Deformation rigidity for Z/2 eigensections

Abstract: We prove a rigidity result for certain critical Z/2 eigensections of the Laplacian on S^2 associated to a flat real line bundle determined by a branch-point configuration. More precisely, we show that every minimal non-degenerate critical eigensection is deformation rigid: any sufficiently small deformation of the configuration that still admits a critical eigensection must come from an SO(3)-rotation. This generalizes the rigidity phenomenon previously discovered in symmetric examples of Taubes-Wu.

• The paper introduces a deformation rigidity theorem showing that minimal, non-degenerate, critical Z/2 eigensections are uniquely determined modulo SO(3) rotations.
• It develops a precise asymptotic analysis near branch points, ensuring that only global rotational symmetries allow persistence of these eigensections under perturbation.
• The results have significant implications for geometric analysis, gauge theory, and calibrated geometry by simplifying the local moduli space of singular geometric structures.

Deformation Rigidity for $\mathbb{Z}/2$ Eigensections: Minimal Non-degenerate Models

Introduction and Motivation

The paper "Deformation rigidity for Z/2 eigensections" (2604.17044) investigates the deformation theory of critical eigensections of the Laplacian on the 2-sphere $S^2$, where the Laplacian acts on sections of a flat real line bundle with nontrivial $\mathbb{Z}/2$ monodromy around a finite set of branch points. Critical $\mathbb{Z}/2$-eigensections have emerged as canonical local models for singularities in a spectrum of geometric settings, including the study of singular gauge-theoretic moduli spaces, coassociative fibrations in $\mathrm{G}_2$-geometry, branched special Lagrangian geometry, and degenerations of harmonic spinors and forms arising, for example, in compactness theory for generalized Seiberg–Witten equations (Taubes, 2014, Taubes, 2013, Taubes et al., 2020, Haydys et al., 2023). The paper addresses the persistence of these eigensections under deformations of the underlying branch point configuration, establishing a rigidity result in the minimal, non-degenerate case.

Structural and Analytical Framework

A $\mathbb{Z}/2$ eigensection on $S^2$ with a configuration $p = \{p_1,\ldots,p_{2n}\}$ of branch points is a section of a flat real line bundle $I_p \to S^2 \setminus p$, characterized by monodromy $-1$ around each $S^2$0. The corresponding Laplacian's spectrum is discrete and varies with the configuration. Critical eigensections, which vanish to order at least $S^2$1 at each branch point, are of primary interest since they capture the local model for branching singularities in various geometric analytic problems. The paper gives a careful description of the local asymptotics, emphasizing that criticality is defined by the vanishing of leading-order terms in their asymptotic expansion, and non-degeneracy by the non-vanishing of the next-to-leading terms.

The eigenspaces for the Laplacian in this setting generically exhibit dimension at least $S^2$2, due to the $S^2$3 symmetry of the sphere, and this lower bound of $S^2$4 is achieved in the so-called minimal eigenspaces. Critical, non-degenerate, minimal eigensections correspond to models with maximal analytic tractability and appear naturally in a wide array of contexts, such as in the work of Taubes–Wu (Taubes et al., 2020, Taubes et al., 2021) and the existence/rididity results recently established for symmetric tetrahedral configurations (Chen et al., 2024).

Main Results: Rigidity Phenomenon

The central result of the paper is the deformation rigidity theorem: any minimal, non-degenerate, critical $S^2$5 eigensection on the sphere is rigid with respect to small deformations of the branch point configuration. More precisely, if such an eigensection persists after a small deformation of the configuration, then the new configuration and corresponding eigensection are $S^2$6-rotates of the original. This assertion generalizes the rigidity previously observed for highly symmetric, tetrahedral Taubes–Wu examples to all minimal non-degenerate models.

The proof leverages a careful local model for the space of configurations, decomposing deformations modulo the $S^2$7 action. A crucial step is the construction of a vector bundle over a slice of the configuration space, where critical eigensections define a section whose vanishing locus would correspond to a preserved critical eigensection under deformation. The linearization of this section is shown to be surjective/transverse to the zero section due to the non-degeneracy condition, and since critical eigensections cannot be perturbed in directions orthogonal to the $S^2$8 orbit, only global symmetries permit their persistence.

No nontrivial deformations of the configuration yield new minimal, non-degenerate, critical eigensections except for those induced by rotations. This is a strong claim: the moduli of such eigensections is discrete modulo rotations, even though the space of all possible configurations has large dimension.

Technical Ingredients

Fundamental to the argument are:

• The precise asymptotic expansion of eigensections near branch points;
• The explicit realization of the action of the rotation group both on configurations and eigensections;
• The smooth dependence of eigenvalues/eigenspaces on configuration parameters, following analytic perturbation theory (cf. (Taubes et al., 2021));
• The calculation and control of the derivative of the map from configuration space to the traces of the eigensection at the singularities;
• The identification of the directions in which nontrivial deformations could act and the exclusion of nontrivial kernel elements outside the $S^2$9-orbit.

The analysis draws on earlier foundational works on the local and global structure of critical eigensections (Taubes et al., 2020, Taubes et al., 2021, Taubes, 2014), and recent results on the existence, index theory, and applications to gauge theory and calibrated geometry (Haydys et al., 2023, He et al., 2024, Parker, 2023).

Implications, Applications, and Future Directions

The rigidity result has significant implications for the structure of moduli spaces of objects modeled by $\mathbb{Z}/2$0 critical eigensections. In geometric analysis, it implies that degenerations leading to such singularities are strictly constrained unless accompanied by global symmetry. For the compactification and perturbative study of gauge-theoretic moduli spaces, coassociative and special Lagrangian deformations, and the analysis of singularities in calibrated and geometric measure theory, minimal non-degenerate singularities do not admit local moduli, ensuring predictability and simplifying gluing/construction arguments (Franceschini et al., 28 Mar 2026, Chen et al., 2024, Haydys et al., 17 Jul 2025).

From a theoretical perspective, the result delineates the precise class of singularities that are robust under deformation, indicating that higher multiplicity or non-minimal eigensections could potentially possess local deformations, a feature largely absent in the minimal case.

Further developments could aim to:

• Classify the locus of configurations leading to critical minimal eigensections and their possible degenerations;
• Analyze the moduli of critical eigensections with non-minimal (higher) multiplicity eigenspaces, where local deformation theory could be richer;
• Extend rigidity results to broader classes of operators or higher-dimensional analogs relevant to calibrated geometry and gauge theory applications.

Conclusion

The paper establishes the deformation rigidity for minimal non-degenerate critical $\mathbb{Z}/2$1 eigensections of the sphere Laplacian: these models are uniquely determined (modulo rotation) by their branch point configuration and cannot be perturbed locally except by $\mathbb{Z}/2$2 actions. This work generalizes prior results for symmetric configurations and gives a comprehensive answer to the local moduli question for this class of singularities, which occupy a central position in the analytic study of branched geometry and singular moduli spaces. The methods and conclusions have substantial reach in geometric analysis, gauge theory, and calibrated geometry, providing a rigorous foundation for ongoing work on singular models and moduli rigidity.