# Why are we interested in the dimensional analysis/power counting in string theory?

I'm learning bosonic strings on my string theory course; here is part of my notes about the dimensional analysis on the world sheet $$\Sigma$$ and the spacetime manifold $$\mathcal{M}$$:

I learned this in the context of figuring out which terms we could add to the Polyakov action:

$$S_p[\gamma_{ab},X^\mu] = -\frac{T}{2}\int_\Sigma d\tau d\sigma\sqrt{-\gamma}\gamma^{ab}\frac{\partial x^\mu}{\partial\xi^a}\frac{\partial x_\mu}{\partial\xi^b}$$

to maintain Weyl invariance. I remember the consequence is the 'cosmological constant term': $$S_C = \lambda\int_{\Sigma} \sqrt{-\gamma}d\tau d\sigma$$ cannot be added, but the Einstein-Hilbert term: $$S_{EH} = \int_{\Sigma}R(\gamma) \sqrt{-\gamma}d\tau d\sigma$$ can be added.

I don't quite understand if this has anything to do with power counting, and if so, how do we interpret the dimensions for $$\Sigma$$ and $$\mathcal{M}$$? I remember in QFT dimensional analysis/power counting is used to determine whether a given Lagrangian is renormalizable. Is the idea of power counting similar here in string theory (like doing renormalization on the world sheet / embedded spacetime)?

You can see using the worldsheet dimensional analysis rules you've written down that the Polyakov action is scale invariant after accounting for the scaling of $$d/d\xi$$ and of the measure $$d\sigma d\tau$$. Alternatively, you can see that the coupling constant $$T$$ associated with this term is dimensionless.