Polyakov action

In physics, the Polyakov action is the action of a 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

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

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