Weyl Covariant Derivative

• Weyl covariant derivative is a geometric operator that extends the Levi–Civita connection by incorporating a Weyl gauge field to enforce local scale invariance.
• It generalizes standard tensor calculus by adding weight-dependent terms, allowing a consistent treatment of conformal transformations in various gravity models.
• This derivative is essential in variational formulations, conservation laws, and the development of gauge-invariant, scale-invariant gravitational theories.

The Weyl covariant derivative is a central geometric construct in Weyl geometry and its modern extensions, providing a systematic method to couple tensor fields and geometric quantities to a local scale (dilatation) gauge symmetry. It generalizes the Levi–Civita connection by incorporating a Weyl gauge field associated with non-metricity, enabling local conformal or scale invariance at the differential geometric level. Different but mathematically equivalent formulations, both metric-compatible and non-metric, exist and are vital for modern gauge theories of gravity, scale-invariant gravitation, modified variational principles, and higher-derivative gravity, as well as supersymmetric and affine-covariant frameworks.

1. Fundamentals of the Weyl Covariant Derivative

The Weyl geometry supplements the standard Riemannian geometric data—namely, a metric $g_{\mu\nu}$—with an independent Weyl gauge (dilatation) field $W_\mu$, $w_\mu$, $\omega_\mu$, or $S_\mu$ depending on conventions. The primary geometric structure is a torsion-free, non-metric connection $\widetilde\Gamma^\rho_{\;\mu\nu}$ defined by

$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$

where $\Gamma^\rho_{\;\mu\nu}$ is the Levi–Civita connection of $g_{\mu\nu}$ (Yuan et al., 2013, Condeescu et al., 2023, Lessa et al., 25 Sep 2025, Condeescu et al., 2024). The associated Weyl covariant derivative $\widetilde\nabla_\mu$ acting on a tensor $W_\mu$0 of Weyl weight $W_\mu$1 is

$W_\mu$2

plus additional connection terms for each index, respecting the weight structure and implementing the local gauge symmetry (Condeescu et al., 2023, Condeescu et al., 2024, Dereli et al., 2019).

The core geometric property is the non-metricity condition: $W_\mu$3 expressing the failure of length preservation under parallel transport—a defining feature of Weyl geometry (Yuan et al., 2013, Condeescu et al., 2023, Lessa et al., 25 Sep 2025).

2. Transformation Properties and Gauge Invariance

Under a local scale transformation parameterized by $W_\mu$4, the metric, Weyl gauge field, and any tensor $W_\mu$5 of Weyl weight $W_\mu$6 transform as: $W_\mu$7 The Weyl covariant derivative preserves the transformation law: $W_\mu$8 ensuring that the action of $W_\mu$9 maintains the Weyl weight of the field (Faci, 2011, Condeescu et al., 2023, Lessa et al., 25 Sep 2025, Condeescu et al., 2024). Gauge invariance is achieved since the inhomogeneous contributions from $w_\mu$0 and $w_\mu$1 cancel in the total variation.

In the torsion-free, non-metric case, the full connection is invariant under Weyl rescalings, with all scale (dilatation) dependence isolated in $w_\mu$2 (Condeescu et al., 2023, Condeescu et al., 2024).

3. Equivalent Formulations, Metric Compatibility, and Duality

There exist two geometric formulations for the Weyl covariant derivative, highlighting a duality between non-metricity and torsion (Condeescu et al., 2023, Sauro et al., 2022):

• Non-metric, torsion-free ("Weyl geometry"):
  • Connection: $w_\mu$3
  • Non-metricity: $w_\mu$4
  • Torsion: $w_\mu$5
• Metric, torsionful ("hat" formalism):
  • Connection: $w_\mu$6
  • Metric compatibility: $w_\mu$7
  • Torsion: $w_\mu$8

These are related by a projective transformation: $w_\mu$9 This duality underlies connections between Weyl-covariant gauge gravity and alternative affine or metric-affine formalisms and is central for understanding their symmetry structures (Condeescu et al., 2023, Sauro et al., 2022).

4. Action on Fields, Curvature, and Bianchi Identities

For a tensor $\omega_\mu$0 of Weyl weight $\omega_\mu$1: $\omega_\mu$2 (Condeescu et al., 2024, Lessa et al., 25 Sep 2025, Condeescu et al., 2023).

The field strength associated with $\omega_\mu$3 is the abelian curvature: $\omega_\mu$4 and the Weyl–Riemann tensor is

$\omega_\mu$5

(Yuan et al., 2013, Lessa et al., 25 Sep 2025, Condeescu et al., 2023).

The Bianchi identities, modified in the presence of non-metricity and $\omega_\mu$6, retain their form but include corrections: $\omega_\mu$7 and

$\omega_\mu$8

(Condeescu et al., 2024).

5. Applications in Variational Principles and Conservation Laws

In variational principles for Weyl-invariant gravity, such as quadratic or higher-derivative actions, the Weyl covariant derivative plays a critical role:

• It enables compact and fully covariant expressions for curvature and Ricci tensors in Weyl geometry, e.g.,

$\omega_\mu$9

where $S_\mu$0 is the Riemann tensor of the Levi–Civita connection and $S_\mu$1 is the additive Weyl tensor (Yuan et al., 2013).

• The Weyl–Palatini identity provides an efficient structure for varying the curvature tensor with respect to $S_\mu$2 and $S_\mu$3, bypassing the complexity of index-level variational calculus (Yuan et al., 2013).
• Conservation laws for the generalized Einstein tensor $S_\mu$4 and Weyl current $S_\mu$5 are defined strictly using the Weyl covariant derivative:

$S_\mu$6

and in the conformal gravity case this implies strict conservation on-shell (Condeescu et al., 2024).

• Trace identities and generalized Nöther identities for dilatations take the form

$S_\mu$7

where $S_\mu$8 is the dilation current (Sauro et al., 2022).

These structures enable variational formulations for locally scale-covariant gravity, including Weyl-invariant extensions of massive gravity, higher-derivative (quadratic) theories, and scale-invariant matter–gravity couplings (Yuan et al., 2013, Dereli et al., 2019, Dereli et al., 2019, Condeescu et al., 2024, Lessa et al., 25 Sep 2025).

6. Weyl Covariant Derivative in Riemann–Cartan–Weyl and Superspace

In Riemann–Cartan–Weyl (RCW) geometry, the Weyl-covariant derivative extends to include torsion and general affinely connected structures. Its definition in exterior form is

$S_\mu$9

where $\widetilde\Gamma^\rho_{\;\mu\nu}$0 is the Weyl gauge 1-form, and $\widetilde\Gamma^\rho_{\;\mu\nu}$1 is the full affine exterior covariant derivative (Dereli et al., 2019, Dereli et al., 2019, Dereli et al., 2019). The connection splits into Levi–Civita, contortion, and non-metricity components. The metric-compatibility and transformation rules maintain local scale covariance: $\widetilde\Gamma^\rho_{\;\mu\nu}$2 Under local rescalings, $\widetilde\Gamma^\rho_{\;\mu\nu}$3, ensuring field strengths $\widetilde\Gamma^\rho_{\;\mu\nu}$4 and curvature 2-forms transform covariantly.

In superspace, the Weyl-covariant derivative (in particular, for 10D or 11D supergravities) is constructed to respect local super-Weyl transformations. The superconnection includes both bosonic (vielbein, Lorentz) and fermionic components, and the Weyl prepotential structure is implemented through superfield representations of the Weyl gauge parameter (Jr. et al., 2020). Component torsion and curvature supertensors acquire Weyl weight and admit covariantly defined transformation laws, with connections to the structure of supergravity prepotentials and field strengths.

7. Physical Implications and Applications in Gauged and Scale-Invariant Gravity

The Weyl covariant derivative is essential in the following domains:

• Gauge theoretical gravity: Forms the geometric basis for Weyl gauge-invariant and quadratic (higher-derivative) gravitational theories, including those where Einstein–Hilbert gravity emerges via spontaneous breaking of scale symmetry (Condeescu et al., 2023, Condeescu et al., 2024, Lessa et al., 25 Sep 2025).
• Scale-invariant and conformal gravity: Enables construction of locally scale-invariant field equations, such as Weyl-gauged Einstein–Hilbert, new massive gravity, and $\widetilde\Gamma^\rho_{\;\mu\nu}$5 (Weyl-tensor squared) actions (Lessa et al., 25 Sep 2025, Dereli et al., 2019, Dereli et al., 2019, Condeescu et al., 2024, Condeescu et al., 2023).
• Construction of conformally invariant field equations: Universal deployment for manifestly conformally invariant operators (Laplacians, Maxwell, and higher-spin equations) (Faci, 2011).
• Supergravity and supersymmetric models: Realization of super-Weyl symmetry in the structure of covariant derivatives and field strengths in superspace (especially 10D/11D) (Jr. et al., 2020).
• Metric-Affine and Affine Gauge Theories: Facilitates the generalization of Nöther identities, conservation laws, and uniquely covariant field strengths in geometric frameworks with independent metric and connection degrees of freedom (Sauro et al., 2022, Condeescu et al., 2023).

A systematic application is realized in the variational construction of Weyl-invariant actions, including dynamical dilaton fields, Weyl gauge kinetic terms, and geometric actions for both Riemannian and RCW geometries (Dereli et al., 2019, Dereli et al., 2019, Condeescu et al., 2024).

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Summary Table: Selected Weyl Covariant Derivative Structures

Context/Geometry	Connection Coefficient $\widetilde\Gamma^\rho_{\;\mu\nu}$6	Metric Compatibility	Weyl Gauge Field
Weyl, torsion-free	$\widetilde\Gamma^\rho_{\;\mu\nu}$7	$\widetilde\Gamma^\rho_{\;\mu\nu}$8	$\widetilde\Gamma^\rho_{\;\mu\nu}$9
Manifestly metric	$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$0	$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$1	$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$2
RCW (exterior forms)	$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$3 (1-form language)	$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$4	$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$5
Superspace (spinor)	Covariant superconnection with Weyl prepotential (Jr. et al., 2020)	Super-Weyl compatibility	$$\widetilde\Gamma^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_{\mu} W_\nu + \delta^\rho_{\nu} W_\mu - g_{\mu\nu} W^\rho$$6

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The Weyl covariant derivative is thus a cornerstone of locally scale-invariant and conformal geometries, unifying metric and connection-based approaches to gauge gravity, and providing internally consistent structures for variational calculus, conservation laws, and physical dynamics for a broad class of gravity theories and related geometric field equations (Yuan et al., 2013, Condeescu et al., 2023, Condeescu et al., 2024, Lessa et al., 25 Sep 2025, Sauro et al., 2022, Faci, 2011, Dereli et al., 2019, Dereli et al., 2019, Dereli et al., 2019, Jr. et al., 2020).