Irreversibility of World-sheet Renormalization Group Flow

TL;DR

This paper proves that certain world-sheet RG flows in string theory are irreversible by constructing entropy functions inspired by Ricci flow mathematics, extending the understanding of flow behavior on noncompact target spaces.

Contribution

It introduces entropy-based methods to demonstrate irreversibility of world-sheet RG flows, adapting Ricci flow techniques to asymptotically flat manifolds in string theory.

Findings

Constructed an entropy increasing along the flow for negative scalar curvature.

Proved the absence of periodic solutions in the RG flow.

Developed a regularized volume entropy for positive scalar curvature cases.

Abstract

We demonstrate the irreversibility of a wide class of world-sheet renormalization group (RG) flows to first order in $\alpha'$ in string theory. Our techniques draw on the mathematics of Ricci flows, adapted to asymptotically flat target manifolds. In the case of somewhere-negative scalar curvature (of the target space), we give a proof by constructing an entropy that increases monotonically along the flow, based on Perelman's Ricci flow entropy. One consequence is the absence of periodic solutions, and we are able to give a second, direct proof of this. If the scalar curvature is everywhere positive, we instead construct a regularized volume to provide an entropy for the flow. Our results are, in a sense, the analogue of Zamolodchikov's $c$-theorem for world-sheet RG flows on noncompact spacetimes (though our entropy is not the Zamolodchikov $C$-function).