# 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?

• What do you mean by 'away from criticality'? Are you allowing the central charge to be non-critical? Relevant operators for the worldsheet CFT are tachyons. These have obviously been discussed a lot, but if you allow them to be turned on, I'm not sure how much is known about the resulting RG flow. Jun 29, 2020 at 16:48
• @4xion What do you mean by the central charge to be non-critical? Do you mean $c \neq 26$ for the bosonic string theory for instance? Jun 30, 2020 at 7:12

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:

[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.

• Thank you for answering my question. You mean the worldsheet CFT other than c=26 cannot be the fixed point of the RG? Also, what is the string theory \beta function? Jun 30, 2020 at 7:19