Discussion:
436 = 436?
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Holger Bomm
2005-09-25 20:43:29 UTC
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Hi!

http://www.pbs.org/wgbh/nova/elegant/program.html says that one fun-
damental success in the development of string theory was a calculation
that Green and Schwarz made in 1973 or so. It is said that they had a
(now famous) equation that could only be correct if both sides spid out
"436" (and it actually did :-)).

Does anyone know anything about this? I found it funny that the com-
plex problems in the evolution of string theory obviously resulted in
an equation like 436 = 436. What was that??

Thanks in advance for any answers!

Best regards
Holger
Lubos Motl
2005-09-25 20:55:56 UTC
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that Green and Schwarz made in 1973 or so. It is said that they had a ...
You probably mean summer 1984.
Does anyone know anything about this? I found it funny that the com-
plex problems in the evolution of string theory obviously resulted in
an equation like 436 = 436. What was that??
You probably mean 496 = 496. It was a classical story from the textbooks
of theoretical physics. In summer 1984, Green and Schwarz were in Aspen,
Colorado. And they were calculating some anomalies in string theory.

Anomaly is a sick quantum effect that destroys a symmetry that is
apparently present in the classical (non-quantum) theory. And such an
anomaly makes the theory meaningless.

It was widely believed that all versions of string theory with non-Abelian
gauge symmetries (in fact, all Yang-Mills theories coupled to supergravity
in ten dimensions) - and gauge symmetries are something that is crucial
for any theory that wants to reproduce reality - must have anomalies.

Green and Schwarz kind of knew that it could not be the case of string
theory because anomalies require some UV divergent (infinite) Feynman
diagrams (one-loop diagrams) and they knew that all one-loop diagrams in
string theory, even in type I superstring theory, were finite if the gauge
group were SO(32). No room for anomalies.

But they needed to explain why the anomalies cancel in terms of low-energy
effective field theory. This proof involves a new unusual transformation
law for some fields under some gauge symmetries - the Green-Schwarz
mechanism; it also requires some trace identities from group theory to
show why the purely gauge anomalies (coming from a hexagon-shaped Feynman
diagram in 10 dimensions) factorize for the SO(32) gauge group and can be
cancelled by the Green-Schwarz mechanism.

After the pure gauge anomalies and mixed anomalies, the last thing that
they needed to check was the cancellation of purely gravitational
anomalies. The gravitino contributes something and it turns out that the
contribution of the gauginos is proportional to the dimension of the
adjoint representation of the gauge group: each basis vector gives a
certain amount. Cancellation requires this dimension to be 496.

The last step they needed to calculate was the dimension of SO(32) because
everything else had been checked. They were in the middle of a
thunderstorm. "We must be getting pretty close because God is attempting
to prevent us from finding the truth," Green remarked. At any rate, they
needed to calculate 32 x 31 / 2 = 31 x 16. Green of them wrote it on the
blackboard and obtained 486. "It does not work," he said in complete
despair.

"Try it again," Schwarz encouraged Green. When he tried it for the second
time, he obtained 496, indeed. String theory has passed this test of
elementary mathematics.

In 1985, another and more realistic gauge group - E_8 x E_8 - was found in
heterotic string theory. Its dimension is 248 + 248. Guess how much it is.
Heterotic string theory was discovered by the Princeton string quartet -
David Gross, Ryan Rohm, Emil Martinec, and Jeff Harvey.

Within weeks or months, the number of researchers in string theory jumped
from a few to about one thousand.
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Holger Bomm
2005-09-25 23:57:30 UTC
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Hi!

Thank you very much for that detailled comment, it was really inte-
resting! I am sorry for recalling 1984 and 496 in a wrong way.

Best
Holger

[Moderator's note: Incidentally, you may have noticed that 1984 = 4 x 496,
and the geometric average 2 x 496 = 992 = 1984 / 2 is the number of
exotic 11-spheres. No explanation offered. ;-) Best, LM]
Holger Bomm
2005-09-26 14:34:38 UTC
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Post by Holger Bomm
[Moderator's note: Incidentally, you may have noticed that 1984 =
4 x 496, and the geometric average 2 x 496 = 992 = 1984 / 2 is
the number of exotic 11-spheres. No explanation offered. ;-)
Best, LM]
Yeah, I remember! That was what I mixed up! :-P

Best regards
Holger

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