The Branson cocycle and its Weyl-scale obstruction

Definitions and mathematical statement

Let $M$ be a closed smooth manifold of even dimension $n$, with a Riemannian conformal class. Treat critical $Q$-curvature as the density $q_g=Q_n(g)\,{\,\mathrm{d}} V_g$. For $\widehat g=e^{2\omega}g$, use the critical transformation law \[ q_{\widehat g}=q_g+(P_g\omega)\,{\,\mathrm{d}} V_g, \qquad (P_{\widehat g}f)\,{\,\mathrm{d}} V_{\widehat g} =(P_gf)\,{\,\mathrm{d}} V_g. \] Here $P$ is the critical GJMS operator, with $P_g1=0$. Define the unmodified functional suggested by Tom Branson, \[ H[\widehat g,g]=\int_M\omega(q_{\widehat g}+q_g), \qquad \omega=\log\Omega. \] The source's displayed $\widehat Q+Q$ must thus use densities or the corresponding separate volume factors; adding two unweighted scalars with no integration convention would not specify this functional.

Problem 24(b). Do Weyl structures help with the preceding, unspecified issue? The extract proposes $H$ and asserts that its three-metric cocycle property is equivalent to self-adjointness of $P$. The antecedent "in this regard" is missing, so no broader objective is inferred.

Supplied exact cocycle test. For $g_0=g$, $g_1=e^{2\omega}g$, $g_2=e^{2(\omega+\phi)}g$, the aid gives \[ H[g_2,g_1]+H[g_1,g_0]-H[g_2,g_0] =\int_M(\phi P_g\omega-\omega P_g\phi)\,{\,\mathrm{d}} V_g. \] Consequently the cocycle identity for all conformal triples is equivalent to formal self-adjointness, namely symmetry of this pairing for all smooth $\omega,\phi$. If tests are restricted to a subspace, they detect only the restricted skew pairing. In particular every one-dimensional conformal-ray test, $\omega=af$ and $\phi=bf$, has zero defect for every linear $P$ and cannot establish full self-adjointness.

Supplied Weyl-scale descent criterion. A Weyl connection is torsion free and preserves the conformal class; in a scale $g$ write $Dg=-2\theta\otimes g$. The connection induced by a metric is its Levi-Civita connection, an exact Weyl connection. Two conformally related metrics induce the same connection precisely when their logarithmic scale ratio is constant on each connected component. For the actual self-adjoint critical GJMS operator, let \[ \mathcal Q_j=\int_{M_j}q_g \] for each connected component $M_j$; these numbers are conformally invariant. Changing the representatives of a pair by $g_0\mapsto e^{2c_j}g_0$ and $g_1\mapsto e^{2d_j}g_1$ changes the functional by \[ 2\sum_j(d_j-c_j)\mathcal Q_j. \] The aid therefore asserts that this unmodified $H$ descends to a function of pairs of metric-induced Weyl connections exactly when $\mathcal Q_j=0$ for every component. On a connected manifold this is simply zero total $Q$; on a disconnected manifold cancellation of the global sum is insufficient. This is a criterion for exact metric-induced Weyl structures, not a construction on all nonexact Weyl connections, nor a claim that no modified functional could remove the obstruction. Boundary versions require boundary conditions and additional terms and are not included.

Short English statement

Branson's proposed functional is a cocycle exactly when the critical operator is formally self-adjoint, provided all conformal triples are tested. The supplied aid also shows that it depends only on metric-induced Weyl connections precisely when total $Q$ vanishes on each connected component. The missing broader aim and nonexact Weyl case remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 20002046, source code catalogue entry; canonical aim-geometry-notes.json, notes[383].
• AIM Conformal structure in geometry, analysis, and physics, Problem 24(b), functional suggested by Tom Branson: https://aimath.org/WWN/confstruct/confstruct.pdf.
• No separate bibliographic citation for the cocycle calculation or Weyl-scale descent criterion is supplied; the latter is the research aid's partial result.
• Registry research attempt 1, dated 2026-08-14: partial result, rigor "proved", novelty confidence low. The explicitly machine-generated, non-peer-reviewed aid requires independent verification before citation.