Weyl, Pontryagin, Euler, Eguchi and Freund

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