Spectral asymmetry via pseudodifferential projections: the massless Dirac operator

Technical Summary: Spectral Asymmetry via Pseudodifferential Projections for the Massless Dirac Operator

Problem Statement
The paper addresses the study of spectral asymmetry for systems of partial differential equations (PDEs) on closed, connected, and oriented Riemannian 3-manifolds. Specifically, it focuses on the massless Dirac operator (also known as the Weyl operator), denoted as $W$. While spectral asymmetry is traditionally quantified by the eta invariant—a global number or local function derived from the spectral zeta function—the authors aim to apply a newer, operator-theoretic approach. This approach seeks to encode asymmetry information within a negative-order pseudodifferential operator, termed the "asymmetry operator," constructed from spectral projections. The central challenge is to compute the principal symbol of this operator for the massless Dirac case, accounting for gauge invariance and the specific geometric properties of the manifold, and to demonstrate that a regularized trace can be defined to yield geometric invariants.

Methodology
The authors employ the framework of pseudodifferential calculus developed in their previous works [9, 10]. The methodology proceeds through the following steps:

1. Spectral Projections: The massless Dirac operator $W$ is self-adjoint and elliptic with a discrete spectrum accumulating at $\pm \infty$. The authors define positive ($P_+$) and negative ($P_-$) spectral projections based on the eigenvalues of $W$. These are pseudodifferential operators of order zero.
2. Construction of the Asymmetry Operator: The asymmetry operator $A$ is defined as the matrix trace of the difference between these projections: $A := \text{tr}(P_+ - P_-)$. While $P_+ - P_-$ is not trace-class, the matrix trace operation yields an operator $A$ of negative order.
3. Symbolic Calculus and Cancellations: The core of the analysis involves the explicit construction of the full symbols of $P_\pm$ using an iterative algorithm (based on [8]). The authors calculate the principal symbols of the components of the asymptotic expansion of $A$. A critical aspect of the methodology is the exploitation of cancellations that occur when taking the matrix trace.
4. Gauge Invariance and Framing: The Dirac operator depends on a choice of global framing. The authors analyze the behavior of the operator under gauge transformations (rotations of the framing). They demonstrate that while intermediate calculations depend on the framing, the spectral invariants do not. To simplify computations, they fix a specific "Levi-Civita framing" generated by parallel transport in geodesic normal coordinates centered at a point $z$. In this specific frame, the torsion of the associated Weitzenböck connection vanishes at $z$, simplifying the symbolic expansion.
5. Regularized Trace: Since the resulting asymmetry operator is of order $-1$ (unlike the order $-3$ found for the curl operator in previous studies), it is not immediately trace-class. The authors define a "regularized local trace" by isolating the continuous part of the Schwartz kernel of $A$ along the diagonal ($x=y$).

Key Contributions and Results
The paper provides explicit computations for the massless Dirac operator on a 3-manifold, yielding the following main results:

• Vanishing of Lower Order Terms: The authors prove that the principal symbol of the asymmetry operator $A$ associated with the Levi-Civita framing vanishes for orders $j=0, -1, -2$. Specifically, the terms $a_0, a_{-1},$ and $a_{-2}$ in the asymptotic expansion of the symbol are zero at the center of the geodesic neighborhood.
• Principal Symbol Calculation: The first non-vanishing term is the principal symbol of order $-3$, denoted $a_{-3}(z, \xi)$. The authors compute this explicitly as:
  $$\tilde{a}_{-3}(z, \xi) = -\frac{1}{12} \frac{E_{\alpha\beta\gamma}(z) \nabla_\alpha \text{Ric}_{\beta\rho}(z) \xi^\gamma \xi^\rho}{|\xi|^5}$$
  where $E_{\alpha\beta\gamma}$ is the Levi-Civita tensor, $\text{Ric}$ is the Ricci tensor, and $\nabla$ is the covariant derivative. This result establishes that the asymmetry operator is of order $-3$ in the Levi-Civita frame, despite being order $-1$ in a generic frame.
• Kernel Decomposition: The paper derives the local integral kernel of the asymmetry operator, showing it can be decomposed into a singular part (dependent on the distance between points and the curvature) and a continuous part.
• Geometric Invariants: Based on the kernel decomposition, the authors define a regularized local trace $\psi_{\text{Dir}}^{\text{loc}}(x)$ and a global trace $\psi_{\text{Dir}}$. They prove that the local trace is a well-defined scalar function independent of the choice of local coordinates, thus constituting a geometric invariant describing spectral asymmetry.

Significance and Claims
The paper positions these results as an extension of the "direct" approach to spectral asymmetry proposed in [9, 10] to the physically significant case of the massless Dirac operator.

• Comparison with Curl: The authors note that the structure of the principal symbol for the Dirac operator is identical to that of the curl operator (up to a factor of 6), reinforcing the universality of the method across different non-semibounded systems.
• Physical Relevance: The study is motivated by applications in high-energy physics (Chern–Simons theory) and condensed matter physics (topological phases, Weyl semimetals), where the eta invariant is a crucial quantity.
• Conjecture on Eta Invariants: The authors conjecture that their newly defined regularized traces coincide with the classical local and global eta invariants evaluated at $s=0$ (i.e., $\psi_{\text{Dir}}^{\text{loc}}(x) = \eta_{\text{Dir}}^{\text{loc}}(0; x)$). However, they explicitly state that a full proof of this correspondence is beyond the scope of the current work and is reserved for future research.
• Modesty of Claims: The paper does not claim to have solved the spectral asymmetry problem entirely but rather provides the necessary computational machinery (the principal symbol and the regularized trace definition) to do so. It emphasizes that the approach yields an "invariantly defined operator" that encodes more information than the eta invariant alone, offering a potentially more accessible tool for the applied mathematical community.