On sigma model RG flow, "central charge" action and Perelman's entropy

Abstract

Zamolodchikov's c-theorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be gradient one to all loop orders. However, the monotonicity of the flow of the target-space metric is not obvious since the metric on the space of metric-dilaton couplings is indefinite. To leading (one-loop) order when the RG flow is simply the Ricci flow the monotonicity was proved by Perelman (math.dg/0211159) by constructing an ``entropy'' functional which is essentially the metric-dilaton action extremised with respect to the dilaton with a condition that the target-space volume is fixed. We discuss how to generalize the Perelman's construction to all loop orders (i.e. all orders in α'). The resulting ``entropy'' is equal to minus the central charge at the fixed points, in agreement with the general claim of the c-theorem.

14 pages. v2: minor comments added, misprints corrected

References in corpus (3)

• A Gradient Flow for Worldsheet Nonlinear Sigma Models
• Irreversibility of World-sheet Renormalization Group Flow
• An eigenvalue problem related to the non-linear sigma-model: analytical and numerical results

Cited by in corpus (36)

• Membranes at Quantum Criticality
• Holographic RG flow and the Quantum Effective Action
• Some Applications of Ricci Flow in Physics
• Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity
• Dirichlet sigma models and mean curvature flow
• Fermionic Kaluza-Klein modes in the string-cigar braneworld
• Black Holes with MDRs and Bekenstein-Hawking and Perelman Entropies for Finsler-Lagrange-Hamilton Spaces
• Gravity localization in a string-cigar braneworld
• Stability of the (two-loop) Renormalization Group Flow for Nonlinear Sigma Models
• The exact $C$-function in integrable $\lambda$-deformed theories
• A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order
• General Covariance from the Quantum Renormalization Group
• On Supersymmetric Geometric Flows and ${\mathcal{R}}^2$ Inflation From Scale Invariant Supergravity
• Stability analysis of the Witten black hole (cigar soliton) under world-sheet RG flow
• Yang Baxter and Anisotropic Sigma and Lambda Models, Cyclic RG and Exact S-Matrices
• Gravitational Kaluza-Klein modes in the String-Cigar Braneworld
• Weyl anomaly and the $C$-function in $\lambda$-deformed CFTs
• Ricci Flow Gravity
• Peculiarities of beta functions in sigma models
• Conformal invariance from scale invariance in non-linear sigma models
• Ricci Flow Approach to The Cosmological Constant Problem
• Lectures in Quantum Gravity
• Strings in compact cosmological spaces
• Spectra of Coset Sigma Models
• Quantum Fluctuations and Geometry: From Graph Counting to Ricci Flow
• Nonequilibrium thermodynamics perspectives for the monotonicity of the renormalization group flow
• Higher order geometric flows on three dimensional locally homogeneous spaces
• Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows
• Integrability of the $\lambda$-deformation of the PCM with spectators
• Mean Curvature Flow on Ricci Solitons
• Scale without conformal invariance from integrable deformations of coset CFTs
• Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear $\sigma$ model
• Universe Decay, Inflation and the Large Eigenvalue of the Cosmological Constant Seesaw
• The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow
• Dynamically restoring conformal invariance in (integrable) $\sigma$-models
• The Higgs mechanism and geometrical flows for two-manifolds