Transcript Analytic Solutions in Open String Field Theory

Analytic Solutions in Open String Field Theory
Puri, 2006
Martin Schnabl (IAS)
Indian Strings Meeting, Puri 2006
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.
Progress of the past 12 months in OSFT
M.S.
hep-th/0511286
Tachyon vacuum constructed ,
Sen’s 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
Fuji, Nakayama, Suzuki hep-th/0609047
Sen’s third conjecture proved
off-shell 4-point amplitude computed
Plan of the talk:
I. Brief review of the CFT techniques in SFT
wedge states,
II. Review of the tachyon solution
Sen’s conjectures
III. Pure gauge like form, partial isometries etc.
multibrane solutions
IV. Marginal deformations
V. Open problems and new directions
Open String Field Theory
(Witten 1986)
We start with a string field
jª i = t(X )c1 j0i + A ¹ (X )®¡¹ 1 c1 j0i + b(X )c0 j0i + ¢¢¢
Write a Chern-Simons-type Witten action
S[ª ] =
1 hª
2
¤ Qª i +
1 hª
3
¤ª ¤ª i
This action has an enormous gauge invariance
±ª = Q¤ + ª ¤ ¤ ¡ ¤ ¤ ª
provided that the star product is associative, BRST charge
Q acts as a derivative, and the bracket like an integration
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
hª 1 ; ª 2 ; ª 3 i =
hf 1 ± ª 1 (0) f 2 ± ª 2 (0) f 3 ± ª 3 (0)i U H P
³
f n (z) = t an
( 2¡ n ) ¼
3
´
+
2
3
arct an z
Simplifying the Witten N-vertex
Let us map the world-sheet from the UHP to a semi-infinite
cylinder via z~ = arct an z (Rastelli et al., 2001)
Create states by inserting local operators on the cylinder,
~
j
Ái
= Ut an jÁi , where
their pullback to UHP is given by
Ut an = e
1
3
L 2¡
1
30
L 4+
11
1890
L 6 + ¢¢¢
is a representation of the conformal map
f (z) = t an z
Simplifying the Witten N-vertex
³ ¼´
³ ¼´
~1 ; Á
~2 ; Á
~3 i = hÁ1
hÁ
Á2 (0)Á3 ¡
i C 3¼
2
2
2
The two-vertex can be similarly written as
~1 ; Á
~2 i = hÁ1
hÁ
¡ ¼¢
2
Á2 (0) i C ¼
Using the two- and three-vertex, one can
introduce the star product
~1 ; Á
~2 i = hÂ
~1 ¤ Á
~2 i ;
hÂ
~; Á
~; Á
8Â
To relate both vertices one has to rescale the three-vertex
cylinder by 2/3, this is generated by
I
I
d~
z
dz
L0 =
z~Tz~z~( z~) =
(1 + z2 ) arct an z Tzz (z)
2¼i
2¼i
X1 2(¡ 1) k + 1
= L0 +
L 2k
2
4k ¡ 1
k= 1
We then easily find
~1 (0)j0i ¤ Á
~2 (0)j0i = U ? U3 Á
~1
Á
3
where
Ur =
¡ 2 ¢L 0
r
;
Ur?
=
¡ ¼¢
4
¡
~2 ¡
Á
¢
¼
4
j0i
¡ 2 ¢L ?
0
r
More generally, star product of n Fock states
looks as
~1 ( x~1 ) : : : Á
~n ( x~n )j0i
Un?+ 1 Un + 1 Á
~1 ( x~1 ) : : : Á
~n ( x~n )j0i ¤ U ? Us Ã~1 ( y~1 ) : : : Ã~m ( y~m )j0i
Ur? Ur Á
s
¡
¢
¡
¢
?
¼
¼
~
~
= Ur + s¡ 1 Ur + s¡ 1 Á1 x~1 + 4 (s ¡ 1) : : : Án x~n + 4 (s ¡ 1)
¡
¢
¡
¢
¼
¼
Ã~1 y~1 ¡ 4 (r ¡ 1) : : : Ã~m y~m ¡ 4 (r ¡ 1) j0i
Manifestly associative !
[wedge states with insertions]
?
L
L
Properties of 0 , 0 and K 1
Useful operators associated to vector fields
@
z~@
¡ z~
¡ z~ +
L0
L ?0
K 1 ´ L¡
1
¼ " (Re z
~)
2
@
@z~
@
@z~
µ(Re z~)
K 1R
µ(¡ Re z~)
and also
[L 0 ; K 1 ] = K 1 ;
@
@z~
(star algebra derivative)
K 1L
Lie brackets give commutators
¢
See RZ (2006)
for generalizations
@
@z~
[L 0 ; L ?0 ] = L 0 + L ?0
£ L
¤
K 1 ; K 1R = 0
Let us introduce
Lb = L 0 + L ?0
Thanks to the commutation relation
£
¤
L 0 ; Lb = Lb
we find rather unexpectedly new class of
L0
eigenstates
Lbn j0i
with eigenvalues n. These states are NOT of
L ¡ n L ¡ n : : : L ¡ n j0i
of the form
1
2
k
These states appear rather
¡ n star
Lb product of
U ? naturally
Un + 2 =inethe
2
n+ 2
Fock states due to
Using the star product formula we find
Lbn j0i ¤ Lbm j0i =
X
MS (2005),
Rastelli, Zwiebach (2006)
Cnk m Lbk j0i
super-additivity
k¸ n+ m
Lbn jI i ¤ Lbm jI i = Lbn + m jI i
L ? n jI i ¤ L ? m jI i =
X
D nk m L ? k jI i
exact additivity
sub-additivity
k· n+ m
Under certain assumptions, these formulas generalize to larger
~
sectors involving modes of primary fields Án
Solving Equations of Motion
Solving Equations of Motion - Toy model
(L 0 ¡ 1)© + © ¤ © = 0
Similar equation studied numerically
in Gaiotto et al. (2002)
Given the algebra, a natural ansatz is
X
©=
fn
n!
¡
¢ n
1 nL
b j0i
2
¡
Simple solution to the recursion is f n = B n , where B n
are the Bernoulli numbers. Can be summed to a closed form
© =
=
Lb=2
1 ¡ e¡
Lb=2
¡
¡
¢
¢
j0i = L ?0 ³ L ?0 + 1 ¡ 1 jI i
X
j1 i ¡
@n jni
n¸ 2
Related by Euler-Maclaurin
formula
Solving Equations of Motion - Witten’s theory
We have seen the power of
L0
as opposed to
L0
Therefore, to solve the equation of motion Qª + ª ¤ ª
leads us to consider B0 ª = 0 instead of the usual
H dz~
Siegel gauge. Here
B0 =
2¼i
z~bz~z~( z~)
Very natural ansatz appears to be
X
f n ;p Lbn c~p j0i +
ª =
n ;p
where
Bb = B0 + B0?
X
n ;p;q
f n ;p;q BbLbn c~p c~q j0i
= 0
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
=
f n ;¡
p;¡ q
=
where
Bn
(¡ 1) n ¼p
B n + p+ 1
n
+
2p+
1
2
n!
(¡ 1) n + q ¼p+ q
B
2n + 2( p+ q) + 3 n! n + p+ q+ 2
p odd
p+q odd
are the Bernoulli numbers
1
1
691
B 0 = 1; B 1 = ¡ ; B 2 = ; : : : ; B 12 = ¡
;:::
2
6
2730
Staring a bit at our solution and Euler-Maclaurin formula
we realized that in fact
"
ª
=
lim
N! 1
Ãn
=
2 c j0i
¼ 1
The derivative
XN
ÃN ¡
#
@n Ãn ;
n= 0
¤ B 1L jni ¤ c1 j0i :
@n acts
on a wedge state
jni
as
¡
¼K L
2 1
And this is how the solution looks like geometrically
X1
ª
0
=
b
T
n= 0
c
P
c
where the distance of the two c-ghost insertions along the two
¼n
¼
connecting arcs is
and 2
respectively.
Discovered independently by Okawa (2006)
Sen conjectures
The tachyon is a manifestation of instability of the D-brane,
on which the open string ends.
1)
2)
3)
V (0) ¡ V (T0) = E
where
E 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
First conjecture
Three ways to show that
1)
2)
3)
V [ª 0 ] = ¡
1
2¼2 go2
Analytically
10¡
Numerically in L 0 -level truncation,
10¡
Numerically in L 0 -level truncation,
5
7
precision
precision
It would be nice to come up with a simpler analytic proof,
and understand why the proof works
Our assumption recently verified by Okawa hep-th/0603159,
and by Fuchs and Kroyter hep-th/0603195
L0
level truncation
ª = tc1 j0i + uc¡ 1 j0i + vL ¡
2 c1 j0i
+ wb¡ 2 c0 c1 j0i + ¢¢¢
The lowest level coefficients are
X1
t
=
n= 2
X1
u
=
n= 2
X1
v
=
n= 2
X1
w
=
n= 2
d
dn
d
dn
d
dn
d
dn
·
µ
µ
¶¶¸
³
´
n
¼
n
2¼
2
sin
¡ 1+
sin
¼
n
2¼
n
·µ
¶µ
µ
¶¶¸
³
´
4
n
¼
n
2¼
¡
sin2
¡ 1+
sin
n¼ ¼
n
2¼
n
·µ
¶
µ
µ
¶¶¸
³
´
4
n
¼
n
2¼
2
¡
sin
¡ 1+
sin
3n¼ 3¼
n
2¼
n
·
µ
¶¶¸
³ ¼´ µ 8
2
2n
n
2¼
sin2
¡
+
sin
n
3n¼ 3¼ 3¼2
n
Numerically they are:
t = 0:553; u = 0:457; v = 0:138; w = ¡ 0:144
Can be easily computed with arbitrary precision
L0
ª
=
level truncation
´
³
´
i
2
1 h³
c~1 j0i +
L 0 + L y0 c~1 j0i ¡ B0 + B0y c~1 c~0 j0i +
¼
· ³ 2¼
¸
´
³
´
³
´
2
1
¼
+
L 0 + L y0 c~1 j0i ¡ 2 B0 + B0y L 0 + L y0 c~1 c~0 j0i +
c~ j0i + ¢¢¢:
24¼
48 ¡ 1
Let us ‘regularize’ the energy
µ
¶ µ
¶
2
¼
4
1
1
1
y
z2 +
hª ; zL 0 QB zL 0 ª i = ¡ 2 2 +
+ 2 ¡
+
¼z
12 3¼
90 1920
µ
¶
2
4
¼
17
11¼
z4 + ¢¢¢
+
¡
¡
5040 17920 193536
But alas, the limit z ! 1 is divergent. Fortunately,
there is a well known technique for summing divergent
series.
Padé approximation to the energy
Third conjecture
Ellwood, MS (2006)
Expanding the SFT around the true vacuum ª 0 produces
a theory which looks just like the original one, but with a new
BRST-like charge
Qª = QB + ª
0
¤² § ² ¤ª
0
We construct a state which obeys
f Qª ; Ag = jI i
Existence of such a state proves that there is no cohomology
Since all Qª closed state are automatically exact
Á = Qª (AÁ)
The solution turns out to be quite simple
B0
A=
jI i = Bb
L0
Z
2
dr jr i
1
Surprisingly it is very close in form to the conjectured
b0 jI i
A
=
of Siegel gauge
L0
A
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:
(i) ‘Non-physical’ cohomology is unphysical, i.e. gauge dependent.
(ii) The Imbimbo’s1states are somewhat sick.
Example:
k
X
(¡ 1) kc¡
k= 0
j0i
2k
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
A = gdg¡
1
also in SFT solutions should be possible
to V
ª = UQ
B
write in the pure-gauge-like
form
UV = V U = I
We do not need
the e.o.m. are satisfied under weaker assumptions,
e.g.:
UV U = U;
QB (UV ) = QB (V U) = 0
Early evidence came from study of the SFT in the large
NS-NS B-field background
Witten factorization:
A = A SF T
A M oy al
(Witten; M.S. 2000)
Allows for simple construction of solutions in the form
P
where
ª = Ã0
n
is a rank
(1 ¡ P )
projector in the Moyal algebra.
y
Interestingly such projectors can be very
S; Sefficiently
constructed using partial isometries
S = j0i h1j + j1i h2j + ¢¢¢
Sn y Sn = 1 ¡ Pn
M.S.; Harvey, Kraus, Larsen (2000)
It was found by Okawa (2006) that the solution can be
written in the pure-gauge-like form
ª
0
= lim (I ¡ ¸ ©) QB (I ¡ ¸ ©) ¡
1
¸! 1
where I = jI i
is the identity and
© = B 1L c1 j0i
How is this possible ?
U= I ¡ ©
is perfectly regular string field,
its inverse however is more tricky
In
L0
V = (I ¡ ©) ¡
level expansion
X1
¸n
;
nk
of the form
1
contains terms
k¸ 3
n= 1
whereas in
L0
expansion it contains terms
1
Xlevel
¸ n nk ; k ¸ 0
n= 1
so it is more singular, and¸ even
= 1 though the singularities
are simpler, the value at
cannot be defined.
This is welcome
UQBsince
V
written as
Ã0 =
2 c j0i
¼ 1
+ ¢¢¢
can never be
Ordinarily the Chern-Simons action gives quantized
values for pure (large) gauge configurations. Similar property
can be shown to hold more generally
Let
ª = UQB V
S[ª ] = ¡
Then
1 hUQ
B
6
V UQB V UQB V i = S0
S[U n QB V n ] = nS0
In particular, formally
S[V QB U] = ¡ S0
V QB U
This suggests that
as a two D-brane solution !
should be interpreted
Ian Ellwood, M.S. in progress
Two D-brane solution in OSFT
ª
How to test the proposal
2D -25
= V QB U
?
(i) Check the energy
(ii) Check the cohomology
Rigorous analytic computation of the energy
ª Nis 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
ª
2D -25
=
1:563 c1 j0i ¡ 0:832 c0 j0i ¡ 0:735 L ¡ 2 c1 j0i
+ 0:348 c¡ 1 j0i ¡ 0:289 b¡ 2 c0 c1 j0i + ¢¢¢
E
6
L
5
10
15
20
10
12
14
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.
Marginal deformations
J (z)
Given any exactly marginal operator
of the
matter CFT we can construct SFT solution
ª
¸
=
1+
¼¸
2
R
1 dr
0
¸
cJ (0)j0i ¤ B L jr i
¤ cJ (0)j0i
1
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.
eX 0
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.
Summary




Open string field theory is simple in the ‘cylinder’
coordinate. B0 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.
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

Compute systematically off-shell amplitudes
deformations)
(some progress in multi-brane solutions, Wilson line
(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)