Pontrjagin forms and invariant objects related to the Q-curvature

Abstract

We clarify the conformal invariance of the Pontrjagin forms by giving them a manifestly conformally invariant construction; they are shown to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the $k$-form operators $Q_k$ of \cite{ddd}, which generalise the Q-curvature, retain a key aspect of the $Q$-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the $Q_k$ give rise to conformally invariant quadratic forms $\Theta_k$ on cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at $k=n/2$) and the Pfaffian (at $k=0$). Using a different construction, we show that the $Q_k$ operators yield a generalisation of the period map which maps conformal structures to Lagrangian subspaces of the direct sum $H^k\oplus H_k$ (where $H_k$ is the dual of the de Rham cohomology space $H^k$). We couple the $Q_k$ operators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalise to $Q_k$-operators the Q-curvature prescription problem.