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Classical solutions of open string field theory. Martin Schnabl Institute of Physics, Prague Academy of Sciences of the Czech Republic. ICHEP, July 22, 2010. What is String Field Theory?. Field theoretic description of all excitations of a string (open or closed) at once.

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classical solutions of open string field theory

Classical solutions of open string field theory

Martin Schnabl

Institute of Physics, Prague

Academy of Sciences of the Czech Republic

ICHEP, July 22, 2010

what is string field theory
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.
open string field theory as the fundamental theory
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 (Witten 1986, Berkovits 1995) might be more fundamental (Sen).Reminiscent of holography (Maldacena) and old 1960’s ideas of Sacharov.
first look at osft
First look at OSFT

Open string field theory uses the following data

Let all the string degrees of freedom be assembled in

Witten (1986) proposed the following action

first look at osft1
First look at OSFT

This action has a huge gauge symmetry

provided that the star product is associative, QB acts as a

graded derivation and < . > has properties of integration.

Note that there is a gauge symmetry for gauge symmetry

so one expects infinite tower of ghosts – indeed they can

be naturally incorporated by lifting the ghost number

restriction on the string field.

witten s star product
Witten’s star product
  • 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:
witten s star product as operator multiplication
Witten’s star product as operator multiplication

We have just seen that the star product obeys

And therefore states obeyThe star product and operator multiplication are thus isomorphic!

simple subsector of the star algebra
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
classical solutions
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 arbitraryM.S. 2005, Okawa, Erler 2006

classes of tachyon vacuum solutions
Classes of tachyon vacuum solutions
  • The space of all such solutions has not been completely classified yet, although we are quite close (Rastelli; Erler; M.S., Murata).
  • Let us restrict our attention to different choices of F(K) only.
  • Let us call a state geometric if F(K) is of the formwhere f(α) is a tempered distribution.
  • Restricting to “absolutely integrable” distributions one gets the notion of L0–safe geometric states.
classes of tachyon vacuum solutions1
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
classes of tachyon vacuum solutions2
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
classes of tachyon vacuum solutions3
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.
examples so far
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

on multiple brane solutions
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.
on multiple brane solutions1
On multiple-brane solutions

The basic ingredient is the correlator

Now insert the identity in the form

on multiple brane solutions2
On multiple-brane solutions

And after some human-aided computer algebra

we find very simple expression

On the second line we have a total derivative which does

not contribute. For a function of the form

we find

on multiple brane solutions3
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 obeylet us calculate the LHS:For we find !

marginal deformation solutions
Marginal deformation solutions
  • Given any exactly marginal matter operator J (with non-singular OPE with itself for simplicity) one can construct a solution (M.S.; Okawa, Kiermaier, Zwiebach; Erler)
  • Particularly nice case is for the special case of J=eX0 the solution interpolates between the perturbative and tachyon vacua.
what next
What next ?
  • We are still missing the ideas and tools for constructing nontrivial solutions describing general BCFT’s
  • Many things need to be done for the superstrings
  • Find a physical applications (early universe, particle physics)
  • Study closed strings