Weyl, Pontryagin, Euler, Eguchi and Freund

Abstract

In a September 1976 PRL Eguchi and Freund considered two topological invariants: the Pontryagin number $P \sim \int d^4x \sqrt{g}R^* R$ and the Euler number $\chi \sim \int d^4x \sqrt{g}R^* R^*$ 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 $\chi$. 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: \[ g^{\mu\nu}\langle T_{\mu\nu}\rangle=\frac{1}{(4\pi)^2}(cF-aG)\] where $F$ is the square of the Weyl tensor and $\int d^4x\sqrt{g} G/(4\pi)^2$ is the Euler number. For free CFTs with $N_s$massless fields of spin $s$ \[ 720c=6N_0 + 18N_{1/2} + 72 N_1~~~~ 720a=2N_0 + 11N_{1/2} + 124N_1 \]