_{μν} = \partial_{μ} X^{a} \partial_{ν} X^{b} g_{ab}$ )

The induced worldvolume metric is the pullback of the target metric tensor by a brane embedding. For a nondegenerate timelike embedding,

- det h

is its invariant worldvolume volume density. The

p = 1

specialization is the induced worldsheet metric.

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BRANE by

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"BRANE" can refer to different things depending on the context: 1. **In Physics and String Theory**: A "brane" (short for "membrane") is a fundamental object in string theory and related theories such as M-theory. Branes can exist in various dimensions, and they can have various properties. For example, a 1-brane is a string, a 2-brane is a surface, and so on.

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