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The Decomposition of Global Conformal Invariants (Am182) (Annals of Mathematics Studies)by Spyros Alexakis
Synopses & ReviewsPublisher Comments: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 invariantssuch as the classical Riemannian invariants and the more recently studied conformal invariantsand 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. Synopsis: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 invariantssuch as the classical Riemannian invariants and the more recently studied conformal invariantsand 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. Synopsis: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 invariantssuch as the classical Riemannian invariants and the more recently studied conformal invariantsand 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. About the AuthorSpyros Alexakis is assistant professor of mathematics at the University of Toronto.
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