# Symmetry of Polyakov action

In every literature and lecture notes on Bosonic String theory I have found that the authors tend to show the diffoemorphism symmetry in terms of definite symmetry i.e. they take an arbitrary transformation and show the invariance. Then they claim that the infinitesimal symmetry would also be there. It is indeed true but when I tried to find the exact derivation that shows invariance of the polyakov action under infinitesimal transformation, I could find no such thing. Everyone just shows the finite case, claims infinitesimal case and then uses the changes produced in fields $$X^{\mu}$$ and the metric $$h{_\alpha}{_\beta}$$ in infinitesimal transformation to find constraints.

Can anybody exactly show the invariance of the Polyakov action under infinitesimal transformations viz.

$$\sigma^{\alpha}$$ ->$$\sigma^{\alpha}+\epsilon^{\alpha}(\sigma,\tau)$$

I have tried various techniques but of no use. If anyone could help that would be great.

• It is a general fact that if some quantity is invariant under finite transformations, it is in particular invariant under infinitesimal transformations. – Qmechanic Sep 8 '19 at 21:06
• It is indeed true as I wrote in the question itself. The problem arises when one tries to prove the same. – MRITYUNJAY NATH Sep 8 '19 at 21:59