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Published online by Cambridge University Press: 28 October 2009
In 1983, L. Alvarez-Gaumé and E. Witten (AGW) wrote a fundamental article in which they calculated the one-loop gravitational anomalies (anomalies in the local Lorentz symmetry of (4k + 2)-dimensional Minkowskian quantum field theories coupled to external gravity) of complex chiral spin-½ and spin- fields and real self-dual antisymmetric tensor fields. They used two methods: a straightforward Feynman graph calculation in 4k + 2 dimensions with Pauli-Villars regularization, and a quantum mechanical (QM) path integral method in which corresponding nonlinear sigma models appeared. The former has been discussed in detail in an earlier book. The latter method is the subject of this book. AGW applied their formulas to N =2B supergravity in 10 dimensions, which contains precisely one field of each kind, and found that the sum of the gravitational anomalies cancels. Soon afterwards, M. B. Green and J. H. Schwarz calculated the gravitational anomalies in one-loop string amplitudes, and concluded that these anomalies cancel in string theory, and therefore should also cancel in N = 1 supergravity in 10 dimensions with suitable gauge groups for the N = 1 matter couplings.
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