Classical solutions of open string field theory

1. Classical solutions of open string field theory Martin Schnabl Institute of Physics, Prague Academy of Sciences of the Czech Republic ICHEP, July 22, 2010

2. 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.

3. 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.

4. 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

5. 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.

6. 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:

7. 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!

8. 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

9. 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

10. 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.

11. 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

12. 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

13. 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.

14. 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

15. 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.

16. On multiple-brane solutions The basic ingredient is the correlator Now insert the identity in the form

17. 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

18. 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 !

19. 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.

20. 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