Wednesday, May 17, 2006

Leonardo Rastelli: generalizing Schnabl gauge

Leonardo Rastelli (formerly from the #1 school in this list, now at YITP in Stony Brook) gave an intriguing talk at MIT about their
on string field theory. It is the first work by really famous authors (except for Yuji Okawa from MIT - sorry other guys if I missed someone) that depends on Martin Schnabl's exact solution for the tachyonic minimum in string field theory. Their main goal is to find the explicit solution in different gauges than Martin used.
The reader is expected to have read the previous article about these issues.

Leonardo started with some introduction to string field theory, surface states (the meaning will be explained below), and other concepts. He often used the level truncation calculation of various quantities which seems as 20th century physics to me. Once exact solutions can be found, the "miracles" that the level truncation works become rather unsurprising.

The fact that Martin's state solves the equations of motion implies infinitely many generalized Euler-Ramanujan identities. Barton Zwiebach showed these identities to Jeffrey Goldstone (MIT) - the father of X different bosons where X is the number of broken generators - and Goldstone said: these are recursive relations equivalent to a differential equation. So they found the right differential equation, generalized it to other values of "s", and solved it for each "s". For each "s", they have something that generalizes Martin's setup. For each "s", they have a closed subalgebra of the star-algebra of string fields.

I must tell you what is "s". Martin's construction depends on operators "Lscript0" and "Lscriptdagger0". Their commutator is
  • [Lscript0,Lscriptdagger0] = Lscript0 + Lscriptdagger0
In order to find such generators, you need to consider various conformal transformations that depend on "arctan" and the corresponding states called wedge states.

Leonardo and Barton look for more general subalgebras of the star-algebra with similar behavior. They must be closed under a generator "L0" and under the star-product. This leads them to look for all surface states whose corresponding generators satisfy
  • [Lscript0,Lscriptdagger0] = s ( Lscript0 + Lscriptdagger0 )
Note that it differs by a factor of "s" only from the previous commutator. The choice "s=2" is related to the butterfly state much like Martin's "s=1" is connected with the wedge states.

Note that these various "surface states" are bra-vectors i.e. gadgets that produce a number for each ket-vector CHI (namely the inner product of the surface state with CHI). The number is defined as the path integral of the CFT over the surface (which must topologically be a disk) with the Neumann boundary conditions at the boundary and with an insertion at a priviliged point of the vertex operator associated with your ket-vector CHI. Such surface states can be written as a pure action of a function of the full Virasoro generators on the vacuum, and the right form may be found after you find the right conformal transformation that maps the surface to the canonical shape, namely the unit half-disk.

If your surface looks like a butterfly, you obtain the butterfly state. If it looks like a squirrel, you obtain a squirrel state. If it looks like Porsche 901, you obtain the Porsche 901 surface state. If it looks like a sliver, you obtain the sliver state, and so on. You got the idea. ;-) Except for squirrels and Porsches, all examples above are connected with rigorous mathematical formulae.

When the dust is settled, they can find their solution in the new generalized gauge as the integral of the minus first power of a certain hypergeometric function of the generators, acting on the identity functional. It's only Martin's case "s=1" for which they know how to evaluate the integral, and they obtain my favorite zeta-function form of Martin's solution. The integral for "s" different from one looks messy and it is likely that you won't be able to evaluate it analytically.

Nevertheless, the coefficients of different excited states in the generalized vacuum solutions are rational for all rational "s". They must be some s-deformations of the Bernoulli numbers but Barton and Leonardo have not found any such generalizations in any tables of special functions and sequences. Although Leonardo says that the hassle is conserved and the complex form of the integral compensates the simple form of the generators "Lscript0" and "Lscriptdagger0" for "s=2" (in this case, the script generators are a simple terminated some of the Roman ones), I would say that the integral is so messy and non-explicit that Martin's "s=1" solution is still infinitely simpler.

Hong and me speculated that the solution could simplify for other cases, too - such as "s" going to infinity and "s" going to zero.

No comments:

Post a Comment