[Paper Review] The Anomaly flow on unimodular Lie groups

D. H. Phong, Sébastien Picard+1 more|arXiv (Cornell University)|May 27, 2017

Geometry and complex manifolds20 references3 citations

TL;DR

This paper investigates the long-time behavior of the Anomaly flow on unimodular Lie groups using general unitary Hermitian connections, revealing diverse dynamics depending on the group and initial data. It establishes that the flow diverges on the Abelian group $\mathbb{C}^3$, fails to converge on nilpotent groups, and exhibits monotonic eigenvalue evolution on solvable and semisimple groups, with convergence or blow-up determined by initial metric symmetries and the parameter $\alpha'\tau$. The analysis reduces the flow to ODEs via left-invariant geometry, providing explicit evolution equations for the metric components.

ABSTRACT

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary points are precisely the solutions of the Hull-Strominger system. Here we examine its long-time behavior on unimodular Lie groups with general unitary Hermitian connections. We find a diverse and intricate behavior, which depends very much on the Lie group and the initial data.

Motivation & Objective

• To understand the long-time behavior of the Anomaly flow on unimodular Lie groups, which are key examples in non-Kähler geometry and string theory.
• To extend the Anomaly flow framework beyond the Chern connection to general unitary Hermitian connections on the Yano-Gauduchon line.
• To analyze the flow's convergence, divergence, or stability on specific 3-dimensional unimodular Lie groups: $\mathbb{C}^3$, the Heisenberg group, the solvable rigid motion group, and $SL(2,\mathbb{C})$.
• To derive explicit ODE systems for the metric evolution under left-invariant assumptions, enabling detailed dynamical analysis.

Proposed method

• The Anomaly flow is formulated as a flow of $(2,2)$-forms, with the evolution equation derived for left-invariant Hermitian metrics on unimodular Lie groups.
• The flow is reduced to a system of ordinary differential equations (ODEs) by exploiting left-invariance and the unimodular condition on structure constants.
• Explicit evolution equations for the metric components $g_{\bar{a}b}(t)$ are derived, involving curvature contractions and the parameter $\alpha'\tau$.
• The analysis uses the norm of the holomorphic $(3,0)$-form $\Omega$ and the Yano-Gauduchon line of connections to generalize the flow beyond the Chern connection.
• The dynamics are studied via eigenvalue analysis of the metric, with monotonicity and blow-up behavior analyzed using Cauchy-Kowalevska theorem and energy-type estimates.
• The flow's behavior is classified by group type: stationary on $\mathbb{C}^3$, non-convergent on nilpotent groups, and with eigenvalue monotonicity on solvable and semisimple groups.

Experimental results

Research questions

• RQ1 How does the Anomaly flow behave on unimodular Lie groups when generalized to arbitrary unitary Hermitian connections on the Yano-Gauduchon line?
• RQ2 Does the Anomaly flow converge to a stationary solution on unimodular Lie groups, and if not, what determines its divergence?
• RQ3 What role do the initial metric's symmetry and eigenvalue structure play in the long-term dynamics of the flow?
• RQ4 How does the parameter $\alpha'\tau$ influence the stability and evolution of the metric under the Anomaly flow?
• RQ5 Can the flow be reduced to a tractable system of ODEs on unimodular Lie groups, and what dynamical features emerge from this reduction?

Key findings

• On $\mathbb{C}^3$, the Anomaly flow is stationary for any initial metric, as all metrics are balanced and satisfy the Hull-Strominger system trivially.
• On the nilpotent Heisenberg group, no stationary points exist, and the flow diverges for any initial metric, with eigenvalues evolving such that the lowest eigenvalue remains constant.
• On the solvable group of rigid motions, the flow preserves diagonal metrics, and eigenvalues evolve monotonically: $\lambda_1(t)$ decreases if $\lambda_1(0) < 2\beta$, and $\lambda_3(t)$ increases if $\lambda_3(0) > 2\beta$.
• On $SL(2,\mathbb{C})$, the flow diverges to infinity when all eigenvalues are equal, and the set where eigenvalues are equal is both open and closed due to analyticity of solutions.
• The ordering $\lambda_1 \geq \lambda_2 \geq \lambda_3$ is preserved along the flow, and eigenvalue equality at any time implies equality for all time due to analytic continuation.
• The evolution equation for $\partial_t \lambda_1$ is explicitly derived and used to prove monotonicity via comparison estimates involving $\beta$ and the eigenvalue ratios.