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The Decomposition of Global Conformal Invariants
Contributor(s): Alexakis, Spyros (Author)

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 ISBN: 0691153477     ISBN-13: 9780691153476
Publisher: Princeton University Press
OUR PRICE: $214.70  

Binding Type: Hardcover - See All Available Formats & Editions
Published: May 2012
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Additional Information
BISAC Categories:
- Mathematics | Geometry - Differential
Dewey: 518
LCCN: 2011037622
Series: Annals of Mathematics Studies (Hardcover)
Physical Information: 1.1" H x 6.2" W x 9.3" L (1.70 lbs) 568 pages
Features: Bibliography, Illustrated, Index
 
Descriptions, Reviews, Etc.
Publisher Description:
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given
manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian
scalar? Deser and Schwimmer asserted that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern-Gauss-Bonnet integrand. This book
provides a proof of this conjecture.The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The
methods introduced here make an interesting connection between algebraic properties of local invariants--such as the classical Riemannian invariants and the more recently studied conformal invariants--and the study of global invariants, in this case conformally invariant integrals. Key tools used to
establish this connection include the Fefferman-Graham ambient metric and the author's super divergence formula.
 
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