Weyl, Pontryagin, Euler, Eguchi and Freund
File(s)2003.02688v2.pdf (148.44 KB)
Accepted version
Author(s)
Duff, MJ
Type
Journal Article
Abstract
In a September 1976 PRL Eguchi and Freund considered two topological invariants: the Pontryagin number $P\sim \int {\mathrm{d}}^{4}x\sqrt{g}{R}^{{\ast}}R$ and the Euler number $\chi \sim \int {\mathrm{d}}^{4}x\sqrt{g}{R}^{{\ast}}{R}^{{\ast}}$ and posed the question: to what anomalies do they contribute? They found that P appears in the integrated divergence of the axial fermion number current, thus providing a novel topological interpretation of the anomaly found by Kimura in 1969 and Delbourgo and Salam in 1972. However, they found no analogous role for χ. This provoked my interest and, drawing on my April 1976 paper with Deser and Isham on gravitational Weyl anomalies, I was able to show that for conformal field theories the trace of the stress tensor depends on just two constants: ${g}^{\mu \nu }\langle {T}_{\mu \nu }\rangle =\frac{1}{{\left(4\pi \right)}^{2}}\left(cF-aG\right)$ where F is the square of the Weyl tensor and $\int {\mathrm{d}}^{4}x\sqrt{g}G/{\left(4\pi \right)}^{2}$ is the Euler number. For free CFTs with N s massless fields of spin s $720c=6{N}_{0}+18{N}_{1/2}+72{N}_{1}720a=2{N}_{0}+11{N}_{1/2}+124{N}_{1}$.
Date Issued
2020-07-31
Date Acceptance
2020-07-01
Citation
Journal of Physics A: Mathematical and Theoretical, 2020, 53 (30)
ISSN
1751-8113
Publisher
IOP Publishing
Journal / Book Title
Journal of Physics A: Mathematical and Theoretical
Volume
53
Issue
30
Copyright Statement
© 2020 IOP Publishing Ltd. This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at https://iopscience.iop.org/article/10.1088/1751-8121/ab956d
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000553710600001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
Science & Technology
Physical Sciences
Physics, Multidisciplinary
Physics, Mathematical
Physics
Weyl
Euler
anomaly
CONFORMAL ANOMALIES
QUANTUM CORRECTIONS
TRACE ANOMALIES
STRESS-TENSOR
C-THEOREM
SUPERGRAVITY
GEOMETRY
GRAVITY
ENTROPY
SCALAR
Publication Status
Published
Article Number
ARTN 301001
Date Publish Online
2020-07-06