Extended Gauss–Bonnet Gravity

• Extended Gauss–Bonnet Gravity is a gravitational theory that extends General Relativity by incorporating nonlinear Gauss–Bonnet terms and additional Weyl geometric structures.
• It introduces novel vector–tensor and scalar–tensor interactions that provide alternative mechanisms for cosmological evolution and dark matter phenomenology.
• In four dimensions, the theory reduces to an Einstein–Proca model with Horndeski-type interactions, ensuring second-order dynamics and ghost-free behavior.

Extended Gauss–Bonnet Gravity is a general term for gravitational theories that generalize General Relativity (GR) by including nonlinear and higher-derivative corrections involving the Gauss–Bonnet (GB) term, either through nontrivial functional dependence or coupling to extra fields, and/or by extending the underlying geometric structure (e.g., to Weyl geometry). Such theories provide a systematic framework for studying quantum corrections, gravitational dynamics in higher dimensions, and scale-invariant extensions of GR. The modifications inherent to extended Gauss–Bonnet models yield new degrees of freedom, novel vector–tensor and scalar–tensor interactions, and alternative mechanisms for cosmological evolution and dark sector phenomenology.

1. Geometric Structure and Principle of Construction

Extended Gauss–Bonnet gravity is typically formulated in a spacetime with a generalized connection, often realized in a Weyl geometric setting. In Weyl geometry, the affine connection $\tilde{\nabla}_\lambda$ is torsionless but is not metric compatible; the nonmetricity is set by a vector field $A_\mu$ and satisfies

$$\tilde{\nabla}_\lambda g_{\mu\nu} = -2A_\lambda g_{\mu\nu}\,.$$

The affine connection $\Gamma^\alpha_{\mu\nu}$ splits into the Levi–Civita part (metric-compatible) and an extra Weyl piece constructed from $A_\mu$. The action encompasses the standard Einstein–Hilbert term $R[g, A]$ and all quadratic in curvature invariants that reduce to the Gauss–Bonnet combination in the limit $A_\mu \to 0$. In $d$ dimensions, the extended action has the form

$$S = \int d^dx \sqrt{-g} [-M_P^2 R(g,A) + \mathcal{A}\,\mathrm{X}\mathcal{G}_{\mathrm{EB}}],$$

where $\mathrm{X}\mathcal{G}_{\mathrm{EB}}$ is a symbolic notation for the extended GB invariant using the full Weyl connection. When $A_\mu=0$, $\mathrm{X}\mathcal{G}_{\mathrm{EB}}$ reduces to the standard Gauss–Bonnet combination, which in $d=4$ is a topological invariant.

2. Vector–Tensor Theories and Parameter Branches

Expanding the action in terms of Riemannian quantities plus the Weyl vector $A_\mu$, the theory becomes a vector–tensor model with Lagrangian terms such as

• $A^2$, $A_\mu \nabla^\mu A_\nu$, and $F_{\mu\nu}^2$ with $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$.

A unique feature is that only a single free parameter $\alpha$ appears in the kinetic sector of $A_\mu$, creating two distinct branches:

• For $\alpha=0$, the kinetic term degenerates, and $A_\mu$ does not propagate (its dynamics is nonpropagating).
• For $\alpha\ne 0$, after normalization, $A_\mu$ acquires Maxwell–Proca–like dynamics with a mass term and nonminimal couplings. This is the physically interesting, propagating branch, describing a massive vector field (Proca field) nonminimally coupled to gravity.

3. Four-Dimensional Theory: Reduction to Einstein–Proca and Horndeski Interactions

In $d=4$ dimensions, nonminimal vector–curvature couplings and higher-derivative interactions reduce or vanish, resulting in significant simplification of the theory:

$$S_{4d} = \int d^4x \sqrt{-g} \left[ \frac{M_P^2}{2} R - \frac{1}{4} F_{\mu\nu}^2 - \frac{1}{2}M^2 A_\mu^2 \right]$$

with $M^2$ proportional to $M_P^{-2}$. Standard Model (SM) matter couples only to $g_{\mu\nu}$ (i.e., the Levi–Civita part), so $A_\mu$ is an "invisible," decoupled massive vector—an automatic dark matter candidate.

Furthermore, a nontrivial cubic term in the curvature (constructed from the full Weyl connection) can exist in $d=4$, which leads (upon recasting into Riemannian variables) to a vector–tensor Horndeski interaction:

$$\mathcal{L}_3 \sim \epsilon^{\mu\nu\alpha\beta} \epsilon^{\rho\sigma\gamma\delta} R_{\mu\nu\rho\sigma} F_{\alpha\beta} F_{\gamma\delta}$$

Such a term preserves the property that no higher than second derivatives appear in the field equations.

4. Extensions to Higher Dimensions and the Stückelberg–Horndeski Connection

In $d>4$, expansion of the action introduces additional nonminimal vector–curvature couplings and derivative interactions, generally breaking the $U(1)$ invariance of the kinetic sector. Canonical analysis, including examining the Hessian structure, reveals a pair of second-class constraints that remove any would-be ghost-like degree of freedom, so the only propagating degrees of freedom are those of a massive vector field (three polarizations).

By introducing a Stückelberg field $\phi$ via $A_\mu\rightarrow A_\mu+\partial_\mu\phi$, the scalar degree of freedom fits into a specific Horndeski sub-class, ensuring the second-order structure of the full equations. Furthermore, in specific Kaluza–Klein reduction schemes, the resulting four-dimensional theory reproduces known Horndeski scalar–tensor actions.

5. Scale Invariance and Weyl Gauging Mechanism

Weyl geometry provides a natural structure for scale (local conformal) invariance. Under the local rescaling

$$g_{\mu\nu}\rightarrow e^{2\sigma(x)}g_{\mu\nu},\qquad A_\mu\rightarrow A_\mu-\partial_\mu \sigma(x)\,,$$

the Weyl–covariant derivative $D_\mu$ is designed such that $D_\mu g_{\alpha\beta}=0$. All tensors (curvature, connection, etc.) acquire definite conformal weights. The Weyl gauge method promotes derivatives to Weyl–covariant derivatives in the Lagrangian. In this process:

• The extended Gauss–Bonnet term (quadratic in curvature) is automatically scale-invariant.
• The vector–tensor Horndeski cubic interaction requires explicit coupling to a scalar (with appropriate Weyl weight) to ensure scale invariance.
• The full scalar–vector–tensor action, including the Weyl vector and a compensator scalar, forms a scale-invariant, second-order, ghost-free structure.

The generic scale-invariant theory in $d=4$ thus includes:

1. A conformally coupled Einstein–Hilbert term plus scalar,
2. Maxwell–Proca kinetic and mass terms for $A_\mu$,
3. Extended vector–tensor Horndeski interactions "dressed" by the compensator scalar.

6. Physical Interpretation and Phenomenological Consequences

In four dimensions, the theory describes standard gravity plus a decoupled massive vector, naturally providing an invisible sector for dark matter phenomenology. The generalized framework (including the cubic Horndeski–type coupling) introduces derivative vector–curvature interactions which may yield distinct cosmological or astrophysical signals in higher dimensions or when considering cosmological reductions.

In higher-dimensional cases, no additional dangerous degrees of freedom propagate, even in the presence of non–$U(1)$–invariant kinetic terms. The Stückelberg–Horndeski embedding eliminates higher-derivative ghosts, supporting the robustness of the vector–tensor sector.

The conformal symmetry built into the Weyl geometric structure underpins attempts to model scale-invariant gravitational theories and serves as a starting point for further generalizations to cosmology and high-energy gravity.

7. Summary Table: Structural Features of Extended Gauss–Bonnet Gravity in Weyl Geometry

Feature	$d=4$ Case	$d>4$ Case	Scale Invariance (Weyl gauging)
Propagating vector (Proca)	Yes	Yes	Yes
Horndeski cubic (second order) term	Constructible	Generalizes with extra terms	Becomes scale-invariant via coupling
Nonminimal vector–curvature coupling	Drops out/total deriv.	Persists	Incorporated via Weyl covariantization
Ghost/extra d.o.f. issue	Absent	Absent (via constraints)	Absent
Candidate dark sector field	Yes (decoupled $A_\mu$)	Possible (depends on reduction)	Yes

This framework for extended Gauss–Bonnet gravity reveals a mathematically controlled route to incorporating higher-curvature and vector degrees of freedom into gravitational dynamics, with explicit connections to Horndeski theory, scale invariance, and potential phenomenological applications related to the dark sector and beyond.