Weylian Boundary Terms in Gravity and Field Theory

• Weylian boundary terms are boundary contributions in gravitational actions where Weyl geometry is promoted from auxiliary to dynamically significant, altering field equations and conservation laws.
• They yield modified Einstein equations with additional Weyl vector terms that exchange energy-momentum with matter, impacting dark energy, inflation, and early-universe nucleosynthesis.
• In holography and conformal anomaly studies, Weylian boundary terms provide essential boundary invariants that ensure a well-posed variational principle and consistent anomaly cancellation.

Searching arXiv for recent and foundational papers on Weylian boundary terms, boundary Weyl anomaly, and Weyl-transverse gravity. Weylian boundary terms are boundary contributions in gravitational or conformal field-theoretic constructions in which Weyl geometry, Weyl-covariant structures, or Weyl-anomalous boundary invariants are promoted from auxiliary boundary data to dynamical or variationally essential ingredients. In the recent cosmological framework of Harko, Lobo, and Shahidi, the Einstein–Hilbert bulk action is kept Riemannian while the boundary is treated as Weylian, so that the boundary variation is evaluated with a Weyl connection and Weyl covariant derivative rather than being canceled by a Gibbons–Hawking–York term (Harko et al., 23 Mar 2026). In quantum field theory and holography, the phrase also refers to Gibbons–Hawking-type boundary companions of bulk Weyl invariants in conformal anomalies, as well as purely boundary conformal invariants built from trace-free extrinsic curvature (Solodukhin, 2015, Astaneh et al., 2017, Chalabi et al., 2021). A distinct but related usage appears in Weyl-transverse gravity, where Weyl-invariant boundary structures organize the variational principle, covariant phase space, and conserved charges in the presence of a fixed background volume form (Odak et al., 22 Jan 2026). Across these settings, the common theme is that boundary geometry is no longer a passive bookkeeping device: it contributes independent Weyl-covariant data, modifies the variational problem, and may generate effective dynamics or anomaly coefficients.

1. Weylian boundary geometry and variational re-interpretation

In the cosmological construction of a Universe with a Weylian boundary, the bulk geometry remains Riemannian and metric-compatible, with Levi–Civita connection $\Gamma^\lambda_{\mu\nu}$ and $\nabla_\mu g_{\alpha\beta}=0$, while the boundary and its neighborhood are assumed to have Weyl-type non-metricity (Harko et al., 23 Mar 2026). The defining condition is

$$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$

where $\omega_\mu$ is the Weyl gauge vector and $\alpha$ is a dimensionless Weyl gauge coupling. The corresponding Weyl connection is

$$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$

with

$$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$

This construction retains the Einstein–Hilbert bulk action

$$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$

but departs from standard GR at the level of the boundary variation. Instead of canceling the variation of the Ricci tensor by adding a Gibbons–Hawking–York term or by imposing $\delta g_{\mu\nu}|_{\partial\Omega}=0$, the boundary part is reinterpreted in Weyl geometry through the replacements

$$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$

This yields a modified variation in which the boundary term becomes a term proportional to $\nabla_\mu g_{\alpha\beta}=0$0 with coefficients built from $\nabla_\mu g_{\alpha\beta}=0$1 and its Weyl-covariant derivatives (Harko et al., 23 Mar 2026). The resulting expression,

$\nabla_\mu g_{\alpha\beta}=0$2

is treated as a genuine contribution to the field equations. In this formulation, a “Weylian boundary” therefore means that the metric $\nabla_\mu g_{\alpha\beta}=0$3 is the same as in the bulk, the non-metricity is confined to boundary-related structures, and the extra field $\nabla_\mu g_{\alpha\beta}=0$4 enters only through boundary-related terms (Harko et al., 23 Mar 2026).

A closely related warm-inflation model adopts the same conceptual move, but specializes to an integrable Weyl geometry $\nabla_\mu g_{\alpha\beta}=0$5, thereby trading the Weyl vector for a scalar $\nabla_\mu g_{\alpha\beta}=0$6 that encodes the boundary effect (Matei et al., 2024). In that setting, the boundary term in the Einstein–Hilbert variation is re-evaluated in Weyl geometry and produces effective scalar contributions in the field equations. This suggests a broader usage of “Weylian boundary terms” for geometric boundary corrections induced by Weyl non-metricity rather than by explicit bulk Weyl actions.

2. Field equations generated by Weylian boundary terms

When the Weylian boundary variation is inserted into the modified action principle and rewritten in Riemannian covariant form, the generalized Einstein equations become (Harko et al., 23 Mar 2026)

$\nabla_\mu g_{\alpha\beta}=0$7

with $\nabla_\mu g_{\alpha\beta}=0$8. The additional terms are purely geometric and involve only the Weyl vector and its first derivatives.

These equations may be read as Einstein equations sourced by an effective geometric sector. The formulation

$\nabla_\mu g_{\alpha\beta}=0$9

isolates the Weyl contribution as an effective stress-energy built from derivative terms and quadratic condensate-like terms in $$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$0 (Harko et al., 23 Mar 2026). The paper does not introduce $$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$1 as a primary tensorial object, but the later cosmological expressions for effective density and pressure are direct projections of these boundary-induced terms.

A significant feature of the construction is that ordinary matter is not conserved. Taking the divergence of the generalized field equations and using the Bianchi identity yields (Harko et al., 23 Mar 2026)

$$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$2

The matter and Weyl sectors therefore exchange energy-momentum. The warm-inflation realization reaches a parallel conclusion: the total effective fluid satisfies a global conservation law, but the radiation, inflaton, and Weyl scalar sectors exchange energy according to how the total conservation equation is split phenomenologically (Matei et al., 2024). In both constructions, non-conservation is not an incidental artifact but a direct consequence of promoting boundary Weyl contributions to dynamical relevance without supplementing them by an independent bulk action for $$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$3 or $$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$4.

3. Cosmological realizations: dark energy, inflation, and BBN

In a spatially flat FLRW geometry,

$$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$5

homogeneity and isotropy force the Weyl vector in the dark-energy model to take the purely temporal form $$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$6 (Harko et al., 23 Mar 2026). The generalized Friedmann equations then become

$$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$7

$$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$8

The effective geometric dark-energy density and pressure are

$$\tilde{\nabla}_\mu g_{\alpha\beta} = -\alpha\,\omega_\mu\, g_{\alpha\beta},$$9

Late-time acceleration is thus attributed to Weylian boundary terms of purely geometric origin, with no scalar potential, no explicit cosmological constant, and no exotic matter in the action (Harko et al., 23 Mar 2026). To close the background system, the effective dark-energy equation of state is imposed phenomenologically through the Barboza–Alcaniz parametrization,

$\omega_\mu$0

so that $\omega_\mu$1 (Harko et al., 23 Mar 2026).

The warm-inflation model uses an integrable Weyl boundary, $\omega_\mu$2, and introduces a dissipative scalar inflaton $\omega_\mu$3 with Lagrangian

$\omega_\mu$4

(Matei et al., 2024). The FLRW equations acquire Weylian boundary contributions through $\omega_\mu$5,

$\omega_\mu$6

$\omega_\mu$7

Three distinct models are then defined by different splittings of the total conservation equation, corresponding respectively to mixed inflaton–Weyl radiation production, radiation driven only by the inflaton, and radiation driven only by the Weyl boundary field (Matei et al., 2024). Numerical solutions show that radiation is created during the very early phases of expansion, reaches a maximum, and is followed by a transition from an accelerating inflationary phase to a decelerating one, with the Weyl vector playing a significant role during radiation creation (Matei et al., 2024).

The post–warm-inflationary consequences of this framework have been constrained at the BBN epoch. In the BBN analysis, the same generalized Friedmann system is specialized to the radiation era and the extra energy density $\omega_\mu$8 coming from the dissipative scalar and the Weyl boundary is treated as a small correction to standard radiation (Matei et al., 1 Sep 2025). The analytic Helium-4 constraint yields

$\omega_\mu$9

which implies

$\alpha$0

at freeze-out (Matei et al., 1 Sep 2025). A numerical implementation based on \texttt{PRyMordial}, together with a Genetic Algorithm and MCMC pipeline implemented in the python code \texttt{genesys}, finds that the Weylian boundary contribution at BBN corresponds to $\alpha$1, so it remains effectively indistinguishable from standard BBN at current observational precision (Matei et al., 1 Sep 2025). This constrains any strong cosmological role of Weylian boundary terms to epochs prior to the MeV era.

4. Observational status and comparison with $\alpha$2CDM

The late-time FLRW realization of Weylian boundary terms has been confronted with cosmic chronometers, Pantheon$\alpha$3, and DESI DR2 BAO data (Harko et al., 23 Mar 2026). In the dimensionless redshift formulation, the Hubble rate is written as

$\alpha$4

with evolution equations for $\alpha$5 and $\alpha$6 closed by the Barboza–Alcaniz equation of state (Harko et al., 23 Mar 2026). The model is then fit jointly in the parameter space $\alpha$7.

For the combined CC + Pantheon$\alpha$8 + BAO dataset, the best-fit parameters are reported as

$\alpha$9

$$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$0

with $$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$1, $$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$2, and $$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$3 Mpc (Harko et al., 23 Mar 2026). For $$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$4CDM on the same dataset, the fit is

$$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$5

The reduced chi-squared values are essentially identical,

$$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$6

while the Bayesian evidence gives

$$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$7

hence

$$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$8

interpreted on Jeffreys’ scale as strong (near “moderate–strong”) evidence in favor of the Weyl boundary model over $$\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \Psi^\lambda_{\mu\nu},$$9CDM (Harko et al., 23 Mar 2026). The best-fit $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$0 is also consistent with Planck 2018 at the level of $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$1, although no SH0ES calibration is included (Harko et al., 23 Mar 2026).

A recurrent misconception is that these models simply repackage a cosmological constant. The data analysis does show that the background expansion can reproduce almost exactly the predictions of the $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$2CDM paradigm, but the mechanism is structurally different: the effective dark energy arises from Weylian boundary terms rather than from an explicit $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$3 term in the action (Harko et al., 23 Mar 2026). A second misconception is that the framework is a fully specified vector-tensor theory. It is not: there is no dynamical equation for $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$4 derived from a bulk action, and the closure of the cosmological system relies on phenomenological assumptions such as the effective equation of state (Harko et al., 23 Mar 2026).

5. Weylian boundary terms in conformal anomaly and boundary CFT

A separate and older line of research uses “Weylian boundary terms” to denote boundary contributions associated with Weyl anomalies on manifolds with boundary. In this setting, the key statement is that each type B bulk invariant $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$5, polynomial in the Weyl tensor, must be supplemented by a Gibbons–Hawking-type boundary companion $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$6 to ensure a well-posed variational principle (Solodukhin, 2015). Solodukhin’s construction introduces a Weylian GH tensor $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$7, built from extrinsic geometry and the metric, with the same algebraic symmetries as the bulk Weyl tensor. This tensor is trace-free and depends only on the trace-free extrinsic curvature $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$8, making it a conformal tensor suitable for boundary anomaly functionals (Solodukhin, 2015).

In four dimensions, the integrated anomaly contains the bulk invariant $$\Psi^\lambda_{\mu\nu} = \frac{\alpha}{2}\left(\delta^\lambda_\mu\,\omega_\nu + \delta^\lambda_\nu\,\omega_\mu - g_{\mu\nu}\,\omega^\lambda\right).$$9 together with the boundary term

$$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$0

and a purely extrinsic boundary invariant $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$1 (Solodukhin, 2015). In six dimensions, the bulk type B invariants $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$2, $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$3, and $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$4 acquire analogous boundary companions built by replacing one occurrence of $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$5 with $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$6, together with purely extrinsic terms such as $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$7 and $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$8 (Solodukhin, 2015). In odd dimensions, there is no local bulk Weyl anomaly density, but the integrated anomaly can still be non-vanishing because of boundary conformal invariants; in $$S_g = -\frac{1}{2\kappa^2}\int_\Omega R(g)\sqrt{-g}\,d^4x + S_m,$$9, for example, it is entirely a boundary effect (Solodukhin, 2015).

The holographic analysis of the integrated conformal anomaly in $\delta g_{\mu\nu}|_{\partial\Omega}=0$0 and $\delta g_{\mu\nu}|_{\partial\Omega}=0$1 confirms the same general structure (Astaneh et al., 2017). In $\delta g_{\mu\nu}|_{\partial\Omega}=0$2, the integrated anomaly contains boundary terms involving $\delta g_{\mu\nu}|_{\partial\Omega}=0$3 and $\delta g_{\mu\nu}|_{\partial\Omega}=0$4, with coefficients interpreted as boundary central charges. For $\delta g_{\mu\nu}|_{\partial\Omega}=0$5 SYM, the minimal-surface holographic prescription reproduces the free-field boundary anomaly for supersymmetric boundary conditions, while the Takayanagi prescription introduces a parameter $\delta g_{\mu\nu}|_{\partial\Omega}=0$6 that changes the boundary charges $\delta g_{\mu\nu}|_{\partial\Omega}=0$7 and $\delta g_{\mu\nu}|_{\partial\Omega}=0$8 (Astaneh et al., 2017).

The classification problem becomes considerably richer for four-dimensional conformal boundaries and defects in ambient dimension $\delta g_{\mu\nu}|_{\partial\Omega}=0$9. A systematic Wess–Zumino-consistent construction finds, for a 4D defect of codimension $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$0, one A-type invariant $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$1, twenty-two parity-even B-type invariants, and six universal parity-odd B-type invariants, with additional parity-odd structures for special codimensions $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$2 (Chalabi et al., 2021). For codimension one, corresponding to a 4D boundary or defect in $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$3, the anomaly reduces to the structure previously identified by Astaneh and Solodukhin, together with three parity-odd terms (Chalabi et al., 2021). Several of these central charges enter the displacement operator two-point function, the stress-tensor one-point function, and the universal part of entanglement entropy (Chalabi et al., 2021). This establishes that Weylian boundary terms in the anomaly context are not merely variational counterterms; they define intrinsic boundary or defect data of the quantum field theory.

6. Boundary conditions, holographic Weyl structures, and Weyl-transverse gravity

The technical treatment of Weylian boundary terms is closely tied to the broader problem of variational principles with boundary. In higher-order theories, adding total divergences is harmless only when the correct higher-order boundary data are fixed. If only the fields are fixed and derivatives are left unconstrained, boundary terms can change the solution space and may even render it empty (Fatibene et al., 2011). This warning is especially relevant for Weyl or metric-affine gravity, because any attempt to treat boundary terms as dynamical must specify boundary conditions consistent with the formal order of the action (Fatibene et al., 2011). A plausible implication is that the cosmological Weylian-boundary construction, being defined through a modified variation rather than a complete covariant bulk action for $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$4, should be interpreted as an effective field-equation-level model rather than a standard higher-order variational theory.

Weyl-transverse gravity provides a more systematic framework in which boundary terms are constructed directly from Weyl-invariant geometric data (Odak et al., 22 Jan 2026). WTG uses a dynamical metric $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$5, a fixed non-dynamical volume form $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$6, and an auxiliary Weyl-invariant metric $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$7 satisfying $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$8 (Odak et al., 22 Jan 2026). The natural boundary Lagrangians are built from the extrinsic curvature $$\Gamma \to \tilde{\Gamma},\qquad \nabla \to \tilde{\nabla},\qquad \delta\Gamma \to \delta\tilde{\Gamma}.$$9 of the auxiliary geometry, for example

$\nabla_\mu g_{\alpha\beta}=0$00

and admit auxiliary Dirichlet, auxiliary Neumann, dynamical Dirichlet, and York boundary conditions (Odak et al., 22 Jan 2026). In particular, York boundary conditions take the form

$\nabla_\mu g_{\alpha\beta}=0$01

which fix the conformal class of the induced dynamical metric and the mean extrinsic curvature defined with $\nabla_\mu g_{\alpha\beta}=0$02 (Odak et al., 22 Jan 2026). The resulting covariant phase-space formalism shows that Weyl transformations lie in the kernel of the presymplectic form, while conserved charges receive a distinctive contribution proportional to $\nabla_\mu g_{\alpha\beta}=0$03 due to the fixed background volume form (Odak et al., 22 Jan 2026).

A holographic realization of boundary Weyl structure appears in the generalized Fefferman–Graham gauge for asymptotically AdS$\nabla_\mu g_{\alpha\beta}=0$04 spacetimes (Arenas-Henriquez et al., 2024). In this gauge, the bulk metric is written as

$\nabla_\mu g_{\alpha\beta}=0$05

and the leading coefficient $\nabla_\mu g_{\alpha\beta}=0$06 transforms inhomogeneously under radial Weyl diffeomorphisms,

$\nabla_\mu g_{\alpha\beta}=0$07

exactly like a boundary Weyl connection (Arenas-Henriquez et al., 2024). The pair $\nabla_\mu g_{\alpha\beta}=0$08 therefore defines a boundary Weyl structure. Holographic renormalization in this framework uncovers new subleading divergences generated by the cross term $\nabla_\mu g_{\alpha\beta}=0$09, which are canceled by a codimension-two counterterm,

$\nabla_\mu g_{\alpha\beta}=0$10

so that the renormalized action remains finite (Arenas-Henriquez et al., 2024). The Weyl anomaly then takes the Weyl-covariant form

$\nabla_\mu g_{\alpha\beta}=0$11

with $\nabla_\mu g_{\alpha\beta}=0$12 the current conjugate to the Weyl connection (Arenas-Henriquez et al., 2024). This explicitly realizes Weylian boundary terms as boundary and corner counterterms required by a Weyl-covariant holographic variational principle.

In Palatini gravity and its dual formulations, Weyl invariance also constrains admissible boundary terms. For the unique Palatini $\nabla_\mu g_{\alpha\beta}=0$13 theory admitting an Eddington reduction in $\nabla_\mu g_{\alpha\beta}=0$14, namely $\nabla_\mu g_{\alpha\beta}=0$15, the bulk action is Weyl invariant and the connection assumes the form of a Weyl connection relative to $\nabla_\mu g_{\alpha\beta}=0$16 (Lindström et al., 2022). The appropriate generalized Gibbons–Hawking–York term is

$\nabla_\mu g_{\alpha\beta}=0$17

with $\nabla_\mu g_{\alpha\beta}=0$18 (Lindström et al., 2022). The paper shows that this boundary term maps consistently under the duality between Weyl-invariant Palatini $\nabla_\mu g_{\alpha\beta}=0$19, Born–Infeld–Einstein gravity, and the Eddington affine action (Lindström et al., 2022). In two dimensions, the corresponding first-order bosonic string formulation requires the boundary term

$\nabla_\mu g_{\alpha\beta}=0$20

which is invariant under the conformal symmetry of the 2D theory (Lindström et al., 2022).

Taken together, these developments show that “Weylian boundary terms” is not a single concept but a family of related constructions. In cosmology, they are dynamical geometric sources induced by a Weylian reinterpretation of the boundary variation (Harko et al., 23 Mar 2026, Matei et al., 2024, Matei et al., 1 Sep 2025). In conformal anomaly theory, they are boundary companions of bulk Weyl invariants and independent conformal invariants of extrinsic geometry (Solodukhin, 2015, Astaneh et al., 2017, Chalabi et al., 2021). In Weyl-transverse and holographic frameworks, they are Weyl-invariant boundary and corner terms that make the action differentiable and render the charge algebra and anomaly structure manifestly Weyl covariant (Odak et al., 22 Jan 2026, Arenas-Henriquez et al., 2024). The unifying idea is that the boundary carries genuine Weyl-covariant information, and once that information is retained rather than discarded, it can alter field equations, anomaly functionals, conserved charges, and cosmological evolution.