Stress-Energy Tensor - derivation

[spaghetti3451]

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, $J^{\alpha}$?

 

[Ambitwistor]

Thank you for bringing up these questions regarding the stress-energy tensor in conformal field theory. I can understand how these concepts may be confusing, so I will try my best to address your concerns.

Firstly, let's clarify the role of the stress-energy tensor in conformal field theory. The stress-energy tensor, also known as the energy-momentum tensor, is a fundamental quantity in any field theory. It encodes the conserved currents associated with the symmetries of the theory, such as translational invariance, rotational invariance, and conformal invariance. In the case of conformal field theory, the stress-energy tensor contains information about the conformal symmetry of the theory.

Now, let's address your first question about why we need to gauge the symmetry in order to derive the stress-energy tensor. The answer lies in the fact that the stress-energy tensor is defined as the matrix of conserved currents. In other words, it is a collection of conserved currents that arise from the symmetries of the theory. In order to obtain these conserved currents, we need to gauge the symmetry. This means that we need to allow the symmetry parameter, $\epsilon$, to vary as a function of the spacetime coordinates. This is what Tong means when he says that we can derive conserved currents by promoting $\epsilon$ to a function of the spacetime coordinates.

To answer your second question, let's take a closer look at the change in the action, $\delta S$. In order for the action to be invariant under the gauge transformation, the change in the action must be proportional to the gauge parameter, $\epsilon$. This is why we can write $\delta S = \int d^{2}\sigma J^{\alpha}\partial_{\alpha}\epsilon$, where $J^{\alpha}$ is a function of the fields. This is known as the Noether current associated with the symmetry transformation.

I hope this clarifies your doubts about the stress-energy tensor and its relation to gauge transformations in conformal field theory. If you have any further questions, please do not hesitate to ask. As scientists, it is important for us to have a clear understanding of these fundamental concepts in order to progress in our research. Thank you for your interest in this topic.