- Art Gallery -

In physics, the Polyakov action is an 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 L. Brink, P. Di Vecchia and P. S. Howe (in "A locally supersymmetric and reparametrization invariant action for the spinning string", Physics Letters B, 65, pp. 369 and 471 respectively), and has become associated with Alexander Polyakov after he made use of it in quantizing the string (in "Quantum geometry of the bosonic string", Physics Letters B, 103, 1981, p. 207). The action reads

\( {\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 )

where T is the string tension, \( g_{\mu \nu } \) is the metric of the target manifold, h a b {\displaystyle h_{ab}} h_{{ab}} is the worldsheet metric, h a b {\displaystyle h^{ab}} h^{ab} its inverse, and h is the determinant of \( h_{{ab}} \). The metric signature is chosen such that timelike directions are + and the spacelike directions are –. The spacelike worldsheet coordinate is called σ {\displaystyle \sigma } \sigma whereas the timelike worldsheet coordinate is called τ {\displaystyle \tau } \tau . This is also known as the nonlinear sigma model.[1]

The Polyakov action must be supplemented by the 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 under spacetime translations and infinitesimal Lorentz transformations:

(i) \( X^{\alpha }\rightarrow X^{\alpha }+b^{\alpha } \)
(ii) \( X^{\alpha }\rightarrow X^{\alpha }+\omega _{\ \beta }^{\alpha }X^{\beta } \)

where \( \omega _{\mu \nu }=-\omega _{\nu \mu } \) and \( b^{\alpha }\) is a constant. This forms the Poincaré symmetry of the target manifold.

The invariance under (i) follows since the action \( {\mathcal {S}} \) depends only on the first derivative of \( X^{\alpha } \). The proof of the invariance under (ii) is as follows:

\( {\mathcal {S}}'\, \) \( ={T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-h}}h^{ab}g_{\mu \nu }\partial _{a}\left(X^{\mu }+\omega _{\ \delta }^{\mu }X^{\delta }\right)\partial _{b}\left(X^{\nu }+\omega _{\ \delta }^{\nu }X^{\delta }\right)\, \)
\( ={\mathcal {S}}+{T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-h}}h^{ab}\left(\omega _{\mu \delta }\partial _{a}X^{\mu }\partial _{b}X^{\delta }+\omega _{\nu \delta }\partial _{a}X^{\delta }\partial _{b}X^{\nu }\right)+O(\omega ^{2})\, \)
\( ={\mathcal {S}}+{T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-h}}h^{ab}\left(\omega _{\mu \delta }+\omega _{\delta \mu }\right)\partial _{a}X^{\mu }\partial _{b}X^{\delta }+O(\omega ^{2})={\mathcal {S}}+O(\omega ^{2}) \)

Local symmetries

The action is invariant under worldsheet diffeomorphisms (or coordinates transformations) and Weyl transformations.
Diffeomorphisms

Assume the following transformation:

\( \sigma ^{\alpha }\rightarrow {\tilde {\sigma }}^{\alpha }\left(\sigma ,\tau \right) \)

It transforms the metric tensor in the following way:

\( {\displaystyle h^{ab}(\sigma )\rightarrow {\tilde {h}}^{ab}=h^{cd}({\tilde {\sigma }}){\frac {\partial {\sigma }^{a}}{\partial {\tilde {\sigma }}^{c}}}{\frac {\partial {\sigma }^{b}}{\partial {\tilde {\sigma }}^{d}}}} \)

One can see that:

\( {\displaystyle {\tilde {h}}^{ab}{\frac {\partial }{\partial {\sigma }^{a}}}X^{\mu }({\tilde {\sigma }}){\frac {\partial }{\partial {\sigma }^{b}}}X^{\nu }({\tilde {\sigma }})=h^{cd}({\tilde {\sigma }}){\frac {\partial {\sigma }^{a}}{\partial {\tilde {\sigma }}^{c}}}{\frac {\partial {\sigma }^{b}}{\partial {\tilde {\sigma }}^{d}}}{\frac {\partial }{\partial {\sigma }^{a}}}X^{\mu }({\tilde {\sigma }}){\frac {\partial }{\partial {\sigma }^{b}}}X^{\nu }({\tilde {\sigma }})=h^{ab}({\tilde {\sigma }}){\frac {\partial }{\partial {\tilde {\sigma }}^{a}}}X^{\mu }({\tilde {\sigma }}){\frac {\partial }{\partial {\tilde {\sigma }}^{b}}}X^{\nu }({\tilde {\sigma }})} \)

One knows that the Jacobian of this transformation is given by:

\( \mathrm {J} =\mathrm {det} \left({\frac {\partial {\tilde {\sigma }}^{\alpha }}{\partial \sigma ^{\beta }}}\right) \)

which leads to:

\( {\displaystyle \mathrm {d} ^{2}{\tilde {\sigma }}=\mathrm {J} \mathrm {d} ^{2}\sigma \,} \)
\( {\displaystyle h=\mathrm {det} \left(h_{ab}\right)\rightarrow {\tilde {h}}=\mathrm {J} ^{2}h\,} \)

and one sees that:

\( {\displaystyle {\sqrt {-{\tilde {h}}}}\mathrm {d} ^{2}{\sigma }={\sqrt {-h({\tilde {\sigma }})}}\mathrm {d} ^{2}{\tilde {\sigma }}} \)
\( σ ~ = σ {\displaystyle {\tilde {\sigma }}=\sigma } {\displaystyle {\tilde {\sigma }}=\sigma } we see that the action is invariant.
Weyl transformation

Assume the Weyl transformation:

\( h_{ab}\rightarrow {\tilde {h}}_{ab}=\Lambda (\sigma )h_{ab} \)

then:

\( {\tilde {h}}^{ab}=\Lambda ^{-1}(\sigma )h^{ab} \)
\( \mathrm {det} (h_{ab})} \mathrm {det} ({\tilde {h}}_{ab})=\Lambda ^{2}(\sigma )\mathrm {det} (h_{ab}) \)

And finally:

\( {\mathcal {S}}'\, \) \( ={T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-{\tilde {h}}}}{\tilde {h}}^{ab}g_{\mu \nu }(X)\partial _{a}X^{\mu }(\sigma )\partial _{b}X^{\nu }(\sigma )\, \)
\( ={T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-h}}\left(\Lambda \Lambda ^{-1}\right)h^{ab}g_{\mu \nu }(X)\partial _{a}X^{\mu }(\sigma )\partial _{b}X^{\nu }(\sigma )={\mathcal {S}} \)

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:

\( {\displaystyle T^{ab}={\frac {-2}{\sqrt {-h}}}{\frac {\delta S}{\delta h_{ab}}}} \)

Let's define:

\( {\displaystyle {\hat {h}}_{ab}=\exp \left(\phi (\sigma )\right)h_{ab}} \)

Because of Weyl symmetry the action does not depend on \( \phi : \)

\( {\displaystyle {\frac {\delta S}{\delta \phi }}={\frac {\delta S}{\delta {\hat {h}}_{ab}}}{\frac {\delta {\hat {h}}_{ab}}{\delta \phi }}=-{\frac {1}{2}}{\sqrt {-h}}\,T_{ab}\,e^{\phi }\,h^{ab}=-{\frac {1}{2}}{\sqrt {-h}}\,T_{\ a}^{a}\,e^{\phi }=0\Rightarrow T_{\ a}^{a}=0,} \)

where we've used the functional derivative chain rule.
Relation with Nambu–Goto action

Writing the Euler–Lagrange equation for the metric tensor \( h^{ab} \)one obtains that:

\( {\frac {\delta S}{\delta h^{ab}}}=T_{ab}=0 \)

Knowing also that:

\( \delta {\sqrt {-h}}=-{\frac {1}{2}}{\sqrt {-h}}h_{ab}\delta h^{ab} \)

One can write the variational derivative of the action:

\( {\frac {\delta S}{\delta h^{ab}}}={\frac {T}{2}}{\sqrt {-h}}\left(G_{ab}-{\frac {1}{2}}h_{ab}h^{cd}G_{cd}\right) \)

where \( G_{ab}=g_{\mu \nu }\partial _{a}X^{\mu }\partial _{b}X^{\nu } \) which leads to:

\( T_{ab}=T\left(G_{ab}-{\frac {1}{2}}h_{ab}h^{cd}G_{cd}\right)=0 \)
\( G_{ab}={\frac {1}{2}}h_{ab}h^{cd}G_{cd} \)
\( G=\mathrm {det} \left(G_{ab}\right)={\frac {1}{4}}h\left(h^{cd}G_{cd}\right)^{2} \)

If the auxiliary worldsheet metric tensor \( {\sqrt {-h}} \) is calculated from the equations of motion:

\( {\sqrt {-h}}={\frac {2{\sqrt {-G}}}{h^{cd}G_{cd}}} \)

and substituted back to the action, it becomes the Nambu–Goto action:

\( S={T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-h}}h^{ab}G_{ab}={T \over 2}\int \mathrm {d} ^{2}\sigma {\frac {2{\sqrt {-G}}}{h^{cd}G_{cd}}}h^{ab}G_{ab}=T\int \mathrm {d} ^{2}\sigma {\sqrt {-G}} \)

However, the Polyakov action is more easily quantized because it is linear.
Equations of motion

Using diffeomorphisms and Weyl transformation, with a Minkowskian target space, one can make the physically insignificant transformation \( {\sqrt {-h}}h^{ab}\rightarrow \eta ^{ab} \), thus writing the action in the conformal gauge:

\( {\mathcal {S}}={T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-\eta }}\eta ^{ab}g_{\mu \nu }(X)\partial _{a}X^{\mu }(\sigma )\partial _{b}X^{\nu }(\sigma )={T \over 2}\int \mathrm {d} ^{2}\sigma \left({\dot {X}}^{2}-X'^{2}\right) \)

where \( \eta _{ab}=\left({\begin{array}{cc}1&0\\0&-1\end{array}}\right) \)

Keeping in mind that \( T_{ab}=0 \) one can derive the constraints:

\( T_{01}=T_{10}={\dot {X}}X'=0 \)
\( T_{00}=T_{11}={\frac {1}{2}}\left({\dot {X}}^{2}+X'^{2}\right)=0. \)

Substituting \( X^{\mu }\rightarrow X^{\mu }+\delta X^{\mu } \) one obtains:

\( \delta {\mathcal {S}}=T\int \mathrm {d} ^{2}\sigma \eta ^{ab}\partial _{a}X^{\mu }\partial _{b}\delta X_{\mu } \)

\( =-T\int \mathrm {d} ^{2}\sigma \eta ^{ab}\partial _{a}\partial _{b}X^{\mu }\delta X_{\mu }+\left(T\int d\tau X'\delta X\right)_{\sigma =\pi }-\left(T\int d\tau X'\delta X\right)_{\sigma =0}=0 \)

And consequently:

\( \square X^{\mu }=\eta ^{ab}\partial _{a}\partial _{b}X^{\mu }=0 \)

With the boundary conditions in order to satisfy the second part of the variation of the action.

Closed strings

Periodic boundary conditions: \( X^{\mu }(\tau ,\sigma +\pi )=X^{\mu }(\tau ,\sigma )\ \)

Open strings

(i) Neumann boundary conditions: \( \partial _{\sigma }X^{\mu }(\tau ,0)=0,\partial _{\sigma }X^{\mu }(\tau ,\pi )=0 \)
(ii) Dirichlet boundary conditions: \( X^{\mu }(\tau ,0)=b^{\mu },X^{\mu }(\tau ,\pi )=b'^{\mu }\ \)

Working in light cone coordinates \( \xi ^{\pm }=\tau \pm \sigma \), we can rewrite the equations of motion as:

\( \partial _{+}\partial _{-}X^{\mu }=0 \)
\( (\partial _{+}X)^{2}=(\partial _{-}X)^{2}=0 \)

Thus, the solution can be written as \( X^{\mu }=X_{+}^{\mu }(\xi ^{+})+X_{-}^{\mu }(\xi ^{-}) \)and the stress-energy tensor is now diagonal. By 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 Virasoro constraints that vanish when acting on physical states.

See also

D-brane
Einstein–Hilbert action

Notes

Friedan, D. (1980). "Nonlinear Models in 2+ε Dimensions" (PDF). Physical Review Letters. 45: 1057. Bibcode:1980PhRvL..45.1057F. doi:10.1103/PhysRevLett.45.1057.

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

String theory
Background

Strings History of string theory
First superstring revolution Second superstring revolution String theory landscape


Calabi-Yau-alternate

Theory

Nambu–Goto action Polyakov action Bosonic string theory Superstring theory
Type I string Type II string
Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism GS formalism

String duality

T-duality S-duality U-duality Montonen–Olive duality

Particles and fields

Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon

Branes

D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition

Conformal field theory

Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model

Gauge theory

Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G2, F4, E6, E7, E8) ADE classification Dirac string p-form electrodynamics

Geometry

Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold
Calabi–Yau manifold Hyperkähler manifold
K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory (physics) Twisted K-theory

Supersymmetry

Supergravity Superspace Lie superalgebra Lie supergroup

Holography

Holographic principle AdS/CFT correspondence

M-theory

Matrix theory Introduction to M-theory

String theorists

Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Cleaver Curtright Dijkgraaf Distler Douglas Duff Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind 't Hooft Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach

Physics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License