Extension of Boundary String Field Theory on Disc and RP2

arXiv:hep-th/0201035v2 28 Aug 2002

January, 2002 OU-HET 404, KEK-TH-794, hep-th/0201035 revised version

Extension of Boundary String Field Theory on Disc and RP 2 Worldsheet Geometries ∗

H. Itoyama†

and

S. Nakamura‡

†

Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan ‡ Theory Division, Institute of Particle and Nuclear Studies, High Energy Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan

Abstract We present a construction of open-closed string field theory based on disc and RP 2 geometries. Finding an appropriate BRS operator in the case of the RP 2 geometry, we generalize the background independent open string field theory (or boundary string field theory) of Witten on a unit disc. The coupling constant flow at the closed string side is driven by the scalar operator inserted at the nontrivial loop of RP 2 . We discuss the off-shell extension of the boundary/crosscap states. Our construction provides an interpolation of orientifold planes of various dimensions as well as that of D-branes.

∗

This work is supported in part by the Grant-in-Aid for Scientific Research (12640272) from the Ministry of Education, Science and Culture, Japan † e-mail: [email protected] ‡ e-mail: [email protected]

–1–

I.

Introduction

For a long time, a number of physicists have suspected that “space of all two-dimensional field theories”, if it is to be defined, should be a natural configuration space of strings on which string field theory is constructed with an appropriate gauge invariant action. Ultraviolet divergences associated with irrelevant operators carrying arbitrarily high dimensions have hampered progresses toward this goal. It was shown by Witten [1]1 , however, that, modulo this difficulty, a clear-cut framework based on Batalin-Vilkovisky formalism (BV) [3, 4] is possible to find in the case of open-string field theory. This framework materializes the idea of “theory space” on a unit disc. String field theory2 of this kind, originally called background independent open string field theory, provides a self-consistent framework for off-shell processes in which relevant and marginal operators are involved. More recently, this theory has been successfully applied to the problem of open string tachyon condensation [6, 7]. It provides an off-shell interpolation between the Neumann and the Dirichlet boundary conditions of an open string. An exact tachyon potential [6, 7] with the right normalization coefficient [8] has been obtained and this has provided a proof of Sen’s conjecture [9] that the difference between the two extrema just cancels the tension of the D-brane in question. It allows us to consider the decay of a higher dimensional brane to a lower one as well. An agreement on tensions of branes of various dimensions has been given. See [10, 11, 12, 13, 14, 15, 16] for some of the subsequent developments. We will refer to the background independent open string field theory of Witten as boundary string field theory (BSFT), following the current nomenclature. An extension of this framework to closed string field theory appears to be formidable as we do not yet understand well enough processes under which the matter central charge changes by the presence of BRS noninvariant operators located on the bulk of a Riemann surface. One may note with this last regard that there are some processes of a closed string and geometries represented by a closed string which do not involve operators living in the entire bulk of a Riemann surface. In particular, orientifold planes of various dimensions are characterized as crosscap states on a complex plane representing the RP 2 worldsheet geometry and can be discussed in parallel to D-brane boundary states representing the disc worldsheet geometry. These points have prompted us to consider a RP 2 generalization of BSFT, an extension of the idea of theory space on the unit disc and hence open-closed string field theory from the disc and the RP 2 worldsheet geometries, which share the same Euler number. For recent discussions on the other aspects of closed string geometries and closed 1 2

See also [2]. For a recent review on string field theory, see, for example, [5].

–2–

string tachyon, see, for example, [17, 18, 19, 20, 21, 22, 23]. In this paper, we will present a construction of open-closed bosonic string field theory3 . Our construction is based on the BV formalism applied both to the correlators on the boundary of the disc and the ones on the nontrivial loop C of RP 2 (π1 (RP 2 ) = Z2 ). The former part is a repetition of [1] except for the insertion of Chan-Paton space. As for the latter, let us first note that this nontrivial loop can be identified as a source of the crosscap contributing to the Euler number when RP 2 is represented as a part of the complex plane. We find that the scalar operators located on this nontrivial loop accomplish the desirable interpolation between the orientifold planes of various dimensions in much the same way as the operators of BSFT do between D-branes of various dimensions. An off-shell extension [26, 27], (see also [28]), of D-brane boundary states and that of crosscap states [29, 30, 31] provide an expedient tool for this. The latter will be given in this paper. We find an appropriate BRS operator which acts on the operators on this loop on RP 2 . Finding such an operator is mandatory for us to materialize the BV formalism on RP 2 . The total action resulting from our discussion is simply a sum of two terms, one from the disc geometry and the other from RP 2 . In this sense, our construction generalizes that of the first quantized unoriented string theory at Euler number one. Each part of the action separately obeys a first order differential equation with respect to the couplings of operators inserted. The relative normalization between the two terms is fixed by appealing to the dilaton tadpole cancellation at Euler number one [33, 34, 35]. The work of [8] tells that the normalization of the disc part is in fact D-brane tension. In our construction, closed string amplitudes are still beyond our reach and our discussion is essentially at the classical level. In the next section, we briefly summarize essential ingredients of BSFT, which are the BV formalism, the action of the BRS operator on the operators located on the boundary of a disc and the defining differential equation for the action S disc . We recall the off-shell boundary states introduced in [26, 27]. In section three, the RP 2 extension of BSFT is given. We find an appropriate BRS operator acting properly on operators on the nontrivial loop represented as half of a unit circle in the complex plane. We compute the action of the BRS 2 operator and derive a differential equation for S RP . We introduce off-shell crosscap states which manage to interpolate states representing orientifold geometries of various dimensions. In section four, we discuss the total action. The total action consists of the contributions from the two geometries. We discuss how to fix the relative normalization. In Appendix A, we construct a holomorphic field with weight p on the disk and that on RP 2 , using the construction on the plane as the double of the surfaces. These are found to 3

[25]

See [24], for a recent discussion. For earlier references on the open -closed mixed system, see, for example,

–3–

be useful in the calculation in the text. In Appendix B, we recall the tachyon vertex operator of an unoriented bosonic string with intercept −4.

II.

Background Independent Open String Field Theory of Witten, and Off-Shell Boundary States

A.

BV Formalism

Let us first recall the BV formalism briefly. The basic ingredients of this formalism can be summarized as a triplet (M , ω , V )

.

(2.1)

Here M is a supermanifold equipped with local coordinates {{uI }}. We denote by ω a nondegenerate odd (fermionic) symplectic two-form which is closed dω = 0 .

(2.2)

Finally, M possesses U(1) symmetry generated by the fermionic vector field V = V I dudI . This U(1) symmetry is identified with ghost number. The action functional (or zero form) S in this formalism is introduced through dS = iV ω ,

(2.3)

where iV is an interior contraction by V . That V generates a symmetry of ω translates into LV ω = 0 ,

(2.4)

where LV ≡ diV + iV d is the Lie derivative. We see that eqs. (2.2) and (2.4) ensure the existence of a scalar functional S obeying eq. (2.3) modulo global problems on “theory space”. That the antibracket {S, S}AB (2.5) be constant provides the nilpotency of V V2 = 0 ,

(2.6)

and V is naturally identified with the BRS charge. In the next section, we will find that this framework is realized on the RP 2 worldsheet geometry as well. Let us first review the realization on the disc by Witten.

–4–

B.

{QBRS , O}

We will first obtain the BRS transformation of a generic operator O with ghost number one located on the boundary ∂Σ of the unit disc Σ. Let O (w, w) ¯ =

1 z c (w)V (w, w) ¯ = cσ (σ) V (w, w) ¯ iw

,

(2.7)

where V is a generic scalar operator with ghost number 0 and the last equality holds only on the unit circle. Note that cσ (σ), the tangent component of the ghost field along the boundary of the disc, is the only nonvanishing component on the boundary. See eq. (A.9) in the Appendix . It is safe to obtain the BRS transformation δBRS O = iǫ{QBRS , O} , using a free open string:

1 = 2πi

QBRS

I

(2.8)

dzjBRS .

(2.9)

3 m jBRS (z) = cz (z)Tzz (z)+ : bzz (z)cz (z)∂z cz (z) : + ∂z 2 cz (z) 2 1 m Tzz (z) = − ′ : ∂z X µ ∂z Xµ : . α

(2.10)

We find !

∂2 + 1 V (w, w) ¯ {QBRS , O (w, w)} ¯ = c ∂σ c (w) α ∂X µ ∂Xµ σ

σ

′

.

(2.11)

We confirm that an on-shell open string tachyon with intercept −1 is represented by the vertex operator V = exp ik · X with α′ M 2 = −α′ k µ kµ = −1.

C.

dS disc = iV ω

The defining differential equation (2.3) for S disc is written as K δS disc = δλα 2

dσ dσ ′ α tr hO (σ) {QBRS , O} (σ ′ )ighost V,disc , 2π 2π

Z Z

O (σ) = cσ (σ) V (σ) =

X

λα cσ (σ) V α (σ) =

α

X

λα Oα (σ) .

(2.12) (2.13)

α

Here h· · ·ighost V,disc is the unnormalized path integral with respect to the worldsheet matter action I

disc

= I

I bulk =

bulk

1 4πα

+

Z

∂Σ

Z q Σ

dσ V (σ) , 2π

det ηη ij ∂i X µ ∂j X ν gµν d2 σ .

(2.14) (2.15)

–5–

Here η ij is a worldsheet metric which is taken to be flat, and gµν is a spacetime metric with Minkowski signature. We denote by tr the trace over the Chan-Paton space. This simply gives us a factor n of so(n) Lie algebra. Separating the ghost Hilbert space, we find δS disc = Kn δλα

Z Z

dσ dσ ′ (1 − cos (σ − σ ′ )) hc1 c0 c−1 idisc 2π 2π !

∂2 hV (σ) α + 1 V (σ ′ )iV,disc . ∂X µ ∂Xµ α

D.

′

(2.16)

Off-shell Boundary State

The unnormalized matter path integral h· · ·iV,disc can be represented as a matrix element h· · ·iV,disc = hB | · · · | 0iV,disc .

(2.17)

The bra vector hB | obeys hB |

!

1 ∂V ∂ ∂ 1 Xµ + z + z¯ ′ 2πα ∂z ∂ z¯ 2π ∂X µ

!

=0 .

(2.18)

z=eiσ , z¯=e−iσ

We refer to hB | as an off-shell boundary state of the disc. The condition (2.18) is a consequence from the correspondence between the matrix element (2.17) and the end point of the path integral. The latter one obeys the boundary condition derived from the worldsheet action I disc on the disc. Eq. (2.18) tells us that, at the initial point of the coupling constant flow, the system obeys the Neumann boundary condition: hB | (αn + α ˜ −n ) = 0 .

(2.19)

∂V The end point of the flow is described by zero of ∂X µ , namely, the Dirichlet boundary condition: hB | (αn − α ˜ −n ) = 0 . (2.20)

An explicit interpolation between the two ends can be done in solvable cases. In the case of quadratic profiles, the proper form of the Green function has been given in [26, 27]. For analyses which utilize the boundary sine-Gordon model and the g function = hB | 0i, see [36, 37].

III.

RP 2 Extension

We now proceed to construct the RP 2 extension of BSFT. RP 2 is a nonorientable Riemann surface of Euler number one with no hole, no boundary and one crosscap. It is a

–6–

geometry swept by a closed string. The external physical states of this nonorientable closed string are represented by the vertex operators inserted at any point on the surface. The physical state conditions are stated as {QBRS , Oclosed} = 0. Here QBRS is the BRS operator on RP 2 and Oclosed is a generic vertex operator. In the case of the ground state scalar of a closed unoriented bosonic string, this leads us to the on-shell condition α′ M 2 = −α′ kµ k µ = −4, namely the closed string tachyon with intercept −4. For completeness of our discussion, we will include its derivation in the Appendix B. Another property of RP 2 , which is of vital importance to us, is π1 (RP 2 ) = Z2 . This permits us to consider a nontrivial loop C which is represented by a path connecting any two conjugate points (for example, 1 and −1) on the complex plane, i .e. the double of this Riemann surface. As a reference loop, we consider half of a unit circle z = eiσ (0 ≤ σ < π).

A.

Glossary of Notation

Let us first summarize our notation in the RP 2 case. (See Appendix for more detail.) The tangential component and the normal component of the ghost field cσ and cr along the loop are respectively c

σ

i ∂σ z ∂σ z¯ c + c =− ≡ ∂z ∂ z¯ 2

cr ≡

cz (z) c¯z¯(¯ z) − z z¯

!

= −i

cz (z) − cz (−z) , 2z

 ∂r z ∂r z¯ cz (z) + cz (−z) 1 z z¯c (z) + z¯ cz¯(¯ z) = c + c = . ∂z ∂ z¯ 2r 2z

(3.1) (3.2)

In both equations, the last equality holds only when |z| = 1. We also note cz (w) − cz (−w) ∂z cz (w) + (∂z cz )(−w) , 2w 2  ∂ cz (w) − cz (−w)  ∂z X µ (w) + (∂z X µ )(−w) V. cσ (w)∂σ V(X µ ) = 2 ∂X µ cσ ∂σ cσ (w) = −i

(3.3) (3.4)

Here the argument −w should be substituted after the derivatives are taken and V is an arbitrary function of X µ .

B.

BRS charge Q(σ)

Our first objective is to obtain a BRS charge which corresponds to the vector field V in the BV formalism for the RP 2 case. Let us consider the following expression: (σ) QBRS

≡

I

dz BRS j (z) , 2πi z

–7–



jzBRS (z) ≡ 2czeven (z) − + : bzz

1 α′



: (∂z X µ )even (z)(∂z Xµ )even (z) :

z z even (z)ceven (z)∂z ceven (z)

: +total derivatives ,

(3.5)

where the subscript even implies that the modes are restricted to the even ones. For examX z z (σ) ple, czeven (z) ≡ cn z −n+1 = c (z)−c2 (−z) . We will now show that our QBRS , when acted n even

upon the conformal fields on |z| = 1, generates the BRS transformations representing the diffeomorphisms in the σ direction. Let us first consider the ghost part of our BRS current:    1 jz(g) (z) ≡ bzz (z) + bzz (−z) cz (z) − cz (−z) ∂z cz (z) + (∂z cz )(−z) . 8 We obtain   z i 1 1 c (z) − cz (−z) ∂z cz (z) + (∂z cz )(−z) jz(g) (z)cσ (w) ∼ − − , 2 z−w z+w 2w 2 so that (I ) dz (g) σ σ δB c (w) ≡ iǫ j (z), c (w) 2πi z

(3.6)

(3.7)

= iǫcσ ∂σ cσ (w).

(3.8)

One can easily show δB cr (w) = 0. These are the desirable BRS transformation laws for the σ diffeomorphisms. (Again |w| = 1 is understood.) Next, we consider the matter part of the BRS current: jz(m) (z)

1 cz (z) − cz (−z) − ′ ≡2 2 α 

We find jz(m) (z)V ′ (X µ (w, w)) ¯



: (∂z X µ )even (z)(∂z Xµ )even (z) : .

cz (z) − cz (−z) 1 1 ∼ + 2 z−w z+w 

α′ cz (z) − cz (−z) − 8 2



: (∂z X µ )even (z)

2 1 1 + + 2 (z − w) (z − w)(z + w) (z + w)2

!

:

(3.9)

∂ V ′ (X µ ) : µ ∂X

∂2 V ′ (X) :,(3.10) µ ∂X ∂Xµ

so that (I ) dz (m) α′ ∂2 ′ µ σ ′ µ σ σ jz (z), V (X (w, w)) ¯ =c ∂σ V (X )− (∂σ c (w)+ 2ic (w)): µ V ′ (X µ ):.(3.11) 2πi 4 ∂X ∂Xµ In particular, setting V ′ (X µ ) = X µ , we obtain µ

δB X ≡ iǫ

(I

)

dz (m) j (z), X µ = iǫcσ ∂σ X µ (w) . 2πi z

(3.12)

This is again the desirable BRS transformation. Thus, the operator I  dz  (g) (σ) jz (z) + jz(m) (z) . (3.13) QBRS = 2πi generates the BRS transformations δB associated with the σ diffeomorphisms on |w| = 1.

–8–

(σ)

{QBRS , O}

C.

It is now immediate to carry out the action of the BRS charge on a generic operator with (σ) ghost number one. This also brings us an operator which is ”on-shell” with respect to QBRS , that is, invariant under δB . Let O be a scalar operator with ghost number one, O(w, w) ¯ = cσ (w)V ′σ (X µ (w, w)) ¯ + cr (w)V ′r (X µ (w, w)) ¯ .

(3.14)

We find n

(σ) QBRS , O(w, w) ¯

o

σ

= c ∂σ c − cr cσ

!

α′ ∂ 2 1+ V ′σ (X) 4 ∂X 2

σ

!

α′ r σ ∂ 2 ′r α′ ∂ 2 ′r V (X) + c ∂σ c V (X).(3.15) ∂σ V ′r (X) − i 2 ∂X 2 4 ∂X 2

The right hand side vanishes when V ′σ (X) = β σ exp{ik · X(w, w)} ¯ , α′ M 2 ≡ −α′ k 2 = −4 , V ′r (X) = β r 1 .

(3.16)

Here β σ and β r are constants and 1 is the identity operator. Eq. (3.16) includes the case V ′σ (X) = β σ exp{ik · X(w, w)} ¯ , α′ M 2 ≡ −α′ k 2 = −4 , V ′r (X) = 0 ,

(3.17)

as a special case. In what follows, we will consider a off-shell deformation of the following kind: O = cσ V ′′ + cr V ′ ,

(3.18)

where V ′ , V ′′ are generic scalar relevant operators.

D.

Nilpotency of δB

The condition we need in our formalism is δB2 O = 0 with O given by eq. (3.18). Using δB2 cσ = 0, we obtain δB2 O = cσ δB2 V ′′ + cr δB2 V ′ .

(3.19)

–9–

On the other hand, for both V = V ′ , V ′′ δB2 V

= iǫδB

(

α′ ∂2 c ∂σ V − (∂σ cσ + 2icσ ) V 4 ∂X 2 σ

)



= iǫ (δB cσ )∂σ V − cσ ∂σ (δB V) −

∂2 α′ (∂σ (δB cσ ) + 2iδB cσ ) V 4 ∂X 2

α′ ∂2 σ σ + (∂σ c + 2ic ) δB V . (3.20) 4 ∂X 2 The terms in the right hand side cancel with one another and we establish δB2 O = 0. 

E.

dS RP

2

The analyses made and the properties established above provide the ingredients necessary for us to define dS in the framework of BV on Σ′ = RP 2 as well. Following the disc case, 2 we introduce a defining differential equation (2.3) for S RP 2

K′ δS RP = δλα 2 K′ = 2 = K

ZZ C

Z Z C

C

dσ dσ ′ α (σ) hO (σ) {QBRS , O} (σ ′ )ighost V ′ ,RP 2 2π 2π

dσ dσ ′ r c (σ) (cr cσ ) (σ ′ ) C 2π 2π 



RP 2



C

V′ ≡

α′ ∂2 V (σ)(−) ∂σ′ V (σ ) − i V ′ (σ ′ ) 2 ∂X µ ∂Xµ ′

′α

′

dσ dσ ′ α′ ∂2 V ′ (σ ′ ) sin(σ − σ ′ )hc1 c0 c−1 iRP 2 V ′α (σ) ∂σ′ V ′ (σ ′ ) − i 2 ∂X µ ∂Xµ C 2π 2π 

ZZ

′

(3.21)

X

λα V ′α .

!

!

V ′ ,RP 2

,(3.22)

V ′ ,RP 2

(3.23)

α

Here the unnormalized path integral I

RP 2

= I

′bulk



···

+

Z



is evaluated with respect to

V′

C

dσ ′ V (σ) , 2π

(3.24)

Z

q 1 det ηη ij ∂i X µ ∂j X ν gµν d2 σ . (3.25) 4πα′ Σ′ −C Note that, in eq. (3.23), the allowed form of the nonvanishing ghost three point function has selected V ′ alone and V ′′ has disappeared.

I ′bulk =

F.

The Off-shell Crosscap States

The unnormalized path integral on RP 2 geometry can be represented as a matrix element between the ket vector of the closed string vacuum and the bra vector of the off-shell crosscap

–10–

state hC |: h· · ·iV ′ = hC | · · · | 0iV ′ .

(3.26)

The derivation of the equation which the bra vector hC | obeys involves a closed and unoriented nature of the Riemann surface RP 2 . The fundamental domain can be represented by Σ′ = {{r < 1}} ∪ {{r = 1, 0 ≤ σ < π}} on the complex plane. The variation of the action 2 I RP must be carried out consistently with the antipodal identification P (z) = − z1¯ . We find 2

δI RP = 1 2πα′

Z

π

1 2πα′

Z

π

0

0



d d + + w¯ )X µ (w, w) ¯ dσδX µ (w dw dw ¯ w=z, w=¯ ¯ z

Z

π

0



d d + + w¯ )X µ (w, w) ¯ dσδX (w dw dw ¯ w=−1/¯ z , w=−1/z ¯ µ



1 ∂V ′ (X µ (w, w)) ¯ dσδX µ 2π ∂X µ w=z, w=¯ ¯ z Z



¯ 1 ∂V ′ (X µ (w, w)) dσδX 2π ∂X µ w=−1/¯ z , w=−1/z ¯

π

µ

0

+ the part proportional to eq. of motion .

(3.2

Here, z = (1 − δ)eiσ and δ is a small but nonvanishing number. Reflecting the fact that RP 2 is nonorientable, we have nonvanishing contributions from an open cover of C = {{r = 1, 0 ≤ σ < π}}, namely, the edge of the fundamental domain Σ′ despite that there is no boundary. Stokes theorem does not apply. From the correspondence between the matrix element and the end point condition of the path integral, we conclude that the bra vector hC | must obey hC | Kµ (z, z¯; δ) = 0 , Kµ (z, z¯; δ) ≡ +

!

1 1 ∂V ′ ∂ ∂ µ X + w + w ¯ 2πα′ ∂w ∂ w¯ 2π ∂X µ !

∂ ∂ 1 Xµ + w + w¯ ′ 2πα ∂w ∂ w¯

! 1 ∂V ′ 2π ∂X µ

!

w=z, w=¯ ¯ z

.

(3.28)

w=−1/¯ z , w=−1/z ¯

Expansion of this condition in δ generates conditions which may be referred to as off-shell crosscap conditions. The initial point of the coupling constant flow is described as the ordinary crosscap condition: hC | (αn + (−)n α ˜ −n ) = 0 .

(3.29)

′

∂V The end point of the flow is described by zero of ∂X µ , which is nothing but the orientifold condition. hC | (αn − (−)n α ˜ −n ) = 0 . (3.30)

We have derived the two point function from eq. (3.28) in the case that V ′ is quadratic in X µ and checked that it takes the proper form on both ends of the flow 4 . 4

These will appear elsewhere together with other related issues.

–11–

Let us finish this section by discussing the properties of orientifold planes in general. They are certainly nondynamical objects in string perturbation theory. Nonperturbatively, they may become dynamical however. It is known in fact that the orientifold planes do become dynamical in Sen’s description of F theory [32] which connects F to orientifolds and the Seiberg-Witten curve of the UV finite 4 dimensional N=2 supersymmetric gauge theory. While we are not able to tell the nonperturbative dynamical origin which causes orientifold planes to get deformed, our construction does provide a proper description once this happens.

IV.

Total Action

Our construction of the action for open-closed string field theory is completed by adding 2 S disc and S RP . 2 S = S disc + S RP . (4.1) 2

The individual parts S disc and S RP obey the first order differential equations eqs. (2.16) and (3.23) respectively. Here we discuss the normalization constants appearing in these two quantities. It has been shown in [1, 2] that S disc on-shell, (that is, with no operator insertion) is actually the matter partition function of a free open string up to an overall normalization. 2 The same applies to S RP as well: on-shell it is the RP 2 matter partition function up to an overall normalization. We write these relations as

S disc

2 S RP

on−shell

on−shell

2

= Z disc /gdisc .

(4.2)

2

(4.3)

= Z RP /gRP 2 .

Here Z disc and Z RP are complete partition functions obtained by carrying out the path integrals over X µ and the two-dimensional metric. (They are actually free(vacuum) energies of a string as a single string can produce only a connected graph.) We denote by gdisc and gRP 2 the contributions in respective geometries from the path integrals of the two dimensional metric in the Polyakov formulation. They are made of the ghost determinant divided by the volume of the conformal killing vector and the order of the disconnected diffeomorphisms. Eq. (4.2) and eq. (4.3) can be related by invoking the cancellation between the dilaton tadpole of disc and that of RP 2 [33, 34, 35]. The operator insertion of a zero momentum tadpole in the first quantized string can be done by taking a derivative of the partition 1 function with respect to the tension of a fundamental string T f und = 2πα ′. On the other hand, the BV formalism guarantees that the right hand side of the basic equation dS = iV ω is a total differential at least locally. The integration constant is,

–12–

2

therefore, an additive one and, up to this constant, S disc/RP is completely determined: S disc = Knf disc (λα ) S RP

2

2

= K ′ f RP (λα ) ,

(4.4)

2

where f disc (λα ) and f RP (λα ) are both scalar functions, and can be determined by using solvable tachyonic profiles5 . Putting all these discussions together, we find that the cancellation of the dilaton tadpoles implies   ∂ RP 2 disc ′ 2 f Kng f + K g =0 . (4.5) disc RP on−shell ∂T f und In [8], K has been determined by demanding that a proper amplitude of a nearly on-shell three point open string tachyon be obtained. There is no argument in the RP 2 case which parallels the disc case as closed string tachyon with intercept −4 lives in the bulk of RP 2 and does not live in our nontrivial closed loop alone. We see that eq. (4.5) determines the normalization K ′ instead. A final remark of this subsection is on unoriented nature of our open and closed strings. The open-closed string field theory constructed above is nonorientable as our construction involves the nonorientable surface RP 2 . On-shell, the state space of an unoriented open string is obtained by choosing O vertex operators and Chan-Paton indices such that the product is even under the twist operation Ω. (We have nothing to say in this paper on the state space of an unoriented closed string.) Let us note, however, that Ω is a symmetry only at the two ends of the coupling constant flow. In the closed string picture, this is easily seen at our construction of the off-shell boundary/crosscap states. Eqs. (2.19), (2.20), (3.29), (3.30) stay invariant under the twist operation as Ω (αn ± α ˜ −n ) Ω−1 = ± (α−n ± α ˜n) , Ω (αn ± (−)n α ˜ −n ) Ω−1 = ± (α−n ± (−)n α ˜n ) .

(4.6)

At a generic value of the coupling, neither of eqs. (2.18), (3.28) stays invariant.

V.

Acknowledgements

We thank Akira Fujii, Nobuyuki Ishibashi, Koichi Murakami and Toshio Nakatsu for useful discussion on this subject. We are grateful to Santo seminar for providing an opportunity for collaboration. 5

We have set hc1 c0 c−1 idisc = hc1 c0 c−1 iRP 2 = 1.

–13–

Appendix

A

In this appendix, the mode expansion of a conformal field with weight p is obtained both on a unit disc D2 and on RP 2 . D2 and RP 2 are constructed from the complex plane by the respective identifications z ′ = z1¯ and z ′ = − z1¯ . q

Let Φ(z) ≡

z}|{

¯ z ) be an antiholobe a holomorphic field with weight p − q and Φ(¯

z···z (z) Φ z···z

|{z} p

morphic field with weight p − q. These mode expansions read respectively Φ(z) =

X

¯ z) = Φ(¯

φn z −n−p+q ,

n∈Z

X

φ¯n z¯−n−p+q .

(A.1)

n∈Z

¯ z ) are related to each other both on D2 and RP 2 and The modes of Φ(z) and those of Φ(¯ this relation can be made clear by extending the domain of Φ(z ′ ) to |z ′ | > 1 through the respective involutions z ′ = z1¯ (D2 ) and z ′ = − 1z¯ (RP 2 ): ∂z ′ ∂ z¯

¯ z) = Φ(¯

!p−q

Φ(z ′ ).

(A.2)

On |z| = 1, this leads to a relation

¯ z ) (¯ z )p−q Φ(¯

|z|=1



= (∓z)p−q Φ(±z)

|z|=1

.

(A.3)

The upper and lower signs refer respectively to the D2 and RP 2 cases. In terms of the modes, this equation reads φ¯n =

 

φ−n e−i(p−q)π , D2  φ−n e−i(p−q)π (−)n , RP 2 .

(A.4)

The radial, tangential and mixed components are constructed as q1

q2

z}|{ z}|{

r···r σ···σ ¯) ≡ Φ r···r σ···σ (z, z

|{z} |{z} p1

∂z ∂r

!p 1

∂z ∂σ

!p 2

∂r ∂z

!q 1

∂σ ∂z

!q 2

Φ(z)

p2

=

∂ z¯ + ∂r

!p 1

  p1

(iz)p2

z r

 p1

z¯ + r

∂ z¯ ∂σ 

!p2

z¯ 2r

(−i¯ z)

p2

∂r ∂ z¯

q1 



z 2r

!q1

1 2iz

q1 

 q2

∂σ ∂ z¯

!q2

¯ z) Φ(¯

(A.5)

¯ z) . Φ(¯

(A.6)

Φ(z)

1 − 2i¯ z

q2

On |z| = 1, we obtain r···r σ···σ Φ r···r σ···σ (z, z¯) |{z} |{z} p p q1

q2

z}|{ z}|{ 1

2

For example,

= (i)p2 −q2 |z|=1

 1 p−q  p2 −q2 p−q Φ(z)| + (−) . (∓) Φ(±z)| z |z|=1 |z|=1 2q

(A.7)

–14–

• p = 1, q = 0: s

αz (z) ≡ i

s

α ¯ z¯(¯ z) ≡ i

X 2 ∂z X(z, z¯) = αn z −n−1 , ′ α n X 2 ¯ ∂ X(z, z ¯ ) = α ˜ n z¯−n−1 , z ¯ α′ n

z z¯ αz (z) + α ¯ z¯(¯ z) , r r ασ (z, z¯) = izαz (z) − i¯ zα ¯ z¯(¯ z) , αr (z, z¯) =

αr (z, z¯)||z|=1 =

 

0, D2 z(αz (z) + αz (−z))||z|=1 , RP 2

ασ (z, z¯)||z|=1 =

 

2izαz (z)||z|=1 , D2 iz(αz (z) − αz (−z))||z|=1 , RP 2





,

, (A.8)

• p = 0, q = 1: cz (z) =

X

cn z −n+1 ,

c¯z¯(¯ z) =

n

cr (z, z¯) =

1 z¯cz (z) + z¯ cz¯(¯ z) 2r

i c (z, z¯) = − 2 σ

cr (σ) ≡ cr (z, z¯)||z|=1 =

σ

σ

c (σ) ≡ c (z, z¯)||z|=1 =

 

      

c¯n z¯−n+1 ,

n



  

X

cz (z) c¯z¯(¯ z) − z z¯



!

0, 1 2z

X

cn e

(cz (z) − cz (eiπ z))||z|=1 = −i

X

z

z

iπ

(c (z) + c (e z))||z|=1 =

−inσ

D2 , RP 2

n odd



1 z c (z) iz |z|=1

1 2iz

Some of these formulas are exploited in the text.

X

= −i

cn e−inσ ,

D2

n

cn e−inσ , RP 2

n even

(A.9)

–15–

Appendix

B

Let O(z, z¯) be a generic scalar operator with ghost number 2 on RP 2 geometry. We write this as O(z, z¯) =

cz (z) c¯(¯z ) (¯ z) V(z, z¯) . z z¯

(B.1)

Let QBRS be the BRS charge obtained from the integration of the holomorphic BRS current. We find 

z

z

[QBRS , O(z, z¯)] = − ∂z c (z) + (∂z c )(−1/¯ z)



!

∂2 α′ O(z, z¯) . 1+ 4 ∂X µ ∂Xµ

(B.2)

The on-shell ground state scalar is represented by the vertex operator V(z, z¯) = eik·X(z,¯z) ,

(B.3)

α′ M 2 = −α′ k 2 = −4 .

(B.4)

This latter condition is that of the tachyon of a closed unoriented bosonic string with intercept −4.

–16–

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