Paper · May 4, 2020

Deformed -models, Ricci flow and Toda field theories

Dmitri Bykov, Dieter Lust

hep-tharXiv:2005.01812
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As on arXiv

Abstract

It is shown that the Pohlmeyer map of a -model with a toric two-dimensional target space naturally leads to the `sausage' metric. We then elaborate the trigonometric deformation of the -model, proving that its -dual metric is Kähler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold -models and Toda field theories.

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Submitted
May 4, 2020
Subjects
hep-th, math.DG
Journal
Lett. Math. Phys. 111:150 (2021)
DOI
10.1007/s11005-021-01484-0

Comments: 40 pages

Ricci flowSurfacesKähler geometry

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

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)7Litvinov 2018 Classical and quantum integrable sigma models. Ricci flow, "nice duality" and perturbed rational conformal field theories (Vladimir Fateev, 2019)6Fateev 2019 Quantum flag manifold σ-models and Hermitian Ricci flow (Dmitri Bykov, 2020)2Bykov 2020 Flag manifold sigma models: spin chains and integrable theories (Ian Affleck, Dmitri Bykov and Kyle Wamer, 2021)1Affleck 2021 Strings in Bimetric Spacetimes (Ziqi Yan, 2021)4Yan 2021 Sigma models as Gross-Neveu models (Dmitri Bykov, 2021)5Bykov 2021 Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry (Georgios Papadopoulos and Edward Witten, 2024)3Papadopoulos 2024 Stress-energy tensor deformations, Ricci flows and black holes (Nicolò Brizio, Tommaso Morone and Roberto Tateo, 2024)8Brizio 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Flag manifold sigma models: spin chains and integrable theories

    Ian Affleck, Dmitri Bykov and Kyle Wamer

    Topics Surfaces · Same author Dmitri Bykov · Shared terms “flag manifold”, “target space”, “deformed”

  2. Quantum flag manifold -models and Hermitian Ricci flow

    Dmitri Bykov

    Topics Ricci flow · Same author Dmitri Bykov · Shared terms “flag manifold”, “deformed”

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

    Georgios Papadopoulos and Edward Witten

    Topics Surfaces · Shared terms “target space”, “field theory”, “relation”

  4. Strings in Bimetric Spacetimes

    Ziqi Yan

    Topics Surfaces · Shared terms “target space”

  5. Sigma models as Gross-Neveu models

    Dmitri Bykov

    Topics Ricci flow · Same author Dmitri Bykov · Shared terms “trigonometric”, “target space”

  6. Classical and quantum integrable sigma models. Ricci flow, "nice duality" and perturbed rational conformal field theories

    Vladimir Fateev

    Topics Ricci flow · Shared terms “field theory”, “deformed”, “relation”

  7. On dual description of the deformed sigma model

    A. V. Litvinov and L. A. Spodyneiko

    Topics Ricci flow · Shared terms “deformed”, “field theory”

  8. Stress-energy tensor deformations, Ricci flows and black holes

    Nicolò Brizio, Tommaso Morone and Roberto Tateo

    Topics Surfaces · Shared terms “field theory”, “deformed”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.