Paper · Feb 4, 2025

Courant Algebroid Relations, T-Dualities and Generalised Ricci Flow

Thomas C. De Fraja, Vincenzo Emilio Marotta, Richard J. Szabo

hep-tharXiv:2502.02318v2
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Abstract

The notion of Courant algebroid relation is used to introduce a definition of relation between divergence operators on Courant algebroids. By introducing invariant divergence operators, a notion of generalised T-duality between divergences is presented through an existence and uniqueness result for related divergence operators on T-dual pairs of exact Courant algebroids, which naturally incorporates the dilaton shift. When combined with the notion of generalised isometry, this establishes circumstances under which generalised Ricci tensors are related, proving that T-duality is compatible with generalised string background equations. This enables an analysis of the compatibility between T-duality and generalised Ricci flow, showing that the T-dual of a solution of generalised Ricci flow is also a solution of generalised Ricci flow. Our constructions are illustrated through many explicit examples.

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Submitted
Feb 4, 2025
Last revised
Feb 20, 2025 (v2)
Subjects
hep-th, math-ph, math.DG

Comments: 49 pages; v2: typos fixed, reference added

Ricci flowGeneralized & G₂ flowsStability & uniqueness

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 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow on Courant algebroids

    Jeffrey Streets, Charles Strickland-Constable and Fridrich Valach

    Topics Generalized & G₂ flows, Stability & uniqueness · Shared terms “Courant algebroids”, “isometry”

  2. On the Generalised Ricci Solitons and Sasakian Manifolds

    Ahmed Mohammed Cherif, Kaddour Zegga and Gherici Beldjilali

    Topics Generalized & G₂ flows · Shared terms “generalised Ricci”

  3. Brane flows

    Georgios Papadopoulos and Kostas Skenderis

    Topics Generalized & G₂ flows · Shared terms “generalised Ricci”

  4. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows · Shared terms “Courant algebroids”, “exact Courant”

  5. Some remarks on strong -structures with torsion

    Anna Fino and Udhav Fowdar

    Topics Generalized & G₂ flows · Shared terms “generalised Ricci”, “relation”, “explicit examples”

  6. The extended quasi-Einstein manifolds with generalised Ricci solitons

    Zhiming Huang, Weijun Lu and Fuhong Su

    Topics Generalized & G₂ flows · Shared terms “generalised Ricci”

  7. Dynamical stability of Pluriclosed and Generalized Ricci solitons

    Kuan-Hui Lee

    Topics Generalized & G₂ flows, Stability & uniqueness

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

    Quang-Tuan Dang

    Topics Generalized & G₂ flows, Stability & uniqueness

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.