---
title: Weyl Covariant Derivative
url: https://www.emergentmind.com/topics/weyl-covariant-derivative
type: topic
---

# Weyl Covariant Derivative

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 $g_{\mu\nu}$ [1301.1316, 2312.13384, 2509.21495, 2412.16548]. The associated Weyl covariant derivative $\widetilde\nabla_\mu$ acting on a tensor $T^{\alpha\cdots}{}_{\beta\cdots}$ of Weyl weight $q$ is
\[
\widetilde\nabla_\mu T^{\alpha\cdots}{}_{\beta\cdots} = \nabla_\mu T^{\alpha\cdots}{}_{\beta\cdots} + q W_\mu T^{\alpha\cdots}{}_{\beta\cdots}
\]
plus additional connection terms for each index, respecting the weight structure and implementing the local gauge symmetry [2312.13384, 2412.16548, 1904.08124].

The core geometric property is the non-metricity condition:
\[
\widetilde\nabla_\mu g_{\alpha\beta} = -2 W_\mu g_{\alpha\beta}
\]
expressing the failure of length preservation under parallel transport—a defining feature of Weyl geometry [1301.1316, 2312.13384, 2509.21495].

## 2. Transformation Properties and Gauge Invariance

Under a local scale transformation parameterized by $\sigma(x)$, the metric, Weyl gauge field, and any tensor $T$ of Weyl weight $q$ transform as:
\[
g_{\mu\nu} \to e^{2\sigma(x)} g_{\mu\nu}, \quad W_\mu \to W_\mu - \partial_\mu \sigma, \quad T \to e^{q \sigma(x)} T
\]
The Weyl covariant derivative preserves the transformation law:
\[
\widetilde\nabla_\mu T \to e^{q \sigma(x)} \widetilde\nabla_\mu T
\]
ensuring that the action of $\widetilde\nabla_\mu$ maintains the Weyl weight of the field [1109.2579, 2312.13384, 2509.21495, 2412.16548]. Gauge invariance is achieved since the inhomogeneous contributions from $\partial_\mu g_{\rho\sigma}$ and $W_\mu$ 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 $W_\mu$ [2312.13384, 2412.16548].

## 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 [2312.13384, 2203.08692]:

- **Non-metric, torsion-free ("Weyl geometry")**: 
  - Connection: $\Gamma^{(w)\rho}_{\;\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_\mu W_\nu + \delta^\rho_\nu W_\mu - g_{\mu\nu} W^\rho$
  - Non-metricity: $\nabla_\mu^{(w)} g_{\alpha\beta} = -2 W_\mu g_{\alpha\beta}$
  - Torsion: $T^\rho_{\;\mu\nu} = 0$
- **Metric, torsionful ("hat" formalism)**: 
  - Connection: $\widehat{\Gamma}^\rho_{\;\mu\nu} = \Gamma^\rho_{\;\mu\nu} + \delta^\rho_\mu W_\nu - g_{\mu\nu} W^\rho$
  - Metric compatibility: $\widehat{\nabla}_\mu g_{\alpha\beta} = 0$
  - Torsion: $T^\rho_{\;\mu\nu} = 2 \delta^\rho_{[\mu} W_{\nu]}$

These are related by a projective transformation:
\[
\widehat{\Gamma}^\rho_{\;\mu\nu} = \widetilde{\Gamma}^\rho_{\;\mu\nu} - \delta^\rho_\nu W_\mu
\]
This duality underlies connections between Weyl-covariant gauge gravity and alternative affine or metric-affine formalisms and is central for understanding their symmetry structures [2312.13384, 2203.08692].

## 4. Action on Fields, Curvature, and Bianchi Identities

For a tensor $T^{\alpha_1\cdots\alpha_r}{}_{\beta_1\cdots\beta_p}$ of Weyl weight $q$:
\[
\widetilde{\nabla}_\mu T^{\alpha_1\cdots}{}_{\cdots\beta_p} = \partial_\mu T^{\alpha_1\cdots}{}_{\cdots\beta_p}
+ \sum_{i=1}^r \widetilde{\Gamma}^{\alpha_i}{}_{\mu\lambda} T^{\cdots \lambda \cdots}{}_{\cdots\beta_p}
- \sum_{j=1}^p \widetilde{\Gamma}^{\lambda}{}_{\mu\beta_j} T^{\alpha_1\cdots}{}_{\cdots\lambda\cdots}
+ q W_\mu T^{\alpha_1\cdots}{}_{\cdots\beta_p}
\]
[2412.16548, 2509.21495, 2312.13384].

The field strength associated with $W_\mu$ is the abelian curvature:
\[
F_{\mu\nu} = \partial_\mu W_\nu - \partial_\nu W_\mu
\]
and the Weyl–Riemann tensor is
\[
\widetilde{R}^\rho{}_{\sigma\mu\nu} = \partial_\mu \widetilde{\Gamma}^\rho_{\nu\sigma} - \partial_\nu \widetilde{\Gamma}^\rho_{\mu\sigma}
+ \widetilde{\Gamma}^\rho_{\mu\lambda} \widetilde{\Gamma}^\lambda_{\nu\sigma}
- \widetilde{\Gamma}^\rho_{\nu\lambda} \widetilde{\Gamma}^\lambda_{\mu\sigma}
\]
[1301.1316, 2509.21495, 2312.13384].

The Bianchi identities, modified in the presence of non-metricity and $F_{\mu\nu}$, retain their form but include corrections:
\[
\widehat{\nabla}_{[\alpha} \widetilde{R}^{\rho}{}_{|\sigma|\mu\nu]} = - F_{[\alpha\mu} \delta^\rho{}_{\nu]} \nabla_\sigma
\]
and
\[
\widehat{\nabla}_{[\alpha} F_{\mu\nu]} = 0
\]
[2412.16548].

## 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.,
  \[
  \widetilde{R}^\rho{}_{\sigma\mu\nu} = R^\rho{}_{\sigma\mu\nu} + \widehat{\nabla}_\mu W^\rho{}_{\nu\sigma} - \widehat{\nabla}_\nu W^\rho{}_{\mu\sigma}
  \]
  where $R^\rho{}_{\sigma\mu\nu}$ is the Riemann tensor of the Levi–Civita connection and $W^\rho{}_{\mu\nu}$ is the additive Weyl tensor [1301.1316].
- The Weyl–Palatini identity provides an efficient structure for varying the curvature tensor with respect to $g_{\mu\nu}$ and $W_\mu$, bypassing the complexity of index-level variational calculus [1301.1316].
- Conservation laws for the generalized Einstein tensor $W_{\mu\nu}$ and Weyl current $B^\mu$ are defined strictly using the Weyl covariant derivative:
  \[
  \widehat{\nabla}^\mu W_{\mu\nu} = \frac{1}{2} F_{\nu\mu} B^\mu
  \]
  and in the conformal gravity case this implies strict conservation on-shell [2412.16548].
- Trace identities and generalized Nöther identities for dilatations take the form
  \[
  T^\mu{}_\mu = -\widehat{\nabla}_\mu \Delta^\mu
  \]
  where $\Delta^\mu$ is the dilation current [2203.08692].

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 [1301.1316, 1906.11552, 1904.08124, 2412.16548, 2509.21495].

## 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
\[
\mathcal{D} \Phi_q = D \Phi_q + q Q \wedge \Phi_q
\]
where $Q$ is the Weyl gauge 1-form, and $D$ is the full affine exterior covariant derivative [1904.08124, 1906.11552, 1904.11255]. The connection splits into Levi–Civita, contortion, and non-metricity components. The metric-compatibility and transformation rules maintain local scale covariance:
\[
\mathcal{D} g = Dg - 2 Q \wedge g = 0
\]
Under local rescalings, $Q \to Q + d\sigma$, ensuring field strengths $dQ$ 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 [2007.05097]. 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 [2312.13384, 2412.16548, 2509.21495].
- **Scale-invariant and conformal gravity**: Enables construction of locally scale-invariant field equations, such as Weyl-gauged Einstein–Hilbert, new massive gravity, and $C^2$ (Weyl-tensor squared) actions [2509.21495, 1906.11552, 1904.08124, 2412.16548, 2312.13384].
- **Construction of conformally invariant field equations**: Universal deployment for manifestly conformally invariant operators (Laplacians, Maxwell, and higher-spin equations) [1109.2579].
- **Supergravity and supersymmetric models**: Realization of super-Weyl symmetry in the structure of covariant derivatives and field strengths in superspace (especially 10D/11D) [2007.05097].
- **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 [2203.08692, 2312.13384].

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 [1904.11255, 1906.11552, 2412.16548].

---

**Summary Table: Selected Weyl Covariant Derivative Structures**

| Context/Geometry        | Connection Coefficient $\Gamma^{\rho}_{\mu\nu}$                | Metric Compatibility                   | Weyl Gauge Field         |
|------------------------|-----------------------------------------------------------------|----------------------------------------|--------------------------|
| Weyl, torsion-free     | $\Gamma^\rho_{\mu\nu} + \delta^\rho_\mu W_\nu + \delta^\rho_\nu W_\mu - g_{\mu\nu} W^\rho$         | $\widetilde\nabla_\mu g_{\alpha\beta} = -2 W_\mu g_{\alpha\beta}$        | $W_\mu$                  |
| Manifestly metric      | $\Gamma^\rho_{\mu\nu} + \delta^\rho_\mu W_\nu - g_{\mu\nu} W^\rho$                                  | $\widehat{\nabla}_\mu g_{\alpha\beta} = 0$               | $W_\mu$                  |
| RCW (exterior forms)   | $\omega^a{}_b+K^a{}_b-q^a{}_b - Q \delta^a_b$ (1-form language)                                      | $\mathcal{D}g = 0$                                       | $Q$                      |
| Superspace (spinor)    | Covariant superconnection with Weyl prepotential [2007.05097]                                        | Super-Weyl compatibility                | $\mathcal{J}^{(+)}$      |

---

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 [1301.1316, 2312.13384, 2412.16548, 2509.21495, 2203.08692, 1109.2579, 1906.11552, 1904.11255, 1904.08124, 2007.05097].

Source: https://www.emergentmind.com/topics/weyl-covariant-derivative