*Spyros Alexakis*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.001.0001
- Subject:
- Mathematics, Geometry / Topology

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 ...
More

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? This book asserts 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. The 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.Less

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? This book asserts 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. The 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.

*M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko*

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199238743
- eISBN:
- 9780191716461
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199238743.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter presents a brief review of the Casimir effect in spaces with nontrivial topology (both flat and curved). As an important application of the numerous results obtained in this field, the ...
More

This chapter presents a brief review of the Casimir effect in spaces with nontrivial topology (both flat and curved). As an important application of the numerous results obtained in this field, the vacuum energy-momentum tensor due to the Casimir effect in the closed Friedmann model is considered. A related subject is the role of the Casimir effect in multidimensional Kaluza–Klein theories, where it provides one of the mechanisms for compactification of extra spatial dimensions. This is also reflected in the chapter, which is concluded with a brief discussion of the Casimir effect for topological defects, such as cosmic strings and domain walls. This problem is of interest for cosmology because some grand unification theories predict the formation of such defects in the early Universe.Less

This chapter presents a brief review of the Casimir effect in spaces with nontrivial topology (both flat and curved). As an important application of the numerous results obtained in this field, the vacuum energy-momentum tensor due to the Casimir effect in the closed Friedmann model is considered. A related subject is the role of the Casimir effect in multidimensional Kaluza–Klein theories, where it provides one of the mechanisms for compactification of extra spatial dimensions. This is also reflected in the chapter, which is concluded with a brief discussion of the Casimir effect for topological defects, such as cosmic strings and domain walls. This problem is of interest for cosmology because some grand unification theories predict the formation of such defects in the early Universe.