Worldsheet CFT away from criticality Can we obtain the worldsheet CFT describing string theory as a fixed point of some renormalization group flow (although I assume it leads breaking of diffeomorphism)? In other words, any irrelevant deformation of the worldsheet CFT have been discussed or known?
 A: Recall that $c=26$ is needed to guarantee the vanishing of the string theory $\beta$ function at linear order in $\alpha{´}$, then if you start with the the construction of a wolrdsheet CFT with $c \neq 26$ then the trace of the energy momomentum tensor no longer vanishes and one should derive the dependence of the new effective action on the zero modes of such trace. This logic is probably the simplest way to construct wolrdsheet CFTs with non trivial $\beta$ functions such as non critical strings, non-linear sigma models and Liouville Theory.
The general case of a non-trivial beta function with general $\Phi$, $G_{\mu \nu}$ and $B_{\mu \nu}$ dependence is discussed in the section 6 of the first edition of the textbook "String Theory in a Nutshell".
Another very interesting example I want to highlight is the $c=1$ string case. Critical bosonic string theory is the starting point in all string theory textbooks despite of its quantum mechanical inconsistency. c=1 string theory is a consistent non-critical string theory with extraordinary properties that have recently recieved a lot of impressive and exciting progress.
References:
Overview talk: Xi Yin - The non-perturbative completion of c=1 string theory.
[1] The c=1 String Theory S-Matrix Revisited https://arxiv.org/abs/1705.07151.
[2] Multi-Instanton Calculus in c = 1 String Theory
https://arxiv.org/abs/1912.07170.
[3] Long String Scattering in c = 1 String Theory
https://arxiv.org/abs/1810.07233.
