Irreversibility of World-sheet Renormalization Group Flow

Abstract

We demonstrate the irreversibility of a wide class of world-sheet renormalization group (RG) flows to first order in ${\alpha }^{\prime }$ 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).

Minor changes, added one citation, version accepted for publication

References in corpus (4)

• Don't Panic! Closed String Tachyons in ALE Spacetimes
• Spacetime Energy Decreases under World-sheet RG Flow
• Can Renormalization Group Flow End in a Big Mess?
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Cited by in corpus (15)

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• On sigma model RG flow, "central charge" action and Perelman's entropy
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• Stability of the (two-loop) Renormalization Group Flow for Nonlinear Sigma Models
• Energy, entropy and the Ricci flow
• Asymptotically Flat Ricci Flows
• The Fixed Points of RG Flow with a Tachyon
• Investigating Off-shell Stability of Anti-de Sitter Space in String Theory
• Conformal invariance from scale invariance in non-linear sigma models
• Monotonicity formulas under rescaled Ricci flow
• On higher order geometric and renormalisation group flows
• A proposal for studying off-shell stability of vacuum geometries in string theory
• Reduction of coupling constants in two-dimensional sigma models
• The Higgs mechanism and geometrical flows for two-manifolds