Paper · Jun 25, 2020

Quantum flag manifold -models and Hermitian Ricci flow

Dmitri Bykov

hep-tharXiv:2006.14124
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Abstract

We show that flag manifold -models (including , Grassmannian models as special cases) and their deformed versions may be cast in the form of gauged bosonic Thirring/Gross-Neveu-type systems. Quantum mechanically the gauging is violated by chiral anomalies, which may be cancelled by adding fermions. We conjecture that such models are integrable and check on some examples that the trigonometrically deformed geometries satisfy the generalized Ricci flow equations.

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Submitted
Jun 25, 2020
Subjects
hep-th, math.DG
Journal
Commun. Math. Phys. 401, 1-32 (2023)
DOI
10.1007/s00220-022-04532-5

Comments: 41 pages, 6 figures

Ricci flowGeneralized & G₂ flows

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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 The continuity equation, Hermitian metrics and elliptic bundles (Morgan Sherman and Ben Weinkove, 2018)7Sherman 2018 Deformed σ-models, Ricci flow and Toda field theories (Dmitri Bykov and Dieter Lust, 2020)3Bykov 2020 Flag manifold sigma models: spin chains and integrable theories (Ian Affleck, Dmitri Bykov and Kyle Wamer, 2021)1Affleck 2021 Sigma models as Gross-Neveu models (Dmitri Bykov, 2021)2Bykov 2021 Integrability and RG flow in 2d sigma models (Nat Levine, 2021)5Levine 2021 On dual regime in Yang-Baxter deformed O(2N) sigma models (Alexey Bychkov and Alexey Litvinov, 2025)4Bychkov 2025 Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds (Quang-Tuan Dang, 2026)8Dang 2026 On the Calabi estimate of geometric flows of Hermitian metrics (Marco Gallo and Luigi Vezzoni, 2026)6Gallo 2026
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 Ricci flow · Same author Dmitri Bykov · Shared terms “flag manifold”, “deformed geometries”, “Grassmannian models”

  2. Sigma models as Gross-Neveu models

    Dmitri Bykov

    Topics Ricci flow · Same author Dmitri Bykov · Shared terms “Gross Neveu”, “integrable”

  3. Deformed -models, Ricci flow and Toda field theories

    Dmitri Bykov and Dieter Lust

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

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

    Alexey Bychkov and Alexey Litvinov

    Topics Generalized & G₂ flows · Shared terms “deformed”, “integrable”

  5. Integrability and RG flow in 2d sigma models

    Nat Levine

    Topics Generalized & G₂ flows · Shared terms “integrable”, “deformed”

  6. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni

    Topics Generalized & G₂ flows · Shared terms “Hermitian”

  7. The continuity equation, Hermitian metrics and elliptic bundles

    Morgan Sherman and Ben Weinkove

    Topics Generalized & G₂ flows · Shared terms “Hermitian”

  8. Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds

    Quang-Tuan Dang

    Topics Generalized & G₂ flows · Shared terms “Hermitian”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009)

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

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    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.