Editing Polyakov action (section)

== 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 (physics)|invariant]] under spacetime [[Translation (geometry)|translations]] and [[infinitesimal]] [[Lorentz transformation]]s{{ordered list
 | list-style-type=lower-roman
 | <math> X^\alpha \to X^\alpha + b^\alpha, </math>
 | <math> X^\alpha \to X^\alpha + \omega^\alpha_{\ \beta} X^\beta, </math>
 }}
where <math> \omega_{\mu \nu} = -\omega_{\nu \mu} </math>, and <math> b^\alpha </math> is a constant. This forms the [[Poincaré group|Poincaré symmetry]] of the target manifold.

The invariance under (i) follows since the action <math> \mathcal{S} </math> depends only on the first derivative of <math> X^\alpha </math>. The proof of the invariance under (ii) is as follows:

: <math>\begin{align}
  \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) + \operatorname{O}\left(\omega^2\right) \\
    &= \mathcal{S} + {T \over 2}\int \mathrm{d}^2\sigma\, \sqrt{-h}\, h^{ab} \left( \omega_{\mu \delta} + \omega_{\delta \mu } \right) \partial_a X^\mu \partial_b X^\delta + \operatorname{O}\left(\omega^2\right) \\
    &= \mathcal{S} + \operatorname{O}\left(\omega^2\right).
\end{align}</math>