Worldsheet functional RG and the $α'$-tower

Abstract

We apply the functional renormalization group to the worldsheet theory of the bosonic string. Imposing the Weyl anomaly cancellation as a condition on the local renormalization group flow, we derive flow equations for the background couplings and the corresponding effective spacetime equations of motion, first in a two-derivative truncation, on a general curved background. We then consider a truncation to all orders in $α'$, which captures only a subset of the possible tensor structures but is analytically tractable, in analogy with $f(R)$-like approximations. Within this truncation, we obtain Tseytlin's spacetime effective action, and we use it to study solutions to all orders in $α'$, including black holes. Our results show that the functional renormalization group can be implemented on the string worldsheet and provide a framework to access the strong-curvature regime. As a longer-term goal, this may allow one to test the scenario of effective asymptotic safety in string theory, in which an approximately scale-invariant regime lies between the low-energy effective field theory and a stringy completion.