Getting GR out of string theory – we’re doing perturbative expansion of the classical action, but that only works at weak curvature. If it were high curvature, we’d need a different approach.
Show how QFT imposes constraints on how the string theory has to interact with the background metric – mathematically beautiful!
Puts up a slide with the weyl transformations on the screen – notes that conformal transformations act on coordinates and weyl transformations act on the metric.
QFT will break weyl invariance…
Now puts up the Polyakov action on the screen and explains the terms – reminds us that we have a worldsheet metric and a target metric (conformal symmetry wrt string metric, not target metric!)
Let’s quantize this action… Perturb the classical solution and move to a quantum description. Transform coordinates until we find Riemannian normal reference system in which everything looks tidy. Perturb the metric and get a term that depends on Riemann tensor. Plug this back into the action and get a divergence. (Offers to demonstrate this, but the audience agree to take her at her word). We want to introduce a length scale to fix this, but we had a conformally invariant theory! Apparently if we understand how this can be the case, we’ll understand how the Ricci flatness condition comes out!
We do a bit of work on the action integral, and the Ricci tensor starts to appear. (This blogger apologises for the fact that her latex equation skills aren’t quite up to the task of typing all this out for the reader.) If we look closely, we see that things aren’t conformally invariant after all!