What is this paper about? The gentlemen claim that
- Alvarez-Gaume and Witten were sloppy in their derivation of gravitational anomalies in 1984
- the textbook of Superstring Theory by Green, Schwarz, Witten is also sloppy
- the sloppiness is based on confusion between two types of a time-ordered product
- these two time-orderings are "T" and "T*"; the former is the literal time-ordering in the Hamiltonian/operator formalism while the latter is the type of time-ordering that one naturally obtains from the covariant path-integral approach; "T" commutes with the time-derivatives while "T*" does not
- as an example, the Japanese physicists argue that what Alvarez-Gaume and Witten considered to be a gravitational anomaly in two dimensions, is actually just the difference between the expressions containing the "T" and "T*" products
- sociological ones: Alvarez-Gaume and Witten are careful (and not just careful) physicists
- the path-integral evaluation of the anomalies is what has been used and tested for a long time, and may be skeptical about the conclusions based purely
- the Feynmanian covariant approach is closely connected with important phenomena of particle physics understood in the past decades while the alternative is closer to axiomatic quantum field theory that is known to have led to many incorrect physical conclusions (the discussion below "Sidneyfest" where someone argued, using AQFT, that the implications of the Weinberg-Witten theorem don't exist is an example of the failures of AQFT)
- sociological ones: these physicists may be careful and picky, and their paper is newer
- the operator formalism is something that many people still find more well-defined than the path integral approach
- more precisely, AQFT is still considered to be "the" rigorous approach to quantum field theory, although its tools have not been refined well to deal easily with the novel phenomena in gauge theory, renormalization, and dualities, among other things
I have personally no experience whatsoever with the Lorentz-non-covariant, operator treatment of the anomalies which is why I prefer to trust Alvarez-Gaume before I look at the details. The anomalies result from the inability to define the integration measure of the path-integral in such a way that all the classical symmetries are preserved. In order to define the measure properly, one needs to gauge-fix and introduce the Faddeev-Popov ghosts etc., and it is mostly unknown to me how their loop effects realize the same tasks in the operator formalism (without Feynman's path integrals).
On the other hand, I find it plausible that there may exist a wrong attempt to define the measure that leads one to believe that there is an anomaly, although there exists a better way in which the anomaly is absent. There are also terminological issues: I would only use the word "anomaly" for a violation of a classical symmetry that can't be fixed by adding a local counterterm. One should also be careful about the local anomalies and global anomalies - anomalies in small or large transformations, respectively.