The Decomposition of Global Conformal Invariants (eBook)

Spyros Alexakis (Autor)

eBook Download: PDF

2012
568 Seiten
Princeton University Press (Verlag)
978-1-4008-4272-8 (ISBN)

Infos

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.

Autor

Spyros Alexakis is assistant professor of mathematics at the University of Toronto.

Produkt-Details

• Erscheint lt. Verlag: 6.5.2012
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: absolute value • Abuse of notation • Addition • Algebraic expression • algebraic propositions • Algebraic Structure • ambient metrics • Asymptote • asymptotic expansion • Atiyah–Singer index theorem • Calculation • Cartan connection • Cauchy–Riemann equations • CauchyВiemann geometry • Charles Fefferman • Clay Mathematics Institute • coefficient • cohomology • combination • Computation • conformal anomalies • Conformal geometry • conformal invariant • conformal invariants • conformally invariant functionals • conjecture • Contact Geometry • Converse (semantics) • corollary • covariant derivative • CR manifold • Curvature • Curvature tensor • Decomposition • Derivative • DeserГchwimmer conjecture • Differential Geometry • Differential operator • Dirac Operator • Einstein Manifold • Equation • existential quantification • fiber bundle • Functional Determinant • fundamental group • fundamental theorem • geometric analysis • Geometry • global conformal invariant • global invariants • grand conclusion • graph theory • Heat kernel • Hermann Weyl • Index Theory • Induction • integral equation • Integration by parts • Invariant theory • Iteration • iterative decomposition • Khler geometry • Lemma • lemmas • Levi-Civita connection • Lie group • Linear combination • Linearization • manifold • Mathematical Induction • Mathematical Proof • Mathematics • Metric tensor (general relativity) • Natural transformation • Notation • Order of magnitude • orthogonal group • Parameter • partially ordered set • Permutation • Pfaffian • polynomial • principal bundle • Quantity • quantum field theory • remainder • renormalization • Requirement • Ricci curvature • Riemannian Geometry • Riemannian invariants • Riemannian manifold • Riemannian metrics • Riemannian scalar • Riemann surface • scalar curvature • Scalar (physics) • Schouten tensor • scientific notation • Simultaneous Equations • small number • Smoothness • Special case • submanifold • Subset • Summation • Symmetric tensor • symmetrization • Tensor • tensor field • tensor product • Theorem • theoretical physics • theory • Topology • Trace (linear algebra) • Variable (mathematics) • vector bundle • Vector field • Weak ordering • Weyl tensor • Without loss of generality
• ISBN-10: 1-4008-4272-7 / 1400842727
• ISBN-13: 978-1-4008-4272-8 / 9781400842728