Invariant Prolongation and Detour Complexes

Abstract

In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as detour complexes. This elaborates on a lecture given at the IMA Summer Programme "Symmetries and overdetermined systems of partial differential equations".

Partially massless spin-2 electrodynamics

“…Actually, all computations in this paper apply also to AdS backgrounds, but partially massless excitations are no longer unitary in that case 2. In fact, the same holds for maximal depth theories of arbitrary spin[8].…”

“…For the remainder of our analysis we retain both parameters (Q, Q ′ ) and the distinction between dynamical and background charged matter will not play any special rôle. Classical consistency of the coupling (11) relies not only on the gauge invariance (4) but also the constraint (8) to ensure that ghost states are non-propagating. In particular one might worry that including the source J µ introduces terms involving covariant derivatives of dynamical fields to the right hand side of (8).…”

“…Classical consistency of the coupling (11) relies not only on the gauge invariance (4) but also the constraint (8) to ensure that ghost states are non-propagating. In particular one might worry that including the source J µ introduces terms involving covariant derivatives of dynamical fields to the right hand side of (8). In particular, the key property that the constraint is only first order in time derivatives of fields 6 , could be violated.…”

CR-invariants and the scattering operator for complex manifolds with CR-boundary

“…To describe our results, we first recall some basic notions of CR geometry and recent results [8], [9] concerning CR-covariant differential operators and CR-analogues of Q-curvature. If M is a smooth, orientable manifold of real dimension (2n + 1), a CR-structure on M is a real hyperplane bundle H on T M together with a smooth bundle map J : H → H with J 2 = −1 that determines an almost complex structure on H. We denote by T 1,0 the eigenspace of J on H ⊗ C with eigenvalue +i; we will always assume that the CR-structure on M is integrable in the sense that [T 1,0 , T 1,0 ] ⊂ T 1,0 .…”

“…We view the operators P k as operators on C ∞ (M ); if one instead views these operators as acting on appropriate density bundles over M they are actually invariant operators. Gover and Graham [9] showed that the CR-covariant differential operators P k are logarithmic obstructions to the solution of the Dirichlet problem (1.3) when X is a pseudoconvex domain in C m with a metric of Bergman type, but did not identify them as residues of the scattering operator.…”