论文标题
Weyl，Pontryagin，Euler，Eguchi和Freund
Weyl, Pontryagin, Euler, Eguchi and Freund
论文作者
论文摘要
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 $χ\sim \int d^4x \sqrt{g}R^* R^*$ and posed the question: to what anomalies do they contribute?他们发现，$ p $出现在轴向费米数电流的综合差异中，因此提供了对木村在1969年发现的异常的新拓扑解释，1972年的德尔堡和萨拉姆在1972年发现。但是，他们没有发现$χ$的相似作用。这激起了我的兴趣，并在1976年4月的纸上用Deser和Isham在重力Weyl异常上吸引了我的兴趣，我能够证明，对于共形场理论，应力张量的痕迹仅取决于两个常数：\ [g^{μNν} \ langle t_ {μν} \ rangle = \ frac {1} {（4π）^2}（cf-ag）\]，其中$ f $是weyl张量的平方，$ \ int d^4x \ sqrt {g} g/（4π）^2 $是Euler编号。免费的CFT，$ n_s $ spin $ s $ \ [720c = 6n_0 + 18n_ {1/2} + 72 N_1 ~~~~~ 720a = 2n_0 + 11n_ {1/2} + 124N_1 \]
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 $χ\sim \int d^4x \sqrt{g}R^* R^*$ 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 $χ$. 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: \[ g^{μν}\langle T_{μν}\rangle=\frac{1}{(4π)^2}(cF-aG)\] where $F$ is the square of the Weyl tensor and $\int d^4x\sqrt{g} G/(4π)^2$ is the Euler number. For free CFTs with $N_s$massless fields of spin $s$ \[ 720c=6N_0 + 18N_{1/2} + 72 N_1~~~~ 720a=2N_0 + 11N_{1/2} + 124N_1 \]
