Wednesday, November 30, 2005

Closed string vacuum solved analytically

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

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 - the second insight in science after the Higgs mechanism that shows that the tachyons are more than just an inconsistency - 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 unified inside 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. Instead of the Siegel gauge that requires the string field to be annihilated by "b0", Martin demands it to be annihilated by the zero mode of the "b"-antighost calculated in a different coordinate system. This smart trick 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 interesting 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". While it remains likely that string field theory will remain an interesting formalism for perturbative string theory only - and it won't answer the physical questions we would like to see answered - it seems that string field theory got a new boost that makes it more mathematically interesting than we used to think.

What does Martin's solution look like?

Because it is not so easy to find what the solution actually is in the 60-page-long paper, let me tell you. First, define the wedge states |r> which satisfy the relation that
  • |r>*|s> = |r+s-1>
so they are some kind of star-product powers of the first one. In this notation, |1> is the identity under the star-product. Confusingly enough, |2> is our symbol for the SL(2,R) invariant vacuum. Expand the wedge states for large values of "r" using a Taylor expansion in "r-2". You will get some new states ||n>>. Martin's solution is then simply
  • |psi> = sum (n=0,infty) B_n ||n>>
where "B_n" are the Bernoulli numbers. The proof that this state satisfies the equations of motion is a 21st century edition of the Euler-Ramanujan identity.

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