Paper · Nov 27, 2016

Ricci flow, Killing spinors, and T-duality in generalized geometry

Mario Garcia-Fernandez

math.DGarXiv:1611.08926v3
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Abstract

We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Hull-Strominger system. Our main result investigates a method to produce new solutions of the Ricci flow and the Killing spinor equations. For this, we consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrate that solutions of the equations are exchanged under this symmetry. As applications, we give a mathematical explanation of the dilaton shift in string theory and prove that the Hull-Strominger system is preserved by T-duality.

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Submitted
Nov 27, 2016
Last revised
Apr 17, 2019 (v3)
Subjects
math.DG, hep-th

Comments: 38 pages, added details to the proof of Proposition 4.1, new Remark 4.6 and Example 4.10, to appear in Advances in Mathematics

Ricci flowRicci-flat & Calabi–YauPhysics

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 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 Ricci flow, Courant algebroids, and renormalization of Poisson–Lie T-duality (Pavol Ševera and Fridrich Valach, 2016)4Ševera 2016 Generalized Ricci Flow (Mario Garcia-Fernandez and Jeffrey Streets, 2020)3Garcia-Ferna 2020 The Gibbons-Hawking ansatz in generalized Kähler geometry (Jeffrey Streets and Yury Ustinovskiy, 2020)7Streets 2020 Robinson-Trautman solutions with scalar hair and Ricci flow (Masato Nozawa and Takashi Torii, 2023)6Nozawa 2023 Ricci flow on Courant algebroids (Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach, 2024)8Streets 2024 The Strominger System and Flows by the Ricci Tensor (Sébastien Picard, 2024)2Picard 2024 Pluriclosed flow and the Hull-Strominger system (Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets, 2024)1Garcia-Ferna 2024 Cohomogeneity one solutions of the IIB system (Lorenzo Foscolo and Mario Garcia-Fernandez, 2026)5Foscolo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pluriclosed flow and the Hull-Strominger system

    Mario Garcia-Fernandez, Raul Gonzalez Molina and Jeffrey Streets

    Topics Ricci flow · Same author Mario Garcia-Fernandez · Shared terms “Hull Strominger”, “Strominger system”, “algebroids”

  2. The Strominger System and Flows by the Ricci Tensor

    Sébastien Picard

    Topics Ricci-flat & Calabi–Yau, Physics · Shared terms “Strominger system”

  3. Generalized Ricci Flow

    Mario Garcia-Fernandez and Jeffrey Streets

    Topics Ricci-flat & Calabi–Yau · Same author Mario Garcia-Fernandez · Shared terms “mathematical”

  4. Ricci flow, Courant algebroids, and renormalization of Poisson–Lie T-duality

    Pavol Ševera and Fridrich Valach

    Topics Physics · Shared terms “Courant”, “algebroids”

  5. Cohomogeneity one solutions of the IIB system

    Lorenzo Foscolo and Mario Garcia-Fernandez

    Topics Ricci-flat & Calabi–Yau · Same author Mario Garcia-Fernandez · Shared terms “Strominger”, “holonomy”

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

    Masato Nozawa and Takashi Torii

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

  7. The Gibbons-Hawking ansatz in generalized Kähler geometry

    Jeffrey Streets and Yury Ustinovskiy

    Topics Ricci-flat & Calabi–Yau, Physics · Shared terms “string theory”

  8. Ricci flow on Courant algebroids

    Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach

    Topics Physics · Shared terms “Courant”, “algebroids”, “extending”

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) 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) 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.