Editing Polyakov action

{{Short description|2D conformal field theory used in string theory}}
{{String theory}}
In [[physics]], the '''Polyakov action''' is an [[action (physics)|action]] of the [[two-dimensional conformal field theory]] describing the [[worldsheet]] of a string in [[string theory]]. It was introduced by [[Stanley Deser]] and [[Bruno Zumino]] and independently by [[Lars Brink|L. Brink]], [[Paolo Di Vecchia|P. Di Vecchia]] and P. S. Howe in 1976,<ref>{{cite journal |last1=Deser |first1=S. |authorlink1=Stanley Deser |last2=Zumino |first2=B. |authorlink2=Bruno Zumino |date=1976 |title=A Complete Action for the Spinning String |url=https://cds.cern.ch/record/201648|journal=Phys. Lett. B |volume=65 |issue= 4|pages=369–373 |doi=10.1016/0370-2693(76)90245-8}}</ref><ref>{{cite journal |last1=Brink |first1=L. |authorlink1=Lars Brink |last2=Di Vecchia |first2=P. |authorlink2=Paolo Di Vecchia |last3=Howe |first3=P. |date=1976 |title=A locally supersymmetric and reparametrization invariant action for the spinning string |url=https://dx.doi.org/10.1016/0370-2693%2876%2990445-7 |journal=Physics Letters B |volume=65 |issue=5 |pages=471–474 |doi=10.1016/0370-2693(76)90445-7|url-access=subscription }}</ref> and has become associated with [[Alexander Markovich Polyakov|Alexander Polyakov]] after he made use of it in quantizing the string in 1981.<ref>{{cite journal |last1=Polyakov |first1=A. M. |authorlink1=Alexander Markovich Polyakov |date=1981 |title=Quantum geometry of bosonic strings |url=https://dx.doi.org/10.1016/0370-2693%2881%2990743-7 |journal=Physics Letters B |volume=103 |issue=3 |pages=207–210 |doi=10.1016/0370-2693(81)90743-7|url-access=subscription }}</ref> The action reads:
: <math>\mathcal{S} = \frac{T}{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>

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, <math>h^{ab}</math> its inverse, 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 the [[nonlinear sigma model]].<ref name="Frie80">{{cite journal |last=Friedan |first=D. |author-link=Daniel Friedan |title=Nonlinear Models in 2+ε Dimensions |journal=[[Physical Review Letters]] |volume=45 |pages=1057–1060 |date=1980 |issue=13 |url=http://www.physics.rutgers.edu/~friedan/papers/PRL_45_1980_1057.pdf |doi=10.1103/PhysRevLett.45.1057 |bibcode=1980PhRvL..45.1057F}}</ref>

The Polyakov action must be supplemented by the [[Liouville field theory|Liouville action]] to describe string fluctuations.

== Global symmetries ==
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.

The action is [[Invariant (physics)|invariant]] under spacetime [[Translation (geometry)|translations]] and [[infinitesimal]] [[Lorentz transformation]]s{{ordered list
 | list-style-type=lower-roman
 | <math> X^\alpha \to X^\alpha + b^\alpha, </math>
 | <math> X^\alpha \to 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.

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:

: <math>\begin{align}
  \mathcal{S}'
    &= {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) \\
    &= \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) + \operatorname{O}\left(\omega^2\right) \\
    &= \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 + \operatorname{O}\left(\omega^2\right) \\
    &= \mathcal{S} + \operatorname{O}\left(\omega^2\right).
\end{align}</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}(\sigma) \rightarrow \tilde{h}^{ab} = h^{cd} (\tilde{\sigma})\frac{\partial {\sigma}^a}{\partial \tilde{\sigma}^c} \frac{\partial {\sigma}^b}{\partial \tilde{\sigma}^d}. </math>
One can see that:
: <math>
  \tilde{h}^{ab} \frac{\partial}{\partial {\sigma}^a} X^\mu(\tilde{\sigma}) \frac{\partial}{\partial \sigma^b} X^\nu(\tilde{\sigma}) =
  h^{cd} \left(\tilde{\sigma}\right)\frac{\partial \sigma^a}{\partial \tilde{\sigma}^c} \frac{\partial \sigma^b}{\partial \tilde{\sigma}^d} \frac{\partial}{\partial \sigma^a} X^\mu(\tilde{\sigma})\frac{\partial}{\partial {\sigma}^b} X^\nu(\tilde{\sigma}) =
  h^{ab}\left(\tilde{\sigma}\right)\frac{\partial}{\partial \tilde{\sigma}^a}X^\mu(\tilde{\sigma}) \frac{\partial}{\partial \tilde{\sigma}^b} X^\nu(\tilde{\sigma}).
</math>

One knows that the [[Jacobian matrix and determinant|Jacobian]] of this transformation is given by
: <math> \mathrm{J} = \operatorname{det} \left( \frac{\partial \tilde{\sigma}^\alpha}{\partial \sigma^\beta} \right), </math>
which leads to
: <math>\begin{align}
  \mathrm{d}^2 \tilde{\sigma} &= \mathrm{J} \mathrm{d}^2 \sigma \\
                            h &= \operatorname{det} \left( h_{ab} \right) \\
        \Rightarrow \tilde{h} &= \mathrm{J}^2 h,
\end{align}</math>
and one sees that
: <math> \sqrt{-\tilde{h}} \mathrm{d}^2 {\sigma} = \sqrt{-h \left(\tilde{\sigma}\right)} \mathrm{d}^2 \tilde{\sigma}. </math>
Summing up this transformation and relabeling <math> \tilde{\sigma} = \sigma </math>, we see that the action is invariant.

=== Weyl transformation ===
Assume the [[Weyl transformation]]:
: <math> h_{ab} \to \tilde{h}_{ab} = \Lambda(\sigma) h_{ab}, </math>
then
: <math>\begin{align}
                                    \tilde{h}^{ab} &= \Lambda^{-1}(\sigma) h^{ab}, \\
  \operatorname{det} \left( \tilde{h}_{ab} \right) &= \Lambda^2(\sigma) \operatorname{det} (h_{ab}).
\end{align}</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> \hat{h}_{ab} = \exp\left(\phi(\sigma)\right) 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 \hat{h}_{ab}} \frac{\delta \hat{h}_{ab}}{\delta \phi} = -\frac12 \sqrt{-h} \,T_{ab}\, e^{\phi}\, h^{ab} = -\frac12 \sqrt{-h} \,T^a_{\ a} \,e^{\phi} = 0 \Rightarrow T^{a}_{\ a} = 0, </math>
where we've used the [[functional derivative]] chain rule.

== 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>\begin{align}
  T_{ab} &= T \left( G_{ab} - \frac12 h_{ab} h^{cd} G_{cd} \right) = 0, \\
  G_{ab} &= \frac12 h_{ab} h^{cd} G_{cd}, \\
       G &= \operatorname{det} \left( G_{ab} \right) = \frac14 h \left( h^{cd} G_{cd} \right)^2.
\end{align}</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 make the physically insignificant transformation <math>\sqrt{-h} h^{ab} \rightarrow \eta^{ab}</math>, thus writing the action in the ''conformal gauge'':
: <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>\begin{align}
  T_{01} &= T_{10} = \dot{X} X' = 0, \\
  T_{00} &= T_{11} = \frac12 \left( \dot{X}^2 + X'^2 \right) = 0.
\end{align}</math>

Substituting <math> X^\mu \to X^\mu + \delta X^\mu </math>, one obtains
: <math>\begin{align}
  \delta \mathcal{S}
    &= T \int \mathrm{d}^2 \sigma \eta^{ab} \partial_a X^\mu \partial_b \delta X_\mu \\
    &= -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.
\end{align}</math>

And consequently
: <math> \square X^\mu = \eta^{ab} \partial_a \partial_b X^\mu = 0. </math>

The boundary conditions to satisfy the second part of the variation of the action are as follows.
* Closed strings:
*: [[Periodic boundary conditions]]: <math> X^\mu(\tau, \sigma + \pi) = X^\mu(\tau, \sigma). </math>
* Open strings:{{ordered list
 | list-style-type=lower-roman
 | [[Neumann boundary conditions]]: <math> \partial_\sigma X^\mu (\tau, 0) = 0, \partial_\sigma X^\mu (\tau, \pi) = 0. </math>
 | [[Dirichlet boundary conditions]]: <math> X^\mu(\tau, 0) = b^\mu, X^\mu(\tau, \pi) = b'^\mu. </math>
 }}
Working in [[light-cone coordinates]] <math>\xi^\pm = \tau \pm \sigma</math>, we can rewrite the equations of motion as

: <math>\begin{align}
          \partial_+ \partial_- X^\mu &= 0, \\
  (\partial_+ X)^2 = (\partial_- X)^2 &= 0.
\end{align}</math>

Thus, the solution can be written as <math>X^\mu = X^\mu_+ (\xi^+) + X^\mu_- (\xi^-)</math>, and the stress-energy tensor is now diagonal. By [[Fourier series|Fourier-expanding]] the solution and imposing [[canonical commutation relations]] on the coefficients, applying the second equation of motion motivates the definition of the Virasoro operators and lead to the [[N = 2 superconformal algebra#Free field construction|Virasoro constraints]] that vanish when acting on physical states.

== See also ==
* [[D-brane]]
* [[Einstein–Hilbert action]]

== References ==
{{Reflist}}

== Further reading ==
* Polchinski (Nov, 1994). ''What is String Theory'', NSF-ITP-94-97, 153&nbsp;pp., [[arXiv:hep-th/9411028v1]].
* Ooguri, Yin (Feb, 1997). ''TASI Lectures on Perturbative String Theories'', UCB-PTH-96/64, LBNL-39774, 80&nbsp;pp., [[arXiv:hep-th/9612254v3]].

{{Quantum field theories}}
{{String theory topics}}

{{DEFAULTSORT:Polyakov Action}}
[[Category:Conformal field theory]]
[[Category:String theory]]