In 1983, L. Alvarez-Gaum´e 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- 1
2 and spin- 3
2 fields and real self-dual antisymmetric tensor
fields1 [1]. 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 [3]. 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 [4] 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. Using the
formulas of AGW, one can indeed show that the sum of anomalies in
N = 1 supergravity coupled to super Yang–Mills theory with gauge group
SO(32) or E8 × E8, though nonvanishing, is in the technical sense exact:
1Just as one can always shift the axial anomaly from the vector current to the axial current
by adding a suitable counterterm to the action or by using a different regularization
scheme, one can also shift the gravitational anomaly from the general coordinate
symmetry to the local Lorentz symmetry [2]. Conventionally one chooses to preserve
general coordinate invariance. AGW chose the symmetric vielbein gauge, so that the
symmetry for which they computed the anomalies was a linear combination of a
general coordinate transformation and a compensating local Lorentz transformation.
However, they used a regulator that manifestly preserved general coordinate invariance,
so that their calculation yielded the anomaly in the local Lorentz symmetry.
it can be removed by adding a local counterterm to the action. These two
papers led to an explosion of interest in string theory