Polyakov action

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, and has become associated with Alexander Polyakov after he made use of it in quantizing the string in 1981. 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.

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