1. D-branes and Closed String Field Theory
N. Ishibashi (University of Tsukuba)
In collaboration with
Y. Baba(Tsukuba) and K. Murakami(KEK)
hep-th/ (JHEP 0605:029,2006 )
hep-th/0702xxx
I am very happy to be here to celebrate Yoneya-san's 60th birthday.
When I was a freshman here in Komaba, Yoneya san taught us
the first physics class for us, classical mechanics.
For many of us, he was the first real physicist we had ever seen.
The phenomenon called “imprinting” happened,
and many of my classmates including me had followed in his footsteps since then.
What I would like to talk about today is what I have done in collaboration with
Baba and Murakami.
Komaba2007 Recent Developments in Strings and Fields
On the occasion of T.Yoneya’s 60th birthday
Feb. 11, 2007

2. D-branes in string field theory
§1 Introduction
D-branes in string field theory
D-branes can be realized as soliton solutions in open string field theory
D-branes in closed string field theory?
A.Sen,……….
Okawa’s talk
It is about how D-branes are realized in closed string field theory. As Okawa san will tell us, D-branes are realized as soliton solutions in open string field theory. Sen’s conjectures are nowadays proved analytically in Witten’s cubic open string field theory.
On the other hand, D-branes in closed string field theory are not studied so much. One of the few works which addresses this question in critical string theory is by Hashimoto and Hata. They consider closed string field theory of HIKKO and show that one can add a BRS invariant source term like this. Here this state is the boundary state for a D-brane and they identified such a source term as the realization of a D-brane in string field theory. Indeed, with such a source term, boundaries are inserted in the Feynman diagrams of strings. One drawback is that the constant here is arbitrary in their formalism, and the tension of the brane cannot be fixed. However, as we will see, what we will do is something similar to this.
Hashimoto and Hata
HIKKO
♦A BRS invariant source term
♦c is arbitrary

3. D-branes in SFT for noncritical strings
SFT for c=0
Kawai and N.I., Jevicki and Rodrigues
String Field l
String Field Theory〜Collective Field Theory in MM
Jevicki and Sakita
For noncritical strings, we more or less understand how D-branes are realized in closed string SFT.
In order to explain this, I first would like to illustrate the SFT for c=0 strings, constructed by these people.
c=0 noncritical strings can be formulated by using one matrix model. Using the matrix M, we can construct what is called the macroscopic loop, like this, which correspond to a string with length l.
SFT for c=0 strings can be constructed as a collective field theory in the matrix model, using this as the collective field. We prepare creation and annihilation operators for the string and the vacuum. We use the formulation of stochastic quantization to the matrix model and introduce so-called fictitious time and the Fokker-Planck Hamiltonian which describes the time evolution.

4. joining-splitting interactions loop amplitudes Virasoro constraints
Using these variables, one can express Fokker Planck Hamiltonian like this. It consists mainly of the joining-splitting interactions like those in the light-cone gauge SFT.
Using this Hamiltonian and operators, the Green’s function for the macroscopic loops can be written like this.
Moreover the Virasoro constraints for c=0 are written in this context as the equation satisfied by this state Psi.
Here T is something which can be expressed as a sum of the Virasoro operators.
Virasoro constraints
FKN, DVV

5. these solitonic operators reproduce amplitudes with ZZ-branes
Fukuma and Yahikozawa
Hanada, Hayakawa, Kawai, Kuroki,
Matsuo, Tada and N.I.
state with D(-1)-brane
ghost D-brane
In this set-up, we can introduce operators which create D-branes. It is something like a vertex operator and satisfies the commutation relation like this. Then, if Psi is a solution to the Virasoro constraints, we can get another solution by acting this operator like this. If one calculate loop amplitudes using this new state, we get amplitudes with ZZ-branes. Indeed psi’s in the exponential have the effect of inserting boundaries in the worldsheet with the right weight.
The coefficients in front of psi’s is fixed by the condition that this commutator yields a total derivative. The operator constructed here can be understood easily in the usual terminology. Usually Virasoro constraints are expressed using a twisted boson and T is the energy momentum tensor for this boson. Then this operator is just the vertex operator with conformal weight 1.
Here we express everything using variables different from the twisted boson because these variables are analogous to the light-cone gauge string field for critical strings. We will use this analogy later.
these solitonic operators reproduce amplitudes with
ZZ-branes
c.f.

6. ♦Can one construct such solitonic operators in SFT
for critical strings?
Similar construction is possible for OSp invariant
string field theory
Plan of the talk
§2 OSp invariant string field theory
§3 Idempotency equation
§4 Solitonic states
§5 Conclusion and discussion
Now we have seen that how D-branes are realized in SFT for noncritical strings. The problem we would like to consider in this talk is if we can do a similar thing in SFT for critical strings. What we will show is that similar construction is possible for what is called OSp invariant SFT.
The plan of the talk is as follows. I will first explain what OSp invariant SFT is and show that it has a structure quite similar to SFT for noncricical strings. Then I will explain what is called idempotency equation satisfied by the boundary states, which plays important roles in our construction. Then we will construct solitonic states which corresponds to D-branes in this SFT.

7. §2 OSp invariant string field theory
light-cone gauge SFT
O(25,1) symmetry
OSp invariant SFT =light-cone SFT with
OSp invariant SFT is a covatiantized version of the light-cone gauge SFT. The light-cone gauge SFT involves these variables with this action. It has O(25,1) symmetry, some of which are nonlinearly realized. OSp invariant SFT is made from the light-cone gauge SFT by adding two bosonic and two fermionic coordinates like this. Here C and C-bar are free fields with conformal weight 0. Then it is obvious that this theory possesses OSp(27,1|2) symmetry. These extra coordinates contribute nothing to the Virasoro central charge, and we can construct some of the OSp generators using this fact.
Grassmann
Siegel, Uehara, Neveu, West,
Zwiebach, Kugo, ….
OSp(27,1|2) symmetry

8. OSp theory 〜 covariant string theory with extra time and length
BRS symmetry
OSp invariant SFT can be looked at from another point of view. Namely, it can be considered as a covariant string field theory with extra time and length. Indeed, C and bar-C can be related to the reparametrization ghost b,c like this. Therefore the variables of OSp theory are those of Euclidean covariant string and time and string length alpha.
In this OSp theory, we consider one of the OSp transformation JC- as the BRS symmetry and define physical observables and states. This symmetry is nonlinearly realized and the transformation law is like this. If we rewrite C, bar-C in terms of b,c, the operator MC- is a sum of BRS operator for strings and a part involving alpha derivative. It is easy to show that the string field Hamiltonian is BRS exact. Therefore if one consider only BRS invariant operators, the time translation essentially has no effect. Hence this theory is quite like Parisi-Sourlas or stochastic quantization in usual field theory. There exists many variations of this kind of theory and we cannot pin down which formulation OSp theory corresponds to.
the string field Hamiltonian is BRS exact

9. OSp theory 〜 Parisi-Sourlas, stochastic quantization
　　　　　　　　　　Marinari-Parisi for c=0
Green’s functions of BRS
invariant observables =
Green’s functions in 26D
S-matrix elements in 26D ∥
S-matrix elements derived from the light-cone gauge SFT
different from the usual formulation, but
・string length
・joining-splitting interaction
・extra time variable
etc.
However we can prove the following. If one calculate green’s functions of some BRS invariant observables, the result essentially depends only on the 26D coordinates and can be considered as Green’s functions in a 26D theory. Then we can derive the S-matrix elements from these Green’s functions which can be proved to coincide with the S-matrix elements derived from the light-cone gauge SFT. Thus we can use this OSp theory to describe bosonic strings in 26D.
Now it is obvious that the OSp theory have many things in common with the SFT for noncritical strings. It involves string length variables, joining-splitting interaction and the extra time variable. Since the structure is very similar, it is conceivable that the construction of solitonic operator is also possible in this theory.
similar to noncritical
SFT
Can we construct solitonic operators?

10. §3 Idempotency equation
we need to consider boundary states
idempotency equation ∝
However there is one crucial difference between c=0 and critical strings. Only variables which characterize the three string vertex for c=0 strings are the length of the strings. On the other hand, for critical strings configurations of X and C, bar-C are involved and the three string vertex is not so simple. In order to construct solitonic operators as in the c=0 case, we should consider the situation when this vertex become simple and only the lengths of the strings matter. Boundary states corresponding to D-branes are exactly what we need.
As these people showed, a boundary state satisfies so-called idempotency equation which is written like this. Namely, if one connect two boundary states using the three string vertex, one gets a boundary state again. Intuitively it is clear that open string boundaries should satisfy such a relation. It is starightforward to show that boundary states corresponding to Neumann and Dirichlet conditions satisfy this equation. These authors argued that this equation is a relation which boundary state in general should satisfy and showed some evidences. In order to calculate this quantity, we should regularize like this. The proportionality constant is divergent when epsilon goes to zero.
Any way, for such states the three string vertex can be described using only the lengths of the strings.
Kishimoto, Matsuo, Watanabe
・regularization

11. Boundary states in OSp theory
the boundary state for a flat Dp-brane
・ BRS invariant regularization
Let us consider the boundary states in OSp invariant SFT. Here we only consider the boundary states for flat Dp-branes which satisfies the following Dirichlet-Neumann conditions. Here we take the boundary states to be an eigenstate of alpha with the eigenvalue l. We should regularize the boundary states like this, and it is a BRS invariant regularization. Then the norm of the boundary state is regularized to be like this. We can define normalized state by dividing the boundary state by a divergent factor.
・ normalized state:

12. light-cone quantization
orthogonal to
light-cone quantization
this part of the theory looks quite similar to SFT for c=0
Using these normalized states, the string field can be expanded like this, and we define phi to be the coefficients of these states. Performing the light-cone quantization, we obtain the commutation relation and see that phi is the annihilation operator and bar-phi is the creation operator.
This part of the theory looks quite similar to the SFT for c=0. We can identify the creation and annihilation operators like this. The three string vertex for phi’s involves only the lengths of the strings because of the idempotency equation satisfied by the boundary state. Moreover, it is a bit more complicated to show but one can see that BRS invariance condition looks quite similar to the Virasoro constraints. Using this analogy, let us construct the solitonic operators for OSp invariant SFT.
we may be able to construct solitonic operators

13. §4 Solitonic states BRS invariant state
these coefficients are fixed by BRS invariance
We can construct a vertex operator using phi and bar-phi. If we act this integrated one on the light-cone vacuum it can be proved that this new state is BRS invariant in the same way as in the noncricical case. The BRS transformation yields a total derivative and if the integration contour is appropriately chosen, the integral vanishes. The condition that the variation becomes a total derivative fixes the coefficients in the exponent. One can act this integrated operator many times and we get BRS invariant states with many D-branes.
more generally
BRS invariant

14. perturbative calculation
amplitudes
S-matrix elements
with D-branes
vacuum amplitude
perturbative calculation
Now let us consider how we can calculate amplitudes using these states. It is not a priori clear how these states should be used to express amplitudes with D-branes. Since the Green’s functions can be expressed by using the light-cone vacuum like this, it is reasonable to expect that with D-branes can be obtained by replacing the vacuum by the solitonic state. We can check that this definition yields the amplitudes with D-branes, for simple cases.
First, let us consider the vacuum amplitude. It should be written like this. We can calculate this amplitude perturbatively. Writing the operator V in terms of psi’s we express this using the Green’s functions of psi’s. Before doing so we integrate over zeta using the saddle point approximation, because there exists string coupling constant in the denominator here. Substituting the saddle point value, we essentially recover the boundary state from the normalized state.
saddle point approximation

15. should be identified with the state with
two D-branes or ghost D-branes
(Okuda, Takayanagi)
Then as the lowest order contribution to the vacuum amplitude, we obtain the cylinder amplitude like this. This implies that there exist two D-branes or ghost D-branes.
The result does not depend on T as expected from the BRS exactness of the Hamiltonian. The variable zeta can be interpreted as the value of open string tachyon because it multiplies the boundary state like this. And we may regard this as the tachyon potential.
It is possible to calculate disk amplitude from the Green’s function and we get amplitudes with two D-branes or ghost D-branes right. Moreover, from the results we can see that Vplus corresponds to D-branes and Vminus corresponds to ghost D-branes. These results are in our forthcoming paper.
・ disk amplitude
Baba, Murakami, N.I.
to appear

16. §5 Conclusion and discussion
♦D-brane
BRS invariant state in OSp invariant
SFT
♦other amplitudes
♦similar construction for superstrings
we need to construct OSp SFT for superstrings first
To summarize, we have shown that we can construct solitonic operators in OSp SFT and we can realize D-branes as BRS invariant states.
There are many problems to study further. We should check whether our proposal is valid for other amplitudes.
An important thing to do is to generalize our results to superstring case. In order to do so, we should construct OSp SFT for superstrings first.
Why we get two branes is a mystery. We should look for a way to realize 1 brane, if possible.
Another thing is , in our formulation, ghost D-branes may be identified as annihilation operator of D-branes and we should consider 2nd quantization of D-branes. Such 2nd quantization may be what Yoneya-san will talk about later.
♦1 D-brane ?
♦ghost D-branes 〜 annihilation operators for D-branes ?
2nd quantization Yoneya-san’s talk