Polyakov action information

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 1976,^[1]^[2] and has become associated with Alexander Polyakov after he made use of it in quantizing the string in 1981.^[3] The action reads:

{\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 ),

where $T$ is the string tension, $g_{\mu \nu }$ is the metric of the target manifold, $h_{ab}$ is the worldsheet metric, $h^{ab}$ 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 $\sigma$ , whereas the timelike worldsheet coordinate is called $\tau$ . This is also known as the nonlinear sigma model.^[4]

The Polyakov action must be supplemented by the Liouville action to describe string fluctuations.

^ Deser, S.; Zumino, B. (1976). "A Complete Action for the Spinning String". Phys. Lett. B. 65 (4): 369–373. doi:10.1016/0370-2693(76)90245-8.
^ Brink, L.; Di Vecchia, P.; Howe, P. (1976). "A locally supersymmetric and reparametrization invariant action for the spinning string". Physics Letters B. 65 (5): 471–474. doi:10.1016/0370-2693(76)90445-7.
^ Polyakov, A. M. (1981). "Quantum geometry of bosonic strings". Physics Letters B. 103 (3): 207–210. doi:10.1016/0370-2693(81)90743-7.
^ Friedan, D. (1980). "Nonlinear Models in 2+ε Dimensions" (PDF). Physical Review Letters. 45 (13): 1057–1060. Bibcode:1980PhRvL..45.1057F. doi:10.1103/PhysRevLett.45.1057.

[1] Deser, S.; Zumino, B. (1976). "A Complete Action for the Spinning String". Phys. Lett. B. 65 (4): 369–373. doi:10.1016/0370-2693(76)90245-8.

[2] Brink, L.; Di Vecchia, P.; Howe, P. (1976). "A locally supersymmetric and reparametrization invariant action for the spinning string". Physics Letters B. 65 (5): 471–474. doi:10.1016/0370-2693(76)90445-7.

[3] Polyakov, A. M. (1981). "Quantum geometry of bosonic strings". Physics Letters B. 103 (3): 207–210. doi:10.1016/0370-2693(81)90743-7.

[Frie80-4] Friedan, D. (1980). "Nonlinear Models in 2+ε Dimensions" (PDF). Physical Review Letters. 45 (13): 1057–1060. Bibcode:1980PhRvL..45.1057F. doi:10.1103/PhysRevLett.45.1057.

Polyakov action information

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