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### Weyl, Pontryagin, Euler, Eguchi and Freund

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2003.02688v2.pdf | Accepted version | 148.44 kB | Adobe PDF | View/Open |

Title: | Weyl, Pontryagin, Euler, Eguchi and Freund |

Authors: | Duff, MJ |

Item 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 $720c=6{N}_{0}+18{N}_{1/2}+72{N}_{1}720a=2{N}_{0}+11{N}_{1/2}+124{N}_{1}$. |

Issue Date: | 31-Jul-2020 |

Date of Acceptance: | 1-Jul-2020 |

URI: | http://hdl.handle.net/10044/1/81879 |

DOI: | 10.1088/1751-8121/ab956d |

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 |

Keywords: | 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 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 hep-th hep-th gr-qc 01 Mathematical Sciences 02 Physical Sciences Mathematical Physics |

Publication Status: | Published |

Article Number: | ARTN 301001 |

Online Publication Date: | 2020-07-06 |

Appears in Collections: | Physics Faculty of Natural Sciences |