This chapter discusses chiral anomalies associated with the gravitational field, namely, gravitational anomalies. It discusses gravitational generalization of chiral U(1) anomalies for ordinary gauge fields. A method for evaluating the Dirac genus and Chern characters in general 2n-dimensional Euclidean space-time is given. It is explained that there appear to be quantum anomalies in general coordinate and local Lorentz transformations for chiral Dirac fields in (4n+2)-dimensional curved space-time. If these anomalies are not cancelled, the basic principles of general relativity and quantum theory become incompatible with each other.
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