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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