# Conformal and CR geometry: Spectral and nonlocal aspects (07rit132)

Arriving in Banff, Alberta Sunday, August 26 and departing Sunday September 9, 2007

## Objectives

In recent work the applicants and collaborators have made several closely linked discoveries. These include: the existence of a new class of conformally invariant elliptic differential complexes on conformally curved structures [math.DG/0309085] (and see also [math.DG/0404004]) new torsion quantities for these ``detour complexes'' which generalise Cheeger's 1/2-torsion [articles in progress]; subcomplexes in Bernstein--Gelfand--Gelfand sequences [math.DG/050834]; operators between differential forms which generalise the Q-curvature. In ongoing joint work, the applicants have developed an efficent version of tractor calculus to exploit the relation between conformal and CR geometry introduced by Fefferman. Broadly the objective of the program is to apply these tools to construct and study new global invariants which are relevant for global analysis and spectral theory on manifolds.

In the differential form setting the detour complexes are based on the usual exterior derivative and ``detour operators'' which are, in an appropriate sense, higher order analogues of the Maxwell operator. The latter includes, at one extreme, the GJMS [Graham, Jenne, Mason and Sparling, J. Lond. Math. Soc. 46 (1992), 557--565] conformally invariant power of the Laplacian that acts on functions. Prelimary work shows that the cohomology of detour complexes is closely related to spectral theory and contains intersting global information. One of our main aims is to explore analogous global invariants on related structures, including quaternionic and CR structures. The latter should be particularly interesting and via the calculus for Fefferman spaces mentioned above, the conformal detour complexes will yield new CR--invariant complexes. The cohomology spaces of these complexes provide new global CR--invariants. Studying sub--ellipticity properties of the complexes and investigating their cohomolgies will be an important part of the activity.

From the case of four--dimensional conformal structures it is known that detour complexes are closely related to subcomplexes in Bernstein--Gelfand--Gelfand sequences, see [math.DG/0606401]. These subcomplexes generalize to CR structures, and are intimately related to the detour complexes arising there. This will provide a link to additional powerful tools from representation theory.

The conformal ``Q-operators'' which generalise the Q-curvature are related to this program in several ways. Firstly the detour operators arise from these as compositions with the exterior derivative and its adjoint. Next the Q-operators lead to both a notion of conformal harmonic spaces and conformally invariant maps from these into the dual de Rham cohomology space. This is one of the main points of [math.DG/0309085] and [math.DG/0511311]. These cohomology maps have an interpretation in terms of a conformally invariant global pairing. While the usual Q-curvature pairs the characteristic functions on a compact manifold, by construction this new prescription problem is tuned to higher cohomology. On conformally flat structures the integral of the usual Q-curvature recovers the Euler characteristic on components. Examples treated thus far suggest that there is topological content in the integral of the analogous semi-local quantities. We plan to explore this as well as its CR analogues further.

Relevance, importance and timeliness? We believe the work to be carried out will have a lasting impact on the geometric analysis/geometric spectral theory program for conformal and CR geometry. There are likely benefits to Physics. So certainly it is important. The timing is ideal. There have been major recent advances in the algebraic aspects of conformal geometry (construction of invariant operators/invariants etc.) so it is an ideal time to apply this to establish new directions for global analysis on manifolds. The setting at Banff provides a stimulating environment away from the distractions of our home institutions.

In the differential form setting the detour complexes are based on the usual exterior derivative and ``detour operators'' which are, in an appropriate sense, higher order analogues of the Maxwell operator. The latter includes, at one extreme, the GJMS [Graham, Jenne, Mason and Sparling, J. Lond. Math. Soc. 46 (1992), 557--565] conformally invariant power of the Laplacian that acts on functions. Prelimary work shows that the cohomology of detour complexes is closely related to spectral theory and contains intersting global information. One of our main aims is to explore analogous global invariants on related structures, including quaternionic and CR structures. The latter should be particularly interesting and via the calculus for Fefferman spaces mentioned above, the conformal detour complexes will yield new CR--invariant complexes. The cohomology spaces of these complexes provide new global CR--invariants. Studying sub--ellipticity properties of the complexes and investigating their cohomolgies will be an important part of the activity.

From the case of four--dimensional conformal structures it is known that detour complexes are closely related to subcomplexes in Bernstein--Gelfand--Gelfand sequences, see [math.DG/0606401]. These subcomplexes generalize to CR structures, and are intimately related to the detour complexes arising there. This will provide a link to additional powerful tools from representation theory.

The conformal ``Q-operators'' which generalise the Q-curvature are related to this program in several ways. Firstly the detour operators arise from these as compositions with the exterior derivative and its adjoint. Next the Q-operators lead to both a notion of conformal harmonic spaces and conformally invariant maps from these into the dual de Rham cohomology space. This is one of the main points of [math.DG/0309085] and [math.DG/0511311]. These cohomology maps have an interpretation in terms of a conformally invariant global pairing. While the usual Q-curvature pairs the characteristic functions on a compact manifold, by construction this new prescription problem is tuned to higher cohomology. On conformally flat structures the integral of the usual Q-curvature recovers the Euler characteristic on components. Examples treated thus far suggest that there is topological content in the integral of the analogous semi-local quantities. We plan to explore this as well as its CR analogues further.

Relevance, importance and timeliness? We believe the work to be carried out will have a lasting impact on the geometric analysis/geometric spectral theory program for conformal and CR geometry. There are likely benefits to Physics. So certainly it is important. The timing is ideal. There have been major recent advances in the algebraic aspects of conformal geometry (construction of invariant operators/invariants etc.) so it is an ideal time to apply this to establish new directions for global analysis on manifolds. The setting at Banff provides a stimulating environment away from the distractions of our home institutions.