Q-Curvature, Spectral Invariants, and Representation Theory

We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on even-dimensional conformal manifolds. The exposition is self-contained, in the sense of giving references sufficient to allow the reader to work through all details.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Translation to Bundle Operators

We give explicit formulas for conformally invariant operators with leading term an $m$-th power of Laplacian on the product of spheres with the natural pseudo-Riemannian product metric for all $m$.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Heat kernel asymptotics with mixed boundary conditions

We calculate the coefficient $a_5$ of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold.Comment: 26 pages, LaTe

Group representations arising from Lorentz conformal geometry

AbstractIt is shown that there exist conformally covariant differential operators D2l,k of all even orders 2l, on differential forms of all orders k, in the double cover n of the n-dimensional compactified Minkowski space n. These act as intertwining differential operators for natural representations of O(2, n), the conformal group of n. For even n, the resulting decompositions of differential form representations of O↑(2, n), the orthochronous conformal group, produce infinite families of unitary representations, the most interesting of which are carried by “positive mass-squared, positive frequency” quotients for 2l ⩾ ¦n − 2k¦. Physically, these generalize unitary representations of the conformal group associated with the modified wave operator D2,0 = □ + ((n − 2)2)2, and the Maxwell operator on vector potentials D2,(n − 2)2 = δd. All the representation spaces produced, unitary and nonunitary, may be viewed as infinite systems of harmonic oscillators. As a by-product of the spectral resolution of the D2l,k, one gets some striking wave propagative properties for all of the equations D2l,k Φ = 0, including Huygens' principle in the curved spacetime n. The operators D2l,k have not been seen before except in the special cases k = 0 or n, and k = (n ± 2)2, l = 1 (the Maxwell operator). Thus much new information is obtained even in the physical case n = 4

Interactions of a String Inspired Graviton Field

We continue to explore the possibility that the graviton in two dimensions is related to a quadratic differential that appears in the anomalous contribution of the gravitational effective action for chiral fermions. A higher dimensional analogue of this field might exist as well. We improve the defining action for this diffeomorphism tensor field and establish a principle for how it interacts with other fields and with point particles in any dimension. All interactions are related to the action of the diffeomorphism group. We discuss possible interpretations of this field.Comment: 12 pages, more readable, references adde

Prolongations of Geometric Overdetermined Systems

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.Comment: 22 pages. In the second version, a comparison with the classical theory of prolongations was added. In this third version more details were added concerning our construction and especially the use of Kostant's computation of Lie algebra cohomolog

Einstein metrics in projective geometry

It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The general theory of such equations singles out a subclass of so-called normal solutions. We prove that non-degerate normal solutions are equivalent to pseudo-Riemannian Einstein metrics in the projective class and observe that this connects to natural projective extensions of the Einstein condition.Comment: 10 pages. Adapted to published version. In addition corrected a minor sign erro