The decomposition of global conformal invariants
 Author/Creator
 Alexakis, Spyros, 1978
 Language
 English.
 Imprint
 Princeton [N.J.] : Princeton University Press, 2012.
 Physical description
 vi, 448 p. : ill ; 25 cm.
 Series
 Annals of mathematics studies ; no. 182.
Access
Available online
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QA1 .A665 NO.182

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QA1 .A665 NO.182
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Contents/Summary
 Bibliography
 Includes bibliographical references and indexes.
 Publisher's Summary
 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 rescalings 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 ChernGaussBonnet 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 FeffermanGraham ambient metric and the author's super divergence formula.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2012
 Responsibility
 Spyros Alexakis.
 Series
 Annals of mathematics studies ; no. 182
 ISBN
 9780691153476 (hardcover)
 0691153477 (hardcover)
 9780691153483 (pbk.)
 0691153485 (pbk.)