Ambient tractor Laplacians and affine conformal laws

Definitions and mathematical statement

The source's actual Problem 1 asks for general or inductive formulae for operators whose extracted name "§2`" is corrupted. The aid recovers the name as the weighted tractor operator $\square_{2\ell}$. The rest of the record begins a different problem's setup about affine conformal transformation laws and should not replace the operator-formula question.

Write $\mathcal T[w]$ for a weighted standard tractor bundle and $\mathcal E[w]$ for conformal densities of weight $w$. In the usual weight convention for the family identified by the aid, its order-$2\ell$ assignment is \[ \square_{2\ell}:\mathcal T[\ell-n/2] \longrightarrow\mathcal T[-\ell-n/2]. \] At the Q-curvature order $\ell=n/2-1$, where this order is defined, this specializes to \[ \square_{n-2}:\mathcal T[-1]\longrightarrow\mathcal T[1-n], \] whose bottom density slot is $\mathcal E[-n]$, the slot of $Q_n$ in the supplied recovery. The admissible dimensions, ambient extension, curvature corrections, normalization and exact operator definition are not supplied. The aid reports general strongly invariant tractor formulae, a curvature-corrected algorithm yielding a pure Q slot, low-order expansions and recursive Q-curvature formulae in primary literature, but explicitly says exact identification with the full ambient operator named in the AIM problem was not verified. Thus this is not a complete formula or a verified solution of Problem 1; the input producing Q and its corrections cannot be invented from the slot statement alone.

The accompanying affine-law discussion has the following precise independent content. On a fixed conformal class write $\widehat g=e^{2\omega}g$, $\omega\in C^\infty(M)$, and identify a geometric tensor-density target $\mathcal V$ consistently across scales. Suppose for every representative and every $\omega$ one has a finite law \[ N_{e^{2\omega}g}=N_g+L_g\omega, \qquad L_g:C^\infty(M)\to\Gamma(\mathcal V)\text{ linear}. \] Comparing two successive conformal changes gives \[ L_{e^{2\omega}g}\eta=L_g\eta \qquad(\omega,\eta\in C^\infty(M)). \] Thus $L$ is conformally invariant in this density-target convention, and \[ C(g,e^{2\omega}g)=L_g\omega,\qquad C(g,g'')=C(g,g')+C(g',g'') \] defines an additive vector-valued cocycle. An infinitesimal first-variation law alone is not the assumed finite law.

If $N_g$ is an integrable scalar density on a closed manifold and $\int_MN_g$ is conformally invariant, then \[ \int_ML_g\omega=0\qquad\text{for every }\omega. \] Writing $L_g\omega=(P_g\omega)\,{\,\mathrm{d}}\mathrm{vol}_g$, this is $P_g^*1=0$ for the formal adjoint. If additionally $P_g$ is formally self-adjoint, it gives $P_g1=0$. Self-adjointness is an extra assumption, not a consequence of the affine law. On a manifold with boundary or a noncompact manifold, support and boundary conditions are needed for this adjoint inference. For scalar Q-curvature, the finite density law is expressed with $N_g=Q_g\,{\,\mathrm{d}}\mathrm{vol}_g$; omitting the density identification from a scalar formula would lose its conformal weight factor.

Conversely, on one fixed conformal class choose a base metric $g_0$, an initial target section $N_0$, and a conformally invariant linear operator $L$ into the fixed target. Then \[ N_{e^{2\omega}g_0}=N_0+L\omega \] satisfies the finite affine law. This construction uses a chosen base and does not by itself provide a natural local Riemannian tensor-density assignment independent of that choice. Locality and naturality are additional reconstruction requirements; conformal-path consistency is already ensured by the cocycle.

Short English statement

The damaged main question concerns tractor-operator formulae, whose exact ambient identification remains unverified. Its appended affine-law setup yields a separate compatibility theorem: the finite law forces operator invariance, while integral invariance and self-adjointness give different annihilation conditions. A base-dependent converse does not produce an intrinsic natural tensor.

Sources

• Permanent UnsolvedMath ID $20002014$, exact public code catalogue entry; local registry aim-geometry-notes.json, notes[351], canonical tag problem.
• AIM workshop "Conformal structure in geometry, analysis, and physics," Problem 1 followed by equation (34): https://aimath.org/WWN/confstruct/confstruct.pdf. No PDF was accessed. The aid's verification note reports using the hard-copy PDF and contemporaneous HTML ALT text to recover the corrupted operator, but the promised recovered sentence is absent from the supplied field.
• Research attempt 1, "Ambient tractor Laplacians and affine conformal laws," labelled partial_result, partially_solved, proved and low novelty confidence. It mentions primary tractor/Q-curvature literature without naming full references and expressly leaves exact full-ambient-operator identification unverified. Original literature, proposer/year metadata and full citations are absent.