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Closed string vacuum solved analytically
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Lubos Motl
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Joined: 02 May 2005
Posts: 38

PostPosted: Wed Nov 30, 2005 6:51 pm    Post subject: Closed string vacuum solved analytically Reply with quote


The most interesting paper on the arXiv today is a paper by

* Martin Schnabl

who is currently at CERN, Switzerland. Around 1998, Ashoke Sen conjectured
that the open string tachyon may get a vev that corresponds to a complete
annihilation of the D-brane onto which the open strings were attached.
This predicts the energy density of this minimum of the tachyonic
potential: it must be equal to minus the tension of the D-brane that has

In the framework of boundary string field theory (BSFT), this fact has
been proved by Kutasov, Marino, and Moore. Of course, there have always
been almost complete physical arguments that assured us that no reasonable
person had any serious doubts that Sen's conjecture was correct.

The formalism of Witten's cubic string field theory of the Chern-Simons
type is however much more well-defined than boundary string field theory.
People wanted to verify Sen's conjecture in this cubic string field
theory, too. They could have done so numerically and they obtained
99.9999% of the right value. Many other facts have been checked
numerically, too. Many physicists also proposed various formal heuristic
solutions and maps between the cubic string field theory and the boundary
string field theory but it was usually hard to give these formulae a
precise meaning.

An exact and rigorous proof was however missing, much like a well-defined
formula for the vev's of all those scalar fields incorporated in the
string field.

At the very same time, many people studied wedge states, sliver states,
and similar states that have a natural physical interpretation but that
can also be expanded as states in the Fock space in a controllable
fashion. Martin Schnabl became one of the five people in the world who
were most familiar with these mathematical objects.

Martin's work is a breakthrough in our understanding of cubic string field
theory because it gives a complete, well-defined, and concrete solution
for the single most important nontrivial classical configuration of the
cubic open string field theory: the so-called closed string vacuum. Much
like other people before him, Martin starts with identifying a convenient
gauge-fixing of the huge open string gauge group. However, he finds a much
more friendly and natural way to gauge-fix the symmetry. This allows him
to get rid of the ghosts in a very cultural manner.

In a level-truncation scheme, he was able to figure out the numerical
values of various coefficients needed for his state. It had to be
exciting. For example, if he ever obtained -691/2730, he had to know: wow,
this is the Bernoulli number B_{12} and we should look at this important
sequence from mathematics more carefully. Bernoulli's numbers and the
Euler-Maclaurin formula (an identity that represents the difference
between a sum and an integral as an expansion involving the Bernoulli
numbers) play a paramount role in Martin's proof - and he derives many
fascinating related identities.

He also analytically proves Sen's conjecture about the energy density of
the closed string vacuum.

Here at Harvard, we widely expect that Martin's paper will revive the
interest in cubic string field theory - at least a little bit. There
should exist some extra solutions analogous to Martin's solution for the
"closed string vacuum".
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