Higher Conformal Yang-Mills Equation

• Higher Conformal Yang-Mills equations are a generalization of classical Yang-Mills theory that incorporate conformal invariance and higher derivatives on even-dimensional, conformally compact manifolds.
• The formulation employs both ordinary-derivative approaches with auxiliary fields and higher-derivative methods ensuring gauge invariance beyond four dimensions.
• These equations reveal boundary anomalies and energy currents, linking gauge theory with advanced tools from conformal geometry and geometric analysis.

The higher conformal Yang-Mills equation generalizes the classical Yang-Mills equations to capture conformally invariant, higher-derivative structures, especially on conformally compact manifolds and in higher even-dimensional spaces. These equations naturally arise as the Euler-Lagrange equations of conformally invariant actions that generalize the Yang-Mills energy, often manifesting as anomalies or obstructions in the analysis of boundary value problems for gauge fields. This unifies the study of gauge theory with sophisticated tools from conformal geometry and is particularly relevant in dimensions $d \geq 6$, where ordinary second-order Yang-Mills theory ceases to be conformally invariant.

1. Conformal Invariance and Higher-Derivative Yang-Mills Structures

The central feature of the higher conformal Yang-Mills equation is its invariance under conformal rescaling of the metric, extending the classical Yang-Mills action to higher-derivative and higher-order equations intrinsically tied to the geometry of the underlying manifold. For a gauge connection $A$ with curvature $F_A$ on a manifold $(M, g)$, the leading candidate for such an action in even dimension $d$ is

$$S_{\text{conf}}[A] = \int_M \mathrm{Tr}\left( -D^{\frac{d}{2}-2}F_{\mu\nu} \, D^{\frac{d}{2}-2}F^{\mu\nu} + g \, \text{(cubic terms)} \right) \, d^{d}x,$$

where $D$ is the covariant derivative and the cubic terms generalize the familiar $F^3$ vertex but at conformally appropriate derivative order. The resulting Euler-Lagrange equations are of order $d-2$ in derivatives and are manifestly gauge- and conformally invariant (Metsaev, 2023, Metsaev, 2024, Gover et al., 2021).

On conformally compact manifolds, the variational analysis of Yang-Mills action functionals reveals that the coefficient of the first logarithmic term in the asymptotic expansion, which is conformally invariant, gives rise to a higher-order boundary current. The variation of this term yields the "higher conformal Yang-Mills equations" as encountered in the study of formally compact manifolds and their boundary value problems (Gover et al., 2023).

2. Lagrangian Formulations and Field Content

Beyond dimension four, classical Yang-Mills action is not conformally invariant; higher-derivative actions are required. The construction proceeds via two equivalent formulations:

• Ordinary-derivative, gauge-invariant formulation: A tower of auxiliary vector fields and Stueckelberg scalar fields is introduced alongside the usual Yang-Mills field, ensuring a conventional (two-derivative) kinetic term for each field and an enlarged gauge symmetry. The fundamental field strengths and interactions are constructed recursively to guarantee invariance under the extended gauge symmetry (Metsaev, 2023, Metsaev, 2024).
• Higher-derivative, manifestly conformal formulation: After integrating out auxiliaries and fixing the Stueckelberg gauge, the resulting Lagrangian contains only the principal connection and its curvature, with higher-derivative kinetic and interaction terms. For $d=6$, the kinetic operator is quartic in derivatives (Gover et al., 2021, Gover et al., 2023).

The following table summarizes field content and formulations:

Dimension ($d$)	Field Content	Lagrangian Type
$d=6,8,10,\dots$	YM field, $\{V^{(n)}_\mu\}$, $\{\varphi^{(n)}\}$	Ordinary/auxiliary
$d=6,8,10,\dots$	YM field only (after elimination/integration)	Higher-derivative

3. Boundary Value Problems, Asymptotics, and Anomalies

On conformally compact manifolds, the interior Yang-Mills energy admits a formal expansion near the boundary. Solutions with prescribed ("magnetic") Dirichlet boundary data are, in general, obstructed by the appearance of a conformally invariant, higher-order boundary current. This current arises as the variation of the conformally invariant log-coefficient (the anomaly) in the renormalized action (Gover et al., 2023). The variation of this coefficient yields the higher conformal Yang-Mills equation, which encodes the obstruction to regular solutions. Correspondingly, global solutions to the magnetic boundary problem determine higher-order "electric" Neumann data, yielding a Dirichlet-to-Neumann map.

Furthermore, conformally invariant, higher transverse derivative boundary operators are constructed. These act on connections and provide obstructions, control the asymptotics of formal solutions, and yield tensors capturing non-local Neumann data, generalizing the role of the classical electric-magnetic boundary map (Gover et al., 2023).

4. Conformally Invariant Operators and Explicit Equations

Explicitly, in six dimensions, the conformally invariant action is constructed using the so-called $Q$-operator: $$Q_A \omega := \delta_A d_A \omega - 4 P \# \omega + 2 J \omega,$$ where $P_{ab}$ is the Schouten tensor, $J$ its trace, and $\delta_A$ is the adjoint covariant derivative. The action takes the form

$S(A) = \int_M (F_A, Q_A F_A)\, d\mathrm{vol}_g,$

with the variation giving the Euler-Lagrange equations

$$D(A) := \delta_A Q_A F_A - [\delta_A F_A, F_A] = 0,$$

which is a 4th-order, source-free, conformally invariant generalization of the Yang-Mills equation (Gover et al., 2021). For the conformal Cartan-tractor connection, this equation reduces to the vanishing of the Fefferman-Graham obstruction tensor, linking gauge field equations directly to the geometry of conformally compact manifolds.

5. Gauge Algebra and Symmetries

The extended field content in the ordinary-derivative approach is associated with a generalized gauge algebra. The full set of gauge transformations depends on an infinite tower of scalar parameters $\xi^{(2n)}$, and their commutator closes into a non-linear "star-commutator" algebra, whose structure constants are determined by combinatorial coefficients. For example: $$[\delta_{\xi},\delta_{\eta}] = \delta_{[\xi,\eta]_*}, \qquad [\xi,\eta]_*^{(2n)} = \sum_{k=0}^n a_{n,k}\, [\xi^{(2n-2k)}, \eta^{(2k)}],$$ ensuring gauge closure at all orders (Metsaev, 2023, Metsaev, 2024). This structure can be further enriched in (A)dS with the extended algebra receiving a Levy–Maltsev decomposition.

6. Relation to Conformal Anomalies and Geometric Invariants

The renormalized Yang-Mills action, upon analysis near the conformal boundary, exhibits an energy anomaly analogous to the renormalized volume for the Poincaré–Einstein metric. The log-coefficient in the expansion of the interior functional encodes a conformally invariant energy, whose variation gives the higher conformal Yang-Mills equations (Gover et al., 2023). This construction parallels the emergence of conformal invariants in other settings, such as the Fefferman-Graham obstruction and conformal Q-curvature, reinforcing the deep connections between conformal geometry and gauge field theory.

7. Generalization and Applications

The higher conformal Yang-Mills equation framework extends to arbitrary even dimensions and can be formulated on general backgrounds, including (A)dS and conformally compact manifolds. In the flat $d$-dimensional limit, the unique conformally invariant Yang-Mills action reduces to

$$S_{\text{conf}}[A] = \int d^d x \, \mathrm{Tr} \left\{ -\partial^{\frac{d}{2}-2} F_{\mu\nu} \, \partial^{\frac{d}{2}-2} F^{\mu\nu} + \cdots \right\},$$

with higher-derivative interaction terms uniquely determined by gauge and conformal symmetry requirements. These structures encode global analytic and geometric features of gauge fields, affecting both physical properties and boundary value problems in geometric analysis (Metsaev, 2023, Metsaev, 2024, Gover et al., 2023, Gover et al., 2021).