Analytic Solutions in Open String Field Theory

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 1990 SFT formulated
Witten, Gross Jevicky, Ohta,
LeClair et al., Kostelecký
Samuel ... 1999
2002 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/0511286 Tachyon
  vacuum constructed ,
• 
  Sens first conjecture proved
  
• Okawa hep-th/0603159
  many details elaborated
• 
  pure-gauge like form,
• Fuchs Kroyter hep-th/0603195 cubic
  term better understood
• Rastelli Zwiebach hep-th/0606131 new
  solutions of SFT-like equations
• See also very recent paper with Okawa and paper
  by Erler
• Ellwood, M.S. hep-th/0606142 Sens
  third conjecture proved
• Fuji, Nakayama, Suzuki hep-th/0609047
  off-shell 4-point amplitude computed

4
Plan of the talk
Brief review of the CFT techniques in SFT
wedge states, Review of the tachyon solution
Sens conjectures Pure gauge like form,
partial isometries etc. multibrane
solutions Marginal deformations Open problems
and new directions
I. II. III. IV. V.
5
Open String Field Theory (Witten 1986)
We start with a string field
Write a Chern-Simons-type Witten action
This action has an enormous gauge invariance
provided that the star product is associative,
BRST charge Q acts as a derivative, and the
bracket like an integration
6
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
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)
Create states by inserting local operators on the
cylinder, their pullback to UHP is given by
, where
is a representation of the conformal map
8
Simplifying the Witten N-vertex
9

The two-vertex can be similarly written as
Using the two- and three-vertex, one can
introduce the star product
To relate both vertices one has to rescale the
three-vertex cylinder by 2/3, this is generated
by
10
We then easily find
where
More generally, star product of n Fock states
looks as
Manifestly associative !
wedge states with insertions
11
Properties of , and
Useful operators associated to vector fields
See RZ (2006) for generalizations
(star algebra derivative)
Lie brackets give commutators and also
12
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
13
Using the star product formula we find
MS (2005), Rastelli, Zwiebach (2006)
super-additivity
exact additivity
sub-additivity
Under certain assumptions, these formulas
generalize to larger sectors involving modes of
primary fields
14
Solving Equations of Motion
15
Solving Equations of Motion - Toy model
Similar equation studied numerically in Gaiotto
et al. (2002)
Given the algebra, a natural ansatz is
Simple solution to the recursion is
, where are the Bernoulli numbers. Can be
summed to a closed form
Related by Euler-Maclaurin formula
16
Solving Equations of Motion - Wittens theory
We have seen the power of as opposed to
Therefore, to solve the equation of motion leads
us to consider instead of
the usual Siegel gauge. Here Very natural
ansatz appears to be
where
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
p odd
pq odd
where are the Bernoulli numbers
18
Staring a bit at our solution and Euler-Maclaurin
formula we realized that in fact
The derivative acts on a wedge state
as
19
And this is how the solution looks like
geometrically
where the distance of the two c-ghost insertions
along the two connecting arcs is and
respectively.
Discovered independently by Okawa (2006)
20
Sen conjectures

• 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

Sen 1999
21
First conjecture
Three ways to show that

• Analytically
• Numerically in -level truncation,
  precision
• Numerically in -level truncation,
  precision

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/0603195
23
level truncation
The lowest level coefficients are
24
Numerically they are
Can be easily computed with arbitrary precision
25
level truncation
Let us regularize the energy
But alas, the limit is divergent.
Fortunately, there is a well known technique for
summing divergent series.
26
Padé approximation to the energy
27
Third conjecture
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
We construct a state which obeys
Existence of such a state proves that there is no
cohomology Since all closed state are
automatically exact
28
The solution turns out to be quite simple
Surprisingly it is very close in form to the
conjectured of Siegel gauge

• 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 Imbimbos states are somewhat sick.
• Example

29
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.
30
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
M.S. Harvey, Kraus, Larsen (2000)
31
It was found by Okawa (2006) that the solution
can be written in the pure-gauge-like form
where is the identity and
How is this possible ?
is perfectly regular string field, its inverse
however is more tricky
32
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
33
Ordinarily the Chern-Simons action gives
quantized values for pure (large) gauge
configurations. Similar property can be shown to
hold more generally Let Then
In particular, formally
This suggests that should be
interpreted as a two D-brane solution !
Ian
Ellwood, M.S. in progress
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 dont know the analog of the
  term.
• We are trying first the level truncation. In the
  Virasoro
• basis the solution takes the form

35
The data are very far from the expected value 1.
The sign switch should occur around
L250,000. 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
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 Js is nontrivial.
37
For the rolling tachyon solution based on
one finds results qualitatively similar to
those of Moeller Zwiebach and Fujita Hata.
The string field doesnt seem to
develop singularity at finite time. To test Sens
tachyon matter conjecture one has to construct
the right energy-momentum tensor.
38
Summary

• Open string field theory is simple in the
  cylinder coordinate. gauge is very
  natural in this context
• Having found the tachyon solution, Sens 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.

39
Open problems and new directions

• Construct lump solution and prove Sens second
  conjecture
• Construct general rolling tachyon solutions and
  prove
• Sens 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)