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ON MULTIPLE-BRANE SOLUTIONS IN OSFT
Martin SchnablInstitute of Physics, Prague
Academy of Sciences of the Czech Republic
SFT 2010, Kyoto, October 18, 2010
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String Field Theory 2011
Prague, September 20-24, 2011organized by
Institute of Physics of the Academy of Sciences of the Czech Republic
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What is String Field Theory? Field theoretic description of all
excitations of a string (open or closed) at once.
Useful especially for physics of backgrounds: tachyon condensation or instanton physics
Single Lagrangian field theory which should around its various critical points describe physics of diverse D-brane backgrounds, possibly also gravitational backgrounds.
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Open String Field Theory as the fundamental theory ?
Closed string field theory (Zwiebach 1990) ― presumably the fundamental theory of gravity ― is given by a technically ingenious construction, but it is hard to work with, and it also may be viewed as too perturbative and `effective’.
Open string field theory (see Kugo’s talk) might be more fundamental (Sen). Related to holography (Maldacena) and old 1960’s ideas of Sacharov.
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First look at OSFTOpen string field theory uses the following data
Let all the string degrees of freedom be assembled in
Witten (1986) proposed the following action
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First look at OSFTThis action has a huge gauge symmetry
provided that the star product is associative, QB acts as agraded derivation and < . > has properties of integration.
Note that there is a gauge symmetry for gauge symmetryso one expects infinite tower of ghosts – indeed they canbe naturally incorporated by lifting the ghost number restriction on the string field.
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Witten’s star productDefined by gluing three strings:
It used to be a very complicated definition…
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The elements of string field star algebra are states in the BCFT, they can be identified with a piece of a worldsheet.
By performing the path integral on the glued surface in two steps, one sees that in fact:
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Witten’s star product as operator multiplication
We have just seen that the star product obeys
And therefore states obey
The star product and operator multiplication are thus isomorphic!
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In case you wonder what e-K is
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Simple subsector of the star algebra
The star algebra is formed by vertex operators and the operator K. The simplest subalgebra relevant for tachyon condensation is therefore spanned by K and c. Let us be more generous and add an operator B such that QB=K.
The building elements thus obey
The derivative Q acts as
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Classical solutions This new understanding lets us construct solutions to
OSFT equations of motion easily.
More general solutions are of the form
Here F=F(K) is arbitrary Okawa 2006 (generalizing M.S. 2005)
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Classes of tachyon vacuum solutions
The space of all such solutions has not been completely classified yet, although we are getting closer (Rastelli; Erler; M.S.).
Let us restrict our attention to different choices of F(K) only.
Let us call a state geometric if F(K) is of the form
where f(α) is a tempered distribution. Restricting to “absolutely integrable” distributions
one gets the notion of L0–safe geometric states.
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Classes of tachyon vacuum solutions
Therefore F(K) must be holomorphic for Re(K)>0 and bounded by a polynomial there. Had we further demanded boundedness by a constant we would get the so called Hardy space H∞ which is a Banach algebra.
Since formally
and
the state is trivial if is well defined
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Classes of tachyon vacuum solutions
There is another useful criterion. One can look at the cohomology of the theory around a given solution. It is given by an operator
The cohomology is formally trivialized by an operator which obeys
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Classes of tachyon vacuum solutions
Therefore in this class of solutions, the trivial ones are those for which F2(0) ≠ 1.
Tachyon vacuum solutions are those for which F2(0) = 1 but the zero of 1-F2 is first order
When the order of zero of 1-F2 at K=0 is of higher order the solution is not quite well defined, but it has been conjectured (Ellwood, M.S.) to correspond to multi-brane solutions.
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Examples so far
… trivial solution
…. ‘tachyon vacuum’ only c and K turned on
…. M.S. ‘05
… Erler, M.S. ’09 – the simplest solution so far
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On multiple-brane solutions
Can one compute an energy of the general solution as a function of F?
The technology has been here for quite some time, but the task seemed daunting. Recently we addressed the problem with M. Murata.
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On multiple-brane solutions
We need to evaluate a general correlator of the form
To achieve this we use
and insert the identity in the form
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On multiple-brane solutions
At the end all correlatorsare given be double integral over z and s.
To compute the energy is rather straightforwad (although requires some human-aided computer algebra) The final expression is simply
The second term is a total derivative and does not contribute provided G does not blow up at infinity.
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On multiple-brane solutions
So the energy simplifies to
The integral can be reduced to a sum of residues, branch cut contributions and possibly a large z surface term by deforming the contour to the right. Note that G(z) has to be holomorphic for Re z>0.
For a function of the form we find
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On multiple-brane solutions
To get further support let us look at the Ellwood invariant (or Hashimoto-Itzhaki-Gaiotto-Rastelli-Sen-Zwiebach invariant) . Ellwood proposed that for general OSFT solutions it would obey
let us calculate the LHS:
For we find !
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Comments on level expansion Is it really sensible to have inverse powers of K ?
To test that we have computed several lowest level coefficients of
The coefficient of c1|0> diverges for negative n≤-4. This is good, as we do not like ‘ghost branes’ as solutions of OSFT. Cases n=-2 and n=-3 should be ruled out separately, n=-1 is allowed (tachyon vacuum).
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Comments on level expansion To study other coefficients we can easily
compute the overlap
where , and V is a matter field of scaling dimension hV. The ghosts contribute 1/t2, so the integral is perfectly convergent. For the integral however diverges for n≥3 asso that ghost coefficients starting with c-2 |0> are divergent.
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Comments on level expansion The most promising candidate for the n+1 brane
solution is however
The reason is that the energy computation is free of surface terms, but perhaps more importantly 1/(1-F2) is holomorphic in Re(z)>0. (This is not true for F~z-n/2.)
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What next ? Find the ‘best’ choice of F Understand in detail level expansion. Is there
some F for which the level expansion is fully regular for all n>0? Is it possible to find regular twist symmetric solutions?
Work out the cohomology around these solutions. (We have seen some hints that we indeed get (n+1)2 copies of the original cohomology.)