Paper · Jun 9, 2020

Conformal invariance from scale invariance in non-linear sigma models

Yu Nakayama

hep-tharXiv:2006.05070
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Abstract

There exists a certain argument that in even dimensions, scale invariant quantum field theories are conformal invariant. We may try to extend the argument in dimensions, but the naive extension has a small loophole, which indeed shows an obstruction in non-linear sigma models in dimensions. Even though it could have failed due to the loophole, we show that scale invariance does imply conformal invariance of non-linear sigma models in dimension from the seminal work by Perelman on the Ricci flow.

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Submitted
Jun 9, 2020
Subjects
hep-th
Journal
Phys. Rev. D 102, 065018 (2020)
DOI
10.1103/PhysRevD.102.065018

Comments: 14 pages

Ricci flowPhysics

Its neighbourhood

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

    Georgios Papadopoulos and Edward Witten

    Topics Physics · Shared terms “conformal invariance”, “scale invariance”, “sigma model”

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

    Mauro Carfora, Claudio Dappiaggi, Nicolò Drago et al.

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

  3. 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”

  4. 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”

  5. On dual regime in Yang-Baxter deformed sigma models

    Alexey Bychkov and Alexey Litvinov

    Topics Physics · Shared terms “sigma model”

  6. Strings in Bimetric Spacetimes

    Ziqi Yan

    Topics Physics · Shared terms “sigma model”, “invariance”

  7. Sigma models as Gross-Neveu models

    Dmitri Bykov

    Topics Physics · Shared terms “sigma model”

  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.