Axial Perturbative Gauge Invariants

• Axial perturbative gauge invariants are combinations of field perturbations that remain invariant under gauge transformations, isolating genuine physical degrees of freedom.
• They are constructed using axial (odd-parity) gauge conditions in theories such as gravity and BF theory to decouple non-physical gauge artifacts.
• Their applications span gravitational wave analysis, quantization in topological field theories, and spectral studies in symmetric spacetimes.

Axial perturbative gauge invariants are distinguished gauge-invariant quantities that arise in the analysis of physical theories—particularly gauge and gravitational field theories—when perturbative expansions are performed about symmetric backgrounds and axial, or odd-parity, gauge-fixing conditions are employed. They encapsulate physical degrees of freedom decoupled from gauge artifacts, allowing the separation, classification, and computation of dynamical observables in the presence of nontrivial residual structure due to the choice of axial or Anosov-type gauge. Such invariants play crucial roles across a range of domains, from the quantization of topological field theories and the study of spectral invariants to the classification of physically meaningful perturbations in black hole spacetimes and cosmological models.

1. Definition and Foundational Principles

Axial perturbative gauge invariants are combinations of field perturbations that are invariant under infinitesimal gauge transformations, constructed within frameworks where "axial" conditions—such as the vanishing of components along certain distinguished vector fields—are enforced by gauge-fixing.

Gauge-invariant constructions in perturbative settings arise generically as follows: given a gauge theory (e.g., BF theory, Yang–Mills, linearized gravity) and a small perturbation about a background, an infinitesimal gauge transformation acts as $h \mapsto h + \mathcal L_\xi g$ or analogous Lie algebraic variations. The subset of perturbations annihilated by these transformations embodies the true, gauge-invariant content. The operators projecting onto this subspace are often constructed using the Green's function for the gauge-fixed operator or via canonical transformations in Hamiltonian language (Garg et al., 2021).

Axial gauge typically imposes the vanishing of a component of the connection or metric along a privileged direction. In the context of gravitational perturbations, the "axial" or "odd-parity" sector indicates those perturbations that change sign under inversion. In topological field theories such as BF theory on contact manifolds, the "axial" gauge is defined with respect to the Reeb (Anosov) vector field $X$ associated to a contact form, by $i_X A = 0$ and $i_X B = 0$, where $i_X$ denotes contraction with $X$ (Schiavina et al., 2023).

2. Axial Gauge Construction and Gauge-Fixing

BF Theory in Anosov-Contact Axial Gauge: In abelian BF theory formulated on a $(2m+1)$-dimensional closed orientable contact manifold $(M, \alpha)$ with unitary flat bundle $(E, \nabla)$, the axial gauge fixing selects the subspace of fields annihilated by contraction with the Anosov vector field $X$: $$\mathrm{L}_{\text{axial}} = \ker i_X \subset \mathcal F_{BF}$$, with $$\mathcal F_{BF} = \Omega^\bullet(M, E)[1] \oplus \Omega^\bullet(M, E)[2m-1]$$ (Schiavina et al., 2023).

This gauge condition is metric-independent, relying solely on the contact (and dynamical) structure given by $X$. Gauge transformations are entirely eliminated if $X$ admits no Pollicott–Ruelle resonance at $0$, as is typical for Anosov flows. In the nontrivial regime, residue symmetries would correspond to resonant states.

Yang–Mills in Generalized Axial Gauge: In 2D Yang–Mills, one can fix an entire family of axial-like gauges interpolating between holomorphic and light-cone gauges by requiring the connection $A$ to have no component along a "minus" direction, $A = A_+ dx_{\alpha,+}$, parametrized by $\alpha \in [0, \pi/2]$, which simplifies the action to be quadratic and renders the path integral exactly solvable perturbatively (Nguyen, 2016).

In both contexts, the axial gauge fixes all gauge freedom, and the construction of perturbative invariants then amounts to identifying polynomial or functional quantities of the remaining variables that are unchanged under the residual (trivial or absent) gauge transformations.

3. Explicit Construction and Characterization of Axial Invariants

Gravitational Axial Master Variables: On a spherically symmetric background such as Schwarzschild or Kantowski–Sachs, axial perturbations take the form of those metric fluctuations with odd parity under coordinate inversion. The analysis decomposes the perturbation into spherical harmonics and constructs the unique (for each $(\ell, m)$) gauge-invariant master variable—typically the Regge–Wheeler or Cunningham–Price–Moncrief (CPM) variable—by combinations of the perturbed metric and its derivatives,

$$Q_{\ell m}^{(\text{axial})}(t, r) = \frac{1}{\Lambda} \left[ r\, h_1(t, r) - r^2 \partial_t \left( \frac{h_2(t, r)}{r^2} \right) \right], \qquad \Lambda = \ell(\ell+1) - 2,$$

which remains invariant under the infinitesimal odd-parity gauge transformations. In the Regge–Wheeler gauge ($h_2 = 0$), this reduces to $Q_{\ell m} = \frac{r}{\Lambda} h_1$ (Garg et al., 2021, Shah et al., 2016).

Hamiltonian Formulation and Darboux Symmetry: In the canonical formalism, one constructs manifestly gauge-invariant canonical pairs $(Q_1, P_1)$ via mode-dependent canonical transformations so that the quadratic perturbative Hamiltonian is diagonal in these variables. These master variables generate the entire gauge-invariant content of the odd sector in both exterior and interior black hole geometries. The existence of an infinite family of Darboux-related master variables corresponds to the non-uniqueness of reduced canonical coordinates, reflecting the freedom in splitting the quadratic form on the reduced phase space (Lenzi et al., 11 Dec 2025).

Axial Invariants in Topological Field Theory: In the context of BF theory quantized perturbatively in axial gauge, the only nontrivial invariants are encoded in functional determinants of the Lie derivative along $X$ shifted by the perturbative parameter $\hbar$, leading directly to the twisted Pollicott–Ruelle resonances (the spectral data of $L_X$ twisted by the flat connection) (Schiavina et al., 2023).

4. Emergence and Computation of Gauge-Invariant Observables

Partition Functions and Spectral Zeta Functions: In Abelian BF theory subjected to an $\hbar$-linear quadratic perturbation in axial gauge,

$$S(A, B; \hbar) = \int_M \langle B \wedge (1 + \hbar L_X^{-1}) d_\nabla A \rangle_E,$$

the perturbative partition function computes the twisted Ruelle zeta function,

$Z(\hbar) = e^{i \phi} \cdot \zeta_\rho(\hbar),$

where $$\zeta_\rho(\lambda) = \prod_k \det(L_{X, k} + \lambda)^{(-1)^k(-1)^m}$$ and $L_{X, k}$ is the Lie derivative restricted to degree $k$ axial forms. The zeros, poles, and residues of the partition function correspond to the set of twisted Pollicott–Ruelle resonances—i.e., the spectrum of correlations’ decay rates along the Anosov flow (Schiavina et al., 2023).

Microlocal Structure and Well-Definedness: The propagators in axial gauge can be of distributional character, with singular support along flow lines (or beyond); rigorous implementation involves detailed microlocal and wavefront set analysis to ensure the perturbative expansion and contractions in Feynman diagrams yield genuine distributions and not ill-defined objects (Schiavina et al., 2023).

Perturbative Observables in 2D Yang–Mills: In generalized axial gauges, expectation values of Wilson loops can be computed exactly up to second order (and in special cases, to all orders) via iterated integrals of closed one-forms, resulting in area law scaling:

$$\langle W_{f,\gamma} \rangle = \exp\left(-\frac{\lambda}{2}\,\text{Area}(\gamma)\right) + O(\lambda^3),$$

where the gauge-fixing kills all cubic and higher-order terms, and the quadratic propagator produces the full result (Nguyen, 2016).

5. Dynamical Equations and Physical Interpretation

Master Wave Equations: The gauge-invariant axial variables universally satisfy decoupled wave equations. For example, in Schwarzschild background, the master variable $Q_{\ell m}$ satisfies the Regge–Wheeler equation:

$$\left[-\partial_t^2 + \partial_{r_*}^2 - V_{RW}(r)\right] Q_{\ell m}(t,r) = 0,$$

with $$V_{RW}(r) = \left(1 - \frac{2M}{r}\right)\left[\frac{\ell(\ell+1)}{r^2} - \frac{6M}{r^3}\right]$$ (Garg et al., 2021, Shah et al., 2016). In anisotropic cosmological models such as Kantowski–Sachs, the analogous master equation for the physical invariant $Q_1$ reads:

$$\ddot Q_1 + 2 \frac{\dot b_\ell}{b_\ell} \dot Q_1 + 4\frac{b_\ell(k_\ell + s_\ell)}{\kappa^2} Q_1 = 0,$$

with $b_\ell, k_\ell, s_\ell$ given by background-dependent expressions (Marugán et al., 13 Feb 2024, Lenzi et al., 11 Dec 2025).

Spectral Invariants and Mixing: In the BF theory context, the spectrum of the (twisted) generator $L_X$ encodes the decay of correlations and mixing rates for dynamics along the Anosov flow. The statistical and spectral features of the dynamics are thus reflected in the axial perturbative gauge invariants.

6. Relations to Other Gauge Invariants and Classification

Comparison with Other Sectors: In metric perturbations, the even (polar) and odd (axial) parity gauge-invariant sectors are decoupled and exhaust the gauge-invariant content when combined. The construction of Chandrasekhar and Detweiler connects the odd and even master variables through algebraic mappings, revealing deep structural relations between different wave equations and the underlying geometry (Shah et al., 2016). In Geroch-Held-Penrose (GHP) language, all master variables, including axial invariants, can be constructed without resorting to coordinate choices, clarifying the transformations under tetrad rotations and other nontrivial symmetries.

Darboux Symmetry: The Darboux transformations correspond to canonical map symmetries between different equivalent master variables in the axial sector. Algorithmic construction of the reduced phase space and its diagonalization exposes this infinite-dimensional symmetry, underlying the apparent redundancy among master functions (Lenzi et al., 11 Dec 2025).

7. Physical and Mathematical Significance

Axial perturbative gauge invariants enable the unequivocal identification of physical content in perturbative expansions, free from gauge ambiguities or coordinate artifacts. Their extraction is essential for:

• Quantization and Spectral Theory: Partition functions and expectation values computed in axial gauge directly realize spectral zeta functions of dynamical generators, linking quantum field theoretic methods to dynamical systems and spectral analysis (Schiavina et al., 2023).
• Gravitational Wave Physics: Only gauge-invariant content such as the axial master variables corresponds to genuinely radiative, physically detectable degrees of freedom (e.g., gravitational waves) (Garg et al., 2021, Shah et al., 2016).
• Topological Field Theory and Quantum Invariants: Pollicott–Ruelle resonances, encoded in the zeros and poles of partition functions in axial gauge, yield new invariants of contact flows and flat bundles, generalizing analytic torsion and related invariants (Schiavina et al., 2023).
• Mathematical Physics and Geometry: The connection between axial gauge-fixing, spectral theory, and periodic-orbit formulas (e.g., via the Guillemin trace formula and Euler product expressions) highlights the intersection of geometric analysis, quantum field theory, and dynamical systems (Schiavina et al., 2023).

These invariants thus represent a robust computational and conceptual toolset, providing essential bridges between different areas of mathematical physics and underpinning a broad array of physical phenomena.