I am reading Polchinski's book on string theory (vol. 1), Chap. 2 subchapter "Ward identities and Noether's current".
At some point, in Eq. (2.3.12), the author claims that under spacetime translations, $\delta X^{\mu}=\epsilon \rho(\sigma)a^{\mu}$, the action varies according to $$\delta S=\frac{\epsilon a_{\mu}}{2\pi\alpha'} \int d^2\sigma\ \partial^aX^{\mu}\partial_a\rho(\sigma)\tag{2.3.12}$$ because the variation reduces to zero when $\rho(\sigma)$ is constant. From that, he defines the conserved current to be $$j^{\mu}_a=\frac{i}{\alpha'}\partial_aX^{\mu}\tag{2.3.13}.$$
Then, he tries to do the same with worldsheet translations $\delta\sigma^a=\epsilon\upsilon^a$, under which $\delta X^{\mu}=-\epsilon\upsilon^a\partial_a X^{\mu}$, and he writes down the conserved current (without writing down the variation of the action) as $$j_a=i\upsilon^bT_{ab},\quad T_{ab}=-\frac{1}{\alpha'} :\Big( \partial_aX^{\mu}\partial_bX_{\mu}- \frac{1}{2}\delta_{ab}\partial_cX^{\mu}\partial^cX_{\mu} \Big):\tag{2.3.15}$$ I guess that first, the author considers a worldsheet translation that is not local and says that if that is a symmetry, then $\delta S=0$, and if it is not local, i.e. $\rho=\rho(\sigma)$, then the transformations become $\delta\sigma^a=\epsilon\upsilon^a\rho(\sigma)$, and hence the fields transforms as $\delta X^{\mu}=-\epsilon\upsilon^a\partial_a X^{\mu}\rho(\sigma)$.
If my guessings are correct, what is the variation of the action and how do we "read off" the worldsheet energy momentum tensor components from this action variation?
