1. Analytic Solutions in Open String Field Theory
Puri, 2006
Martin Schnabl (IAS)
Indian Strings Meeting, Puri 2006

2. Open Bosonic String Field Theory has had a long history
1986 – SFT formulated
Witten, Gross & Jevicky, Ohta,
LeClair et al., Kostelecký & Samuel
…..….
1999 – SFT applied to tachyon condensation
Sen, Zwiebach, Taylor, Rastelli, Hata,
………
With little or no activity in the mean time. Hopefully now we are
entering new period in which SFT becomes a valuable tool
and many new exciting things can be studied analytically.

3. Progress of the past 12 months in OSFT
M.S hep-th/ Tachyon vacuum constructed ,
Sen’s first conjecture proved
Okawa hep-th/ many details elaborated
pure-gauge like form,
Fuchs & Kroyter hep-th/ cubic term better understood
Rastelli & Zwiebach hep-th/ new solutions of SFT-like equations
[See also very recent paper with Okawa and paper by Erler]
Ellwood, M.S hep-th/ Sen’s third conjecture proved
Fuji, Nakayama, Suzuki hep-th/ off-shell 4-point amplitude computed

4. Plan of the talk: I. II. III. IV. V.
Brief review of the CFT techniques in SFT
wedge states,
Review of the tachyon solution
Sen’s conjectures
Pure gauge like form, partial isometries etc.
multibrane solutions
Marginal deformations
Open problems and new directions

5. Open String Field Theory (Witten 1986)
We start with a string field j i = t ( X ) c 1 + A b
Write a Chern-Simons-type Witten action S [ ] = 1 2 h Q i + 3
This action has an enormous gauge invariance = Q +
provided that the star product is associative, BRST charge
Q acts as a derivative, and the bracket like an integration

6. h ª ; i = f ± ( ) f ( z ) = t a ³ + r c ´
In the CFT language (LeClair et al., Rastelli et al.) the integration
of a star product of N factors (N-vertex) is given by a CFT
correlation function on glued world-sheet like here
Normally we map the strips
to half-disks h 1 ; 2 3 i = f ( ) U H P f n ( z ) = t a 2 3 + r c

7. Simplifying the Witten N-vertex
Let us map the world-sheet from the UHP to a semi-infinite
cylinder via (Rastelli et al., 2001) ~ z = a r c t n
Create states by inserting local operators on the cylinder,
their pullback to UHP is given by , where j ~ Á i = U t a n U t a n = e 1 3 L 2 4 + 8 9 6
is a representation of the conformal map f ( z ) = t a n

8. Simplifying the Witten N-vertex~ Á 1 ; 2 3 i = ( ) C

9. h ~ Á ; i = ¡ ¢ ( ) h ~ Â ; Á i = ¤ 8 L = I d ~ z 2 ¼ i T ( ) 1 + a r
The two-vertex can be similarly written as h ~ Á 1 ; 2 i = ( ) C
Using the two- and three-vertex, one can
introduce the star product h ~ Â ; Á 1 2 i = 8
To relate both vertices one has to rescale the three-vertex
cylinder by 2/3, this is generated by L = I d ~ z 2 i T ( ) 1 + a r c t n X k 4

10. ~ Á ( ) j i ¤ = U ¡ ¢ U ~ Á ( x ) : j i U = ¡ ¢ ; We then easily find1 ( ) j i 2 = U ? 3 4
where U r = 2 L ; ?
More generally, star product of n Fock states
looks as U ? n + 1 ~ Á ( x ) : j i U ? r ~ Á 1 ( x ) : n j i s à y m = + 4
Manifestly associative !
[wedge states with insertions]

11. Properties of , and L L K ~ z L ¡ ~ z + " ( R e ) ¢ L K ´ L K µ ( R eL ? K 1
Useful operators associated to vector fields
See RZ (2006)
for generalizations ~ z @ L ~ z + 2
" ( R e ) @ L ? @ ~ z K 1 L
(star algebra derivative) K L 1 ( R e ~ z ) @ K R 1 ( R e ~ z ) @
Lie brackets give commutators
and also [ L ; ? ] = + [ L ; K 1 ] = R

12. b L = + £ L ; b ¤ = L b L j i L : j i U = e Let us introduce
Thanks to the commutation relation
we find rather unexpectedly new class of eigenstates
with eigenvalues n. These states are NOT of
of the form
These states appear rather naturally in the star product of
Fock states due to b L = + ? L ; b = L b L n j i L n 1 2 : k j i U ? n + 2 = e b L

13. b L j i ¤ = X C b L j I i ¤ = L j I i ¤ = X D ~ Á
Using the star product formula we find
MS (2005),
Rastelli, Zwiebach (2006) b L n j i m = X k + C
super-additivity b L n j I i m = +
exact additivity L ? n j I i m = X k + D
sub-additivity
Under certain assumptions, these formulas generalize to larger
sectors involving modes of primary fields ~ Á n

14. Solving Equations of Motion

15. Solving Equations of Motion - Toy model
Similar equation studied numerically
in Gaiotto et al. (2002) ( L 1 ) + =
Given the algebra, a natural ansatz is = X f n ! 1 2 b L j i
Simple solution to the recursion is , where
are the Bernoulli numbers. Can be summed to a closed form f n = B B n = b L 2 1 e j i ? + I X n @
Related by Euler-Maclaurin
formula

16. Solving Equations of Motion - Witten’s theory
We have seen the power of as opposed to L L
Therefore, to solve the equation of motion
leads us to consider instead of the usual
Siegel gauge. Here
Very natural ansatz appears to be Q + = B = B = H d ~ z 2 i b ( ) = X n ; p f b L ~ c j i + q B
where b B = + ?

17. Thanks to the super-additivity of the star product, the e.o.m.
leads to a solvable recursion. With a little bit of luck and
help by Mathematica we discovered f n ; p = ( 1 ) 2 + ! B q 3
p odd
p+q odd
where are the Bernoulli numbers B n B = 1 ; 2 6 : 9 7 3

18. ¡ K @ j n i ª = l i m " Ã ¡ X @ # ; c j ¤ B :
Staring a bit at our solution and Euler-Maclaurin formula
we realized that in fact = l i m N ! 1
" Ã X n @ # ; 2 c j B L :
The derivative acts on a wedge state as 2 K L 1 @ n j n i

19. And this is how the solution looks like geometrically= 1 X n b T c P c
where the distance of the two c-ghost insertions along the two
connecting arcs is and respectively. 2 n
Discovered independently by Okawa (2006)

20. Sen conjectures V ( ) ¡ T = E E
The tachyon is a manifestation of instability of the D-brane,
on which the open string ends.
where is the D-brane tension
There are nontrivial classical solutions corresponding to D-branes of lower dimensions.
At the minimum, there are no perturbative degrees of freedom V ( ) T = E E
Sen 1999

21. First conjecture V [ ª ] = ¡ L 1 L 1 Analytically
Three ways to show that V [ ] = 1 2 g o
Analytically
Numerically in -level truncation, precision L 1 5 L 1 7
It would be nice to come up with a simpler analytic proof,
and understand why the proof works

22. Our assumption recently verified by Okawa hep-th/0603159,
and by Fuchs and Kroyter hep-th/

23. L ª = t c j i + u v L w b ¢ level truncation
level truncation = t c 1 j i + u v L 2 w b
The lowest level coefficients are t = 1 X n 2 d s i + u 4 v 3 w 8

24. t = : 5 3 ; u 4 7 v 1 8 w ¡ Numerically they are:: 5 3 ; u 4 7 v 1 8 w
Can be easily computed with arbitrary precision

25. L 1 z ! level truncation Let us ‘regularize’ the energy
level truncation = 2 ~ c 1 j i + h L y B 4 8 :
Let us ‘regularize’ the energy h ; z L y Q B i = 4 2 + 1 3 9 7 5 6
But alas, the limit is divergent. Fortunately,
there is a well known technique for summing divergent
series. z ! 1

26. Padé approximation to the energy

27. Third conjecture ª Q = + ¤ ² § f Q ; A g = j I i Q Á = Q ( A )
Ellwood, MS (2006)
Expanding the SFT around the true vacuum produces
a theory which looks just like the original one, but with a new
BRST-like charge Q = B +
We construct a state which obeys f Q ; A g = j I i
Existence of such a state proves that there is no cohomology
Since all closed state are automatically exact Q Á = Q ( A )

28. A = B L j I i b Z d r A A = j I i X ( ¡ ) c j i
The solution turns out to be quite simple A = B L j I i b Z 2 1 d r
Surprisingly it is very close in form to the conjectured
of Siegel gauge A A = b L j I i
Recently there has been a paper by C.Imbimbo who finds by level truncation
non-zero cohomology in Siegel gauge at ghost numbers zero and three.
Possible explanations are:
‘Non-physical’ cohomology is unphysical, i.e. gauge dependent.
The Imbimbo’s states are somewhat sick.
Example: 1 X k = ( ) c 2 j i

29. A = g d ª = U Q V U V = I U V = ; Q ( )
Pure gauge like form, partial isometries etc.
It has long been suspected that just as in ordinary
Chern-Simons theory any solution to the equations of
motion can be brought locally to the pure gauge form
also in SFT solutions should be possible to
write in the pure-gauge-like form
We do not need
the e.o.m. are satisfied under weaker assumptions,
e.g.: A = g d 1 = U Q B V U V = I U V = ; Q B ( )

30. A = ­ ª = Ã ­ ( 1 ¡ P ) P n S ; S = j i h 1 + 2 ¢ S = 1 ¡ P
Early evidence came from study of the SFT in the large
NS-NS B-field background
Witten factorization:
(Witten; M.S. 2000)
Allows for simple construction of solutions in the form
where is a rank projector in the Moyal algebra.
Interestingly such projectors can be very efficiently
constructed using partial isometries A = S F T M o y a l = Ã ( 1 P ) P n S ; y S = j i h 1 + 2 S n y = 1 P
M.S.; Harvey, Kraus, Larsen (2000)

31. ª = l i m ( I ¡ © ) Q © = B c j i I = j i U = I ¡ ©
It was found by Okawa (2006) that the solution can be
written in the pure-gauge-like form = l i m ! 1 ( I ) Q B
where is the identity and = B L 1 c j i I = j i
How is this possible ?
is perfectly regular string field,
its inverse however is more tricky U = I

32. L V = ( I ¡ © ) X ¸ ; 3 L X ¸ ; ¸ = 1 Ã = c j i + ¢ U Q V
In level expansion contains terms
of the form
whereas in level expansion it contains terms
so it is more singular, and even though the singularities
are simpler, the value at cannot be defined.
This is welcome since can never be
written as L V = ( I ) 1 1 X n = k ; 3 L 1 X n = k ; = 1 Ã = 2 c 1 j i + U Q B V

33. ª = U Q V S [ ª ] = ¡ h U Q V i S [ U Q V ] = S [ V Q U ] = ¡ V Q U
Ordinarily the Chern-Simons action gives quantized
values for pure (large) gauge configurations. Similar property
can be shown to hold more generally
Let
Then = U Q B V S [ ] = 1 6 h U Q B V i S [ U n Q B V ] =
In particular, formally S [ V Q B U ] =
This suggests that should be interpreted
as a two D-brane solution !
Ian Ellwood, M.S. in progress V Q B U

34. Two D-brane solution in OSFT
How to test the proposal ?
Check the energy
Check the cohomology
Rigorous analytic computation of the energy is hard
because we don’t know the analog of the term.
We are trying first the level truncation. In the Virasoro
basis the solution takes the form 2 D - 5 = V Q B U N 2 D - 5 = 1 : 6 3 c j i 8 7 L + 4 9 b

35. The data are very far from the expected value +1.
The sign switch should occur around L=250,000.
With the current level of accuracy we are able to ‘predict’
that the asymptotic value as L goes to infinity is between
-1 and +3.

36. Marginal deformations
Given any exactly marginal operator of the
matter CFT we can construct SFT solution J ( z ) = 1 + 2 R d r c J ( ) j i B L
One can also easily work out formally the spectrum
of fluctuations around the solution. The real challenge,
however, is to make the solution explicit in some basis
especially when the OPE between two J‘s is nontrivial.

37. e For the rolling tachyon solution based on one finds
results qualitatively similar to those of Moeller & Zwiebach
and Fujita &Hata. The string field doesn’t seem to develop
singularity at finite time. To test Sen’s tachyon matter
conjecture one has to construct the right energy-momentum
tensor. e X

38. Summary
Open string field theory is simple in the ‘cylinder’ coordinate gauge is very natural in this context
Having found the tachyon solution, Sen’s first and third conjectures were proved.
Work is currently under progress to find out whether the multi brane solutions really exist.
Marginal deformation solutions found and are being analyzed. B

39. Open problems and new directions
Construct lump solution and prove Sen’s second conjecture
Construct general rolling tachyon solutions and prove
Sen’s rolling tachyon conjectures
Find more solutions (some progress in multi-brane solutions, Wilson line deformations)
Compute systematically off-shell amplitudes
(see e.g. recent paper by Fuji, Nakayama, Suzuki )
Study closed strings (open-closed duality, boundary states, etc.)
Everything above in super-OSFT (e.g. the Berkovits’ theory)