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Flows of conformally coclosed $G_2$-structures with dilaton

Spiro Karigiannis, Sébastien Picard, Caleb Suan

TL;DR

This work develops a dimensional-reduction framework that lifts natural complex-geometric flows to seven-dimensional $G_2$-geometry, emphasizing the $G_2$-Laplacian coflow as a lift of the Kähler–Ricci flow and a seven-dimensional anomaly-flow lift yielding conformally coclosed $G_2$-structures with a dilaton. It derives curvature identities for conformally coclosed $G_2$-structures, analyzes fixed points of the modified $G_2$-anomaly flow, and proves short-time existence for a broad family of flows with parameter $C<1$, using DeTurck-type gauges and symbol-positivity arguments. The fixed-point analysis reveals generalized Ricci solitons and torsion-free or nearly parallel cases depending on $C$, and the metric evolution aligns with the generalized Ricci flow in the distinguished case $C= frac{4}{3}$. By connecting to complex Monge–Ampère flows and heterotic $G_2$ geometry, the paper provides a unified picture of seven-dimensional dilaton flows and their potential physical interpretations.

Abstract

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the Kähler--Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.

Flows of conformally coclosed $G_2$-structures with dilaton

TL;DR

This work develops a dimensional-reduction framework that lifts natural complex-geometric flows to seven-dimensional -geometry, emphasizing the -Laplacian coflow as a lift of the Kähler–Ricci flow and a seven-dimensional anomaly-flow lift yielding conformally coclosed -structures with a dilaton. It derives curvature identities for conformally coclosed -structures, analyzes fixed points of the modified -anomaly flow, and proves short-time existence for a broad family of flows with parameter , using DeTurck-type gauges and symbol-positivity arguments. The fixed-point analysis reveals generalized Ricci solitons and torsion-free or nearly parallel cases depending on , and the metric evolution aligns with the generalized Ricci flow in the distinguished case . By connecting to complex Monge–Ampère flows and heterotic geometry, the paper provides a unified picture of seven-dimensional dilaton flows and their potential physical interpretations.

Abstract

We study flows of -structures guided by the principle of dimensional reduction: natural geometric flows in -geometry reduce to natural flows in complex geometry. Our main examples are the -Laplacian coflow, which lifts the Kähler--Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The -lift of the anomaly flow deforms conformally coclosed -structures. We compare the -anomaly flow to the -Laplacian coflow, and investigate short-time existence and fixed points.

Paper Structure

This paper contains 19 sections, 17 theorems, 222 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Lifting flows of complex geometry
  3. Kähler--Ricci flow
  4. Anomaly flow
  5. Ascension of the complex Monge--Ampère equation
  6. Remarks on the hypersymplectic flow
  7. Curvature of conformally coclosed G2-structures
  8. Preliminary identities for conformally coclosed G2-structures
  9. The Ricci curvature of a conformally coclosed G2-structure
  10. Modified G2-anomaly flows
  11. Notation and setup
  12. Fixed points of the flow
  13. Evolution of the metric
  14. Comments on heterotic G2
  15. Short-time existence of the flow
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $M^7$ be a compact 7-manifold with initial positive 4-form $\psi_0$ satisfying the conformally coclosed condition $d ( e^{-2f_0} \psi_0)=0$ for some function $f_0$. If $C<1$, then the flow where $e^{4f} = e^{4f_0} \frac{ \text{\normalfont{vol}} _\varphi}{ \text{\normalfont{vol}} _0}$, admits a unique short-time solution. Note that for all time for which the solution exists, the cohomology cla

Theorems & Definitions (43)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • ...and 33 more