Polyakov action

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History Glossary
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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:

${\displaystyle {\mathcal {S}}={\frac {T}{2}}\int \mathrm {d} ^{2}\sigma \,{\sqrt {-h}}\,h^{ab}g_{\mu \nu }(X)\partial _{a}X^{\mu }(\sigma )\partial _{b}X^{\nu }(\sigma ),}$

where ${\displaystyle T}$ 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 the nonlinear sigma model.^[4]

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