Stress-Energy Tensor - derivation

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In page 64 of David Tong's notes (http://www.damtp.cam.ac.uk/user/tong/string/four.pdf) on conformal field theory, Tong mentions that

1. the stress-energy tensor is defined as the matrix
of conserved currents which arise from translational invariance,
$$\delta\sigma^{a} = \epsilon^{a},$$
where ##\sigma^{a}## are the worldsheet coordinates, and that

2. in flat spacetime, a translation is a special case of a conformal transformation.

This I understand. But then he goes on to say that

we can derive conserved currents by promoting the constant parameter ##\epsilon## that appears in the symmetry to a function of the spacetime coordinates.

I do not follow this. If the translational invariance of the worldsheet coordinates under ##\delta\sigma^{a} = \epsilon^{a}## give us the stress-energy tensor of the theory, why do we need to gauge the symmetry in order to derive the stress-tensor of the theory?

How do you then go ahead to show that the change in the action must then be of the form,
$$\delta S = \int d^{2}\sigma J^{\alpha}\partial_{\alpha}\epsilon$$
for some function of the fields, ##J^{\alpha}##?
 

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