Paper · Feb 27, 2024

The Strominger System and Flows by the Ricci Tensor

Sébastien Picard

math.DGarXiv:2402.17770v4
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Abstract

This is a survey on the Strominger system and a geometric flow known as the anomaly flow. We will discuss various aspects of non-Kähler geometry on Calabi-Yau threefolds. Along the way, we discuss balanced metrics and balanced classes, the Aeppli cohomology class associated to a solution to the Strominger system, the equations of motion of heterotic supergravity, and a version of Ricci flow in this special geometry.

Comments: v4 typos corrected. 45 pages

Ricci flowRicci-flat & Calabi–Yau3-manifoldsKähler geometryPhysics

Its neighbourhood

Connected papers

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 2021, 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 Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity (A. A. Varshovi, 2022)7Varshovi 2022 Flows of G₂-Structures associated to Calabi-Yau Manifolds (Sébastien Picard and Caleb Suan, 2022)2Picard 2022 The Heterotic-Ricci flow and its three-dimensional solitons (Andrei Moroianu, Ángel J. Murcia and C. S. Shahbazi, 2023)5Moroianu 2023 Robinson-Trautman solutions with scalar hair and Ricci flow (Masato Nozawa and Takashi Torii, 2023)8Nozawa 2023 Pluriclosed flow and the Hull-Strominger system (Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets, 2024)3Garcia-Ferna 2024 Chern-Ricci flow and t-Gauduchon Ricci-flat condition (Eder M. Correa, Giovane Galindo and Lino Grama, 2025)6Correa 2025 Flows of conformally coclosed G₂-structures with dilaton (Spiro Karigiannis, Sébastien Picard and Caleb Suan, 2025)1Karigiannis 2025 Cohomogeneity one solutions of the IIB system (Lorenzo Foscolo and Mario Garcia-Fernandez, 2026)4Foscolo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Flows of conformally coclosed -structures with dilaton

    Spiro Karigiannis, Sébastien Picard and Caleb Suan PDF

    Topics 3-manifolds, Kähler geometry · Same author Sébastien Picard · Shared terms “anomaly flow”, “threefold”, “geometric flows”

  2. Flows of -Structures associated to Calabi-Yau Manifolds

    Sébastien Picard and Caleb Suan PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same author Sébastien Picard

  3. Pluriclosed flow and the Hull-Strominger system

    Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets PDF

    Topics Kähler geometry · Shared terms “Strominger system”, “Yau”

  4. Cohomogeneity one solutions of the IIB system

    Lorenzo Foscolo and Mario Garcia-Fernandez PDF

    Topics Ricci-flat & Calabi–Yau, 3-manifolds, Kähler geometry · Shared terms “Strominger”, “balanced”

  5. The Heterotic-Ricci flow and its three-dimensional solitons

    Andrei Moroianu, Ángel J. Murcia and C. S. Shahbazi PDF

    Topics Physics, 3-manifolds · Shared terms “Heterotic”, “supergravity”

  6. Chern-Ricci flow and t-Gauduchon Ricci-flat condition

    Eder M. Correa, Giovane Galindo and Lino Grama PDF

    Topics Ricci-flat & Calabi–Yau, 3-manifolds, Kähler geometry · Shared terms “balanced”, “threefold”, “geometric flows”

  7. Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity

    A. A. Varshovi PDF

    Topics Physics, 3-manifolds

  8. Robinson-Trautman solutions with scalar hair and Ricci flow

    Masato Nozawa and Takashi Torii PDF

    Topics Ricci-flat & Calabi–Yau, Physics · Shared terms “Ricci tensor”, “supergravity”

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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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.