# Is the Polyakov action invariant under general coordinate transformation on the target space?

Tags:
1. Oct 1, 2014

### 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, $\partial_\alpha X^i\partial_\beta X^j$.

2. Oct 2, 2014

### Ben Niehoff

Apply the chain rule as usual, it should work.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook