 in physics, the polyakov action, invented by leonard susskind Â§ http://www.youtube.com/watch?v=v8xbspipkdg is the two-dimensional action of a 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 physics letters b65, pages 369 and 471 respectively), and has become associated with alexander polyakov after he made use of it in quantizing the string. 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_{ab} is the worldsheet metric 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 nonlinear sigma model.  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^\alpha_{\ \beta} x^\beta where \omega_{\mu \nu} = - \omega_{\nu \mu} and b^\alpha is a constant. this forms the poincarÃ© 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^\mu_{\ \delta} x^\delta \right) \partial_b \left( x^\nu + \omega^\nu_{\ \delta} 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} + 
Conformal_field_theory String_theory