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{{String theory}}
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In [[physics]], the '''Polyakov action''' is the two-dimensional [[action (physics)|action]] of a [[conformal field theory]] describing the [[worldsheet]] of a string in [[string theory]]. It was introduced by [[Stanley Deser]] and [[Bruno Zumino]] and independently by L. Brink, P. Di Vecchia and P. S. Howe (in ''Physics Letters'' B65, pages 369 and  471 respectively), and has become associated with [[Alexander Markovich Polyakov|Alexander Polyakov]] after he made use of it in quantizing the string. The action reads


:<math>\mathcal{S} = {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-h} h^{ab} g_{\mu \nu} (X) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma)</math>
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where <math> T </math> is the string [[Tension (mechanics)|tension]], <math>g_{\mu \nu}</math> is the metric of the [[target manifold]], <math>h_{ab}</math> is the worldsheet metric and <math>h</math> is the determinant of <math>h_{ab}</math>. The [[metric signature]] is chosen such that timelike directions are + and the spacelike directions are -. The spacelike worldsheet coordinate is called <math> \sigma </math> whereas the timelike worldsheet coordinate is called <math> \tau </math>. This is also known as [[nonlinear sigma model]].<ref name="Frie80">
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{{cite journal|last=Friedan|first=D.|authorlink=Daniel Friedan|title=Nonlinear Models in 2+ε Dimensions|journal=[[Physical Review Letters]]|volume=45|issue=|pages=1057|publisher=|location=|year=1980|url=http://www.physics.rutgers.edu/~friedan/papers/PRL_45_1980_1057.pdf|doi=10.1103/PhysRevLett.45.1057|id=|accessdate=|bibcode=1980PhRvL..45.1057F}}</ref>


The Polyakov action must be supplemented by the  
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[[Liouville field theory|Liouville action]] to describe string fluctuations.
 
 
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== Global symmetries ==
 
 
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N.B. : Here, a symmetry is said to be local or global from the two dimensional theory (on the worldsheet) point of view. For example, Lorentz transformations, that are local symmetries of the space-time, are global symmetries of the theory on the worldsheet.
 
 
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The action is [[Invariant (physics)|invariant]] under spacetime [[Translation (geometry)|translations]] and [[infinitesimal]] [[Lorentz transformation]]s:
 
:(i) <math> X^\alpha \rightarrow X^\alpha + b^\alpha </math>
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:(ii) <math> X^\alpha \rightarrow X^\alpha + \omega^\alpha_{\ \beta} X^\beta </math>
 
where <math> \omega_{\mu \nu} = - \omega_{\nu \mu} </math> and <math> b^\alpha </math> is a constant. This forms the [[Poincaré group|Poincaré symmetry]] of the target manifold.
  <li>[http://42.121.2.33:100/bbs/viewthread.php?tid=3270052&extra= http://42.121.2.33:100/bbs/viewthread.php?tid=3270052&extra=]</li>
 
 
The invariance under (i) follows since the action <math> \mathcal{S} </math> depends only on the first derivative of <math> X^\alpha </math>. The proof of the invariance under (ii) is as follows:
</ul>
 
::{|
| <math> \mathcal{S}' \, </math>
| <math> = {T \over 2}\int \mathrm{d}^2 \sigma  \sqrt{-h} h^{ab} g_{\mu \nu} \partial_a \left( X^\mu + \omega^\mu_{\ \delta} X^\delta \right) \partial_b \left( X^\nu + \omega^\nu_{\ \delta} X^\delta \right) \, </math>
|-
|
|<math> = \mathcal{S} + {T \over 2}\int \mathrm{d}^2 \sigma  \sqrt{-h} h^{ab} \left( \omega_{\mu \delta} \partial_a X^\mu \partial_b X^\delta + \omega_{\nu \delta} \partial_a X^\delta \partial_b X^\nu \right) + O(\omega^2) \, </math>
|-
|
|<math> = \mathcal{S} + {T \over 2}\int \mathrm{d}^2 \sigma  \sqrt{-h} h^{ab} \left( \omega_{\mu \delta} + \omega_{\delta \mu } \right) \partial_a X^\mu \partial_b X^\delta + O(\omega^2) = \mathcal{S} + O(\omega^2) </math>
|}
 
== Local symmetries ==
 
The action is [[invariant (physics)|invariant]] under worldsheet [[diffeomorphism]]s (or coordinates transformations) and [[Weyl transformation]]s.
 
=== Diffeomorphisms ===
Assume the following transformation:
::<math> \sigma^\alpha \rightarrow \tilde{\sigma}^\alpha\left(\sigma,\tau \right) </math>
It transforms the [[Metric tensor]] in the following way:
::<math> h^{ab} \rightarrow \tilde{h}^{ab} = h^{cd} \frac{\partial \tilde{\sigma}^a}{\partial \sigma^c} \frac{\partial
\tilde{\sigma}^b}{\partial \sigma^d} </math>
One can see that:
::<math> \tilde{h}^{ab} \frac{\partial}{\partial \tilde{\sigma}^a} X^\mu \frac{\partial}{\partial \tilde{\sigma}^b} X^\nu =  h^{cd} \frac{\partial \tilde{\sigma}^a}{\partial \sigma^c} \frac{\partial \tilde{\sigma}^b}{\partial \sigma^d} \frac{\partial}{\partial \tilde{\sigma}^a} X^\mu \frac{\partial}{\partial \tilde{\sigma}^b} X^\nu = h^{ab} \partial_a X^\mu \partial_b X^\nu </math>
One knows that the [[Jacobian]] of this transformation is given by:
::<math> \mathrm{J} = \mathrm{det} \left( \frac{\partial \tilde{\sigma}^\alpha}{\partial \sigma^\beta} \right) </math>
which leads to:
::<math> \mathrm{d}^2 \sigma \rightarrow \mathrm{d}^2 \tilde{\sigma} = \mathrm{J} \mathrm{d}^2 \sigma \, </math>
::<math> h = \mathrm{det} \left( h_{ab} \right) \rightarrow \tilde{h} = \mathrm{J}^{-2} h \, </math>
and one sees that:
::<math> \sqrt{-\tilde{h}} \mathrm{d}^2 \tilde{\sigma} = \sqrt{-h} \mathrm{d}^2 \sigma </math>
summing up this transformation leaves the action invariant.
 
=== Weyl transformation ===
Assume the [[Weyl transformation]]:
::<math> h_{ab} \rightarrow \tilde{h}_{ab} = \Lambda(\sigma) h_{ab} </math>
then:
::<math> \tilde{h}^{ab} = \Lambda^{-1}(\sigma) h^{ab} </math>
::<math> \mathrm{det} ( \tilde{h}_{ab} ) = \Lambda^2(\sigma) \mathrm{det} (h_{ab}) </math>
And finally:
::{|
| <math> \mathcal{S}' \, </math>
| <math> = {T \over 2}\int \mathrm{d}^2 \sigma  \sqrt{-\tilde{h}} \tilde{h}^{ab} g_{\mu \nu} (X) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma) \, </math>
|-
|
|<math> = {T \over 2}\int \mathrm{d}^2 \sigma  \sqrt{-h} \left( \Lambda \Lambda^{-1} \right) h^{ab} g_{\mu \nu} (X) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma) = \mathcal{S} </math>
|}
And one can see that the action is invariant under [[Weyl transformation]]. If we consider n-dimensional (spatially) extended objects whose action is proportional to their worldsheet area/hyperarea, unless n=1, the corresponding Polyakov action would contain another term breaking Weyl symmetry.
 
One can define the [[Stress-energy tensor]]:
::<math> T_{ab} = \frac{2}{\sqrt{-h}} \frac{\delta S}{\delta h^{ab}} </math>
Let's define:
::<math> h_{ab} = \exp\left(\phi(\sigma)\right) \hat{h}_{ab} </math>
Because of [[Weyl symmetry]] the action does not depend on <math> \phi </math>:
::<math> \frac{\delta S}{\delta \phi} = \frac{\delta S}{\delta h^{ab}} \frac{\delta h^{ab}}{\delta \phi} = \frac12 \sqrt{-h} T_{ab} h^{ab} = \frac12 \sqrt{-h} T_a^{\ a} = 0 \rightarrow T_a^{\ a} = \mathrm{tr} \left( T_{ab} \right) = 0</math>
 
== Relation with Nambu-Goto action ==
 
Writing the [[Euler-Lagrange equation]] for the [[metric tensor]] <math> h^{ab} </math> one obtains that:
::<math> \frac{\delta S}{\delta h^{ab}} = T_{ab} = 0 </math>
Knowing also that:
::<math> \delta \sqrt{-h} = -\frac12 \sqrt{-h} h_{ab} \delta h^{ab} </math>
One can write the variational derivative of the action:
::<math> \frac{\delta S}{\delta h^{ab}} = \frac{T}{2} \sqrt{-h} \left( G_{ab} - \frac12 h_{ab} h^{cd} G_{cd} \right) </math>
where <math> G_{ab} = g_{\mu \nu} \partial_a X^\mu \partial_b X^\nu </math> which leads to:
::<math> T_{ab} = T \left( G_{ab} - \frac12 h_{ab} h^{cd} G_{cd} \right) = 0 </math>
::<math> G_{ab} = \frac12 h_{ab} h^{cd} G_{cd} </math>
::<math> G = \mathrm{det} \left( G_{ab} \right) = \frac14 h \left( h^{cd} G_{cd} \right)^2 </math>
 
If the auxiliary [[worldsheet]] [[metric tensor]] <math>\sqrt{-h}</math> is calculated from the equations of motion:
::<math> \sqrt{-h} = \frac{2 \sqrt{-G}}{h^{cd} G_{cd}} </math>
and substituted back to the action, it becomes the [[Nambu-Goto action]]:
::<math> S = {T \over 2}\int \mathrm{d}^2 \sigma  \sqrt{-h} h^{ab} G_{ab} = {T \over 2}\int \mathrm{d}^2 \sigma \frac{2 \sqrt{-G}}{h^{cd} G_{cd}} h^{ab} G_{ab} = T \int \mathrm{d}^2 \sigma \sqrt{-G}</math>
 
However, the Polyakov action is more easily [[quantization (physics)|quantized]] because it is [[linear]].
 
== Equations of motion ==
 
Using [[diffeomorphism]]s and [[Weyl transformation]], with a [[Minkowski space|Minkowskian target space]], one can transform the action into the following form:
::<math> \mathcal{S} = {T \over 2}\int \mathrm{d}^2 \sigma  \sqrt{-\eta} \eta^{ab} g_{\mu \nu} (X) \partial_a X^\mu (\sigma) \partial_b X^\nu(\sigma) = {T \over 2}\int \mathrm{d}^2 \sigma \left( \dot{X}^2 - X'^2 \right) </math>
where <math> \eta_{ab} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) </math>
 
Keeping in mind that <math> T_{ab} = 0 </math> one can derive the constraints:
::<math> T_{01} = T_{10} = \dot{X} X' = 0 </math>
::<math> T_{00} = T_{11} = \frac12 \left( \dot{X}^2 + X'^2 \right) = 0 </math>.
 
Substituting <math> X^\mu \rightarrow X^\mu + \delta X^\mu </math> one obtains:
::<math> \delta \mathcal{S} = T \int \mathrm{d}^2 \sigma \eta^{ab} \partial_a X^\mu \partial_b \delta X_\mu = </math>
 
:::<math> = -T \int \mathrm{d}^2 \sigma \eta^{ab} \partial_a \partial_b X^\mu \delta X_\mu + \left( T \int d \tau X' \delta X \right)_{\sigma=\pi} - \left( T \int d \tau X' \delta X \right)_{\sigma=0} = 0</math>
 
And consequently:
::<math> \square X^\mu = \eta^{ab} \partial_a \partial_b X^\mu = 0 </math>
 
With the boundary conditions in order to satisfy the second part of the variation of the action.
* Closed strings
: [[Periodic boundary conditions]]: <math> X^\mu(\tau, \sigma + \pi) = X^\mu(\tau, \sigma)\ </math>
* Open strings
:(i) [[Neumann boundary conditions]]: <math> \partial_\sigma X^\mu (\tau, 0) = 0, \partial_\sigma X^\mu (\tau, \pi) = 0 </math>
:(ii) [[Dirichlet boundary conditions]]: <math> X^\mu(\tau, 0) = b^\mu, X^\mu(\tau, \pi) = b'^\mu \ </math>
 
== See also ==
* [[D-brane]]
* [[Einstein-Hilbert action]]
 
==Notes==
{{Reflist|2}}
 
== References ==
* Polchinski (Nov, 1994). ''What is String Theory'', NSF-ITP-94-97, 153pp, [[arXiv:hep-th/9411028v1]]
* Ooguri, Yin (Feb, 1997). ''TASI Lectures on Perturbative String Theories'', UCB-PTH-96/64, LBNL-39774, 80pp, [[arXiv:hep-th/9612254v3]]
 
{{DEFAULTSORT:Polyakov Action}}
[[Category:Conformal field theory]]
[[Category:String theory]]

Latest revision as of 07:07, 13 July 2014

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