Symmetries of Polyakov Action

1. Poincaré Invariance:

   ${\tilde{X}}^{\mu }={\Lambda }_{\nu }^{\mu }{X}^{\nu }+a^{\mu }$

   ${\tilde{h}}^{ab}=h^{ab}$

   where ${\Lambda }_{\nu }^{\mu }{\Lambda }_{\sigma }^{\rho }{\eta }_{\mu \rho }={\eta }_{\nu \sigma }$. The generators ${\Lambda }_{\nu }^{\mu },b^{\mu }$ are independent of the world-sheet coordinates ${\sigma }^a$.

2. Reparametrisation Invariance:

   Consider coordinate transformations ${\sigma }^a\to {\tilde{\sigma }}^a(\sigma )$. Then,

   ${\tilde{X}}^{\mu }(\sigma )=X^{\mu }(\sigma )$

   $\frac{\partial {\tilde{\sigma }}^c}{\partial {\sigma }^a}\frac{\partial {\tilde{\sigma }}^d}{\partial {\sigma }^b}{\tilde{h}}_{cd}(\tilde{\sigma })=h_{ab}(\sigma )$.

3. Weyl Scaling

   ${\tilde{X}}^{\mu }(\sigma )=X^{\mu }(\sigma )$

   ${\tilde{h}}^{ab}(\sigma )=e^{\Theta (\sigma )}{h}^{ab}(\sigma )$

   where $\Theta (\sigma )$ is an arbitrary function.

Poincaré and reparametrisation invariances are manifest in the notation, while invariance under Weyl transformations follows immediately from $\sqrt{-\tilde{h}}=e^{-\Theta }\sqrt{-\tilde{h}}$.

Poincaré invariance is a global symmetry whereas the world-sheet coordinate transformations and Weyl invariance are gauge symmetries or redundancies that simply reparametrise classical solutions.