Chaudhuri on the Hagedorn "myth"
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Lubos Motl
2005-06-20 00:46:00 UTC
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This short note is closely related to a previous article about the work of
Dienes and Lennek. Tonight, it is Shyamoli Chaudhuri who is "dispelling
the Hagedorn myth" (incidentally, it was already in 1965 when Hagedorn
suggested that at high enough temperatures, open strings merge into a gas
of chaotic long closed strings):


She calculates the thermal free energy - apparently in a different way
than we are used to (from Atick and Witten and related works) - to
conclude that the exponential growth of the states with the energy does
not exist. In section 2.1 she argues that the growth of the number of
states with the level does not imply the same growth of free energy as a
function of temperature (or the density of states with the total energy).
The true growth is slower, she says, making the full expression
convergent. Nevertheless, she finds a first order phase transition at the
T-self-dual temperature.

Her basic argument is the same as in the Dienes and Lennek's paper: the
correct one-loop torus path integral only goes over the fundamental region
of the modular group which removes the dangerous region with small
"Im(tau)" and makes, according to her beliefs, the integral convergent for
any temperature.

I encourage everyone for whose research and thinking the Hagedorn behavior
is important to decide about the fate of the transition without any
prejudices. After checking various things, I personally believe that the
Hagedorn "folklore" will survive and both of the recent anti-Hagedorn
papers are misled. (Chaudhuri is more radical because she seems to believe
that the transition would be absent even in type 0 and other strings.)

The integral over the fundamental region combined with the summation over
the two winding numbers that count how both circles of the worldsheet
torus wind around the thermal circle in spacetime may be replaced by a
full integral over the upper "tau" half-plane, which re-introduces the
dangerous region with small "Im(tau)" and revives the "Hagedorn myth".

Technically, I think that her error is the step from (15) to (16) in her
paper where she uses the Hardy-Ramanujan formula, assuming that the
excitation of the string is very large, which removes by hand the actual
divergence that would, in this calculational procedure, emerge from the
thermal tachyon (the ground state of the winding sector "w=1" around the
thermal circle in spacetime - in this sector the GSO projection is
reversed) - a contribution that she neglects because the Hardy-Ramanujan
formula is definitely not applicable for low-lying states such as this
thermal tachyon.

Note that once you admit that the relevant CFT has a thermal tachyon, the
discussion simply ends. With a thermal tachyon, the Hagedorn divergence
arises from the region with large values of "Im(tau)", not small ones. And
this "infrared" region is definitely not removed in string theory. To
summarize, I now believe that if one defines the thermal stringy
amplitudes in the most obvious stringy extension of the thermal
path-integral rules of QFT, one finds the thermally wound tachyon whose
mass determines the Hagedorn temperature, and the new critical papers

Feel free to disagree.
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2005-06-20 18:17:55 UTC
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I believe that the error is in (25), which is the first equation that
differs from the standard understanding, by the unmotivated introduction
of the phase factor e^{i \pi w}. This manifestly destroys modular
invariance, as it breaks the symmetry between w and the Poisson
resummation variable m.

Similar equations appeared in several earlier and now-withdrawn preprints
by this author, in each case with the assertion that it was the unique
modular-invariant partition function. In fact, the unique
modular-invariant partition function is the usual one, with Hagedorn

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