Hexagon Anomaly
HEXAGON ANOMALY
[111] "Explicit Evaluation of Anomalies in Higher Dimensions"
by P.H. Frampton and T.W. Kephart.
Phys. Rev. Lett. 50 1343 (1983).
[113] "Analysis of Anomalies in Higher Dimensions"
Phys. Rev. D28, 1010 (1983).
After I first came to Chapel Hill as Assistant Professor January 1, 1981,
the first postdoc hired was Tom Kephart (now full professor at Vanderbilt
University) in fall of 1981. The two of us investigated the generalization of
the chiral triangle anomaly in spacetime dimension d = 4 to arbitrary even
dimensionality d = 2n. Odd spacetime dimensions permit no chirality. A special
target was the hexagon anomaly in d = 10 because of the Type I superstring
with gauge group G = O(N) which has d = 10 as its critical dimension.
At first sight, it appeared that d>10 Yang-Mills theory was meaningless because
of non-renormalizability at short distance. Nevertheless, we pursued the
calculation which was technically complicated by traces of products
of long strings of Dirac matrices. Kephart and I worked it out in [111]
and discovered that the anomaly in d = 10 pure gauge theory is calculable,
finite and unique. For N=32 the connected part vanishes[113].
There was one snag: the disconnected part of the hexagon anomaly in pure
d=10 gauge theory is non-zero and positive definite.
For O(32), however, gravity miraculously contributes with a minus sign
to cancel: Green and Schwarz, Phys. Lett. B149, 117 (1984).
Papers [111,113] expedited the "First Superstring Revolution".
Open questions about hexagon anomaly:
None because it is understood.
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