Weyl-Covariant Structures

• Weyl-covariant structures are geometric, analytic, or algebraic constructs that obey systematic transformation laws under local scale rescalings, ensuring conformal invariance across diverse fields.
• They provide precise methodologies for constructing invariant field equations and analyzing anomalies in differential geometry, gravitational theories, and quantum field settings.
• Applications include holographic renormalization, operator-theoretic quantization, and special holonomy decompositions, bridging insights between mathematics and physics.

A Weyl-covariant structure is a geometric, analytic, or algebraic object equipped with a systematic transformation law under local scale (Weyl) rescalings. Such structures arise across differential geometry, mathematical physics, and quantum theory wherever local conformal symmetry—or its substructures—play a central role. Weyl-covariant structures provide a unified language to encode scale invariance, organize the geometry of connections with prescribed non-metricity, construct invariant field equations, and analyze anomalies, both classically and quantum-mechanically.

1. Fundamental Definitions and Classical Geometric Formulation

A Weyl structure on a smooth manifold $M$ is defined either as an equivalence class of pairs $(g,\omega)$ under local rescalings ($g \mapsto e^{2\sigma(x)}g$, $\omega \mapsto \omega + d\sigma$), or as a torsion-free affine connection $\nabla$ compatible with a conformal class $[g]$ in the sense that

$$\nabla_\alpha g_{\mu\nu} = \omega_\alpha g_{\mu\nu}\,,$$

where $\omega$ is a globally defined $1$-form called the Weyl potential or Lee form. The unique torsion-free affine connection compatible with $(g,\omega)$ has connection coefficients

$$\Gamma^\alpha_{\mu\nu} = \{^\alpha_{\mu\nu}\}_g - \tfrac12\left(\omega_\mu\delta^\alpha_\nu + \omega_\nu\delta^\alpha_\mu - \omega^\alpha g_{\mu\nu}\right)\,.$$

This connection is invariant under simultaneous Weyl rescaling and shift of $\omega$:

$$g_{\mu\nu} \mapsto e^{2\phi(x)}g_{\mu\nu}\,,\quad \omega_\mu \mapsto \omega_\mu + \partial_\mu \phi\,.$$

Thus, a "Weyl-covariant" structure is any geometric entity (connection, tensor, operator, etc.) for which a well-defined transformation law is induced by this rescaling.

Generalizations include Weyl structures with arbitrary (not necessarily vectorial) non-metricity $Q_{\alpha\mu\nu}$, where the torsion-free connection is uniquely fixed by requiring $\nabla_\alpha g_{\mu\nu} = Q_{\alpha\mu\nu}$ and $Q$ transforms under conformal rescalings so that $\Gamma^\alpha_{\mu\nu}$ remains invariant (Delhom et al., 2019).

2. Weyl-Covariant Connections and Bundles

A Weyl-covariant derivative acts on a field $\psi$ of conformal weight $w$ via

$$D_\mu \psi = \nabla_\mu \psi + w\,\omega_\mu\,\psi\,,$$

where $\nabla_\mu$ is the metric (Levi-Civita) derivative for $g \in [g]$. For vector and tensor fields, $D_\mu$ extends in the natural manner to include the action on indices through the Weyl connection. The Weyl-covariant exterior derivative on forms is

$$\mathcal{D} \psi = d\psi + w\,\omega \wedge \psi\,.$$

Such connections provide the basic tool for constructing Weyl-covariant field equations, Laplacians, and invariants on both Riemannian and Lorentzian manifolds (Wheeler, 2018, Faci, 2011, Attard et al., 2015).

The generalization to parabolic geometries yields an affine bundle of Weyl structures $A\rightarrow M$, sections of which correspond to choices of Weyl structure. The change-of-scale law for Weyl structures in this context encodes shifts of the representative connection via affine translations by $1$-forms, and Weyl-covariant derivatives pick up additional algebraic terms reflecting the underlying projective or conformal structure (1908.10325).

In holographic contexts (e.g., AdS/CFT), the Weyl-Fefferman-Graham (WFG) gauge elevates the boundary $1$-form to a physical Weyl connection, enabling the renormalization and construction of boundary stress tensors and anomalies in a manifestly covariant fashion (Arenas-Henriquez et al., 19 Nov 2024).

3. Curvature Decomposition and Weyl Invariants

The curvature tensor of a Weyl connection decomposes into conformally invariant and covariant pieces:

$$\hat{R}_{\mu\nu\rho\sigma} = C_{\mu\nu\rho\sigma} + 2\big(g_{\mu[\rho} P_{\sigma]\nu} + g_{\nu[\sigma} P_{\rho]\mu}\big)\,,$$

where $C_{\mu\nu\rho\sigma}$ is the (metric) Weyl tensor and $P_{\mu\nu}$ is the Schouten tensor. The Weyl tensor transforms homogeneously under Weyl rescaling, and is the conformally invariant part; the Schouten tensor transforms inhomogeneously and encodes the trace data (Wheeler, 2018, Faci, 2011).

In $3$ dimensions (and other cases where the Weyl tensor vanishes identically), the Cotton tensor, constructed through the Weyl-covariant structure, becomes the fundamental conformal invariant.

4. Cartan and Generalized Geometric Structures

Weyl-covariant structures are naturally embedded in the language of Cartan geometry and parabolic geometry. For conformal gravity, the second-order conformal Cartan connection $\omega$ on a principal bundle $P(M, H)$ encodes metric, Lorentz, dilation, and special conformal data in matrix block form. The "dressing-field" approach allows the local elimination of the Weyl (dilation) gauge field as a Stueckelberg field, isolating the physically meaningful content (Attard et al., 2015).

Torsionful generalizations—Riemann–Cartan–Weyl spaces—are parametrized by an affine connection with both torsion and trace non-metricity $Q$, the latter serving as the geometric realization of the Weyl potential. The Weyl-covariant exterior calculus and Bianchi identities persist, providing a manifestly local scale-invariant framework for gravity and matter couplings (Dereli et al., 2019, Dereli et al., 2019, Dereli et al., 2019).

5. Special Holonomy, Reductions, and Classification

Closed Weyl connections (i.e., with closed Lee form $d\omega=0$) on compact conformal manifolds admit a classification: flat, irreducible with special holonomy, or reducible (locally conformal product, LCP). The Kourganoff decomposition theorem shows that for a closed, non-exact, reducible Weyl connection $D$, the universal cover splits isometrically as $(\mathbb{R}^q, g_{\mathrm{flat}})\times (N, g_N)$, $N$ irreducible but incomplete (Madani et al., 2017, Belgun et al., 2023).

In the case $\dim N=2$, $M$ is diffeomorphic to the mapping torus of an Anosov torus automorphism, but the possible $\mathbb{R}^q$ factors are severely constrained by number theory to $q=1,2$. This provides concrete links to locally conformally Kähler (LCK), Vaisman, and special holonomy geometries.

Adapted metrics in LCP structures are characterized by a Lee form vanishing on the flat factor, and it is shown that neither Kähler nor Einstein metrics admit non-trivial LCP Weyl structures, enforcing strong restrictions on possible geometries (Belgun et al., 2023).

6. Quantum and Operator-Theoretic Weyl-Covariant Structures

Weyl covariance governs a broad range of quantum constructions:

Quantum Channels and Operator Algebras

• A quantum channel $\Phi$ is Weyl-covariant if it commutes with conjugation by Weyl operators $W(g)$:

$$\Phi\left(W(g) A W(g)^\dagger\right) = W(g)\, \Phi(A)\, W(g)^\dagger\,.$$

Such channels are completely classified via positive-definite functions $f(g)$ (Fourier transforms of probability measures) or probability kernels—in both finite and continuous phase spaces (Haapasalo, 2019).

• Compatibility (broadcasting/joint measurability) of Weyl-covariant quantum channels can be exactly characterized in terms of positive kernels or Schur product criteria, with explicit results for Gaussian (continuous) and finite (discrete) Weyl-covariant channels (Haapasalo, 2019, Siudzińska et al., 2017).

Discrete Weyl–Heisenberg Quantization

Covariant integral quantization of systems with discrete phase space $\mathbb{Z}_d \times \mathbb{Z}_d$ proceeds via a discrete Weyl–Heisenberg group, resulting in operator mappings $f(m,n) \mapsto A^w_f$ satisfying strict Weyl-covariance:

$$U(k,\ell)\, A^w_f\, U(k,\ell)^\dagger = A^w_{f_{(k,\ell)}}\,.$$

Canonical Wigner and Husimi functions, quantization by weight functions, and the representation theory are all under pauline control (Murenzi et al., 24 Dec 2024).

Weyl Operators and Covariant Quantum Fields

In relativistic quantum field theory, Weyl operators implement canonical commutation relations (CCR) and their transformation laws under the full Poincaré group. Explicit constructions of the symmetric Fock space, annihilation and creation operators, and systems of imprimitivity lead to covariant quantum fields whose stochastic calculus is grounded in Weyl symmetry (Balu, 2019).

7. Weyl Substructures, Groupoids, and Anomaly Structures

The full local Weyl group is the Abelian group of all smooth, point-dependent rescalings $g \mapsto \Omega^2(x)g$. Substructures arise by imposing metric-dependent differential constraints on $\Omega$, leading to partial associative groupoids rather than subgroups:

• Harmonic (Restricted Weyl) Groupoid: $\Omega$ satisfies $\Box_g \Omega=0$.
• Light-cone Constraint: $\Omega$ satisfies $g^{\mu\nu}\partial_\mu\Omega\partial_\nu\Omega=0$.
• Higher-order Generalizations: E.g., the $Q$-curvature constraint and higher-derivative shift operators.

These substructures have profound consequences for the trace of the energy-momentum tensor and the quantum anomaly structure. In restricted Weyl invariance ($\Box_g\sigma=0$), the trace anomaly can acquire terms ($\Box R$) which are trivial in the full Weyl group but nontrivial in the restricted setting. BRST gauge-fixing methodology reveals these structures as partial gauge fixings of full Weyl invariance, inducing a corresponding Slavnov–Taylor identity at the quantum level (Martini et al., 8 Apr 2024).

Table: Canonical Weyl-Covariant Entities Across Contexts

Structure Type	Defining Equation or Property	Reference
Weyl connection (metric/1-form)	$$\nabla_\alpha g_{\mu\nu} = \omega_\alpha g_{\mu\nu}$$	(Wheeler, 2018, Delhom et al., 2019)
Transformation law	$g_{\mu\nu} \to e^{2\phi}g_{\mu\nu}$, $\omega_\mu\to\omega_\mu + \partial_\mu\phi$	(Wheeler, 2018, Faci, 2011)
Weyl-covariant derivative	$D_\mu \psi = \nabla_\mu \psi + w\,\omega_\mu\,\psi$	(Attard et al., 2015, Dereli et al., 2019)
Riemann curvature decomposition	$$\hat{R}_{\mu\nu\rho\sigma} = C_{\mu\nu\rho\sigma} + 2(g_{\mu[\rho}P_{\sigma]\nu} + g_{\nu[\sigma}P_{\rho]\mu})$$	(Wheeler, 2018)
Affine bundle of Weyl structures	$A = \mathcal{G}\times_P (P/G_0) \to M$, modeled on $T^*M$, sections correspond to Weyl structures	(1908.10325)
Quantum channel Weyl covariance	$$\Phi(W(g) A W(g)^\dagger) = W(g) \Phi(A) W(g)^\dagger$$	(Haapasalo, 2019, Siudzińska et al., 2017)
Harmonic subgroup constraint	$\Box_g\Omega = 0$	(Martini et al., 8 Apr 2024)
LCP decomposition	Universal cover splits: $(\mathbb{R}^q, g_{\mathrm{flat}}) \times (N, g_N)$, $N$ irreducible, incomplete	(Madani et al., 2017, Belgun et al., 2023)
Cartan connection for gravity	Block-matrix with Weyl potential $W$, translational and Lorentz parts; field strength encodes generalized torsion, curvature, dilation	(Attard et al., 2015)

8. Applications and Physical Significance

• Conformal Geometry and Invariant Operators: Weyl-covariant structures are foundational in constructing conformally invariant operators (e.g., conformal Laplacian, conformal Dirac) and for systematically encoding conformally invariant field equations of all spins (Faci, 2011).
• Gravity and Gauge Theories: Weyl-covariant gravity provides a manifestly scale-invariant version of general relativity or higher-derivative gravities. In 3D, New Massive Gravity and Minimal Massive Gravity admit Weyl-covariant generalizations via Riemann–Cartan–Weyl geometry (Dereli et al., 2019, Dereli et al., 2019).
• Holography: The Weyl-Fefferman–Graham gauge yields a boundary structure fully compatible with local scale symmetry, allowing for the unambiguous definition of holographic stress tensors and anomalies in AdS/CFT (Arenas-Henriquez et al., 19 Nov 2024).
• Quantum Theory and Information: Weyl-covariant quantum channels and operator algebras are constrained by positive-type functions, facilitating Fourier-theoretic analyses of compatibility, broadcastability, and symmetry-reduced state manipulation (Haapasalo, 2019).
• Supersymmetric Theories: 10D $\mathcal{N}=1$ supergravity admits a Weyl-covariant off-shell superspace formulation, with explicit transformation laws for all geometric superfields and a finite classification of possible prepotentials and superfield strengths (Jr. et al., 2020).

9. Future Directions and Generalizations

The flexibility of Weyl-covariant structures—allowing arbitrary non-metricity, extensions to disformal symmetry, higher-derivative groupoid substructures, and their compatibility with additional geometric (torsion, special holonomy) or algebraic (quantum, operator-algebraic) frameworks—continues to fuel advances in:

• Geometric analysis of elliptic and parabolic conformal invariants,
• Extensions to parabolic and more general Cartan geometries,
• Refined quantum anomaly structures under restricted Weyl symmetry,
• Applications to non-Riemannian, "metric-affine" and teleparallel gravity theories,
• Systematic construction of higher-dimensional and higher-spin Weyl-covariant field equations.

The Weyl-covariant formalism provides a conceptually unifying infrastructure, linking local geometric symmetries to the analytic, topological, and quantum structures encountered across mathematical physics and geometry.