$\mathbb Z_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds

Technical Summary: $Z_2$-Harmonic Spinors and 1-Forms on Connected Sums and Torus Sums of 3-Manifolds

Problem Statement
The paper addresses the construction of explicit examples of $Z_2$-harmonic spinors and $Z_2$-harmonic 1-forms on closed, oriented Riemannian 3-manifolds. These objects arise as singular limiting solutions to various gauge-theoretic equations, including the Seiberg-Witten equations and the study of $SL(2, \mathbb{C})$ representation varieties. While Doan and Walpuski [DW21] established a general existence theorem for manifolds with $b_1(Y) > 1$, their proof was non-constructive. Furthermore, explicit examples were previously scarce, limited mostly to symmetric cases. A primary technical barrier is that the singular Dirac operator associated with these objects fails to be Fredholm on standard function spaces due to infinite-dimensional obstructions related to the singular set $Z$. Additionally, the behavior of $Z_2$-harmonic 1-forms is constrained by $L^2$-Hodge theory, distinguishing them from the spinor case.

Methodology
The authors employ a gluing argument to construct solutions on connected sums ($Y_1 \# Y_2$) and torus sums ($Y_1 \natural Y_2$) from solutions on the constituent manifolds. The core technical framework relies on a parameterized version of the Nash-Moser implicit function theorem, established by Donaldson [Don21] and Parker [Par26], which is necessary to handle the "loss of regularity" inherent in the problem.

The methodology proceeds in three main stages:

1. Approximate Solutions: The authors construct approximate solutions on the glued manifold by patching together solutions from the summands.
   • For spinors on connected sums, they utilize the conformal invariance of the spin Dirac operator and a "neck-stretching" construction, where the neck length $T \to \infty$.
   • For 1-forms and torus sums, conformal invariance fails. Instead, they employ "neck-pinching" arguments with model metrics (spherical or toroidal) and construct approximate solutions using locally exact forms (primitives) to ensure the closedness condition ($d\nu = 0$) is maintained.
2. Uniform Estimates and Parametrix Patching: The authors establish uniform elliptic estimates for the degenerating Dirac operators on the neck regions. This involves analyzing model operators on cylindrical ends (spherical or toroidal) and constructing parametrices by patching inverses from the summands with inverses on the neck. A critical component is the analysis of spectral flow and the behavior of the operator near the singular set, which reveals infinite-dimensional obstructions.
3. Correction via Nash-Moser Theory: To correct the approximate solutions to true solutions, the authors apply the Nash-Moser implicit function theorem in the category of tame Fréchet manifolds. This allows them to deform the singular set $Z$ and introduce finite-dimensional perturbations (parameters) to cancel the infinite-dimensional obstructions. The theorem guarantees the existence of a true solution for parameters in a specific open neighborhood.

Key Contributions and Results

• Gluing Theorems:

  • Theorem 1.3 (Connected Sums): Given smooth, weakly non-degenerate $Z_2$-harmonic spinors (or 1-forms) on $Y_1$ and $Y_2$, the authors construct $Z_2$-harmonic spinors on $Y_1 \# Y_2$. The singular set is the disjoint union of the original singular sets, and the solution is a linear combination of the original spinors.
  • Theorem 1.5 (Torus Sums): For the spinor case, the authors construct solutions on torus sums $Y_1 \natural Y_2$ (formed by identifying knot neighborhoods). This requires matching spin structures and framing. The result does not generally hold for 1-forms due to topological constraints on the $L^2$-cohomology of the double branched cover.

• Abundance of Examples on Seifert-Fibered Spaces:

  • Using Seifert-fibered manifolds as building blocks, the authors prove Theorem 1.7, showing that for every $k \geq 1$, there exist metrics admitting smooth, non-degenerate $Z_2$-harmonic spinors with singular sets consisting of disjoint fibers.
  • Corollary 1.8 provides explicit examples on $S^3$ (Hopf links), $S^1 \times S^2$, and the Poincaré homology sphere.

• Global Existence and Isotopy Classes:

  • Corollary 1.10: By combining the gluing results with Bär's theorem on harmonic spinors, the authors prove that every closed, oriented 3-manifold admits infinitely many parameters yielding smooth, non-degenerate $Z_2$-harmonic spinors. Crucially, the singular sets represent infinitely many distinct isotopy classes of embedded links. This strengthens the Doan-Walpuski existence result by removing the $b_1(Y) > 1$ restriction and providing explicit constructions.
  • Corollary 1.11: Using torus sums, they construct examples where the singular set represents non-trivial homology classes (specifically, $2k$ disjoint copies of a knot $K$).

• Applications to Seiberg-Witten Theory:

  • Corollary 1.12: The authors demonstrate that if $b_1(Y) > 0$, there exist infinitely many $\text{Spin}^c$ structures on $Y$ such that the moduli space of solutions to the two-spinor Seiberg-Witten equations is non-empty and non-compact. This contrasts sharply with the standard Seiberg-Witten theory, where such moduli spaces are typically empty for negative formal dimensions or compact.

• Results for 1-Forms and $L^2$-Cohomology:

  • The paper highlights a fundamental distinction between spinors and 1-forms. While spinors can be glued freely, 1-forms are constrained by the $L^2$-cohomology of the double branched cover.
  • Proposition 1.13 constructs $Z_2$-harmonic 1-forms on Seifert-fibered spaces satisfying specific topological conditions (e.g., Brieskorn homology spheres with $n \geq 4$).
  • Conjecture 1.14: The authors conjecture that $Z_2$-harmonic 1-forms with non-trivial singular sets do not exist on manifolds where the $SL(2, \mathbb{C})$ character variety is zero-dimensional (e.g., $S^3$).
  • Theorem 1.15: Describes the connected sum behavior for 1-forms in terms of $L^2$-cohomology, showing an additional $\mathbb{R}$ factor in the cohomology of the double cover, which corresponds to new boundary strata in the representation variety.

Significance
The paper claims to provide the first systematic method for constructing explicit, abundant examples of $Z_2$-harmonic spinors and 1-forms on general 3-manifolds. By moving beyond non-constructive existence proofs and symmetric examples, the work elucidates the relationship between these singular objects and the global topology of the manifold.

Specifically, the results:

1. Strengthen Existence Theorems: They extend the existence of $Z_2$-harmonic spinors to all closed 3-manifolds, not just those with $b_1 > 1$.
2. Clarify Moduli Space Structure: They reveal that the moduli spaces of two-spinor Seiberg-Witten equations can be non-compact for infinitely many $\text{Spin}^c$ structures, a phenomenon linked to the non-compactness of the $SL(2, \mathbb{C})$ representation variety.
3. Distinguish Spinor vs. 1-Form Behavior: The work rigorously demonstrates that while spinors are robust under gluing, 1-forms are subject to stricter topological constraints arising from Hodge theory and the structure of the double branched cover.
4. Provide a Framework for Future Study: The explicit construction of singular sets representing distinct isotopy classes offers a new tool for investigating the geometry of gauge-theoretic moduli spaces and the compactification of representation varieties.