Paper · Sep 20, 2018

Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

Mauro Carfora, Claudio Dappiaggi, Nicolò Drago, Paolo Rinaldi

math-pharXiv:1809.07652v2
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Abstract

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati and Moretti for vector valued free fields. As a by-product of such classification, we prove that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.

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Submitted
Sep 20, 2018
Last revised
Aug 31, 2019 (v2)
Subjects
math-ph, hep-th, math.DG
DOI
10.1007/s00220-019-03508-2

Comments: 24 pages

Ricci flowPhysics

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On dual description of the deformed O(N) sigma model (A. V. Litvinov and L. A. Spodyneiko, 2018)3Litvinov 2018 Scaling and Entropy for the RG-2 Flow (Mauro Carfora and Christine Guenther, 2018)2Carfora 2018 Classical and quantum integrable sigma models. Ricci flow, "nice duality" and perturbed rational conformal field theories (Vladimir Fateev, 2019)1Fateev 2019 Integrable 2d sigma models: quantum corrections to geometry from RG flow (Ben Hoare, Nat Levine and Arkady A. Tseytlin, 2019)8Hoare 2019 Conformal invariance from scale invariance in non-linear sigma models (Yu Nakayama, 2020)5Nakayama 2020 Strings in Bimetric Spacetimes (Ziqi Yan, 2021)4Yan 2021 Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry (Georgios Papadopoulos and Edward Witten, 2024)6Papadopoulos 2024 Functional Renormalization Group as a Ricci Flow: An 𝓕-Entropy Perspective on Information Metric Dynamics (Jin Mo Bok and Ki-Seok Kim, 2026)7Bok 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classical and quantum integrable sigma models. Ricci flow, "nice duality" and perturbed rational conformal field theories

    Vladimir Fateev

    Topics Physics · Shared terms “sigma model”, “field theory”, “quantum field”

  2. Scaling and Entropy for the RG-2 Flow

    Mauro Carfora and Christine Guenther

    Topics Physics · Same author Mauro Carfora · Shared terms “nonlinear Sigma”, “sigma model”, “group flow”

  3. On dual description of the deformed sigma model

    A. V. Litvinov and L. A. Spodyneiko

    Topics Physics · Shared terms “sigma model”, “quantum field”, “field theory”

  4. Strings in Bimetric Spacetimes

    Ziqi Yan

    Topics Physics · Shared terms “sigma model”, “nonlinear Sigma”, “group flow”

  5. Conformal invariance from scale invariance in non-linear sigma models

    Yu Nakayama

    Topics Physics · Shared terms “sigma model”, “quantum field”, “field theory”

  6. Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry

    Georgios Papadopoulos and Edward Witten

    Topics Physics · Shared terms “sigma model”, “quantum field”, “perturbation theory”

  7. Functional Renormalization Group as a Ricci Flow: An -Entropy Perspective on Information Metric Dynamics

    Jin Mo Bok and Ki-Seok Kim

    Topics Physics · Shared terms “quantum field”, “renormalization group”, “field theory”

  8. Integrable 2d sigma models: quantum corrections to geometry from RG flow

    Ben Hoare, Nat Levine and Arkady A. Tseytlin

    Topics Physics · Shared terms “sigma model”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.