Weyl, Pontryagin, Euler, Eguchi and Freund

In a September 1976 PRL Eguchi and Freund considered two topological invariants: the Pontryagin number P similar to integral d(4)x root gR*R and the Euler number chi similar to integral d(4)x root gR*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 byKimura in 1969 andDelbourgo 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 > = 1/(4 pi)(2) (cF - aG) where F is the square of the Weyl tensor and integral d(4)x root gG/(4 pi)(2) is the Euler number. For free CFTs with N-s massless fields of spin s 720c = 6N(0) + 18N(1/2) + 72N(1)720a = 2N(0) + 11N(1/2) + 124N(1).