Is Diffeomorphism on Target Space a Symmetry of Polyakov's Action?

[synoe]

The Polyakov action,

$$S=\frac{1}{4\pi\alpha^\prime}\int d^2\sigma\sqrt{-h}h^{\alpha\beta}G_{ij}(X)\partial_\alpha X^i\partial_\beta X^j$$

has the local symmetries, diffeomorphism on world sheet and the Weyl invariance.
But is diffeomorphism on the target space also a symmetry?
The target space metric transforms

$$G_{ij}(X)\to G_{ij}^\prime(X^\prime)=\frac{\partial X^k}{\partial X^i}\frac{\partial X^l}{\partial X^j}G_{kl}(X).$$

But I don't know how to deal with the part, $\partial_\alpha X^i\partial_\beta X^j$.

 

[Ben Niehoff]

Apply the chain rule as usual, it should work.