p -Brane

A p -bridge (pronounced [ piːˈbɹeɪn ]) is a p -dimensional object in string theory . The word is from the membrane ( English membrane derived), a membrane according to point particles ( p = 0 ) and string ( p = 1 ) the object with the next higher dimension ( p = 2 ).

The p -Branes are considered in a d -dimensional space; for supersymmetric string theory, d = 10 with nine space dimensions and one time dimension. If some of the d = 10 dimensions are compacted , some of the p dimensions of a p- bridge may also be reduced , so that the effective theory contains lower- dimensional objects in d <10 dimensions instead of the p- bridge.

The D-branes (D according to Dirichlet, sometimes also referred to as Dp -branes) are a special case . In M-theory , p -branes are also called M-branes.

meaning

The five string theories and 11-dimensional supergravity as borderline cases of the M-theory.

p -Branes were in the 1990s as part of the standardization of the five string theories Type I , Type IIA and IIB and the two heterotic string theories , referred to in the picture with E8 and SO (32), and the 11-dimensional supergravitation to M-theory in more detail considered. The string theories and supergravity are understood as the limit values ​​of the M-theory with a small string coupling constant. As the coupling constant increases, one-dimensional strings can become two-dimensional membranes in M-theory.

The investigations went in different directions: Gary Horowitz and Andrew Strominger found that higher-dimensional objects can also have an event horizon like black holes . Joseph Polchinski looked at type II superstring theories that were expanded to include open strings. Given the open strings in p + 1 dimensions, Neumann boundary conditions , i.e. H. the derivative vanishes at the endpoints, and Dirichlet boundary conditions in the 9-p other dimensions , i.e. H. the field disappears at the end points, so the end points move on p -dimensional objects, the D-branes.

effect

World line, world area and world volume in a 2 + 1 dimensional space-time

In the simplest case, the equations of motion of a p- bridge are based on the effect

${\displaystyle S_{p}=-T_{p}\int d^{p+1}\xi {\sqrt {-\det(g_{ab})}}}$

derived, where is the induced metric and the coordinates are the spatial coordinates in the d -dimensional space. This is precisely the volume that the p -dimensional object spans with its temporal development, the so-called world volume . The Euler-Lagrange equations are currently looking for a minimum of the world volume. ${\displaystyle g_{ab}={\frac {\partial X^{\mu }}{\partial \xi ^{a}}}{\frac {\partial X^{\nu }}{\partial \xi ^{b}}}g_{\mu \nu }(X(\xi ))}$ ${\displaystyle X^{\mu }}$

Special case p = 0

For p = 0 and constant metric one obtains with ${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }}$ ${\displaystyle T_{0}=m}$

${\displaystyle S_{0}=-m\int d\tau {\sqrt {-{\frac {dX^{\mu }}{d\tau }}{\frac {dX^{\nu }}{d\tau }}\eta _{\mu \nu }}}=-m\int d\tau {\sqrt {-{\dot {X}}^{2}}}\ .}$

In d = 4 dimensions (three space dimensions and one time dimension) this is precisely the effect of a point particle known from the theory of relativity . It is the length of the world line spanned by the point particle , the point particle moves on a path on which the length of this world line is as small as possible.

The point particle is therefore the p- brane for p = 0 and is therefore also referred to as the 0-brane or zero-brane.

Special case p = 1

→ Main article : String theory

For p = 1 and a constant metric , the Nambu-Goto effect is obtained ${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }}$

${\displaystyle S_{NG}=-T\int d^{2}\sigma {\sqrt {-\det \left({\frac {\partial X^{\mu }}{\partial \sigma ^{\alpha }}}{\frac {\partial X^{\nu }}{\partial \sigma ^{\beta }}}\eta _{\mu \nu }\right)}}=-T\int dA\ ,}$

thus the content of the world area spanned by a boson string . The dynamic of the string minimizes this area. The string is the 1 brane.

General effect

The general p -Brane effect is

${\displaystyle S_{p}=-T_{p}\int d^{p+1}\xi e^{-\Phi }{\sqrt {-\det(G_{ab}+B_{ab}+2\pi \alpha 'F_{ab})}}\ ,}$

where the p -bridge is coupled to the dilaton , the induced metric , the antisymmetric tensor and a field tensor . ${\displaystyle \Phi }$ ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle F}$

cosmology

→ Main article : Bran cosmology

While many of the considerations on p -Branes assume that the additional dimensions are not perceived by us because they have shrunk to small scales and can only be recognized in the context of elementary particle physics , there are also other approaches that our entire universe is actually in embedded in a higher-dimensional space and in this there is currently a brane. The original considerations by Gunnar Nordström and Theodor Kaluza can thus be formulated in such a way that our 3 + 1-dimensional space-time is a 3-brane that is embedded in a 5-dimensional space.

literature

• Brian Greene : The Elegant Universe . Siedler Verlag, 2015, ISBN 978-3-641-18565-7 ( limited preview in Google book search).
• Dieter Lüst : Quantum fish: The string theory and the search for the world formula . CHBeck, 2012, ISBN 978-3-406-62286-1 ( limited preview in Google book search).

Individual evidence

1. ^ Brian Greene : The Elegant Universe . WW Norton & Company, 2010, ISBN 978-0-393-07134-4 , pp. 316 ( limited preview in Google Book Search).
2. ^ GT Horowitz, A. Strominger: Black strings and p-branes . In: Nuclear Physics B . tape 360 , no. 1 , 1991, p. 197-209 , doi : 10.1016 / 0550-3213 (91) 90440-9 .
3. ^ Joseph Polchinski: Dirichlet Branes and Ramond-Ramond Charges . In: Physical Review Letters . tape 75 , no. 26 , December 25, 1995, pp. 4724-4727 , doi : 10.1103 / PhysRevLett.75.4724 , arxiv : hep-th / 9510017 .
4. ↑ Joseph Polchinski : String Theory, vol. I . Cambridge University Press, 1998, ISBN 1-139-45740-3 , pp. 270 ( limited preview in Google Book search).