The Polyakov action is invariant under arbitrary transformations of the sort:

$$ \sigma^{\alpha} \rightarrow \tilde{\sigma}^{\alpha}(\sigma). $$

How do I show that the metric will transform like a World Sheet 2 tensor.

$$ g_{\alpha \beta}=\frac{\partial \sigma^{\mu}}{\partial \sigma^{\alpha}} \frac{\partial \sigma^{\prime v}}{\partial \sigma^{\beta}} g_{\mu \nu}^{\prime}\left(\sigma^{\prime}, \tau^{\prime}\right) =\mathcal{L}_{X} g_{a b}~?$$

Thus the answer being of the form:

$$ \mathcal{L}_{X} g_{a b}=\nabla_{X} g_{a b}+g_{c b} \nabla_{a} X^{c}+g_{a c} \nabla_{b} X^{c}. $$


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