Affine conformal densities from form Q-operators

Definitions and mathematical statement

Problem $34$ asks for other natural tensor-densities satisfying an absent transformation law $(43)$, observing that each gives a conformally invariant natural operator $L$. The aid reconstructs this as the affine law \[ N_{\widehat g}=N_g+L_g\omega,\qquad\widehat g=e^{2\omega}g. \] The original sentence ends "Solutions to"; no further target or classification is supplied. The following is the reported Branson--Gover construction, not an unrestricted completion of that fragment.

In the form-$Q$ framework, closed $k$-forms satisfy, with its weighted-form identifications and normalizations, \[ Q_k^{\widehat g}u=Q_k^gu+\delta Q_{k+1}^g d(\omega u). \] Let $\tau$ be a natural closed conformally invariant Pontryagin form of degree $k$ for which this construction applies. Pairing gives the natural top-form density \[ N_g=\langle\tau,Q_k^g\tau\rangle_g\,{\,\mathrm{d}} V_g, \] with affine variation \[ L_{\tau,g}(\omega)= \langle\tau,\delta Q_{k+1}^g d(\omega\tau)\rangle_g\,{\,\mathrm{d}} V_g. \] The aid reports that this variation is a formally self-adjoint conformally invariant natural operator, and that $\int_MN_g$ is conformally invariant on closed manifolds. Closedness of $\tau$, formal self-adjointness of the form operator and the degree/weight normalization are substantive parts of the construction. An arbitrary chosen closed form would not automatically give a natural density.

The aid's explicit nonzero family is: for $r\geq1$, take a Riemannian product $X^{4r}\times Y^{4r+2}$, where $X$ is Einstein and locally symmetric, $Y$ has constant curvature, and \[ \operatorname{Ric}_X=\lambda_X g_X,\qquad \operatorname{Ric}_Y=\lambda_Y g_Y. \] Assume the pulled-back degree-$4r$ Pontryagin character form $\tau_r$ of $X$ is nonzero. It is parallel in this setting, and in the aid's standard near-middle normalization, \[ Q_{4r}\tau_r= \frac{2((4r+2)\lambda_Y-(4r+1)\lambda_X)}{8r+1}\,\tau_r. \] In particular $X=\mathbb{CP}^{2r}$ with its standard locally symmetric Einstein metric and $Y=S^{4r+2}$ with a round metric give a nontrivial $Q$-like affine density in real dimension $8r+2$, except when \[ (4r+2)\lambda_Y=(4r+1)\lambda_X. \] The nonzero-form assumption is necessary for a general $X$; not every locally symmetric factor has the required Pontryagin form. The exceptional scaling ratio concerns this product value and does not classify all possible affine densities. The source's open-ended construction request is answered in the reported framework by examples, not by a uniqueness theorem for every natural tensor-density.

Short English statement

The aid reconstructs the missing law as an affine conformal transformation and supplies Branson--Gover form-$Q$ constructions paired with natural Pontryagin forms. A product family on $\mathbb{CP}^{2r}\times S^{4r+2}$ has an explicit eigenvalue and is nonzero except at one scaling ratio. No complete classification is claimed.

Sources

• Local UnsolvedMath registry, permanent ID $20002058$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[395].
• AIM workshop "Conformal structure in geometry, analysis, and physics," Problem $34$: https://aimath.org/WWN/confstruct/confstruct.pdf. Formula $(43)$ and the continuation after "Solutions to" are absent. No PDF was opened in this session.
• The aid attributes the form-$Q$ construction to Branson and Gover and uses natural Pontryagin character forms. No full authoritative bibliography, endpoint or normalization conventions are supplied; original remarks and literature fields are empty, with no proposer or year.
• Supplied research attempt $1$, result full_solution, run status "solved," rigor label "proved," novelty confidence "low." Registry status "solved," classification SOLVED-IN-LITERATURE, updated 14 August 2026. The claimed explicit contribution is the product eigenvalue family within the known construction.