Fermionic Ghosts in Moyal String Field Theory

USCHEP/0304ib2 UT-03-08

hep-th/0304005 April, 2003

arXiv:hep-th/0304005v3 20 Jul 2003

Fermionic Ghosts in Moyal String Field Theory

I. Barsa,∗ , I. Kishimotob,† and Y. Matsuob,‡

a)

Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA b)

Department of Physics, Faculty of Science, University of Tokyo Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan

Abstract We complete the construction of the Moyal star formulation of bosonic open string field theory (MSFT) by providing a detailed study of the fermionic ghost sector. In particular, as in the case of the matter sector, (1) we construct a map from Witten’s star product to the Moyal product, (2) we propose a regularization scheme which is consistent with the matter sector and (3) as a check of the formalism, we derive the ghost Neumann coefficients algebraically directly from the Moyal product. The latter satisfy the Gross-Jevicki nonlinear relations even in the presence of the regulator, and when the regulator is removed they coincide numerically with the expression derived from conformal field theory. After this basic construction, we derive a regularized action of string field theory in the Siegel gauge and define the Feynman rules. We give explicitly the analytic expression of the off-shell four point function for tachyons, including the ghost contribution. Some of the results in this paper have already been used in our previous publications. This paper provides the technical details of the computations which were omitted there.

∗

e-mail address: [email protected] e-mail address: [email protected] ‡ e-mail address: [email protected] †

Contents 1 Introduction

2

2 Moyal’s star from Witten’s star in fermionic ghost sector

6

2.1

Half string formalism and regularization . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.1

Dirichlet at end point, Neumann at midpoint (DN)

9

2.1.2

Dirichlet at end point, Dirichlet at midpoint (DD) . . . . . . . . . . . . . . . 10

2.1.3

Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4

GL(N |N ) supergroup property of the regulator

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . 14

2.2

Moyal ⋆ from Witten’s ∗ for fermionic modes . . . . . . . . . . . . . . . . . . . . . . 16

2.3

Moyal ⋆ product in the bc ghost sector

2.4

Oscillators

. . . . . . . . . . . . . . . . . . . . . . . . . 17

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1

Oscillators on the fields in MSFT

. . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2

Oscillator as a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.3

L0 and L0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Monoid and Neumann coefficients

26

3.1

Monoid structure for gaussian elements

. . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2

Neumann coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3

Numerical comparison of Neumann coefficients . . . . . . . . . . . . . . . . . . . . . 34

4 Applications

35

4.1

Regularized MSFT action and equation of motion . . . . . . . . . . . . . . . . . . . . 35

4.2

Computing Feynman graphs including fermionic ghost sector . . . . . . . . . . . . . 37

5 Discussion

42

A Brief review of MSFT in matter sector

43

A.1 Half-string for cosine modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A.1.1 Dirichlet at midpoint

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.1.2 Neumann at midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1

A.1.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A.2 Some results in matter sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 A.2.1 Oscillators in MSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 A.2.2 Butterfly projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A.2.3 L0 and L0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.2.4 n-string vertex and Neumann coefficients . . . . . . . . . . . . . . . . . . . . 50 B Derivation of regularized matrix formula

53

C bc-ghost sector in position space

54

D Moyal ⋆ for (DD)b (N N )c split strings

56

E Ghost butterfly projector with even modes

57

F Algebra of gaussian operators

58

G Neumann coefficients

60

G.1 Neumann coefficients from CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 G.2 Ratios of MSFT-regulated and CFT Neumann coefficients . . . . . . . . . . . . . . . 61 H Twist and SU (1, 1) in the Siegel gauge

1

62

Introduction

Witten’s open bosonic string field theory [1], its operator version [2], and its split string reformulations [3, 4, 5, 6, 7], led to the development of the Moyal star formulation of string field theory (MSFT) during the past two years [7]-[12]. This formulation has the following features: 1. The star product in Witten’s string field theory is mapped to the Moyal star product [7] after an appropriate change of variables. The string field A (¯ x, xe .pe ) in the new basis is a function of the string midpoint x ¯ and the phase space (xe , pe ) of even string modes e = 2, 4, 6, · · · . While Witten’s star product, written in terms of Neumann coefficients [2], is very complicated to manipulate in computations, the Moyal star product in MSFT, which is diagonal in mode space labelled by e, is the simplest form of the star product that occurs in standard noncommutative geometry. This feature vastly simplifies the structure of string 2

field theory and is very helpful in explicit computations. 2. The change of variables introduces a set of simple but infinite matrices Teo , Roe , vo , we , labelled by e = 2, 4, 6, · · · , and o = 1, 3, 5, · · · , which contain basic information about even (e) and odd (o) string modes. These matrices obey a matrix algebra that has an associativity anomaly, which in turn feeds into an associativity anomaly among string fields [8]. The origin of the anomaly is the infinite number of string modes that cause the appearance of ambiguous terms of the form ∞/∞. In order to resolve this problem, we proposed [8, 9] a regularization by truncating the number of the oscillators to finite 2N , and defined a deformed set of finite matrices T, R, v, w as functions of the oscillator frequencies κe , κo of the 2N modes. After such regulation, the associativity is restored and all manipulations become well-defined. The original open string field theory is restored by taking the original frequencies, κe = e, κo = o, and the large N limit at the end of the computation. Through explicit computation of specific examples, it has been shown that this regulation procedure correctly reproduces results computed independently in conformal string theory. 3. In the regularized basis, we computed the Neumann coefficients analytically by using only the Moyal product. These coefficients are not needed for computations in MSFT, but they provide a check of MSFT relative to the operator formulation given in [2]. We have shown that the Neumann coefficients derived in the regularized MSFT framework satisfy the GrossJevicki nonlinear relations, and thus provide a generalization of Neumann coefficients for any set of frequencies κe , κo and any N. This provided the first consistency check of MSFT [9]. Furthermore, we found that the Neumann coefficients for any n-point vertex are all simple 1/2 −1/2 functions of a single matrix teo = κe Teo κo . Diagonalizing the matrix t diagonalizes all the Neumann coefficients simultaneously for all n-point vertices [9]. At large N our diagonal form agrees with the one given in [13][14], and explains in particular why there is Neumann spectroscopy for the 3-point vertex [13], and generalizes it to any n-point vertex [9]. 4. One of the nice features of MSFT is that the star product is diagonal in mode space (i.e. independent and same for each mode). The only cost for this simplification is that the kinetic term given by the Virasoro operator L0 becomes off-diagonal in mode space (xe , pe ) [9]. However, this has not hindered explicit computations. In particular, we have derived the Feynman rules, including the propagator (L0 − 1)−1 , and shown that we can evaluate efficiently and explicitly the Feynman graphs in open string field theory [10]. 5. The off-diagonal part of the kinetic term depends on a specific combination of momentum P modes, namely pˆ = (1 + ww) ¯ −1/2 e we pe which we refer to as the “anomalous midpoint mode” [8][11]. This mode appears in the kinetic operator in the form L0 = γ + · · · , with γ ∼ pˆ2 , while the remaining part of the kinetic term is diagonal in mode space. We named the term γ the “midpoint correction” . We found that if it were not for this midpoint correction, the rest of the kinetic plus interaction terms would define a theory, equivalent to an infinite matrix theory, that is vastly simpler and completely solvable [12]. However we 3

determined that the γ term is essential for the correct definition of string field theory. Thus we have isolated the hard part of string field theory in the form of the quadratic “midpoint correction” term γ. We have shown that all classical solutions, including the true vacuum, of the interacting string theory, are obtained analytically by first solving the nonlinear equation explicitly by ignoring γ, and then including the effect of γ in a closed formal expression that can be evaluated to any order in a perturbative expansion in powers of γ [12]. In our published work so far, we have demonstrated all of the above results explicitly mainly in the matter sector. We have implied, and sometimes explicitly included, the corresponding contribution of ghosts either in the bosonic [9] or fermionic [10, 12] version but the details were not given explicitly. The purpose of the present paper is to provide all the relevant material on fermionic ghosts which we used previously, in an organized and comprehensive manner. In this sense, this paper completes the basic formulation of MSFT. Because we will try to be quite explicit, the content of this paper will be rather technical. The construction of the Moyal product for fermionic ghosts is basically parallel to the bosonic case [7, 8, 9], except that we need to be careful in some minor differences, including midpoint issues, which appear in the ghost fields b, c. The first point relates to the boundary conditions. While we needed to consider Neumann type boundary conditions for the matter fields (open strings on the D25 brane), Dirichlet type boundary conditions appear for the ghost field. Therefore, the Fourier basis is different. The regularization method developed in [8, 9] was based on the Fourier modes, hence some care is needed to make the regularization compatible in the matter and ghost sectors. The regularization developed in this paper can be applied also to the treatment of the matter sector for lower Dp branes (p < 25). The second point relates to the overlapping conditions of split strings. For the matter fields, we have to treat only overlapping conditions for the split string degrees of freedom. For the ghost fields, on the other hand, we should also consider the anti-overlapping conditions. This induces some changes in the mapping of Witten’s star to Moyal’s star. The third point is the fermionic nature of Moyal variables. The usual bosonic derivatives which appear in the definition of the Moyal product get replaced by derivatives of fermionic variables, and care is needed in the ordering and signs. The fourth point is the treatment of the midpoint mode. This is a rather delicate issue since, as in the matter sector, it cannot be determined from the split string formulation. The fermionic midpoint mode is not part of the Moyal ⋆ product, and it is integrated in the definition of the action. In the Siegel gauge, the dependence on this extra fermionic variable becomes trivial, and it drops out in actual computations. We mention the work of Erler [16] where he defined the Moyal star formulation for the ghost system mainly in the continuous basis [14].1 There are overlaps of the current paper with his work 1

There are some works on Moyal structure of Witten’s string field theory.[15]

4

especially in the second and third points mentioned above. The correct treatment of the midpoint appeared first in our work [10], and his paper was modified subsequently. Beyond the preliminary level, for correct computation with ghosts, the results of the present paper are needed. The definition of the fermionic Moyal product outlined above is given in section 2. Because of its technical nature, we first summarize the fundamental formulae at the beginning of the section, and explain the full detail in the subsections. Readers who wish to skip the details of the derivation can proceed to later sections by skipping the latter part. In subsection 2.1, we first discuss the issue of boundary conditions in general for fermions and the corresponding regularization scheme. We also give a brief review of [9] in appendix A. Together with it, this paper provides the basic formulae for both the ghost and matter sectors in a compact form. We then discuss the mapping from Witten’s star to Moyal’s star in subsection 2.2. Finally we apply these techniques to the actual bc ghost system in subsection 2.3, and derive the correspondence between the conventional oscillators and Moyal variables in subsection 2.4. After this preparation, in section 3, we define the monoid algebra among gaussian string fields, which provides a useful tool for computations (this is almost a group, except for inverse). We compute the star product of n monoid elements, which as a by-product give the Neumann coefficients in the ghost sector. These were conjectured in [9] by using a nontrivial relation with the Neumann coefficients in the matter sector. In this paper, we derive them directly from the fermionic Moyal product. They satisfy the Gross-Jevicki- nonlinear relations exactly for any frequencies κe , κo and any N . This fact confirms the consistency of our construction including the midpoint prescription. As in the matter sector, they are simple functions of the matrix teo for any n-point vertex, and they can be related to the corresponding matter Neumann coefficients by the simple procedure of replacing the matrix t by its inverse. We also give the direct numerical comparison between the analytic form of the Neumann coefficients M obtained from conformal field theory and our algebraic expression. We confirmed that the approximate value for finite N converges to its exact value as N → ∞ with the following universal behavior, Mnm (N )/Mnm (cf t) ∼ 1 + anm N −α where the exponent is approximately α ∼ 1.33 for matter sector and α ∼ 0.67 for the ghost sector for any components of the Neumann coefficients. While this analysis is of different nature from the other parts in this paper, it is included here since it gives strong support on the consistency of MSFT in [9] and this paper. It also provides the basis for numerical computation of MSFT in our future study. In section 4 we apply the formalism. First, we present the derivation of the open string field action in the Siegel gauge by including both matter and ghost fields. This action was the starting point in our recent work [10, 12] where we used the results of the present paper without providing the details. Finally, we also compute the ghost contribution to the Feynman graphs for the fourpoint scattering amplitude for off-shell tachyons, whose matter sector was discussed in [10].

5

2

Moyal’s star from Witten’s star in fermionic ghost sector

In this section we construct the map from Witten’s star product to the Moyal star product for fermionic variables. The fermionic version of the Moyal ⋆ product was called “anti-Moyal star product” in the literature [18]. More general star products have also been considered in the context of deformation quantization of super-Poisson brackets [19]. The anti-Moyal star product will simply be referred to as the Moyal star product in the following. A first basic construction for the ghost bc system, following the one in the matter sector [7], was previously discussed in [16], while the correct treatment of the midpoint was first given in [10]. In this section we give the complete treatment, including the consistent regularization with the matter sector. We also define the even basis of ghost modes that is most transparent for our computations, after defining some other bases as well. We first summarize the main results of this rather lengthy and technical section: • The Moyal ⋆ acts on fermionic ghost modes ξ ≡ (xo , po , yo , qo ) (o = 1, 3, 5, · · · , 2N − 1) as ← − − ← − − ← − − ← − − → → → → !! θ′ X ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (A ⋆ B)(xo , po , yo , qo ) = A exp + + + B, 2 ∂xo ∂po ∂yo ∂qo ∂po ∂xo ∂qo ∂yo o>0 (2.1) where θ ′ is a parameter which absorbs units, and if desired, could be absorbed away by a rescaling of the variables which amounts to a choice of units. We note the canonical structure {xo , po′ }⋆ = θ ′ δoo′ = {yo , qo′ } . This odd basis (o) of ghost modes, which was used in [10], is naturally defined in the process of mapping the Witten star to the Moyal star, including the treatment of the midpoint. However, a more transparent basis that is more parallel to the matter sector, which simplifies the overall formalism, is obtained by rewriting the odd basis, through the following linear canonical transformation, in terms of an even basis xbe , pbe , xce , pce (e = 2, 4, · · · , 2N ), where the labels b, c refer to the modes of the usual b, c ghosts X X X X Seo po , xce := Teo yo , pce := qo Roe . (2.2) Seo xo , pbe := κe xbe := κ−1 e o>0

o>0

o>0

o>0

This even basis is different than the one defined in [16]. The Moyal product is rewritten as ← − − → ← − − → ← − − → ← − − → !! ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ θ′ X b b c c + + + B. (A ⋆ B)(xe , pe , xe , pe ) = A exp 2 e>0 ∂xbe ∂pbe ∂xce ∂pce ∂pbe ∂xbe ∂pce ∂xce (2.3) c c b b ′ We note the canonical structure xe , pe′ ⋆ = θ δee′ = xe , pe′ ⋆ . The linear transformation matrices, T, R (and matrices U and vectors v, w which appear in the following) were defined in [9], while the matrix S appears for the first time in this paper. Their properties are derived explicitly in subsection 2.1. They play a central role in MSFT since they define a Bogoliubov transformation from the oscillators in the operator formalism to the Moyal coordinates, and thus carry essential information about string theory. For instance, see Eqs.(2.4,2.6) in the 6

next paragraph. For the moment, we just mention that they satisfy the inverse properties ¯ = 1 which prove that the two versions of the star product (2.1) and T R = RT = S S¯ = SS (2.3) are canonically equivalent. Throughout this paper, the bar (¯) means the transpose of matrices or vectors. The relation of split strings to the Moyal star is discussed in subsection 2.2. • In the operator formulation, the ghost sector of the string field |Ψi is represented in the Fock space of the bc ghost oscillators. The transformation to the Moyal field A (ξ) as a function of noncommutative coordinates ξ = (x, p) is obtained through the Fourier transformation [7] of the coordinate representation of |Ψi. The whole procedure is more neatly expressed, as in the matter sector [9], by the inner product with a particular bra state hξ0 , ξ1 , ξ2 | in the Fock space, where ξ¯1 = (xo , po ) , ξ¯2 = (yo , qo ) are the noncommutative fermionic coordinates, and ξ0 is a fermionic variable related to the zero mode dependence of |Ψi 1

hξ0 , ξ1 , ξ2 | = −2−2N (1 + ww) ¯ − 4 hΩ|ˆ c−1 e−ξ0 (ˆc0 −

√

(o)

coˆbo −2iξ¯1 M0 ξ2 −ξ¯1 λ1 −ξ¯2 λ2 2wˆ ¯ ce ) cêˆbe −ˆ

e

ˆ 0 , ξ1 , ξ2 ) = hξ0 , ξ1 , ξ2 |Ψi . A(ξ

, (2.4) (2.5)

Here hΩ| is the SL(2, R) invariant bra Fock vacuum, ˆbn , cˆn are the conventional ghost oscilla(o) tors, and the matrices M0 and λ are defined as ! ! ! √ √ 1 co −i 2ˆ − 2ˆbo 1o 0 (o) 2 √ √ , λ2 = , λ1 = M0 = . 2i ¯ 2 ¯¯ ′ 2ˆbe + we ξ0 0 − 2 ′ 2 Roe cê ′ Soe − 2 SR θ′

oo

θ

θ

(2.6) The matrices we , Seo , Roe are functions of the oscillator frequencies κe , κo as given below. Through Eqs.(2.1,2.4) we map Witten’s star (⋆W ) into the Moyal’s star ⋆ as follows hξ0 , ξ|Ψ1 ⋆W Ψ2 i ∼ hξ0 , ξ|Ψ1 i ⋆ hξ0 , ξ|Ψ2 i , W

|Ψ1 ⋆

Ψ2 i3 =

1 hΨ1 |2 hΨ2 |V3 i123 ,

1 hΨ1 |

(2.7) = 14 hV2 |Ψ1 i4 , 2 hΨ2 | = 25 hV2 |Ψ2 i5 . (2.8)

We note that the product is local in ξ0 , while ξ0 plays a similar role to the midpoint coordinate x ¯µ in the matter sector. The Moyal star reproduces correctly the three string vertex |V3 i and the reflector |V2 i of the operator formalism [2]. The details are given in subsection 2.3. The precise correspondence including the zero mode is in section 3. • Eq.(2.4) is enough to derive the connection between the conventional operator formalism and the Moyal star formalism. For example, the action of the standard oscillators on the Fock space field |Ψi can be rewritten in terms of their Moyal images acting on the field Aˆ = hξ0 , ξ1 , ξ2 |Ψi through the star product, as follows ˆb0 |Ψi ↔ −ξ0 Aˆ , θ′ ∂ ∂ Aˆ , + v¯ cˆ0 |Ψi ↔ − ∂ξ0 2 ∂qo ˆbo |Ψi ↔ √1 β b ⋆ Aˆ − (−1)|A| Aˆ ⋆ β b o −o , 2 7

cô |Ψi ↔ ˆbe |Ψi ↔ cê |Ψi ↔

1 c √ βoc ⋆ Aˆ + (−1)|A| Aˆ ⋆ β−o , 2 1 b √ βeb ⋆ Aˆ + (−1)|A| Aˆ ⋆ β−e + we′ ξ0 Aˆ , 2 1 c ˆ c √ βe ⋆ A − (−1)|A| Aˆ ⋆ β−e , 2

(2.9)

where βob , βoc are fields in Moyal space which obey oscillator relations under the ⋆ product 2 1 2 1 b c q − iǫ(o)x|o| , βo := y|o| − i ′ ǫ(o)p|o| , {βob , βoc′ }⋆ = δo+o′ . βo := (2.10) 2 θ ′ |o| 2 θ On the other hand βeb , βec are not independent from the βob , βoc . Rather, they are their Bogoliubov transforms which obey the following relations βeb =

X o

−1 , βec = βob U−o,e

X o

Ue,−o βoc , {βeb , βec′ }⋆ = δe+e′ ,

(2.11)

b and β−o , βec ⋆ = Ue,−o where the matrix Ue,−o will be given below. With these formulas, one can directly translate operators in Fock space into their images which act in Moyal space. The proof of these formulas and some variants are discussed in subsection 2.4. In this way we derive the explicit form of the Virasoro operator L0 which acts in Moyal space L0 =

2N X

k=1 2N X

c ˆb b ˆc βk ) κk (βˆ−k βk + βˆ−k

(2.12)

! X ∂ ∂ θ′2 ∂ ∂ 4 = κk + i po q o + κo xo yo + +i κo ∂xo ∂yo 4 ∂po ∂qo θ′2 o>0 o>0 k=1 ! ! ! X X X 4i 2i vo κo po ξ0 . ¯ + ′ 2 (1 + ww) κo vo po ¯ vo′ qo′ + ′ (1 + ww) θ θ ′ X

o>0

o >0

o>0

This was used in [10, 12]. Here βˆkb , βˆkc are not the Moyal fields βob , βoc or βeb , βec given in Eqs.(2.10,2.11); rather, they are differential operators that obey the standard oscillator relations, and which are derived from the star products of the fields βob , βoc or βeb , βec as will be shown below.

2.1

Half string formalism and regularization

We start from full string functions ψ(σ), 0 ≤ σ ≤ π with Dirichlet boundary conditions at σ = 0, π, and discuss their split string formulation in the interval 0 ≤ σ ≤ π/2. Such functions have a Fourier expansion with only sine modes in the full string formalism. By contrast, the corresponding problem in the matter sector involved only cosine modes because of Neumann boundary conditions at σ = 0, π. We collect the basic formulae in the appendix A.1. The essential step was the construction of the regularization [8, 9] which is needed to avoid the associativity anomaly. We

8

give a regularization of the ghost sector for the sine mode expansion which is compatible with the previous results. A full string function ψ(σ) which satisfies Dirichlet boundary conditions ψ(0) = ψ(π) = 0 is expanded as √ Z π ∞ √ X 2 ψ(σ) = 2 dσψ(σ) sin nσ . (2.13) ψn sin nσ , ψn = π 0 n=1

We decompose such a field ψ(σ) into left half l(σ) and right half r(σ) as follows ( √ Z π l(σ) 0 ≤ σ ≤ π2 2 2 ψ(σ) = , ψ = dσ(l(σ) − (−1)n r(σ)) sin nσ . n π π r(π − σ) 0 2 ≤σ ≤π

(2.14)

As we have seen in [7, 8], we have some arbitrariness in the choice of the boundary condition for the split string functions l(σ), r(σ) at the midpoint σ = π/2. They can be expanded using either odd or even modes according to the two possible choices of the boundary condition at the midpoint σ = π2 as discussed below. 2.1.1

Dirichlet at end point, Neumann at midpoint (DN)

First we consider Neumann boundary conditions at the midpoint σ = π/2, while we have Dirichlet boundary conditions at the end point for the split string functions l(σ), r(σ): l(0) = r(0) = 0 ,

l′ (π/2) = r ′ (π/2) = 0 .

We can expand them using odd sine modes o = 1, 3, 5, · · · √ Z π ∞ √ X 2 2 2 lo sin oσ , lo = l(σ) = 2 dσ l(σ) sin oσ , π 0 o=1 √ Z π ∞ √ X 2 2 2 ro sin oσ , ro = r(σ) = 2 dσ r(σ) sin oσ . π 0 o=1

(2.15)

(2.16) (2.17)

By comparing the mode expansions Eqs.(2.13,2.16,2.17) with (2.14), we obtain the correspondence between the full and split string variables, lo = ψo + S¯oe ψe ,

ro = ψo − S¯oe ψe ,

(2.18)

or the inverse

1 1 Seo (lo − ro ) , ψo = (lo + ro ), 2 2 where e = 2, 4, 6, · · · , and the matrix Seo is given by Z π 4 2 4io−e+1 e . Seo = dσ sin eσ sin oσ = π 0 π(e2 − o2 ) ψe =

(2.19)

(2.20)

The above mappings are consistent because Seo is an orthogonal matrix: ¯ = S S¯ = 1 . SS 9

(2.21)

The continuity condition at the midpoint ψ(π/2) = l(π/2) = r(π/2) is ∞ X

w õ ψo =

∞ X

w õ lo =

o=1

o=1

∞ X

w õ ro

(2.22)

n=1

where we defined the odd vector w ˜ associated with the midpoint √

oπ √ o−1 . (2.23) = 2i 2 P Eq.(2.22) holds thanks to the identity (S w) ˜ e= ∞ õ = 0. This equation implies that S has a o=1 Seo w ¯ as stated in (2.21). singular eigenvector even though it has an inverse, which is just its transpose S, This esoteric relation is possible because S is an infinite matrix. However it causes an associativity anomaly with respect to matrix products, which in turn feeds into associativity anomaly of string field star products, as discussed in [8]. w õ =

2 sin

¯ w) ¯ w An example of the associativity anomaly is S(S ˜ = 0, but (SS) ˜ = w. ˜ Each single sum indicated by the parentheses has a unique answer, but the double sums are ambiguous. The reason is that, due to infinite sums, in the first expression there are terms of the form ∞/∞ which are ambiguous. Since these matrices appear in many physical computations we must give an unambiguous definition of the matrix product. We will resolve this ambiguity in computations successfully by a regularization procedure. 2.1.2

Dirichlet at end point, Dirichlet at midpoint (DD)

We consider another possibility: we define the midpoint of the string ψ¯ := ψ(π/2), and impose Dirichlet boundary conditions at both σ = 0, π/2 on the split string functions l(σ), r(σ), l(0) = r(0) = 0 ,

¯ l(π/2) = r(π/2) = ψ.

(2.24)

We note that the midpoint value ψ¯ is an additional degree of freedom in the split string basis and cannot be chosen arbitrarily. We expand l(σ), r(σ) using even sine modes, e = 2, 4, 6, · · · ∞ √ X 2 l(σ) = σ ψ¯ + 2 le sin eσ , π e=2 ∞

√ X 2 re sin eσ , r(σ) = σ ψ¯ + 2 π e=2

√ Z π 2 2 2 2 le = dσ l(σ) − σ ψ¯ sin eσ , π 0 π √ Z π 2 ¯ 2 2 2 dσ r(σ) − σ ψ sin eσ . re = π 0 π

(2.25) (2.26)

Again, by comparing the mode expansions Eqs.(2.13,2.25,2.26) with (2.14), the correspondence between split and full string variables is obtained ψ¯ = w õ ψo , or the inverse ψe =

le = ψe + T˜eo ψo ,

1 (le − re ) , 2

re = −ψe + T˜eo ψo ,

1 ψo = u õ ψ¯ + S¯oe (le + re ) , 2 10

(2.27)

(2.28)

where 1 4o2 io−e+1 = Seo + v˜e w õ = 2 Seo o2 , πe(e2 − o2 ) e √ Z π √ 4 2 2 4 2 io−1 = dσσ sin oσ = , π2 0 π 2 o2 √ e √ Z π 2 2i 4 2 2 . dσσ sin eσ = = − 2 π eπ 0

T˜eo = u õ v˜e

(2.29) (2.30) (2.31)

¯ le , re ) ↔ (ψn ) are consistent by the relation The maps (ψ, õ = 1, u õ w

T˜S¯ = 1 ,

T˜u ˜ = 0,

Sw ˜ = 0.

(2.32)

We can prove the following relations among infinite matrices S, T˜ and vectors u ˜, v˜, w ˜ by straightforward computation: S = κ2e T˜ κ−2 o ,

¯ S = T˜ − v˜w ˜,

¯ = S S¯ = T˜S¯ = 1, SS

¯v = −˜ S˜ u,

¯˜ , S¯T˜ = 1 − u ˜w

Su ˜ = −˜ v,

Sw ˜ = T˜u ˜ = 0,

1 ˜, (2.33) T¯˜v˜ = −˜ u+ w 3 1 ¯˜ u = 1 , v¯˜v˜ = u ¯˜u w˜ ˜ = , (2.34) 3

where κe , κo are the diagonal matrices κe = diag(2, 4, 6 · · · ) and κo = diag(1, 3, 5, · · · ). This algebra is similar to the one among the infinite matrices T, R, v, w which appeared in the matter sector [8], where a full string function ψ(σ) is expanded in terms of cosine modes (see §A.1). 2.1.3

Regularization

In the split string formulation given in §A.1.1 §A.1.2 §2.1.1 §2.1.2, we encountered a set of infinite dimensional matrices T, R, S, T˜ and vectors w, v, w, ˜ v˜, u ˜. These represent Bogoliubov transformations between odd and even modes, with (T, R, w, v) appearing when the full string is expanded in terms of cosine modes, and (S, T˜, w, ˜ v˜, u ˜) appearing when the full string is expanded in terms of sine modes. Such transformations are essential in the Moyal formulation since they carry basic information about string theory. We note that, the Moyal star product itself, which is applied independently for each mode, has no specific information about string theory, and as such is a more general structure. In the analysis of the matter sector [8] as well as the sine mode expansion given so far, there appears an associativity anomaly in the matrix algebra of these matrices. This originates from the infinite dimensionality of these matrices. It produces ambiguities in computations in string field theory. To have a well defined theory, it is mandatory to define a deformed, unambiguous, associative algebra that preserves the basic matrix algebraic structure of these matrices [8, 9]. Such a deformation contains a parameter N, that corresponds to the rank of the matrices. The original definition of these matrices given above is reproduced by taking the limit N → ∞ of this parameter. All computations in the open string field theory are performed unambiguously with finite N, and 11

the correct value in string theory is obtained at the end of the computation by taking the limit N → ∞. This is the basic strategy for practical computations in the MSFT proposal. The deformed set of matrices for T, R, w, v that have correctly reproduced string theory was proposed in [8, 9]. Since an additional set of matrices S, T˜, w, ˜ v˜, u ˜ have appeared in the ghost sector we need to obtain their deformation consistently with the matter sector. For that purpose, we start from the infinite dimensional matrix U and vectors w′ , v ′ defined in [9], U−e,o =

2 io−e−1 , π o−e

−1 = U−o,e

2 e io−e−1 , πo o−e

we′ = i−e+2 ,

vo′ =

2 io−1 , π o

(2.35)

where e (o) now run over both positive and negative integers ±2, ±4, · · · (resp. ±1, ±3, · · · ). In single sums these matrices satisfy X o

X

−1 U−e,o U−o,e ′ = δe,e′ ,

e6=0

−1 U−e,o′ = δo,o′ , U−o,e

(2.36)

which we denote U U −1 = 1e , U −1 U = 1o for short in the following. More importantly, there exists the following matrix relations among them, U −1 = κ′o

−1 ¯ ′ U κe ,

¯ + v′ w U −1 = U ¯′ ,

¯ w′ , v′ = U

¯ −1 v ′ , w′ = U

(2.37)

where κ′e = diag(· · · , −4, −2, 2, 4, · · · ), and κ′o = diag(· · · , −3, −1, 1, 3, · · · ) are the diagonal matrices which specify the spectrum. These relations will be used as the defining relations. From them it is possible to derive the matrices themselves, as given in Eq.(2.35). Therefore, they will be used as the basic relations that are also satisfied by the deformed matrices, as given below. The first relation implies that U defines an invertible Bogoliubov transformation between even and odd spectra (see also Eq.(2.11)). The second relation shows that the transformation U is almost orthogonal except for the vectors v ′ , w′ which are associated with the midpoint mode. Finally the last two define the relation between the vectors. On the other hand, the matrices (2.35) also satisfy ¯ = 1e , UU

U v ′ = 0,

v¯′ v ′ = 1 ,

(2.38)

which break the associativity [8, 9]. Therefore, these relations will be deformed in the regulated theory, as seen below. Of course these equations will hold when the regulator is removed. All the other matrices are written in terms of U, w′ , v ′ , (for e, o > 0), Teo = U−e,o + Ue,o ,

−1 −1 + Uo,e , Roe = U−o,e

we =

√

2we′ ,

−1 −1 − Uo,e , T˜eo = κ−1 Seo = U−e,o − Ue,o = U−o,e e T κo , π 2 2 w ˜ = κo v , v˜ = − κ−1 w, u ˜ = κ−1 o v, π 2 π e

vo =

√

2vo′ ,

(2.39) (2.40) (2.41)

where κo and κe are restrictions of κ′o and κ′e to the positive sector. These definitions in terms of U, w′ , v ′ together with the relations (2.37) among U, w′ , v ′ are sufficient to derive all the relations 12

among T, R, w, v, S, T˜, u˜, v˜, w. ˜ Therefore, we may use these relations as the definitions of these matrices and vectors even when we use the regularization of U, w′ , v ′ . In [8, 9, 10], the regularization of U, v ′ , w′ is given explicitly. We truncate the size of U, v ′ , w′ to 2N while keeping their property of Bogoliubov transformation between even and odd spectrum. It turns out that one may take the even and odd frequencies κe , κo as arbitrary functions of the positive integers (e, o) , while keeping the reflection property of κ′e,o to extend the definition to negative integers κ′−e = −κ′e , κ′−o = −κ′o . Therefore we put κ′e = ǫ(e)κ|e| ,

κ′o = ǫ(o)κ|o| .

(2.42)

We suppose implicitly that κ′e 6= 0, κ′o 6= 0 and that these are not degenerate. The matrices U, U −1 and vectors w′ , v ′ can then be derived from the defining relations (2.37) as functions of the arbitrary spectral parameters κe , κo as (see appendix B for details of the derivation) we′ vo′ κ′o , κ′e − κ′o

we′ vo′ κ′e −1 −1 , = Uo,−e , U−e,o = Ue,−o , U−o,e κ′e − κ′o 1 Q 2 2 − 1 2 κ /κ ′ ′ √ ′ o >0 o |e| ′ ′ 2 we = w|e| = i2−e 1 , we = w−e , Q 2 2 2 e′ >0,e′ 6=|e| κ|e| /κe′ − 1 1 Q 2 /κ2 2 − κ √ ′ e′ >0 1 e′ |o| ′ ′ 2 vo = v|o| = i|o|−1 1 , vo = v−o Q 2 2 2 o′ >0,o′ 6=|o| 1 − κ|o| /κo′

U−e,o =

−1 = U−o,e

(2.43)

(2.44)

(2.45)

where now the indices (e, o) run over the finite set e = ±2, ±4, . . . , ±2N and o = ±1, ±3, . . . , ±(2N − 1). It is easy to check explicitly that in the limit N → ∞ and κe = e, κo = o, these expressions reduce to Eq.(2.35). The regulated expressions for these matrices look considerably more complicated than their large N limit. Therefore, it may appear that this would create a problem in analytic computations. Actually this is not the case at all, because in analytic computations one uses the matrix relations satisfied by these matrices rather than the explicit matrices themselves. The relations are preserved in the regularized version for any N, and they look the same as their N = ∞ counterpart. Therefore analytically the expressions in any computation look the same in the regulated or infinite versions as long as they are written in terms of these matrices without using their explicit form. The explicit construction of the regulated version insures associativity and eliminates the ambiguity of the associativity anomaly as explained below. Thus, in addition to the basic defining relations (2.37) which are the same for any N, including N = ∞, there are more relations among U, U −1 , v ′ , w′ that can now be derived from the defining relations alone for any N, κ′e , κ′o : ¯ −1 U −1 = 1 + w′ w ¯ U = 1 − v ′ v¯′ , U −1 U = 1 , U ¯′ , U ′ ¯′ w′ w ¯ ′ w′ ′ ′ ′ ¯ =1− ww UU , U v = , v ¯ v = , 1+w ¯ ′ w′ 1+w ¯ ′ w′ 1+w ¯ ′ w′ U U −1 = 1 ,

13

(2.46) (2.47)

¯ −1 = 1 + v ′ v¯′ (1 + w U −1 w′ = v ′ (1 + w ¯′ w′ ) , U −1 U ¯ ′ w′ ) , Q Q ′ κ2 κ 1+w ¯ ′ w′ = Qe e′ = Qe>0 e2 . o κo o>0 κo

(2.48) (2.49)

In particular note that Eqs.(2.38) are now deformed into Eqs.(2.47), and that w ¯ ′ w′ → 2N → ∞ in the large N limit. Thus, the deformed algebra actually holds also at N = ∞; it simply makes explicit the behavior as a function of N. With this, the associativity anomaly hidden in the original algebra is now resolved and the matrix algebra for all the matrices U, U −1 , T, R, S, T˜, v, w, v˜, w, ˜ u ˜ defined through Eqs.(2.39–2.45) becomes associative. In particular, from Eq.(2.39–2.41) we obtain the regularized matrices, such as T, R, S, Teo =

we vo κ2o , κ2e − κ2o

Roe =

we vo κ2e , κ2e − κ2o

Seo =

we vo κe κo . κ2e − κ2o

(2.50)

From the relations among U, w′ , v ′ we can derive the relations among T, R, v, w, S, T˜, u ˜, v˜, w. ˜ The deformed algebra among T, R, w, v (A.18) is already given in [8], while the deformed algebra among S, T˜, u ˜, v˜, w ˜ which replaces (2.33,2.34) is2 , S = κ2e T˜κ−2 o ,

¯˜ , S = T˜ − v˜w

¯v = −˜ S˜ u,

Su ˜ = −˜ v, −1 ¯ e ¯ = 1 , S S¯ = 1 , S¯T˜ = 1 − u ¯˜ , T˜S¯ = 1 − κe wwκ SS ˜w , 1 + ww ¯ 2 2 κe w 2 2 κ−1 ¯ e w ˜ ˜ Sw ˜= , Tu ˜= , T v˜ = −˜ u+ wκ ¯ −2 ˜, e ww π 1 + ww ¯ π 1 + ww ¯ π 2 2 2 2 ww ¯ −2 ¯ ¯ ¯ wκ ¯ e w, u ˜u ˜= v¯κ−2 , v˜v˜ = w˜ ˜u = v¯v = o v. 1 + ww ¯ π π

(2.51) (2.52) (2.53) (2.54)

Furthermore, note that for any N, κe, κo we have wκ ¯ −2 ¯κ−2 e w =v o v =

X o>0

κ−2 o −

X

κ−2 e .

(2.55)

e>0

2

The right hand side converges to π12 if we take the open string limit κe = e, κo = o, N = ∞. More relations of this type can be found in the next subsection. 2.1.4

GL(N |N ) supergroup property of the regulator

In this subsection we take a small detour to make an observation on the regulator whose significance for computations in MSFT is not yet fully apparent, but which is mathematically interesting, and could be useful in future applications. Many computations in MSFT boil down to expressions of the form wf ¯ (κ) w where f (κ) is a matrix constructed from the frequencies κe , κo through the regulated matrices we discussed above. Therefore, we are interested in developing analytic methods of computation involving such expressions, in particular for arbitrary frequencies κe , κo . In such 2

Note that for finite N , the continuity condition at the midpoint (2.22) is not satisfied because S w ˜ 6= 0. However, we recover it, as well as all other infinite matrix relations by taking the open string limit κe = e, κo = o, N = ∞.

14

computations the properties of the supergroup GL(N |N ) mysteriously makes an appearance, as follows. By using explicitly the expression for we given in Eq.(2.44), we have I X deto κ2e κ−2 dz f (z) det 1 − zκ−2 o −1 o 2 2 = , wf ¯ κe w = f κe 2πi z det 1 − zκ−2 dete′ 6=e κ2e κ−2 e e′ − 1 e

(2.56)

where the contour encircles only the poles at z = κ2e . The contour may then be deformed to evaluate the integral. When f κ2e = 1 or κ−2 e the results have already been given in Eqs.(2.49,2.55). We note that these may be written in the form ww ¯ = −1 + Sdet κ2 ,

−2 wκ ¯ −2 , e w = −Str κ

where we used the superdeterminant (Sdet) and supertrace (Str) by treating the matrix ! κe 0 κ= 0 κo

(2.57)

(2.58)

as if it is a graded GL(N |N ) super matrix. When f κ2e = κ−2n , with n = 1, 2, · · · , we note e that the contour integral is precisely the integral representation of the supercharacter of GL(N |N ) for the representation of GL(N |N ) described by a Young supertableau with a single row with n superboxes [17] (2.59) wκ ¯ −2n w = −χn κ−2 . e

The expression for the supercharacter χn (M ) for any supermatrix M, can also be written in terms of supertraces of powers of M in the fundamental representation, as given in [17]. By taking advantage of this observation we evaluate χn κ−2 in terms of the supertraces of powers of κ−2 . For example, χ2 (M ) = 21 Str M 2 + 12 (Str (M ))2 , which gives the following interesting expression for the sum 1 2 1 −2 T r κ−2 − T r κ−4 − T r κ−2 = − T r κ−4 − . wκ ¯ −4 e e o o e w = −χ2 κ 2 2

(2.60)

We can check the correctness of Eq.(2.59) in the limit κe = e, κo = o, N = ∞. In this limit, √ since we → 2 i2−e , we can evaluate the sum on the left side directly in terms of the zeta funcP −2n 2 tion, wκ ¯ −2n w→2 ∞ = 22n ζ (2n) , and then compare it to the value generated by the e k=1 (2k) −2 supercharacter −χn κ in the limit.

For example, for n = 1 the right hand side of Eq.(2.59) is already given following Eq.(2.55), and this agrees with the left hand side which is 222 ζ (2). Similarly, for n = 2 the left hand side of 1 7 1 Eq.(2.60) gives 224 ζ (4) = 720 π 4 while the right hand side gives 16 ζ (4) − 18 (ζ (2))2 = 720 π 4 which agree. The GL(N |N ) supergroup property of these sums is intriguing. It may be the signal of an underlying mathematical structure that could be helpful in computations in MSFT.

15

2.2

Moyal ⋆ from Witten’s ∗ for fermionic modes

The second step in constructing the map from Witten’s star to Moyal’s star is to perform the Fourier transformation from position space to momentum space for a subset of string modes [7]. We recall the definition of Witten’s ∗-product for functions of split strings in the ghost sector Z Ψ1 ∗ Ψ2 [l(σ), r(σ)] ∼ Dz(σ)Ψ1 [l(σ), z(σ)]Ψ2 [±z(σ), r(σ)] . (2.61) For the bc ghost sector, we consider two types of overlapping conditions [2]. The anti-overlapping condition (resp. overlapping condition) is defined by choosing the minus (resp. plus) sign in this formula. In both cases the Witten ∗ product is mapped to the Moyal ⋆ product as follows. ˆ p] in the Moyal We denote the string field as Ψ[l, r] in the split string formulation and as Ψ[x, formulation. The variables l, r, x, p are all fermionic and we consider the simplified situation where each of them represents a single degree of freedom. The generalization to multiple variables is straightforward. For the simplified setup we define Witten’s star product in the split string formalism (ignoring the midpoint for the time being) as Z |Ψ1 | Ψ1 ∗ Ψ2 [l, r] = (−1) dw Ψ1 [l, w] Ψ2 [±w, r] . (2.62) The sign factor (−1)|Ψ1 | (Grassmann parity of Ψ1 ) is needed to make the ∗ product associative. We define the mapping from a string field in the split string picture Ψ[l, r] to the Moyal picture ˆ p] by using the Fourier transform3 Ψ[x, Z h yi y ˆ , (2.63) Ψ[x, p] = ± dy e−py Ψ ±x + , x ∓ 2 2 Z ˆ r ± l,p . Ψ[l, r] = ± dp ep(l∓r) Ψ (2.64) 2 ˆ Witten’s star for Ψ is then mapped to Moyal’s star for Ψ: ˆ1 ⋆ Ψ ˆ 2 [x, p] = Ψ ˆ 1 [x, p] exp ∓ 1 Ψ 2

← −− ← −− → → !! ∂ ∂ ∂ ∂ ˆ 2 [x, p] . Ψ + ∂x ∂p ∂p ∂x

(2.65)

The derivation of this correspondence is completely parallel to the bosonic case [7]: 1

← − ∂

→ − ∂

− ← − → ∂ ∂

ˆ 1 [x, p]e∓ 2 ∂x ∂p + ∂p ∂x Ψ ˆ 2 [x, p] Ψ Z ← Z − − − → ← − → h h ∂ ∂ ∂ ∂ y1 i ∓ 21 ∂x y2 i y2 y1 + ∂p ∂p ∂x e ± dy2 e−py2 Ψ2 ±x + , x ∓ = ± dy1 e−py1 Ψ1 ±x + , x ∓ 2 2 2 2 ← Z − → ← − → − − h i h ∂ ∂ ∂ ∂ y2 y1 −py1 ∓ 12 ∂x y2 i y1 + ∂p ∂p ∂x e e e−py2 Ψ2 ±x + , x ∓ = (−1)|Ψ1 | dy1 dy2 Ψ1 ±x + , x ∓ 2 2 2 2 Z i 1← i h h − → − ∂ 1 ∂ y y y y 1 2 1 2 e± 2 ∂x y2 e−p(y1 +y2 ) e∓ 2 y1 ∂x Ψ2 ±x + , x ∓ = (−1)|Ψ1 | dy1 dy2 Ψ1 ±x + , x ∓ 2 2 2 2 3

If l and r consist of N variables, the sign factor on the right hand sides of Eqs.(2.62)(2.63)(2.64) become (−1)|Ψ1 |N , (±1)N , (±1)N respectively. In particular, they are trivial in the case that N is even.

16

y1 + y2 y1 − y2 y1 + y2 y1 − y2 = (−1) dy1 dy2 e Ψ1 ±x + ,x ∓ ,x ∓ Ψ2 ±x − 2 2 2 2 Z Z i h i h y y = ±(−1)|Ψ1 | dye−py dz Ψ1 ±x + , z Ψ2 ±z, x ∓ 2 2 Z h i y y ˆ1 ⋆ Ψ ˆ 2 [x, p] . = ± dye−py (Ψ1 ∗ Ψ2 ) ±x + , x ∓ =: Ψ (2.66) 2 2 |Ψ1 |

Z

−p(y1 +y2 )

The form of the product in Eq.(2.65) is similar to the ordinary Moyal product although the derivatives in the exponential are for fermionic variables. This Moyal ⋆ product is associative and noncommutative. From Eq.(2.64), we also obtain the correspondence between the definition of trace in the splitstring formulation, which we take with the anti-periodic condition, and that of Moyal one which is given by an integration in “phase space” Z Z ˆ p] =: ±Tr Ψ ˆ. Tr Ψ := dz Ψ[±z, z] = ± dxdp Ψ[x, (2.67)

2.3

Moyal ⋆ product in the bc ghost sector

In this section we define the Moyal ⋆ product which represents Witten’s star product using the results in §2.1, §2.2, §A.1. We first review the conventional operator formalism to fix the notation in the bc ghost sector. We take the ghost coordinates b(σ), c(σ) and their conjugates πb (σ), πc (σ) ±

b (σ) =

∞ X

n=−∞

ˆbn e±inσ = πc (σ) ∓ ib(σ) ,

±

c (σ) =

∞ X

n=−∞

cˆn e±inσ = c(σ) ± iπb (σ) .

(2.68)

We introduce the fermionic variables x ˆn , yˆn and their conjugates pˆn , qˆn as follows b(σ) = −

∞ X

∞ √ X (ˆbn − ˆb−n ) sin nσ = i 2 x ˆn sin nσ ,

n=1

c(σ) = cˆ0 +

∞ X

(ˆ cn + cˆ−n ) cos nσ = cˆ0 +

n=1

πb (σ) =

∞ X

∞ √ X yˆn cos nσ , 2

(2.70)

n=1

∞ √ X pˆn sin nσ , (ˆ cn − cˆ−n ) sin nσ = −i 2

(2.71)

n=1

n=1

πc (σ) = ˆb0 +

(2.69)

n=1

∞ X

(ˆbn + ˆb−n ) cos nσ = ˆb0 +

n=1

∞ √ X qˆn cos nσ . 2

(2.72)

n=1

The nonzero modes x ˆn , yˆn , pˆn , qˆn are related to ˆbn , cˆn : i x ˆn = √ (ˆbn − ˆb−n ) , 2

1 yˆn = √ (ˆ cn + cˆ−n ) , 2

i pˆn = √ (ˆ cn − cˆ−n ) , 2

1 qˆn = √ (ˆbn + ˆb−n ) , (2.73) 2

and the canonical commutation relation {ˆbn , cˆm } = δm+n,0 can be rewritten as {ˆ xn , pˆm } = δn,m ,

{ˆ yn , qˆm } = δn,m , 17

n, m = 1, 2, · · · .

(2.74)

We represent the string field by treating c0 and xn , yn as the “position” coordinates. The translation between Fock space representation and position representation is made through Ψ(c0 , xn , yn ) = hc0 , xn , yn |Ψi .

(2.75)

Here we introduced the bra state hc0 , xn , yn | (and the corresponding ket state) as states in Fock space which satisfy the eigenvalue conditions for the operators cˆ0 , x ˆn , yˆn , hc0 , xn , yn |ˆ c0 = hc0 , xn , yn |c0 , hc0 , xn , yn |ˆ xn = hc0 , xn , yn |xn , hc0 , xn , yn |ˆ yn = hc0 , xn , yn |yn , cˆ0 |c0 , xn , yn i = c0 |c0 , xn , yn i , x ˆn |c0 , xn , yn i = xn |c0 , xn , yn i , yˆn |c0 , xn , yn i = yn |c0 , xn , yn i . Explicitly these are given by ! ∞ X √ √ −ˆ cnˆbn − i 2ˆ hc0 , xn , yn | = hΩ|ˆ c−1 cˆ0 exp c0ˆb0 + , cn xn + 2ynˆbn + iyn xn

(2.76)

n=1

|c0 , xn , yn i = exp ˆb0 c0 +

! ∞ X √ √ ˆ ˆ cˆ0 cˆ1 |Ωi (2.77) −b−n cˆ−n + i 2xn cˆ−n + 2b−n yn − ixn yn

n=1

where hΩ|, |Ωi represents the conformal vacuum4 normalized as hΩ|ˆ c−1 cˆ0 cˆ1 |Ωi = 1. These bras and kets satisfy the normalization and completeness relations hc0 , xn , yn |c′0 , x′n , yn′ i = −(c0 − c′0 ) −

Z

∞ Y

n=1

−2i(xn − x′n )(yn − yn′ ) ,

Z Y ∞ dxn dyn dc0 |c0 , xn , yn ihc0 , xn , yn | = 1 . 2i

(2.78) (2.79)

n=1

Witten’s star product for the ghost sector is defined by the (anti-)overlapping conditions [2], b±(r) (σ) − b±(r−1) (π − σ) = 0 , for

c±(r) (σ) + c±(r−1) (π − σ) = 0 ,

(2.80)

r = 1, 2, 3 mod 3 , σ ∈ [0, π/2], or equivalently b(r) (σ) − b(r−1) (π − σ) = 0 ,

(r) πb (σ)

+

(r−1) (π πb

− σ) = 0 ,

c(r) (σ) + c(r−1) (π − σ) = 0 ,

πc(r) (σ) − πc(r−1) (π − σ) = 0 .

(2.81) (2.82)

These (anti-)overlapping conditions for bc ghost will be used to define the mapping from Witten’s ∗ to Moyal’s ⋆ by using Eq.(2.63) defined in the previous section. To apply the formulation in §2.2, we need to specify the boundary conditions of the split string variables, since we have to use the Bogoliubov transformation given in §2.1/§A.1 accordingly. At σ = 0, π, b(σ) (resp. c(σ)) satisfies the Dirichlet (resp. Neumann) boundary condition. On the other hand, at the midpoint, there are two options, namely Neumann or Dirichlet type boundary 4

We take the convention that |Ωi is Grassmann even and hΩ| is odd.

18

conditions. In the following we choose the Neumann condition for b(σ) and Dirichlet condition for c(σ)5 . In the split string language, the left and right halves of b(σ), lb (σ), r b (σ), satisfy Dirichlet at σ = 0 and Neumann at σ = π/2, while the left and right halves of c(σ), lc (σ), r c (σ), satisfy Neumann at σ = 0 and Dirichlet at π/2. With this choice, lb (σ), r b (σ) are expanded by using odd sine modes: {sin oσ, o = 1, 3, 5, · · · }, and lc (σ), r c (σ) by using odd cosine modes: {cos oσ, o = 1, 3, 5, · · · } : ∞ ∞ √ X √ X lob sin oσ , r b (σ) = i 2 rob sin oσ , lb (σ) = i 2 o=1

(2.83)

o=1

∞ ∞ √ X √ X c c c l (σ) = c¯ + 2 lo cos oσ , r (σ) = c¯ + 2 roc cos oσ . o=1

(2.84)

o=1

From Eqs.(2.69)(2.70)(2.18)(A.7), we have the relations between split- and full-string variables ¯ e + xo , lob = Sx c¯ = c0 − wy ¯ e,

¯ e + xo , rob = −Sx

loc = Rye + yo ,

(2.85)

roc = Rye − yo

(2.86)

where we used a matrix notation. Witten’s ∗ product for the split string formulation is written as6 ˜ c, lob , loc , rob , roc ) = A˜ ∗ B(¯

Z Y ˜ c, ηob , −ηoc , rob , roc ) . ˜ c, lob , loc , ηob , ηoc )B(¯ idηob dηoc A(¯

(2.87)

o>0

The string field in the split-string formulation is identified with the usual position representation Ψ(x (σ)), which is written in terms of modes ˜ c, lob , loc , rob , roc ) ∼ Ψ(c0 , xn , yn ) := hc0 , xn , yn |Ψi . A(¯

(2.88)

In order to map it to the Moyal formulation, we compare Eqs.(2.85)(2.86) with Eq.(2.63), and note the similarities (we add prime ′ to distinguish the variables with anti-overlapping condition from the variables with overlapping conditions) ¯ e + xo ∼ x + y , Sx 2

¯ e + xo ∼ x − −Sx

y , 2

Rye + yo ∼ −x′ +

y′ , 2

Rye − yo ∼ x′ +

y′ (2.89) 2

or equivalently7 xo ∼ x ,

¯ e∼y, Sx 2

yo ∼ −x′ ,

5

Rye ∼

y′ . 2

(2.90)

The other choice (Neumann for b and Dirichlet for c at the midpoint) is discussed in the appendix D. It gives equivalent but more complicated expression for the Moyal formulation. 6 Here we consider naive overlapping condition. To obtain the conventional Witten’s star product, as we will show, we should treat midpoint variable c¯ more carefully. The phase factor i in the measure is only convention so that it is “real” (idxe dye )† = idxe dye . Here we define complex conjugate for fermionic variables ξ, ξ ′ as (ξξ ′ )† = (ξ ′ )† (ξ)† . 7 We note that the odd modes correspond to the “x-variable” of phase space in the Moyal formulation of ghosts. By contrast, in the matter sector, the even modes played the corresponding rˆ ole (see Eq.(30) in [7]).

19

Thus, using Eqs.(2.63)(2.88), we obtain the map from the field in the position representation to the Moyal representation Z Y 1 ¯ c + wy ¯ e , xn , yn ) . (2.91) A(¯ c, xo , po , yo , qo ) := 2−2N (1 + ww) ¯ −4 i−1 dxe dye e−2po Sxe −2qo Rye Ψ(¯ e>0

At this point, we used the MSFT regularization scheme, by truncating the ghost modes xn , yn to n ≤ 2N , and using the parameters (N, κe , κo ) given in the previous section. Thus, w, R, S are redefined in Eqs.(A.17)(A.16)(2.50) and 22N , ww ¯ are finite. We fixed the normalization factor − 1 1 2 = 2−2N (1 + ww) ¯ det(16S¯R) ¯ − 4 consistently with the trace that will be given later. Hence, from Eqs.(2.65)(2.90), the Moyal ⋆ product that corresponds to Witten’s ∗ product is A ⋆ B(¯ c, xo , po , yo , qo ) − 12

= A(¯ c, xo , po , yo , qo ) e

P

o>0

← − → ← − → ← − → ← − → − − − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂y + ∂p + ∂q ∂xo ∂po o ∂qo o ∂xo o ∂yo

B(¯ c, xo , po , yo , qo ) .

(2.92)

We introduced an arbitrary parameter θ ′ which is the analog of θ in the matter sector[9] to absorb units. We note that the product is local as a function of the midpoint c¯, while c¯ is related to the center or mass variable c0 by c0 = c¯ + wy ¯ e. (2.93) By rescaling variables and performing Fourier transformation with respect to c¯, we arrive at the definition of the string field in MSFT ˆ ,x ,p ,y ,q ) A(ξ Z 0 o o o o = d¯ ce−ξ0 c¯A(¯ c, xo , −po /θ ′ , yo , −qo /θ ′ ) Z Y ¯ e + 2′ qo Rye ¯ e + θ2′ po Sx −2N − 14 θ dc0 = 2 (1 + ww) ¯ i−1 dxe dye e−ξ0 c0 +ξ0 wy Ψ(c0 , xn , yn ) . (2.94) e>0

ˆ 0 , xo , po , yo , qo ) By this Fourier transformation with respect to zero mode, the Grassmann parity of A(ξ and the corresponding |Ψi coincide. The Moyal ⋆ product which is modified after Eqs.(2.92)(2.94) is →! ← − ← − − ← − − → − → !! ∂ ′ ∂ ∂ ′ ∂ 1 1 ∂ ∂ Σ σ + σ , (2.95) = exp ⋆ := exp 2 ∂ξ ∂ξ 2 ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ2 ! ! ! ! ! ξ1 xo yo 0 1 σ′ 0 ξ= , ξ1 = , ξ2 = .(2.96) , Σ= , σ′ = θ′ ξ2 po qo 1 0 0 σ′ We define the trace in MSFT as integration over the “phase space”: Z Z Y ′ N ′ 2N ˆ ˆ 0 , ξ) . ˆ Tr A(ξ0 , ξ) = det σ (dxo dpo dyo dqo ) A(ξ dξ A(ξ0 , ξ) = (−1) θ

(2.97)

o>0

ˆ 0 , ξ) which is mapped from |Ψi by (2.94) is normalized as The Moyal field A(ξ Z † ∂ θ′ ∂ ˆ 0 , ξ) = hΨ|ˆ ˆ 0 , ξ) ⋆ A(ξ − v¯ A(ξ c0 |Ψi , dξ0 Tr ∂ξ0 2 ∂qo 20

(2.98)

where

∂ ∂ξ0

−

θ′ ¯ ∂q∂ o 2v

corresponds to −ˆ c0 (2.107).

It is convenient to introduce the bra hξ0 , ξ| as a state in Fock space such that the Moyal field is ˆ 0 , ξ) = hξ0 , ξ|Ψi. This can be obtained from the bra related directly to the Fock space field via A(ξ state hc0 , xn , yn | (2.77) by Fourier transformation (2.94), hξ0 , ξ| = hξ0 , ξ1 , ξ2 | = hξ0 , xo , po , yo , qo | Z Y 1 ¯ e + 2′ qo Rye ¯ e + θ2′ po Sx θ = 2−2N (1 + ww) ¯ − 4 dc0 hc0 , xn , yn | i−1 dxe dye e−ξ0 c0 +ξ0 wy e>0

−2N

= −2

− 14

(1 + ww) ¯

hΩ|ˆ c−1 e−ξ0 (ˆc0 −

√ (o) be −ˆ coˆbo −2iξ¯1 M0 ξ2 −ξ¯1 λ1 −ξ¯2 λ2 2wˆ ¯ ce ) cêˆ

(2.99)

e

where we used notation: (o) M0

1 2

=

0

0 2 ¯¯ SR θ′ 2

!

, λ1 =

! √ −i 2ˆ co √ , λ2 = ¯ 0 − 2 ′2i S¯ˆbe + 2i′ Swξ θ

θ

√ − 2ˆbo √

ce − 2θ′ 2 Rˆ

!

.

(2.100)

This is the result given in the summary at the beginning of this section. Examples Here we give some examples of string fields in MSFT. For the conventional ghost number 1 vacuum cˆ1 |Ωi and SL(2, R) invariant vacuum |Ωi, the corresponding fields are given by ¯ 1 −ix y −i q ′ p S¯R Aˆ0 (ξ0 , ξ) = hξ0 , ξ|ˆ c1 |Ωi = 2−2N (1 + ww) ¯ − 4 ξ0 e o o θ′ 2 o ( )oo′ o , 1 1 ¯¯ ¯ − 4 ξ x e−ixo yo −i4po (S R)oo′ qo′ . Aˆ (ξ , ξ) = hξ , ξ|Ωi = −i 2−2N + 2 (1 + ww) 4

Ω

0

0

0

1

(2.101) (2.102)

We note that in the open string limit κe = e, κo = o, N = ∞, these expressions become singular. For example, the coefficient of po qo′ is divergent, ′ ∞ 16 io+o +2 X e3 ¯ ¯ (S R)oo′ = = ±∞ . π 2 o′ (e2 − (o′ )2 )(e2 − o2 ) e=2

(2.103)

Therefore, it is advisable not to take the open string limit at the level of the state, but wait for the end of a computation. We note that the physical quantities such as the scattering amplitude become regular in this limit [10]. P

nˆ ˆ −n b−n c ˜ = N˜ e ∞ n=1 (−1) c For the identity-like state in the Siegel gauge: |Ii ˆ1 |Ωi, which is a delta I Q function with respect to the even modes in position basis, ΨI˜(c0 , xn , yn ) = NI˜ e>0 (−4iδ(xe )δ(ye )), the corresponding field is 1 AÎ˜(ξ0 , xo , po , yo , qo ) = (−1)N (1 + ww) ¯ − 4 NI˜ ξ0 .

Except for the zero mode and the normalization factor, this AÎ˜ is the identity element with respect to the Moyal ⋆ product. The conventional identity state |Ii (C.1) [2] (which is BRST invariant, and not in the Siegel gauge) becomes more complicated in MSFT. 21

2.4

Oscillators

In this subsection, we obtain the Moyal images of the conventional oscillators which are used in applications in MSFT. In this way we can write various operators in oscillator language and in particular discuss the form of L0 and the butterfly state which came up in our work in [12]. 2.4.1

Oscillators on the fields in MSFT

ˆ which consists of ˆbn , cˆn , acting on a state |Ψi in Fock space, we define its Moyal For an operator O image βÔˆ , which is a differential operator acting on the Moyal field AˆΨ (ξ0 , ξ) = hξ0 , ξ|Ψi, as follows ˆ βÔˆ AˆΨ (ξ0 , ξ) = hξ0 , ξ|O|Ψi .

(2.104)

For the basic operators, cˆ0 , ˆb0 , x ô , yô , x ê , yê , pô , qô , pê , qê , this rule gives the corresponding operators in MSFT ∂ θ′ ∂ βˆcˆ0 = − + v¯ , βˆˆb0 = −ξ0 , (2.105) ∂ξ0 2 ∂qo θ′ ∂ θ′ ∂ βˆxê = S βˆyô = yo , βˆxô = xo , , βyê = T , (2.106) 2 ∂po 2 ∂qo ∂ ∂ 2 2 ¯ βˆpô = , βˆqô = , βˆpê = ′ Spo , βˆqê = ′ R qo + we ξ0 . (2.107) ∂xo ∂yo θ θ The nonzero modes of the oscillators ˆbn , cˆn become X 1 X 2 ∂ θ′ 1 −1 b ˆ βe = √ + we′ ξ0 , β¯ob U−o,e qo Ro|e| − iǫ(e) S|e|o + √ w|e| ξ0 = ′ 2 ∂po 2 o>0 θ 2 o ∂ 1 − iǫ(o)x|o| , βôb = √ 2 ∂y|o| X 1 X θ′ ∂ 2 c ˆ βe = √ T|e|o − iǫ(e) ′ S|e|o po = Ue,−o β¯oc , ∂qo θ 2 o>0 2 o 1 ∂ βôc = √ . y|o| − iǫ(o) ∂x|o| 2

(2.108) (2.109) (2.110) (2.111)

In the first and third equations we introduced the symbols β¯ob ,β¯oc to denote the differential operators ′ 2 θ ∂ θ′ ∂ 2 1 1 c b ¯ ¯ √ √ q|o| − i ǫ(o) − iǫ(o) ′ p|o| . , βo = βo = 2 ∂p|o| θ 2 θ′ 2 2 ∂q|o| The even and the odd sets satisfy canonical anti-commutation relations {βêb , βêc′ } = δe+e′ ,

{βôb , βôc′ } = δo+o′ ,

{β¯ob , β¯oc′ } = δo+o′ .

(2.112)

(2.113)

Using the above maps of operators, we can translate operators in the usual oscillator representation into MSFT language. For example, the ghost number operator (we assigned ghost number 1 to cˆ1 |Ωi) X Ngh = (ˆ c−nˆbn − ˆb−n cˆn ) + cˆ0ˆb0 + 1 , (2.114) n≥1

22

is mapped to its MSFT image X ∂ ∂ ∂ ∂ ∂ Ngh = − xo + po − qo + 2. − ξ0 yo ∂yo ∂xo ∂po ∂qo ∂ξ0

(2.115)

o>0

2.4.2

Oscillator as a field

It is often useful to rewrite differential operators in Moyal space in terms of the star product. The idea is to replace the derivative by the ⋆ -(super)commutator:8 ∂ ˆ ˆ ⋆. A = Σ−1 [ξ, A} ∂ξ

(2.116)

More concretely 1 ∂ ˆ ˆ ⋆, A = ′ [po , A} ∂xo θ

1 ∂ ˆ ˆ ⋆, A = ′ [qo , A} ∂yo θ

1 ∂ ˆ ˆ ⋆, A = ′ [xo , A} ∂po θ

1 ∂ ˆ ˆ ⋆. A = ′ [yo , A} ∂qo θ

(2.117)

This observation leads to the star product representation of the differential operators as follows 1 b , βôb Aˆ = √ βob ⋆ Aˆ − (−1)|A| Aˆ ⋆ β−o 2 1 c , βôc Aˆ = √ βoc ⋆ Aˆ + (−1)|A| Aˆ ⋆ β−o 2

1 b β¯ob Aˆ = √ βob ⋆ Aˆ + (−1)|A| Aˆ ⋆ β−o , (2.118) 2 1 c β¯oc Aˆ = √ βoc ⋆ Aˆ − (−1)|A| Aˆ ⋆ β−o , (2.119) 2

where we defined the fieldsβob,c in Moyal space that play the fundamental role of oscillators βob :=

1 i q|o| − ǫ(o)x|o| , ′ θ 2

βoc :=

1 i y|o| − ′ ǫ(o)p|o| . 2 θ

(2.120)

The odd β¯ob,c and even βêb,c differential operators are related to each other as in Eqs.(2.108,2.110). Therefore we also define fields with even labels via the Bogoliubov transformation βeb :=

X o

−1 , βob U−o,e

βec :=

X

Ue,−o βoc

(2.121)

o

where the sum runs over odd integers from −2N + 1 to 2N − 1. These give the star product representation of the even differential operators βêb,c 1 b + we′ ξ0 Aˆ , βêb Aˆ = √ βeb ⋆ Aˆ + (−1)|A| Aˆ ⋆ β−e 2 1 c . βêc Aˆ = √ βec ⋆ Aˆ − (−1)|A| Aˆ ⋆ β−e 2

(2.122) (2.123)

The fields βeb,c or βob,c satisfy the oscillator anticommutation relations with respect to the star product (2.124) {βob , βoc′ }⋆ = δo+o′ , {βeb , βec′ }⋆ = δe+e′ . 8

← −

→ −

ˆ. ˆ2 }⋆ := Aˆ1 ⋆ Aˆ2 − (−1)|A1 ||A2 | Aˆ2 ⋆ Aˆ1 . Note that Aˆ ∂ = −(−1)|A| ∂ A We define the supercommutator as [Aˆ1 , A ∂ξ ∂ξ

23

But they do not anticommute with each other c b −1 , {βob , β−e }⋆ = U−e,o {β−e , βoc }⋆ = Uo,−e

(2.125)

since they are related to each other by the Bogoliubov transformation given above. One may regard the fields β b,c as the harmonic oscillators in Moyal space which act on the string field A from either side by the star product. It is then natural to introduce the vacuum field associated with the odd (and similarly the even) oscillators, as the field that satisfies the following conditions under the star product b c = AˆB ⋆ β−o = 0, βob ⋆ AˆB = βoc ⋆ AˆB = AˆB ⋆ β−o

∀o > 0 .

(2.126)

By definition, this field is the Moyal image of the Fock space operator AˆB ∼ |0ih0|, where |0i is the vacuum state with respect to the oscillators βob,c . These are first order differential equations whose solution is the gaussian ! X 4i AˆB = ξ0 2−2N exp − ixo yo + ′ 2 po qo . (2.127) θ o>0 If we write AˆB = ξ0 AB , we see that AB is a projector (3.24) with respect to the Moyal ⋆ product: AB ⋆ AB = AB . It turns out this is the butterfly projector that came up in other formulations of string field theory [25, 26, 27] as shown in appendix E. 2.4.3

L0 and L0

In string field theory computations the zeroth Virasoro operator L0 plays a critical role since it defines the propagator. In this section, we derive various forms of L0 in MSFT. In the usual oscillator representation acting on Fock space, L0 is given by9 L0 =

∞ X k=1

k(ˆb−k cˆk + cˆ−k ˆbk ) .

(2.128)

In MSFT which is regularized by (N, κe , κo ), we truncate the number of oscillators to 2N and replace the frequencies by κe,o . Then the Moyal image of L0 becomes the following differential operator L0 =

2N X

k=1 2N X

c ˆb b ˆc βk ) κk (βˆ−k βk + βˆ−k

! 4 θ′2 ∂ ∂ ∂ ∂ + po q o + = κk + i κo xo yo + ∂xo ∂yo θ ′ 2 4 ∂po ∂qo o>0 k=1 ! ! ! X X X 4i 2i vo κo po ξ0 . (2.130) ¯ κo vo po + ′ 2 (1 + ww) ¯ vo′ qo′ + ′ (1 + ww) θ θ ′ X

o>0

9

(2.129)

o >0

o>0

We take the convention such that L0 (ˆ c1 |Ωi) = 0 fixes the constant that comes from normal ordering.

24

The last two terms come from the identities X

¯ eR ¯ = κo RR ¯ = κo + (1 + ww)κ Sκ ¯ v, o v¯

Seo κe we = κo Rwe = κo v(1 + ww) ¯ .

(2.131)

e>0

The similarity to the matter sector is enhanced by introducing an even basis xbe , pbe , xce , pce which is related to the odd basis through the following linear canonical transformation10 xbe := κ−1 e Sxo ,

pbe := κe Spo ,

xce := T yo ,

¯ o. pce := Rq

(2.132)

The Moyal ⋆ product (2.95) and trace (2.97) are invariant under this canonical transformation. With the new variables, L0 is rewritten as ! 2N X X ∂ ∂ 4 b c θ′2 2 ∂ ∂ 2 b c κk + i κe xe xe + b c + ′ 2 pe pe + L0 = κ ∂xe ∂xe θ 4 e ∂pbe ∂pce e>0 k=1 ! ! ! X X 2i X ∂ ∂ i we b (2.133) we′ c + ′ − we pbe ξ0 . 1 + ww ¯ ∂x ∂x θ ′ e e ′ e>0 e>0 e >0

Under this change of variables, the usual perturbative vacuum that was given in the odd basis (2.101) becomes: ¯ Rxc −i 4 pb κ−1 pc 1 −ixb Rκ (2.134) Aˆ0 = 2−2N (1 + ww) ¯ − 4 ξ0 e e o e θ ′ 2 e e e . Then the apparent divergence (2.103) of the coefficient at the limit κe = e, κo = o, N = ∞ does not occur, since ∞ 16 ie+e′ (ee′ )2 X 1 ¯ ′ |(Rκo R)ee | = (2.135) < ∞. π2 o(e2 − o2 )(e′2 − o2 ) o=1

Following the ideas of the previous subsection, the differential operator L0 can be represented in terms of star products by introducing the field L0 and the “midpoint correction” γ as follows L0 Aˆ = L0 ⋆ Aˆ + Aˆ ⋆ L0 + γ Aˆ , X κ2 2 1 e b c xe xe + ′ 2 pbe pce + L0 = i 2 2 θ e>0 =

X e>0

b c ⋆ βec + β−e ⋆ βeb κe β−e

i γ = − 1 + ww ¯

X

∂ we b ∂xe e>0

!

X e>0

1 − 2

X

e′ >0

κe +

X e>0

∂ we′ c ∂xe′

X

κo

o>0

κe − !

X o>0

! κo

+ !

i θ′ i + ′ θ

X

!

(2.136)

we pbe ξ0

(2.137)

!

ξ0 , (2.138)

e>0

X e>0

we pbe

(2.139)

where βeb,c can be rewritten in terms of the even mode variables in Eq.(2.132) βeb = 10

i 1 c p − ǫ(e)κe xb|e| , θ ′ |e| 2

1 i b βec = xc|e| − ′ ǫ(e)κ−1 e p|e| . 2 θ

(2.140)

We used these even modes in [12]. They are just a linear transformation of the odd modes in §2.3 and different from the even modes which are considered in appendix D.

25

The ξ0 -dependent term vanishes when L0 acts on the fields in the Siegel gauge. The field L0 is multiplied with the star product, but γ is still a differential operator. It is possible to rewrite γ in terms of star products (a double supercommutator) γ Aˆ = −

X i ˆ ⋆ }⋆ , we we′ [pbe , [pce′ , A} θ ′ 2 (1 + ww) ¯ e,e′ >0

(2.141)

but this does not have the single star product structure as the star products with L0 and therefore γ cannot be absorbed into a redefinition of L0 . As discussed in [12], γ depends only on a single combination of modes in the direction of the vector we , which is closely related to the midpoint. If it were not for the “midpoint correction” term γ, string field theory would reduce to a matrix-like theory that would be exactly solvable, as shown in [12]. The above form of writing L0 focuses on the γ term as the remaining difficult aspect of string field theory. A similar structure exists in the canonically equivalent odd basis by using L′0 , γ ′ L0 Aˆ = L′0 ⋆ Aˆ + Aˆ ⋆ L′0 + γ ′ Aˆ , X 1 1 2 ′ L0 = i κo xo yo + ′ 2 po qo + 2 2 θ o>0

γ′

3

!

! i X ¯ κe + κo + (1 + ww) vo κo po ξ0 θ ′ o>0 e>0 o>0 ! ! 1 X X X i X b c c b ¯ κe − κo + (1 + ww) vo κo po ξ0 , = κo β−o ⋆ βo + β−o ⋆ βo + 2 θ′ e>0 o>0 o>0 o>0 ! ! X X 4i = κo vo po (1 + ww) ¯ vo′ qo′ . (2.142) θ′2 o>0 o′ >0 X

X

Monoid and Neumann coefficients

The perturbative states and nonperturbative squeezed states in Fock space (perturbative vacuum, sliver state, butterfly state and so on) are mapped to gaussian functions in Moyal space [9]. In fermionic Moyal space, such as the basis ξ = xbe , pbe , xce , pce , the generic form of such a gaussian is written as, ¯ ¯ ¯ = −M . AN ,M,λ (ξ) = N e−ξM ξ−ξλ , M (3.1) As in the bosonic case, such shifted gaussians form a monoid algebra [9].11 A monoid is almost a group except for the property of inverse. This implies that under star products the shifted gaussians satisfy the properties of closure, identity and associativity. Although the generic gaussian has also an inverse under star products, not all of them do (for example, projectors such as Eq.(2.127) do not have an inverse). The identity element under star products is the natural number 1, which corresponds to the trivial gaussian. The monoid structure is an effective tool for computations in MSFT. In particular it was used to calculate the product of n gaussian functions, whose trace gives the n-point vertex. A corollary 11

Some issues on Moyal product are discussed in [20].

26

of this result is the determination of the Neumann coefficients for the n-string vertex. These coefficients were computed in [2] from conformal field theory, but now they can be determined by using only the Moyal product. The MSFT approach gives simple expressions for all Neumann 1/2 −1/2 coefficients in terms of a single matrix teo = κe Teo κo . Neumann coefficients are not needed for computations in MSFT, but the computation can be used to test the MSFT formalism. This was used as successful test of MSFT in the matter sector [9]. In this section we will carry out a similar program in the ghost sector. While the treatment of the unity elements becomes more subtle because of the zero mode, the closure of gaussian functions (3.1) under the star product will be proved exactly the same way as in the bosonic sector. In particular, the algebraic structure is formally the same. With this information, one can compute the product of n fermionic gaussians and derive the Neumann coefficients by using this algebraic machinery. In particular, we verify consistency of the Moyal ⋆ product (2.95) with the conventional reflector and the 3-string vertex in oscillator language. We will show the correspondence by including the zero mode part, Z hΨ1 |Ψ2 i ↔ dξ0 Tr Aˆ1 (ξ0 , ξ) ⋆ Aˆ2 (ξ0 , ξ) , (3.2) Z (3) (2) (1) (1) (2) (3) hΨ1 |Ψ2 ⋆W Ψ3 i ↔ dξ0 dξ0 dξ0 Tr Aˆ1 (ξ0 , ξ) ⋆ Aˆ2 (ξ0 , ξ) ⋆ Aˆ3 (ξ0 , ξ) . (3.3)

3.1

Monoid structure for gaussian elements

We first derive the structure of the monoid (3.1) under the fermionic star product defined by {ξa , ξb } = Σab for a general symmetric Σab . This matrix takes a block diagonal form Σ = diag (σ ′ , σ ′ ) in the bases we discussed so far, but it is useful to develop the formalism for any Σ. For the moment, we suppress the ξ0 -dependence because it is not relevant to the Moyal ⋆-product (2.95). The structure of the monoid is summarized in the following algebra, AN1 ,M1,λ1 ⋆ AN2 ,M2 ,λ2 = AN12 ,M12 ,λ12 , mi := Mi Σ ,

m ¯ = −ΣmΣ−1 ,

(3.4) (3.5) −1

m12 = M12 Σ = (1 + m2 )m1 (1 + m2 m1 ) −1

λ12 = (1 − m1 )(1 + m2 m1 ) 1 2

+ (1 − m1 )m2 (1 + m1 m2 ) −1

λ2 + (1 + m2 )(1 + m1 m2 ) − 14

N12 = N1 N2 det (1 + m2 m1 ) e Kab =

−1

P2

a,b=1

¯ a ΣKab λb λ

−1 (m−1 (1 + m2 m1 )−1 2 + m1 ) −1 −(1 + m1 m2 )−1 (m2 + m−1 1 )

!

λ1 ,

,

.

To prove this formula, it is convenient to use Fourier transformation. We define Z Z ¯ ¯ ˜ ˜ A(η) := dξ eξη A(ξ) , A(ξ) = dη e−ξη A(η) . 27

,

(3.6) (3.7) (3.8) (3.9)

(3.10)

The ⋆ product is rewritten in terms of Fourier coefficients, Z 1 ¯ ¯ A1 ⋆ A2 (ξ) = dη1 dη2 e− 2 η¯1 Ση2 −ξη1 −ξη2 A˜1 (η1 )A˜2 (η2 ) .

(3.11)

For the gaussian AN ,M,λ (3.1), the Fourier transform is also gaussian: 1¯

1

− λM g AN ,M,λ (η) = N (det(2M )) 2 e 4

−1 λ

1

e− 4 η¯M

−1 η+ 1 η ¯M −1 λ 2

.

(3.12)

The main result (3.4–3.9) follows by carrying out the gaussian integration over fermionic variables. We note that Eq.(3.1) have exactly the same form as the bosonic case (Eqs.(3.11–3.17) in [9]) if we put d = −2. Similarly, the formula for the trace also related to the bosonic case as if d = −2 1¯

1

Tr(A(ξ)) = N (det(2m)) 2 e− 4 λM

−1 λ

.

(3.13)

With this observation, we find that all the algebraic manipulations in [9] which use the structure of the star product for monoids also apply in the ghost sector with no other modification. In particular the product of n monoids with the same M, but different λi , Ni is one of the useful results that is used to compute the n-point vertex. The result in [9] is now adopted to the ghost sector as follows AN12···n ,M (n) ,λ12···n := AN1 ,M,λ1 ⋆ AN2 ,M,λ2 ⋆ · · · ⋆ ANn ,M,λn , m(n) = M (n) Σ =

Jn− Jn+

Jn± :=

,

(1 +

m)n

± (1 − 2

m)n

(3.14)

,

(3.15)

n 1 X (1 − m)r−1 (1 + m)n−r λr , Jn+ r=1 1 + 12 N12···n = N1 N2 · · · Nn (det Jn ) exp − Kn (λ) , 4 n n − s−r−1 (1 + m)n+r−s−1 X X ¯ r Σ Jn−1 λr + 2 ¯ r Σ (1 − m) Kn (λ) = λ λ λs , Jn+ Jn+ r0

1 −1 M (tanh(τ κ ˜))−1 = 4 0 1 −1 M (sinh(τ κ ˜ ))−1 = 4 0

!− d−2 2

(1 − e−2τ κo )

e

ls2 −1 2κe (tanh(τ κe ))

0 ls2 2κe

(sinh(τ κe ))−1 0

tanh(τ κe /2) 2 wls p , κe

−

e tanh( τ κ τ 2 )w +w ¯ 2 κe

ls2 p2

,

! 0 , θ2 ¯ Rκo (tanh(τ κo ))−1 R 8ls2 ! 0 , θ2 ¯ −1 R Rκ (sinh(τ κ )) o o 2 8l s

(4.19) 38

F gh (τ ) =

1 εM0gh−1 (tanh τ κ ˜gh )−1 = ε 4

Ggh (τ ) =

1 εM0gh−1 (sinh τ κ ˜gh )−1 = ε 4

−1 ¯ − 2i T κ−1 o (tanh τ κo ) T −1 ¯ − 2i T κ−1 o (sinh τ κo ) T

′2

!

− iθ8 κe (tanh τ κe )−1 ! ′2

− iθ8 κe (sinh τ κe )−1

,

.

The ghost structure of the quadratic term in the exponent is similar to the matter one. 1-loop vacuum amplitude By taking the trace of the propagator Eq.(4.18), we have Z Z Z dd p Tr e−τ (L0 −1) = dd p d2N d η d4N η gh eτ ∆(η, η, p, τ ) Y Y d d (1 − e−τ κo )−(d−2) . (1 − e−τ κe )−(d−2) = (2π) 2 ls−d τ − 2 eτ

(4.20)

o>0

e>0

This reproduces the expected partition function, with the correct spectrum, including the ghost contribution. If we take the open string limit κe = e, κo = o, N = ∞ naively in the formula of L0 (A.47,2.133), namely at the Lagrangian level, we lose the information on odd spectrum. This is one of the indications that the γ term plays a nontrivial role. Of course, we obtain the correct limiting partition function by taking the limit at the last stage of the computation, which is given above. External state As external states in Feynman graphs it is enough to consider monoid elements such as ¯

¯

¯gh M gh ξ gh −ξ¯gh λgh

AN ,M,λ,M gh ,λgh = N eip¯x e−ξM ξ−ξλ e−ξ

.

(4.21)

In fact, we can compute various perturbative diagrams by preparing a particular class of gaussian external states given by M = M0 , M gh = εM0gh , where these matrices were given in Eqs.(4.10). If ¯ = (−iwe p, 0), λgh = 0, this external field represents perturbative vacuum cˆ1 |p, Ωi we also take λ with momentum p, which represents the perturbative tachyon with proper normalization Ap (ξ) = 2N (d−2) (1 + ww) ¯ −

d+2 8

¯gh εM gh ξ gh

¯

eip¯x e−ξM0 ξ+ipwexe e−ξ

0

,

Tr(Ap (ξ)† ⋆ Ap (ξ)) = 1 .(4.22)

We omitted an overall ξ0 because it drops out in the Siegel gauge action (4.14). Excited states (that ¯ ¯gh gh correspond to polynomials multiplying this Ap (ξ)) can be obtained by differentiating Ap (ξ)e−ξλ−ξ λ with respect to general λ, λgh appropriately, and then setting λ = (−iwe p, 0), λgh = 0. Therefore an explicit computation of Feynman graphs with general λ, λgh , M, M gh has many physical applications. τ -evolved monoid element It is convenient to have a τ -evolved formula for a gaussian Eq.(4.21) to compute Feynman graphs in ξ-basis [10]. We can derive it explicitly by evaluating: e−τ L0 AN ,M,λ,M gh ,λgh (ξ) 39

=

Z

2N d

d

¯ ξ¯gh ηgh 4N gh −iξη

ηd

η

e

AÑ ,M,λ,M gh,λgh (η) =

e Z

Z

2N d ′

d

4N ′ gh

η d

η

′ ˜ ∆(η, η , τ, p) AN ,M,λ,M gh ,λgh (η ) , ′

d2N d ξ 4N gh −iξη ¯ ξ¯gh ηgh d ξ e e AN ,M,λ,M gh ,λgh (ξ) . (2π)2N d

(4.23)

By gaussian integration we have the following formula ¯

¯

¯gh M gh (τ )ξ gh −ξ¯gh λgh (τ )

e−τ L0 AN ,M,λ,M gh ,λgh (ξ) = N N m (τ )N gh (τ ) eip¯x e−ξM (τ )ξ−ξλ(τ ) e−ξ where

h −1 i (cosh τ κ ˜)−1 M0 , M (τ ) = sinh τ κ ˜ + sinh τ κ ˜ + M0 M −1 cosh τ κ ˜ h i −1 λ(τ ) = cosh τ κ ˜ + M M0−1 sinh τ κ ˜ (λ + iwp) − iwp , i h 1 2 2 ¯ + ipw ¯ (M + coth τ κ ˜ M0 )−1 (λ + iwp) e− 2 ls p τ exp 14 λ N m (τ ) = d/2 det 21 1 + M M0−1 + 12 1 − M M0−1 e−2τ κ˜

for the matter sector [10]18 and −1 gh gh gh gh gh−1 gh M (τ ) = sinh τ κ ˜ + sinh τ κ ˜ + εM0 M cosh τ κ ˜ (cosh τ κ ˜gh )−1 εM0gh ,

(4.24)

(4.25) (4.26) (4.27)

(4.28)

i−1 h λgh (τ ) = cosh τ κ ˜gh − M gh εM0gh−1 sinh τ κ ˜gh λgh ,

(4.29)

i−1 h ′ λgh (τ ) = cosh τ κ ˜gh + M gh M0gh−1 sinh τ κ ˜gh λgh ,

(4.32)

1 ¯ gh

gh

gh

gh −1 gh

N gh (τ ) = e− 4 λ (M +coth(τ κ˜ )εM0 ) λ 1 2 1 1 gh−1 gh−1 −2τ κ gh gh ˜ gh (1 − M εM0 ) + (1 + M εM0 )e × det (4.30) 2 2 for ghost sector. When we consider a class of monoid such that ξ¯gh M gh ξ gh is SU (1, 1)-symmetric and twist even19 then the evolved M (τ ) also has this symmetry. We can see this explicitly by noting that f (˜ κ)M0gh is a block diagonal and symmetric matrix (where f (x) is an arbitrary function). In ′ ′ this case with M gh = εM gh , where M gh is a 2N × 2N matrix, the above formula becomes −1 gh gh gh gh gh′ −1 gh M (τ ) =ε sinh τ κ ˜ + sinh τ κ ˜ + M0 M cosh τ κ ˜ (cosh τ κ ˜gh )−1 M0gh , (4.31) 1 ¯ gh

N gh (τ ) = e− 4 λ

× det

′

ε(M gh +coth(τ κ ˜ gh )M0gh )−1 λgh

1 1 gh ′ ′ (1 + M gh M0gh−1 ) + (1 − M gh M0gh−1 )e−2τ κ˜ 2 2

.

(4.33)

We can use this reduced monoid of n-th product of the pertur formula to compute the τ -evolved ¯ 1 ¯ n (n) − ξλ − ξλ −τ L 0 because the coefficient matrix M0 (3.15) A0 (ξ)e ⋆ · · · ⋆ A0 (ξ)e bative vacuum: e in the quadratic term in the exponent is proportional to ε. With the above preparation, we have all that is needed to compute the ghost contribution in various amplitudes, by following the methods that were developed in the matter sector in [10]. 18 19

λgh

We supposed that M is Lorentz symmetric. In §H, we defined SU (1, 1)-symmetric and twist even monoid element which has λgh = 0 (H.9). Here we permit 6= 0 case.

40

4-tachyon amplitude The 4 point amplitude for tachyons is computed by putting together several diagrams that are related to each other by permutations of the external legs. For a typical 4-pt diagram 21 > − 0

o>0

As noted in [8], there is an ambiguity in naive computation using these matrices and vectors. For example, T has an inverse matrix given by R, and yet it has a zero eigenvalue T v = 0. This is possible only because they are infinite dimensional matrices. It causes associativity anomalies. To avoid the ambiguous results that come from the associativity anomaly, a finite matrix regularization is proposed in [8]. We define N × N matrices T, R and N -vectors v, w by Roe = (κo )−2 T¯oe (κe )2 ,

Roe = T¯oe + vo w ¯e ,

vo = T¯oe we ,

¯ eo vo , we = R

(A.15)

where a bar means transpose, and we introduced a set of 2N frequencies κe , κo . These relations are identical to the ones satisfied by the infinite matrices in Eq.(A.14), but we now use them as defining relations for finite dimensional matrices and arbitrary frequencies. We can solve the equations in Eq.(A.15) explicitly in term of the frequencies we vo κ2o we vo κ2e , Roe = 2 , 2 2 κe − κo κe − κ2o 1 1 Q 2 2 Q 2 2 2 2 o′ κe /κo′ − 1 e′ 1 − κo /κe′ 2−e o−1 we = i 1 . 1 , vo = i Q Q 2 /κ2 − 1 2 2 /κ2 2 κ 1 − κ e o e′ 6=e o′ 6=o e′ o′

(A.17)

¯ = 1 + ww, RT = 1o , RR ¯ T¯T = 1 − v¯ v, w ww ¯ w w ¯ , Tv = , v¯v = , T T¯ = 1 − 1 + ww ¯ 1 + ww ¯ 1 + ww ¯ ¯ = 1 + v¯ Rw = v(1 + ww), ¯ RR v (1 + ww) ¯ .

(A.18)

Teo =

(A.16)

By using only the defining relations we can show the following further relations for the regularized version of T, R, v, w. T R = 1e ,

The original T, R, v, w in Eq.(A.10) are reproduced by setting the open string limit: κe = e , κo = o , N → ∞ . 45

(A.19)

We note that at this limit ww ¯ diverges 1 + ww ¯ =

A.2

N Y κ2n κ n=1 2n−1

!2

→

√

πΓ (N + 1) Γ N + 12

!2

→ ∞.

(A.20)

Some results in matter sector

Here we summarize notation and conventions in the matter sector in MSFT.21 A.2.1

Oscillators in MSFT

• Mode expansion in matter sector (µ = 0, 1, · · · , d − 1): ∞ ∞ X √ √ X 1 µ x ˆµn cos nσ = x ˆµ0 + i 2α′ 2 (A.21) (α − αµ−n ) cos nσ , n n n=1 n=1 ! ! ∞ ∞ √ X 1 1 X ν ν pˆnµ cos nσ = pˆ0µ + 2 pˆ0µ + √ ηµν (αn + α−n ) cos nσ . π 2α′ n=1 n=1

X µ (σ) = x ˆµ0 + Pµ (σ) =

1 π

Nonzero modes in matter sector √ √ µ ν µν µν κn a ˆµn , αµ−n = κn a ˆ†µ , [ˆ aµn , a ˆ†ν , n , [αn , αm ] = ǫ(n)κn δn+m,0 η m ] = δn,m η r i κn ηµν ν ls (ˆ aµn − a ˆ†µ xµn = √ (ˆ a +a ˆ†ν xµn , pˆmν ] = iδn,m δνµ , n ) , pnµ = n ) , [ˆ 2 ls n 2κn κ|n| 1 x ˆ|n| , αn = √ ls pˆ|n| − iǫ(n) (A.22) ls 2 αµn =

where we define the symbols ls2 = 2α′ , κn = n. Zero mode in matter sector, with aµ0 := ls pˆµ0 i √ µ 1 µν x ˆµ0 = ls b(ˆ a0 − a ˆ†µ ˆ0µ = √ ηµν (ˆ aν0 + a ˆ†ν xµ0 , pˆ0ν ] = iδνµ , [ˆ aµ0 , a ˆ†ν 0 ), p 0 ) , [ˆ 0 ]=η 2 ls b

(A.23)

where b is some positive constant. • Position eigenstates hx0 , xn |ˆ xn = hx0 , xn |xn ,

hx0 , xn |ˆ x0 = hx0 , xn |x0 ,

x ˆn |x0 , xn i = xn |x0 , xn i ,

x ˆ0 |x0 , xn i = x0 |x0 , xn i,

(A.24)

are given as squeezed states in Fock space P

hx0 , xn | = hx0 |e

n>0

√ 1 α2 + il 2 xn αn − κn2 x2n 2κn n s 2ls

Y κn 4

d

n>0

21

πls2

In this subsection, we use the same symbols for matter as we did for the ghosts in the main text for some quantities, such as positions and momenta. We can avoid confusion from the context. We omit some definitions and details because we can refer to Ref.[9] for them.

46

P

= hx0 |e hx0 | = h0|e

n>0

√ 1 2 ˆn − κn2 x2n a ˆ + i l2κn xn a 2 n s 2ls

1 2 a ˆ +i 2√ x0 a ˆ0 − 21 x20 2 0 ls b ls b

Y κn 4 πls2

n>0

Y κn 4 πls2 d

n>0

|x0 i =

2 πls2 b

4

P

n>0

P

n>0

e

d

=

Y κn 4

d

n>0

d

|x0 , xn i =

e

2 πls2 b

d 4

πls2

,

√ 1 α2 − i l 2 xn α−n − κn2 x2n 2κn −n s 2ls

√ 1 †2 i 2κn xn a ˆ†n − κn2 x2n a ˆ − l 2 n s 2ls

1 †2 ˆ†0 − 21 x20 a ˆ0 −i 2√ x0 a ls b ls b

e2

,

|x0 i

|x0 i ,

|0i .

(A.25)

They satisfy normalization and completeness conditions hx0 , xn |x′0 , x′n i = δd (x0 − x′0 ) Z

dd x0

Y

n>0

Y

n>0

δd (xn − x′n ) ,

dd xn |x0 , xn ihx0 , xn | = 1 .

(A.26)

• Oscillators as differential operators in position space κ|m| i ∂ hx0 , xn |αm |Ψi = − √ x + ls hx0 , xn |Ψi , ǫ(m) ls |m| ∂x|m| 2 ∂ hx0 , xn |Ψi . hx0 , xn |α0 |Ψi = −ils ∂x0

(A.27)

• Transformation from position space to Moyal space Z 2i A(¯ x, xe , pe ) = (det(2T ))d/2 dxo e− θ pe T xo Ψ(x0 , xn ) Z Nd d 2i = 2 2 (1 + ww) ¯ − 4 dxo e− θ pe T xo hx0 , xn |Ψi =: h¯ x, xe , pe |Ψi , (A.28) where the midpoint x ¯µ := X µ (π/2) is related to the center of mass x0 through Eq.(A.21) ! X X µ µ µ µ (A.29) x ¯ := X (π/2) = x0 − xe we , h¯ x| = hx0 | exp iˆ p· xe we . e

e

The Moyal space h¯ x, xe , pe | is given by a squeezed state α2 e

α2 o

¯

¯

d

h¯ x, xe , pe | = h¯ x|e 2κe − 2κo −ξM0 ξ−ξλ det(4κe 1/2 T κ−1/2 )2 o α2 e

M0 =

α2 o

¯

¯

d

= h¯ x|e 2κe − 2κo −ξM0 ξ−ξλ 2N d (1 + ww) ¯ −8 , ! √ ! κe 2 0 −i α − iˆ p w 2 e e 2ls ls √ , λ= . 2 2 2ls ¯ − T κ−1 0 2l2s T κ−1 T o αo o θ

θ

47

(A.30)

• Moyal ⋆ product and trace: ⋆ = exp

− − → 1← ∂ξ σ ∂ξ 2

= exp − d2

Tr A(¯ x, ξ) = | det(2πσ)|

Z

iθ ←−−→ ←−−→ ∂xe ∂pe − ∂pe ∂xe , 2

σ = iθ

−N d

dxe dpe A(¯ x, ξ) = (2πθ)

Z

0 1 −1 0

!

, (A.31)

dxe dpe A(¯ x, ξ) . (A.32)

The normalization of a field A in Moyal space coincides with the normalization of its image in Fock space (A.28) Z hΨ|Ψi = dd x ¯ Tr A† (¯ x, ξ) ⋆ A(¯ x, ξ) . (A.33) • Oscillators as differential operators in Moyal space ∂ h¯ x, xe , pe |α0 |Ψi = −ils h¯ x, xe , pe |Ψi =: β0 h¯ x, xe , pe |Ψi , ¯ ∂x w|e| x, xe , pe |Ψi = β¯ex − we′ β0 h¯ h¯ x, xe , pe |αe |Ψi = β¯ex − √ β0 h¯ x, xe , pe |Ψi 2 = βex h¯ x, xe , pe |Ψi , X h¯ x, xe , pe |αo |Ψi =: β p h¯ x, xe , pe |Ψi = β¯p h¯ x, xe , pe |ΨiU−e,o , o

e

e6=0

X X 1 θκ|o| ∂ 2ls √ pe Te|o| = R|o|e + ǫ(o) β¯ep U−e,o , = 2l ∂p θ 2 s e e>0 e6=0 θκ|e| κ ∂ ∂ 2ls 1 i |e| p x ¯ ¯ √ √ x + ls ǫ(e) + p , βe = ǫ(e) .(A.34) βe = − ls |e| ∂x|e| ∂p|e| θ |e| 2 2 2ls

βop

These satisfy ordinary commutation relation: [β¯ex , β¯ex′ ] = ǫ(e)κ|e| δe+e′ ,

[βex , βex′ ] = ǫ(e)κ|e| δe+e′ ,

[β¯ep , β¯ep′ ] = ǫ(e)κ|e| δe+e′ ,

[βop , βop′ ] = ǫ(o)κ|o| δo+o′ ,

[β¯ex , β¯ep′ ] = 0 , [βex , βop′ ] = 0 .

• Oscillators as fields in Moyal space22 r r κ|e| κ|e| x p ¯ ¯ βe A = (βe ⋆ A − A ⋆ β−e ) , βe A = (βe ⋆ A + A ⋆ β−e ) , 2 2 ls i 1 − ǫ(e)κ|e| x|e| + p|e| , βe := √ κ|e| 2ls θ [βe , βe′ ]⋆ = ǫ(e)δe+e′ .

(A.35)

(A.36)

We can also define odd mode fields through a Bogoliubov transformation

22

p

κ|o| βo :=

Xp e6=0

κ|e| βe U−e,o .

(A.37)

Theqconvention for βn here is the same as [12] βnBKM = βn , but differs by a factor from the convention in [9] κ|n| βn . βnBM = 2

48

The following relations hold r βop A =

κ|o| (βo ⋆ A + A ⋆ β−o ) , 2 1

A.2.2

−1

2 [β−e , βo ]⋆ = −ǫ(e)κ|e| U−e,o κ|o|2 .

[βo , βo′ ]⋆ = ǫ(o)δo+o′ ,

(A.38)

Butterfly projector

The momentum independent butterfly state AB (ξ) satisfies βe ⋆ AB = AB ⋆ β−e = 0 ,

∀e > 0 .

(A.39)

There is a unique solution in monoid AB (ξ) = 2dN

X 1 2ls2 1 exp − x ¯e κe xe + 2 p¯e pe 2ls2 θ κe e>0

!

.

(A.40)

It satisfies AB ⋆ AB = AB ,

T r (AB ) = 1.

(A.41)

In ordinary oscillator language, Eq.(A.39) means αe |ΨB i = 0 ,

X o>0

−1 −1 + α−o Uo,e αo U−o,e |ΨB i = 0 ,

∀e > 0

for zero momentum state. Now we take the ansatz   X 1 B † a†m Vmn an |Ωi |ΨB i = N exp − 2

(A.42)

(A.43)

m,n≥1

B : which corresponds to a monoid element in MSFT. Then we have constraints for Vmn

X

o′ >0

√ −1 −1 B√ κo′ U−o κo Uo,e , Voo ′ ′ ,e =

o, e > 0 ,

B B B Veo = Voe = Vee ′ = 0.

(A.44)

B which was At the open string limit κe = e, κo = o, N = ∞, we can show that the matrix Vmn obtained in [27] :  √ ]Γ[ n ] Γ[ m mn  −(−1) m+n 2 2 2 m+n πΓ[ m+1 ]Γ[ n+1 ] for m and n odd B 2 2 (A.45) Vmn =  0 for m or n even

satisfies Eq.(A.44). Namely, we have obtained the correspondence: 1 AB ↔ |ΨB i = exp − L−2 |Ωi , for κe = e, κo = o, N = ∞ . 2

49

(A.46)

A.2.3

L0 and L0

In MSFT L0 in matter sector is defined as L0 = =

1 2 X x x X p p β + β β + β β 2 0 e>0 −e e o>0 −o o X ∂ dX 1 2 κn il w (1 + ww)β ¯ + β − s e 0 0 2 ∂xe 2 n>0 e>0

X l2 ∂ 2 2ls2 θ2 2 ∂ 2 1 2 2 2ls2 2 1 s + − κ + κ x + p − − 2 ∂x2e 8ls2 e ∂p2e 2ls2 e e θ2 e 1 + ww ¯ θ2 e>0

X

we pe

e>0

!2

. (A.47)

The operator L0 can be rewritten in terms of a field L0 using Moyal star product, plus a remnant γ called the “midpoint correction term” [12], which is multiplied with an ordinary product L0 Aβ0 = L0 (β0 ) ⋆ Aβ0 + Aβ0 ⋆ L0 (−β0 ) + γAβ0 , X l2 κ2e 2 ls dX 1 s 2 2 κn , pe + 2 xe − we pe β0 + (1 + ww)β ¯ L0 (β0 ) := 0 − 2 θ 4ls θ 4 4 n>0 e>0 !2 1 2ls2 X γ=− we pe . 1 + ww ¯ θ2

(A.48) (A.49)

(A.50)

e>0

The γ term formally goes to zero as κn = n, N → ∞, since ww ¯ → ∞. However, this is not true in computations due to contributions of the form ∞/∞ that are related to the associativity anomaly. In fact, γ is indispensable to reproduce the correct spectrum of L0 [9][10]. The γ term depends P only on one special momentum mode pˆ = (1 + ww) ¯ −1/2 e>0 we pe which we call the anomalous midpoint momentum mode [9][11]. We can rewrite L0 in terms of oscillators ! X X d X 1 ls 2 L0 (β0 ) = κe β−e ⋆ βe + κe − κo + (1 + ww)β ¯ (w ¯e pe ) β0 , (A.51) 0 − 4 4 θ e>0 e>0 o>0 and then, acting on the butterfly projector (A.39), we have L0 AB = γAB . A.2.4

(A.52)

n-string vertex and Neumann coefficients

Here we give brief review of the correspondence of n-string vertex and Neumann coefficients in MSFT. We note the properties of the momentum state and coherent states in Fock space, together with the corresponding Moyal images. • Momentum eigenstate (zero mode part): |p0 i = 2πbls2

d4

1 †2

e− 2 aˆ0

50

√ bl2 ˆ†0 p0 − 4s p20 + bls a

|0i ,

1 2

hp0 | = h0|e− 2 aˆ0 +

hp0 |p′0 i

√

d d

= (2π) δ

bls a ˆ 0 p0 −

bl2 s p2 4 0

2πbls2

(p0 − p′0 ) ,

hp0 |x0 i = e−ip0 x0 ,

d4

,

hx0 |p0 i = eip0 x0 .

(A.53)

• Coherent state :23 ∗a ˆ

hΨ|ˆ a† = hΨ|µ∗ , 2 hΨ|V2 i12

hΨ| = hp|eµ

−µ∗ Cˆ a†(1)

=e

,

˜ 1, | − pi1 =: |Ψi

d 1 ∗2 1 ∗2 ¯ ¯ ˜ = 2N d (1 + ww) A˜ := h¯ x, xe , pe |Ψi ¯ − 8 e−i¯xp e 2 µe − 2 µo −ξM0 ξ−ξλ ,   √ 1 i 2 2 ∗ κe µ + ipwe  = 2K ∗ (µ∗ + W p) , λ =  ls √ e − 1 2 2ls 2 ∗ − θ T κo µo ! ! p κe i √ls w 0 ls 2 ∗ q 2κ e , K = , W = 0 − lθs T κ2o 0

(A.54)

where we used the reflector Z d (1) d (2) P n (1) (2) d p d p hV2 | = h0, p(1) |h0, p(2) |e− n≥1 (−1) aˆn aˆn (2π)d δd (p(1) + p(2) ) , d d (2π) (2π) Z d (1) d (2) P †(1) †(2) d p d p ˆn a ˆn d d (1) (2) − n≥1 (−1)n a (2π) δ (p + p )e |0, p(1) i|0, p(2) i . (A.55) |V2 i = d d (2π) (2π) In particular we have the bra-ket correspondence for eigenstates of xn : 12 hV2 |x0 , xn i2

= 1 hx0 , (−1)n xn | .

(A.56)

• Compute the n-string vertex for coherent states in terms of unknown Neumann coefficients rs , V rs , V rs V(n) 0(n) 00(n) |Vn i =

Z

dd p(1) dd p(n) · · · (2π)d δd (p(1) + · · · + p(n) ) (2π)d (2π)d rs a rs a rs − 21 a ˆ†(r) V(n) ˆ†(s) −p(r) V0(n) ˆ†(s) − 21 p(r) V00(n) p(s)

×e

hΨ1 | · · · hΨn |Vn i = (2π)d δd (p(1) + · · · + p(n) )

|p(i) i ,

rs µ(s)∗ −p(r) V rs µ(s)∗ − 1 p(r) V rs p(s) − 12 µ(r)∗ V(n) 0(n) 2 00(n)

×e

.

(A.57)

(A.58)

• Compute the trace of the Moyal images of n coherent states in MSFT Z dd x ¯Tr A˜1 (¯ x, ξ) ⋆ A˜2 (¯ x, ξ) ⋆ · · · ⋆ Añ (¯ x, ξ) = (−1)

23

Nd 2

d

(det((1 + m0 )n − (1 − m0 )n ))− 2 2nN d (1 + ww) ¯ −

× (2π)d δd (p(1) + · · · + p(n) ) eE

(n)

nd 8

,

(A.59)

Here we introduce the bra coherent state. This is a different convention from that in [9].

51

1 X (r)∗ µ C 2K ∗−1 m0 O(s−r) (m0 )K ∗ − δr,s µ(s)∗ 2 r,s X X ¯ K ∗−1 m0 O(s−r) (m0 )K ∗ W p(s) , ¯ K ∗−1 m0 O(s−r) (m0 )K ∗ µ(s)∗ − 1 p(r) 2W −2 p(r) W 2 r,s r,s ! iθ 0 2ls2 κe m0 := M0 σ = (A.60) 2 0 − 2ils T κ−1 T¯ E (n) = −

θ

o

where we used ¯ ∗σ , CK ∗−1 m0 = −K

CK ∗−1 m0 K ∗ = −K ∗−1 m0 K ∗ C ,

We note



m ˜ ∗0 := K ∗−1 m0 K ∗ = 

0 1 −1 −κo 2 T¯κe2

O(s−r) (m0 ) = −O(r−s) (−m0 ) . (A.61)

1 2

−1 −κe T κo 2

0



 = −m ˜0.

(A.62)

The sign is changed compared to that in [9] because we used bra coherent state hΨc | to define ˜ the Moyal field A. • The Neumann coefficients in the matter sector are obtained by identifying the Fock space and MSFT expressions and comparing the exponents24 Z dd x ¯ Tr A˜1 (¯ x, ξ) ⋆ A˜2 (¯ x, ξ) ⋆ · · · ⋆ Añ (¯ x, ξ) = ρhΨ1 |hΨ2 | · · · hΨn |Vn i , (A.63) ρ = (−1)

Nd 2

d

(det((1 + m0 )n − (1 − m0 )n ))− 2 2nN d (1 + ww) ¯ −

nd 8

(A.64)

and using momentum conservation δd (p(1) + p(2) + · · · + p(n) ). Then one has the Neumann coefficients rs V(n) = C 2K ∗−1 m0 O(s−r) (m0 )K ∗ − δr,s , 2 V rs0(n) = −2K ∗−1 m0 O(s−r) (−m0 )K ∗ W − W , n 2 rs ∗−1 ∗ ¯ K m0 O(s−r) (m0 )K W − W ¯W. V00(n) = 2W n

(A.65)

They satisfy Neumann matrix algebra as in Ref.[9]. For the 3-string vertex we write them explicitly 2±m ˜ ∗0 m ˜ ∗2 r,r±1 (±) 0 −1 = , M := CV , (3) m ˜ ∗2 m ˜ ∗2 0 +3 0 +3 −2m ˜ ∗2 ˜ ∗0 − 21 wls 4m ˜ ∗2 − 1 wls 0 ∓ 6m 0 κe 2 √ , V (±) := V r,r±1 κe √ , = V (0) := V rr0(3) = ∗2 ∗2 0(3) 3(m ˜ 0 + 3) 3(m ˜ 0 + 3) 2 2 1 1 ¯ t t − − rr κe 2 w V00 := V00(3) = ls2 wκ ¯ e2 tt¯ + 3 rr M(0) := CV(3) =

24

(A.66)

Here we defined the Witten’s ∗ product using the ket |V3 i as Eq.(2.8) which is different convention from that in [9]. Also, here we have included the overall normalization ρ which does not play a role in the computation of the Neumann coefficients.

52

r,s where we redefined as V00 = V00 δr,s using momentum conservation. We used the notation 1/2 −1/2 w ¯ = (we , 0) , and t = κe T κo .

• Fermionic ghost Neumann coefficients for the 3-vertex can be derived from matter Neumann coefficients for the 6-vertex √ 1 r,s r,s+3 − V(6) )√ , X rs := (−1)r+s κn (V(6) κn r,s r,s+3 −1 rs r+s √ (A.67) X 0 := (−1) κn (V 0(6) − V 0(6) )ls . This gives m ˆ ∗2 ˆ ∗2 ˆ ∗2 2m ˆ ∗0 + 2m 2m ˆ ∗0 − 2m 0 −1 0 0 (+) (−) , X = −C , X = C ∗2 + 1 ∗2 + 1 , 3m ˆ ∗2 3 m ˆ 3 m ˆ + 1 0 0 0 ∗2 w ∗ − 2m ∗ + 2m w 4m ˆ ∗2 ˆ ˆ ∗2 2 m ˆ 2 m ˆ (+) (−) 0 0 0 w 0 0 √ √ √ , (A.68) = , X = , X = − 0 0 ∗2 ∗2 ∗2 3m ˆ0 +1 2 3m ˆ0 +1 3m ˆ0 +1 2 2

X (0) = C X

(0) 0

where we defined m ˆ ∗0 :=

B

√

√ 1 1 κn m ˜ ∗0 √ = κn K ∗−1 m0 K ∗ √ . κn κn

(A.69)

Derivation of regularized matrix formula

Here we sketch a derivation of fundamental formulas for regularized matrices presented in §2.1.3. We begin from the defining relations in Eq.(2.37). The first two equations imply Eq.(2.43) and then from the remaining equations we have X X Qeo (vo′ )2 = (κ′e )−1 , Qeo (we′ )2 = (κ′o )−1 , (B.1) o

e

where

1 , (B.2) κ′e − κ′o with e = ±2, ±4, · · · ± 2N, and o = ±1, ±3, · · · ± (2N − 1). Now we regard Q = (Qeo ) as a 2N × 2N matrix and compute its inverse Q Q (κ′ − κ′o′ ) e′ 6=e (κ′o − κ′e′ ) −1 ′ ′ Qo′ 6=o e Q . (B.3) (Q )oe = (κe − κo ) ′ ′ ′ ′ e′ 6=e (κe − κe′ ) o′ 6=o (κo − κo′ ) Qeo :=

To prove the above formula, it is convenient to define a rational function f (z) which is determined by the setup (N, κ′e , κ′o ) uniquely: Q ′ ′ (z − κo′ ) . (B.4) f (z) := Qo ′ e′ (z − κe′ )

Next we compute (Q−1 Q)oo′′ . We use contour integration and residues, where we denote the residue of f (z) at z = z0 as Resz=z0 f (z) and assume that the frequencies κ′e , κ′o are nondegenerate and finite Q Q Q ′ ′ X κ′ − κ′ (κ′ − κ′o′ ) e′ 6=e (κ′o − κ′e′ ) X −Resz=κ′e f (z) o Qo′ 6=o e e e′ (κo − κe′ ) Q Q = κ′e − κ′o′′ e′ 6=e (κ′e − κ′e′ ) o′ 6=o (κ′o − κ′o′ ) (κ′e − κ′o )(κ′e − κ′o′′ ) o′ 6=o (κ′o − κ′o′ ) e e 53

Q ′ ′ X f (z) ′ (κ − κe′ ) Resz=κ′e = −Q e o′ ′ ′ (z − κo )(z − κ′o′′ ) o′ 6=o (κo − κo′ ) e Q I ′ ′ f (z) dz ′ (κ − κe′ ) = Q e o′ ′ ′ ′ o′ 6=o (κo − κo′ ) z=κ′o ,κ′o′′ 2πi (z − κo )(z − κo′′ ) Q ′ ′ f (z) e′ (κo − κe′ ) Q = δo,o′′ = δo,o′′ . Resz=κ′o ′ − κ′ ) (z − κ′o )2 (κ o′ 6=o o o′

(B.5)

This shows that we have the correct inverse matrix Q−1 . Similarly, we can obtain (vo′ )2 , (we′ )2 Eqs.(2.45,2.44) as follows:25 Q Q ′ ′ X X 1 o′ 6=o (κ′e − κ′o′ ) ′ (κo − κe′ ) ′ 2 −1 ′ −1 e Q (vo ) = (Q )oe (κe ) = − Q ′ ′ ′ ′ ′ o′ 6=o (κo − κo′ ) e κe e′ 6=e (κe − κe′ ) e Q Q ′ ′ ′ ′ X f (z) f (z) e′ (κo − κe′ ) e′ (κo − κe′ ) Q Q ′ Res Resz=0 = = − z=κ e ′ ′ −κ ) ′ − κ′ ) ′) (κ (κ z(z − κ z(z − κ′o ) o o′ 6=o o o′ 6=o o o′ o′ e Q Q Q Q 2 2 2 ′ ′ ′ o′ >0,o′ 6=|o| κo′ e′ >0 (κe′ − κ|o| ) e′ (κe′ − κo ) o′ 6=o κo′ Q , (B.6) = Q = Q ′ Q ′ ′ 2 e′ >0 κ2e′ o′ >0,o′ 6=|o| (κ2o′ − κ2|o| ) o′ 6=o (κo′ − κo ) e′ κe′ Q Q ′ ′ X X 1 e′ 6=e (κ′o − κ′e′ ) ′ (κe − κo′ ) ′ 2 −1 ′ −1 o Q (we ) = (Q )oe (κo ) = Q ′ ′ ′ ′ ′ κ ′ 6=e (κe − κe′ ) o e o′ 6=o (κo − κo′ ) o o Q Q ′ ′ ′ ′ X f (z) f (z) ′ (κe − κo′ ) o′ (κe − κo′ ) o Q ′ = − Res Resz=0 = Q z=κ ′ ′ o ′ ′ ′ z(z − κe ) z(z − κ′e ) e′ 6=e (κe − κe′ ) o e′ 6=e (κe − κe′ ) Q Q Q Q 2 2 2 ′ ′ ′ o′ >0 (κ|e| − κo′ ) e′ >0,e′ 6=|e| κe′ e′ 6=e κe′ o′ (κe − κo′ ) Q Q = Q ′ Q (B.7) = ′ ′ 2 o′ >0 κ2o′ e′ >0,e′ 6=|e| (κ2|e| − κ2e′ ) e′ 6=e (κe − κe′ ) o′ κo′

where we assumed κ′e , κ′o are nonzero and Eqs.(2.42). We note that the above formula (B.6,B.7) can also be rewritten as (vo′ )2 =

1 f (0) , Resz=κ′o ′ κo f (z)

(we′ )2 =

f (z) 1 . Resz=κ′e ′ κe f (0)

(B.8)

Now we consider the open string limit (A.19). By setting the open string limit κ′e = e, κ′o = o, N → ∞, the rational function f (z) (B.4) becomes Q∞ z2 πz πz −1 πz 2 f (z) n=1 (1 − (2n−1)2 ) = = Q∞ sin (B.9) = cos πz . z2 f (0) 2 πz 2 2 tan 2) 2 n=1 (1 − 4n

With this formula and Eq.(B.8) we can show that the regularized quantities reduce to the original ones in Eq.(2.35).

C

bc-ghost sector in position space

Here we consider the ghost position space representation of the n-point string vertices for n = 1, 2, 3, starting with the Fock space formalism. 25

We choose the sign convention of we′ , vo′ such that they are consistent with Eq.(2.35).

54

Identity state X

|Ii =

(−1)

o−1 2

o>0

ˆb−o

!

ˆb0 + 2

X e>0

X o−1 (−1) 2 xo

i hc0 , xn , yn |Ii = √ 2

o>0

e 2

(−1) ˆb−e

!

!

P∞

nˆ ˆ −n b−n n=1 (−1) c

e

cˆ0 cˆ1 |Ωi ,

Y √ √ e (i 2xe )( 2ye − 2(−1) 2 c0 ) .

(C.1) (C.2)

e>0

This is BRST invariant[2] although the form is rather complicated. Reflector 12 hV2 |

(1)

(2)

= 1 hΩ|ˆ c−1 2 hΩ|ˆ c−1 e− (1)

P∞

(2)

|V2 i12 = (ˆb0 − ˆb0 )e

P∞

(2) (2) (1) n c(1) ˆ cn ˆbn ) n bn +ˆ n=1 (−1) (ˆ

(2) (2) (1) n c(1) ˆ c−nˆ b−n ) n=1 (−1) (ˆ −n b−n +ˆ

(2) (2) (2) (1) (1) (1) 1 hc0 , xn , yn |2 hc0 , xn , yn |V2 i12

=

(1)

(2)

(ˆ c0 + cˆ0 ) ,

(1) (1)

(C.3)

(2) (2)

cˆ0 cˆ1 |Ωi1 cˆ0 cˆ1 |Ωi2 , (C.4) ∞ Y (1) (2) (2) n (1) (2) −2i((−1)n x(1) = (c0 + c0 ) n + xn )((−1) yn + yn )

n=1 (2) (1) (1) (1) n (2) hc0 , xn , yn | − c0 , −(−1) xn , −(−1)n yn(2) i .

(C.5)

3-string vertex P3

|V3 i123 = e

=

r,s=1

(s)

b0 −ˆ c†(r) X rsˆ b†(s) −ˆ c†(r) X r 0s ˆ

(1) (1)

(2) (2)

(3) (3)

cˆ0 cˆ1 |Ωi1 cˆ0 cˆ1 |Ωi2 cˆ0 cˆ1 |Ωi3 ,

(1) (1) (1) (2) (2) (2) (3) (3) (3) 1 hc0 , xn , yn |2 hc0 , xn , yn |3 hc0 , xn , yn |V3 i123 (1) (2) (3) rs − det (δr,s δn,m + Xnm )(c0 − wy ¯ e(1) )(c0 − wy ¯ e(2) )(c0 r,s,n,m

− wy ¯ e(3) )ei

P

(C.6) rs

r,s

y (r) ( 1−X x(s) 1+X )

(C.7)

where we used the relation[21][22][23], w X rs0 = (δrs + X rs ) √ . 2

(C.8)

Witten’s star product in ghost position space Ψ1 ⋆W Ψ2 (c0 , xn , yn ) =

=

Z

Z

(2)

(2) (2) (3) dxn dyn

dc0 dc0

(3)

(3)

dxn dyn −2i

−2i (2) g (2) (2) (3) g (3) (3) (2) (3) (3) (3) (2) ×1 hc0 , xn , yn |2 hc0 , xn , yn |3 hc0 , xn , yn |V3 i123 Ψ1 (c0 , x(2) n , yn )Ψ1 (c0 , xn , yn ) (2)

(2) (2) (3) dxn dyn

dc0 dc0

−2i

(2)

(3)

(3)

dxn dyn −2i

(2) (2) (2) (3) (3) (3) 1 hc0 , xn , yn |2 hc0 , xn , yn |3 hc0 , xn , yn |V3 i123 (3)

n (2) n (3) n (3) ×Ψ1 (−c0 , −(−1)n x(2) n , −(−1) yn )Ψ2 (−c0 , −(−1) xn , −(−1) yn )

(C.9)

where xn , yn | 1 hc0 , g

:= 12 hV2 |c0 , xn , yn i2 = 1 h−c0 , −(−1)n xn , −(−1)n yn | .

55

(C.10)

D

Moyal ⋆ for (DD)b (N N )c split strings

In this appendix, we examine the remaining choice of the midpoint boundary condition for the split string variables as compared with our discussion in section 2.3. Namely, we consider Dirichlet boundary condition for b(σ) and Neumann for c(σ) at the midpoint σ = π/2. In this case the left and right half of b(σ) : lb (σ), r b (σ) satisfy Dirichlet boundary condition at both σ = 0, π/2, and those of c(σ) : lc (σ), r b (σ) satisfy Neumann at σ = 0, π/2. The lb (σ), r b (σ) and lc (σ), r c (σ) are expanded in terms of even sine/cosine modes respectively lb (σ) =

∞ √ X 2 ¯ leb sin eσ , σb + i 2 π

r b (σ) =

e=2

lc (σ) = c¯ +

∞ √ X lec (cos eσ − ie ) , 2

∞ √ X 2 ¯ reb sin eσ , σb + i 2 π

r c (σ) =

e=2

e=2 ∞ √ X c c¯ + 2 re (cos eσ e=2

− ie ) .

(D.1) (D.2)

From Eqs.(2.69)(2.70)(2.27) (A.12), we have relations between split and full string variables: ¯b = wx ¯ ˜ o,

leb = xe + T˜xo ,

c¯ = c0 − wy ¯ e,

lec

reb = −xe + T˜xo ,

= ye + T yo .

rec

= ye − T yo .

With this setup Witten type product in the split string formulation becomes: Z Y b c ˜ ′ (¯b, c¯, η b , −η c , r b , r c ) . ˜′ ¯ ¯, lb , lc , η b , η c )B ˜ ′ (¯b, c¯, lb , lc , r b , r c ) = idη dη A˜′ ∗ B e e e e e e e e e e A (b, c e e e e

(D.3) (D.4)

(D.5)

e>0

where the split string and full string fields in position space are the same A˜′ (¯b, c¯, leb , lec , reb , rec ) ∼ Ψ(c0 , xn , yn )

(D.6)

by substituting on the right hand side the inverse maps obtained in Eqs.(D.3)(D.4) 1 1 õ¯b + S¯oe (leb + reb ) , xe = (leb − reb ) , xo = u 2 2 1 1 c 1 c c c0 = c¯ + w ¯e (le + re ) , ye = (le + rec ) , yo = Roe (lec − rec ) . 2 2 2

(D.7) (D.8)

These relations are valid only when ww ¯ = ∞ in the limit: κe = e, κo = o, N = ∞. As the next step, we consider the Moyal formulation, including the regularization with (N, κe , κo ). Comparing Eqs.(D.3)(D.4)(D.5) with Eq.(2.63), we identify x ∼ T˜xo ,

y ∼ 2xe ,

x′ ∼ −T yo ,

y ′ ∼ 2ye ,

and define new variables with even index E by xE := T˜xo ,

yE := T yo .

56

(D.9)

At the limit κe = e, κo = o, N = ∞, T˜ has a zero mode u ˜ (2.34) and we meet as usual the associativity anomaly, but at finite N everything is well-defined. From Eq.(2.63)(D.6) we obtain the Moyal image A′ (¯b, c¯, xE , pE , yE , qE ) of the position space field Ψ(c0 , xn , yn ) : A′ (¯b, c¯, xE , pE , yE , qE ) Z Y = 2−2N i−1 dxe dye e−2pE xe −2qE ye A˜′ (¯b, c¯, xe + xE , ye + yE , −xe + xE , ye − yE ) e>0

= 2−2N

Z Y

e>0

¯ E , ye , RyE ) , c + wy ¯ e , xe , u ˜¯b + Sx i−1 dxe dye e−2pE xe −2qE ye Ψ(¯

(D.10)

and the corresponding Moyal ⋆ product becomes − 21

⋆=e

← − → − ∂ ∂ ∂xE ∂pE

← −

→ − ∂ E ∂qE

+ ∂y∂

← −

→ − ∂ E ∂xE

+ ∂p∂

← −

→ − ∂ E ∂yE

+ ∂q∂

.

(D.11)

In this case, the above formula is more complicated than our previous choice due to the additional b-ghost midpoint mode ¯b.

E

Ghost butterfly projector with even modes

The butterfly projector in Eq.(2.126) is based on the odd mode oscillators in Eq.(2.10). There is another choice using even mode oscillators Eq.(2.140) c b = Aˆ′B ⋆ β−e = 0, βeb ⋆ Aˆ′B = βec ⋆ Aˆ′B = Aˆ′B ⋆ β−e

∀e > 0 .

(E.1)

Explicitly, we have the even butterfly state Aˆ′B

−2N

= ξ0 2

exp −

X

ixbe κe xce

e>0

4i c + ′ 2 pbe κ−1 e pe θ

!

(E.2)

in the Siegel gauge. As we will see in the following, the even butterfly is the one defined by Gaiotto-Rastelli-SenZwiebach(GRSZ) using twisted ghosts [25]. The conditions in Eq.(E.1) correspond to ˆbe |ΨB i = 0 , cê |ΨB i = 0 , X −1 −1 ˆ ˆbo U −o,e + b−o Uo,e |ΨB i = 0 , o>0

X o>0

(Ue,−o cô + Ue,o cˆ−o ) |ΨB i = 0

for all e > 0 in ordinary oscillator language. If we take a gaussian ansatz   X B ˆ  |ΨB i = N exp  cˆ−m V˜mn b−n cˆ1 |Ωi ,

(E.3)

(E.4)

n,m≥1

the above conditions become

B B B = V˜ee V˜eo = Võe ′ = 0,

57

e, e′ , o > 0 ,

(E.5)

X

o′ >0

X

−1 −1 VõB′ o U−o ′ ,e = −Uo,e , B Võo ′ Ue,−o′ = Ue,o ,

e > 0,

(E.6)

e > 0.

(E.7)

o′ >0

We can solve Eq.(E.6) by comparing it with the matter one (A.44) as √ 1 B √ B B 1 √ V˜mn = − √ Vnm m = − mVmn n n  m n m+n Γ[ 2 ]Γ[ 2 ]  (−1) 2 m m+n πΓ[ m+1 ]Γ[ n+1 ] for m and n odd 2 2 = .  0 for m or n even

(E.8)

We can check that this V˜ B satisfies Eq.(E.7). The ghost butterfly which we have obtained in MSFT as above can be identified with GRSZ’s (twisted) ones [25]26 : 1 ′ ′ ˆ AB ↔ |ΨB i = exp − L−2 |Ω′ i for κe = e, κo = o, N = ∞ . 2 The relation between the matter and (twisted) ghost generating functions is obtained by using [28] ∂ ˜ S(w, z) = S(z, w) ∂z where we defined the generating functions S(z, w) := ˜ w) := S(z,

∞ X √

mn(−z)m−1 (−w)n−1 Smn =

m,n=1 ∞ X

f ′ (z) w f (w) + , z(w − z) f (z) f (w) − f (z)   ∞ X ˜ = exp  |Si cˆ−m S˜mnˆb−n  |Ω′ i

(−z)m−1 (−w)n S˜mn = −

m,n=1



namely

f ′ (z)f ′ (w) 1 − , (z − w)2 (f (z) − f (w))2

 ∞ X 1 |Si = exp − a ˆ† Smn a ˆ†n  |Ωi , 2 m,n=1 m

butterfly state e

F

(E.10)

(E.11)

m,n=1

√ √ 1 1 S˜mn = − √ Snm m = − mSmn √ . n n

(E.12)

√ z 1+z 2

which represents (canonical)

B = S Here we have Vmn mn for the conformal mapping f (z) = − 12 L−2

(E.9)

|Ωi [27] and the relation Eq.(E.8).

Algebra of gaussian operators

We discuss algebraic relations for gaussians constructed from bosonic and fermionic oscillators which we used to compute the propagator (4.17) in MSFT. 26

There is a correspondence for the vacuum: cˆ1 |Ωi ∼ |Ω′ i .

58

For bosonic oscillators: a, a† , [a, a† ] = 1, we can prove a formula † Aa† +aBa

ea

√

1

= e− 2 Tr log(cos(2

AB))

1 †

e2a

√ −a† log(cos(2 AB ))a

×e

e

√ √ tan(2 AB ) ABB −1 a†

√ tan(2 AB ) 1 √ aB a 2 AB

(F.1)

¯ = B. using a similar method to Appendix A in [32]. Here A, B are symmetric matrices: A¯ = A, B Then we have ∂

∂

√

1

eηAη+ ∂η B ∂η eiξη = e− 2 Tr log(cos(2 AB)) √ √ √ tan(2 AB ) 1 1 √ √ ξ η tan(2 AB ) ABB −1 η− 12 ξB ξ+iη 2 AB cos(2 AB ) ×e where we used the relation

∂ , η = 1, ∂η

∂ ∂ ηC ∂η iξη −ηC ∂η

e

e

e

(F.2)

C

= eηe ξ .

(F.3)

We obtain the formula for the propagator Z ∂ ∂ ′ η′ Aη′ + ∂η ′ B ∂η ′ iξη e dM ξ e−iξη e = (2π)

√ √ √ √ sin(2 AB) AB AB AB √ √ √ − 12 η B −1 η− 12 η′ B −1 η′ +η′ B −1 η − 21 Tr log B √ tan(2 AB) tan(2 AB) sin(2 AB)

M 2

e

e

AB

. (F.4)

When the momentum is nonzero in (4.17), we need the modified version of the above formula: Z ∂ ∂ ′ ′ η′ Aη′ + ∂η ′ B ∂η ′ +Cη iξη e dM ξ e−iξη e M 2

√ √ √ √ sin(2 AB) AB AB AB √ √ √ B −1 η− 12 η′ B −1 η′ +η′ − 12 η B −1 η − 21 Tr log B √ tan(2 AB) tan(2 AB) sin(2 AB)

= (2π) e

e

AB

√ √ √ AB C+ 1 (η+η ′ ) tan √ AB C − 14 CA−1 1− tan 2 AB AB

×e

.

(F.5)

For fermionic oscillators: a, a† , {ai , a†j } = δij , we have a similar formula † Aa† +¯ aBa

ea¯

√

1

= e 2 Tr(log cosh(2

AB))

1 †

e 2 a¯

√ −¯ a† log(cosh(2 AB))a

×e

e

√ √ tanh(2 AB) ABB −1 a† √ tanh(2 AB) 1 √ a ¯ B a 2 AB

,

(F.6)

¯ = −B. Noting where A, B are antisymmetric matrices A¯ = −A, B ∂ ∂ C ¯ −¯ η¯C ∂ η C ∂η = 1 , e ∂η e−ξη e = eη¯e ξ , η, ∂η

(F.7)

we obtain ¯ ∂ ∂ ¯ B ∂η η¯Aη+ ∂η −ξη

e

e

=e

√

1 Tr(log cosh(2 2

AB))

e

√

1 η¯ tanh(2 2

√ √ ¯ tanh √2 AB ξ+¯ AB) ABB −1 η+ 12 ξB η AB

1√ ξ cosh 2 AB

By integration we have Z ∂ ∂ ¯ η¯′ Aη′ + ¯ ¯ ′ B ∂η ′ −ξη ∂η ′ dξ eξη e e ! √ √ √ √ AB AB AB 1 1 sinh 2 AB √ √ √ η¯ B −1 η′ −¯ η′ B −1 η B −1 η+ 12 η¯′ tanh 2 AB sinh 2 AB . e 2 tanh 2 AB = det 2 B √ AB 59

.

(F.8)

G G.1

Neumann coefficients Neumann coefficients from CFT

In this subsection, we give a short summary of the analytic expression of Neumann coefficients given in [2] which are obtained from conformal field theory. We introduce a set of numbers An , Bn (n = 0, 1, 2, · · · ) which appear in the Taylor expansion, X X 1 + ix 1/3 X 1 + ix 2/3 X = Ae xe + i Ao xo , = Be x e + i Bo x o . (G.1) 1 − ix 1 − ix o>0 o>0 e≥0

e≥0

(0,±) (0,±) ñm From these data, the Neumann coefficients (Nnm for matter sector and N for ghost are written as follows. First when n, m > 0 and n 6= m, ( (−1)n An Bm +Bn Am An Bm −Bn Am n + m = even + (0) 3 (n+m) (n−m) Nnm = , 0 n + m = odd  −(−1)n An Bm +Bn Am An Bm −Bn Am  n + m = even +  (n+m) (n−m)  √6 (±) ± 3 An Bm −Bn Am m +Bn Am Nnm = n + m = odd , + An B(n−m) 6 (n+m)    ( −(−1)n An Bm +Bn Am An Bm −Bn Am n + m = even − (0) 3 (n+m) (n−m) ñm , N = 0 n + m = odd  (−1)n An Bm +Bn Am An Bm −Bn Am  n + m = even −  (n+m) (n−m)  √6 (±) ˜ A B −B A A B +B A ∓ 3 n m n m m n m Nnm = n + m = odd . − n (n−m) 6 (n+m)   

For the diagonal components (n = m > 0), they are replaced by, ! n X 1 n (0) 2(−1) (1 + Nnn = (−1)k A2k ) − (−1)n − A2n , 3n k=1

sector)

(G.2)

(G.3)

(G.4)

(G.5)

(0)

(±) Nnn

(−1)n Nnn =− − , (G.6) 2n 2

2(−1)n An Bn (−1)n 1 ˜ (0) (0) (0) (±) ñn ñn N = Nnn − , N =− − Nnn . (G.7) 3n 2n 2 For the zero mode, we use ( ( −1 2 A m = even A m = even 1 33 (±) (0) 3m 3m m √ m , N = − ln , (G.8) , N0m = N0m = 00 ∓ 3 2 42 0 m = odd m = odd 3m Am ( ( 1 −2 B m = even B m = even m (±) (0) 3m 3m ˜ ˜ = √ m , N = . (G.9) N 0m 0m ∓ 3 0 m = odd m = odd 3m Bm There are some differences in the convention to make direct comparison of these quantities with the corresponding ones obtained in Moyal language which are given in (A.66, A.68). We summarize them as follows, √ (0,±) (0,±) n√ mnNmn , Vn(0,±) (cf t) ≡ −ls nN0n , V00 (cf t) ≡ −ls2 N00 ,(G.10) M(0,±) nm (cf t) ≡ −(−1) 60

(0,±) ˜ (0,±) . Xm0 (cf t) ≡ mN 0m

(0,±) (0,±) ñm Xnm (cf t) ≡ mN ,

(G.11)

The sign factor in M(0,±) (cf t) comes in because we include the multiplication of Cnm = (−1)n δn,m in the MSFT definition.

G.2

Ratios of MSFT-regulated and CFT Neumann coefficients

The MSFT-regulated Neumann coefficients are discussed in sections 3.2,3.3 and A.2.4. The numer(0) (0) (0) (0) ical ratios Mee (N ) /Mee′ (cf t) and Xee (N ) /Xee′ (cf t) for e, e′ = 2, 4, 6, 8 at N = 5, 20, 100, 400, shows the convergence of these to the CFT values in the large N limit. (0)

(0)

Mee′ (5)

2

4

6

8

Mee′ (20)

2

4

6

8

2

1.15355

1.27359

1.43938

1.71214

2

1.02373

1.04035

1.05899

1.07957

4

1.27359

1.41879

1.61307

1.92691

4

1.04035

1.05982

1.08099

1.10391

6

1.43938

1.61307

1.84084

2.20463

6

1.05899

1.08099

1.10438

1.12937

8

1.71214

1.92691

2.20463

2.64473

8

1.07957

1.10391

1.12937

1.15628

Mee′ (100)

2

4

6

8

Mee′ (400)

2

4

6

8

2

1.00272

1.00459

1.00664

1.00885

2

1.00043

1.00071

1.00103

1.00137

4

1.00459

1.00675

1.00907

1.01151

4

1.00071

1.00105

1.0014

1.00178

6

1.00664

1.00907

1.0116

1.01426

6

1.00103

1.0014

1.00179

1.0022

8

1.00885

1.01151

1.01426

1.01709

8

1.00137

1.00178

1.0022

1.00263

Xee′ (5)

2

4

6

8

Xee′ (20)

2

4

6

8

2

1.28946

1.42714

1.60523

1.89334

2

1.1113

1.15228

1.19097

1.22861

4

1.42714

1.56675

1.75409

2.06284

4

1.15228

1.18898

1.22491

1.26065

6

1.60523

1.75409

1.9584

2.29906

6

1.19097

1.22491

1.25898

1.29343

8

1.89334

2.06284

2.29906

2.69597

8

1.22861

1.26065

1.29343

1.32697

2

4

6

8

Xee′ (400)

2

4

6

8

(0)

Mee′ (cf t)

(0)

Mee′ (cf t)

(0)

(0) Mee′ (cf t)

(0)

(0)

Mee′ (cf t)

(0)

(0) Xee′ (cf t)

(0)

(0) Xee′ (cf t)

(0)

Xee′ (100) (0)

Xee′ (cf t)

(0)

(0)

Xee′ (cf t)

2

1.03837

1.052

1.06441

1.07597

2

1.01529

1.02071

1.02562

1.03019

4

1.052

1.06393

1.07517

1.08587

4

1.02071

1.02544

1.02988

1.03409

6

1.06441

1.07517

1.08556

1.09557

6

1.02562

1.02988

1.03397

1.0379

8

1.07597

1.08587

1.09557

1.10504

8

1.03019

1.03409

1.0379

1.0416

61

H

Twist and SU (1, 1) in the Siegel gauge

ˆ = (−1)L0 . In MSFT this becomes The twist operator is usually given by Ω ˆ x, xe , pe ; ξ0 , xo , po , yo , qo ) = A(¯ ˆ x, xe , −pe ; ξ0 , −xo , po , −yo , qo ) . βˆΩ A(¯ This follows from D D gh gh gh ˆ gh gh gh gh x0 , xe , xo ; c0 , xgh , x , y , y Ω = x , x , −x ; c , x , −x , y , −y 0 e o 0 e e o e o o e o .

(H.1)

(H.2)

When we use the even variables xbe , pbe , xce , pce , we have the expression

ˆ x, xe , pe ; ξ0 , xbe , pbe , xce , pce ) = A(¯ ˆ x, xe , −pe ; ξ0 , −xbe , pbe , −xce , pce ) . βˆΩ A(¯

(H.3)

The SU (1, 1) generators are given by Gˆ :=

∞ X

(ˆ c−nˆbn − ˆb−n cˆn ) (= Ngh − (ˆ c0ˆb0 + 1)) ,

n=1

ˆ := − X

∞ X

n=1

Yˆ :=

nˆ c−n cˆn ,

∞ X 1ˆ ˆ b−n bn n

(H.4)

n=1

by oscillator representation [30][31]. In MSFT we translate them as X X ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ b b c c βˆGˆ = yo − xo + po − qo = xe c − xe b + p e b − p e b , ∂yo ∂xo ∂po ∂qo ∂xe ∂xe ∂pe ∂pe o>0 e>0 X X ∂ ∂ ∂ ∂ − po κo (H.5) =i xce b − pbe c , βˆXˆ = i yo κo ∂xo ∂qo ∂xe ∂pe e>0 o>0 X X ∂ c ∂ −1 ∂ b ∂ βˆYˆ = i − p − q κ = i xo κ−1 x o o e o e ∂yo ∂po ∂xce ∂pbe o>0

e>0

on the fields in the Siegel gauge. These operators satisfy the su(1, 1) algebra: [βˆXˆ , βˆYˆ ] = −βˆGˆ ,

[βˆGˆ , βˆXˆ ] = 2βˆXˆ ,

[βˆGˆ , βˆYˆ ] = −2βˆYˆ .

(H.6)

They are derivations with respect to the Moyal ⋆ product: βÔˆ (A1 ⋆ A2 ) = (βÔˆ A1 ) ⋆ A2 + A1 ⋆ (βÔˆ A2 ) ,

ˆ = G, ˆ X, ˆ Yˆ . O

(H.7)

In fact, the above su(1, 1) generators (H.5) are inner derivations ˆ = G, ˆ X, ˆ Yˆ , βÔˆ A = [βOˆ , A]⋆ , O 1 X 1 X c c βGˆ = ′ (yo qo − xo po ) = ′ xe pe − xbe pbe , θ θ o>0 e>0 X X i i X i X b c i yo κo po = ′ xce pbe , βYˆ = ′ xo κ−1 q = x p . βXˆ = ′ o o θ o>0 θ e>0 θ o>0 θ ′ e>0 e e 62

(H.8)

By Eqs.(H.3,H.7), we can restrict solutions of the equations of motion (4.15) to the twist even and SU (1, 1) singlet sector: βˆΩˆ A(ξ) = A(ξ) ,

βÔˆ A(ξ) = 0 ,

ˆ = G, ˆ X, ˆ Yˆ O

(H.9)

in the Siegel gauge consistently. In fact, we note that in the Siegel gauge [βÔˆ , L0 ] = 0 ,

ˆ = Ω, ˆ G, ˆ X, ˆ Yˆ , O

βˆΩˆ (A ⋆ A) = (βˆΩˆ A)˜⋆(βˆΩˆ A) = (βˆΩˆ A) ⋆ (βˆΩˆ A) ,

˜⋆ := ⋆|−θ,−θ′

(H.10)

from Eqs.(4.10,4.11). This condition (H.9) can be used to search for the nonperturbative tachyon vacuum.[30] In MSFT, it is convenient to note the monoid structure (§3). It consists of gaussian Moyal ¯ ¯ fields: AN ,M,λ = N e−ξM ξ−ξλ . We can consider the twist and SU (1, 1) symmetric class within the monoid. From Eqs.(H.3,H.5) the restriction by this symmetry (H.9) of a monoid element AN ,M,λ (ξ) is given by ! ! ′ 0 M A 0 ¯ =B, M = εM ′ := , M′ = , A¯ = A , B λ=0 (H.11) −M ′ 0 0 B in the basis ξ¯ = (xbe , pbe , xce , pce )27 . Namely, the coefficient matrix of the quadratic term in the exponent becomes block diagonal and symmetric. For example, the perturbative vacuum and butterfly states (4.22,E.2) are of the form of (H.11). Their τ -evolved gaussians are also in this class (4.31). We note that this class of gaussian is not closed within the monoid because of twist operator which changes the sign of noncommutative parameters θ, θ ′ in the Moyal ⋆ product, while the SU (1, 1)-symmetry is conserved in the Siegel gauge by Eq.(H.7).

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65