Concerning the ghost contribution to the one-loop integrands in open string field theory
CConcerning the ghost contribution to the one-loop integrands inopen string field theory
Albin James
University of Southern California,Los Angeles , CA 90089, USAElite Prep, Arcadia, CA 91007, USA
E-mail: jamesalbin44 @ gmail.com
Abstract
We examine the ghost contribution to the one-loop integrands in open string field theory usingthe Moyal representation of the star product. We primarily focus on the open string tadpoleintegrand, which is an intrinsically off-shell quantity. Due to the closed string tachyon, the fullamplitude is badly divergent from the closed string degeneration region t → + of the Schwingerparameter. We obtain expansions for the finite factors from the squeezed state matrix R ( t )characterizing the ghost part of the tadpole in Siegel gauge. The analytic structure of theintegrands, as a function of the Schwinger parameter, captures the correct linear order behaviournear both the closed and open string degeneration limits. Using a geometric series for the matrixinverse, we obtain an approximation for the even parity matrix elements. We employ an expansionbased on results from the oscillator basis to construct Pad´e approximants to further analyse hintsof non-analyticity near this limit. We also briefly discuss the evaluation of ghost integrands forthe four string diagrams contributing to the one-loop 2-point function in open string field theory. a r X i v : . [ h e p - t h ] M a r ontents t → + . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Illustrations for geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Expansions near t = ∞ in the continuous κ -basis . . . . . . . . . . . . . . . . . . . . 394.4 A convergent expansion in q using the oscillator expression . . . . . . . . . . . . . . 44 I ( s )12 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 The planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 nm ( q )
57C The bc system and ˆ β oscillators 60D The twisted ghost butterfly case 62 String field theories provide an off-shell formulation of string theory that is conceptually simple andvery similar in structure to conventional gauge field theories. By construction [1–4], these furnisha field theory of strings with a spacetime action, and aspire to describe different regions of theparameter space of string theory using a universal set of degrees of freedom, encoded in the stringfield | Φ (cid:105) . The best understood covariant string field theory is the bosonic open string field theory(OSFT) with Witten type [4] cubic vertices. Remarkably, starting from a few axioms, this OSFTdefines an interacting theory for an infinite number of fields by virtue of the underlying worldsheetconformal symmetry—which is closely tied to its spacetime gauge invariance.In this paper, we revisit the perturbative structure of OSFT [5–9] at the one-loop level, withonly one or two external states. This requires evaluating the one-loop 1-point function (tadpole)and the one-loop 2-point function (string propagator). The latter receives contributions from fourdiagrams—three planar and one non-planar—due to the rigid nature of the Witten vertex. We havemade analytical and numerical progress primarily on the ghost sector contribution to the tadpole1ntegrand, which also appears as a subdiagram in two of the planar one-loop 2-point functions, andis an intrinsically off-shell quantity.In [7], Ellwood et al. carry out a careful study of the (open string) tadpole state using boundaryconformal field theory (BCFT) and oscillator methods. In the Siegel gauge b | Φ (cid:105) = 0 that we shallbe working in, the integrand is expressible as a function of the Schwinger parameter t (the lengthof the propagator loop in Fig. 1) and exhibits essential singularities at its limiting values, namely0 and ∞ . Physically, these divergent pieces may be understood in terms of degenerating stringdiagrams from the boundary of moduli space [2, 6] and arise from the open string tachyon ( t → ∞ ),the closed string tachyon, and massless closed string states ( t →
0) propagating in the loop.One method to study the tadpole diagram near t = 0 is to approximate it by using an appro-priate boundary state |B(cid:105) in a BCFT analysis. This explicitly includes the closed string oscillators c n , ˜ c n , b n , ˜ b n , a n , ˜ a n and can be organized into levels. The chain of conformal maps employed re-produce the correct divergence structure; we refer the interested reader to [7, §
3] where the leadingdivergence (for the D25 brane case) was carefully derived to be: |T ( t ) (cid:105) ∼ e +2 π /t t exp (cid:20) − a † n C nm a † m − c † n C nm b † m (cid:21) ˆ c | ˆΩ (cid:105) . (1.1)Here, the ket state on the RHS is the Shapiro-Thorn closed string tachyon state that arose in thework of [6] and later analyzed in detail in [2, 7, 8] and C is the twist matrix ( − ) n δ nm . As discussedin [7, App A] (see also [5, 10]), it also contributes to a BRST anomaly Q B |T (cid:105) (cid:54) = 0, that is alsopresent in the bosonic OSFTs based on the lower dimensional (unstable) Dp-branes. For a genericvalue of the parameter t , however, the expressions could only be represented in terms of implicitline integrals (which may however be inverted numerically). See also the earlier treatment in [6]using off-shell conformal theory. Additionally, as described in [7] there is operator mixing inducedby conformal transformations, since the boundary state is not a conformal primary, and this leadsto mixing of divergences from the massless and tachyon sectors.In the oscillator construction of the 3-vertex, using squeezed state methods for inner products[11], the state was shown to be [7, § |T (cid:105) = (cid:90) ∞ dt e t det(1 − S (cid:101) X )( Q det(1 − S (cid:101) V )) exp (cid:20) − a † M a † − c † R b † (cid:21) ˆ c | ˆΩ (cid:105) , (1.2)where the constituent matrices are expressible in terms of Neumann matrices, as we shall describelater in § e t factors contribute to divergences in the t → t → ∞ limits respectively. There are also additional subleading IR divergences from the masslessfields. The somewhat complicated nature of the Neumann matrices makes analytic study of thematrix R ( t ) difficult; it also suffers from an order of limits issue while considering expansions around t = 0 (see [7, App B] or § Moyal ∗ (star) representation of the vertex[12–17] developed by Bars et al. Although Witten’s formulation of OSFT is very elegant and onlyrequires a cubic interaction, explicit calculations are made difficult by the somewhat complicatedstructure of the 3-string vertex that encodes the gluing condition of strings. By choosing a convenientdiagonal basis ξ := ( x n , p n ) for the degrees of freedom (matter + ghosts), the formalism redefinesthe interactions in terms of the simple “Moyal product” [12, 18] between string fields. As suggestedby Ellwood et al. in [7], it would therefore be interesting to explore the analytic structure of thetadpole state in the Moyal/diagonal basis, where the interaction term simplifies.Although the BCFT analysis in [7] reveals a lot of information about the structure of the state |T (cid:105) , it is also a useful exercice to understand it purely from the open string perspective as we doin the somewhat algebraic approach here. Another motivation for our work has been to test thevalidity of the Moyal representation at the one-loop level by extending the off-shell tree level results[13–15] and the computation of Neumann coefficients [13, 14].2ince the tachyonic divergences are artefacts of the bosonic theory, we shall limit our attentionin this paper to the finite factors from the squeezed state matrix R ( t ) characterizing the Fock spacestate in the ghost sector. Its matrix elements may be extracted by taking inner products with pureghost excited states: (cid:104) ˆΩ | ˆ c m ˆ b n |T ( t ) (cid:105) = − R nm ( t ) × S ( t ) , (1.3)where S ( t ) would be a scalar piece, dependent on the Dp brane system. We will be interested inhints of non-analyticities in R nm ( t ), such as the exponentially suppressed sub-leading terms from(4.23) that are expected from closed string physics. Furthermore, in Siegel gauge, it is consistent torestrict to twist even and SU (1 ,
1) singlets [19] for the test states, which translates to R n, m − = 0 , m R nm = n R mn . (1.4)The Feynman rules in non-commutative ξ space ( § e iξ (cid:62) η − ξ gh (cid:62) η gh . The evaluation in the Moyal basis then involves transforming certaincoefficient matrices having substructure in terms of some simple matrices—which in turn obey aset of rather simple relations (See [14] or § consistent truncation for numerical checks, although we lack gaugeinvariance and are limited to machine precision due to the size of the constituent matrices and theirsubstructure. Since gravity is an inconvenience, we shall concern ourselves with only the flat D25brane background.We show that the formal expressions involving matrix inverses correctly capture the linear orderbehaviour near both limits t → t → ∞ of R ( t ). The qualitative difference near the closedstring region between the Moyal and oscillator expressions is that, in case of the oscillators anintermediate matrix becomes singular but in the Moyal case the matrix becomes singular triviallydue to the whole matrix vanishing. Interestingly enough, the peculiar nature of the Virasoro operator L in the diagonal basis (3.9) leads to a pole-zero cancellation and results in R nm ( t ) = C nm − n C nm t + O ( t ) , (1.5)whose linear term carries information about the conformal mappings used for the incoming externalstates in the BCFT prescription, and serves as a consistency check on the expansions that follow.Again, the monoid algebra renders the treatment of excited states more tractable and allows one toextend the existing (very detailed) results from the tachyon case [6], at least numerically. Somewhatsurprisingly, associativity is also seen to hold to this order § R ( t ) in terms of special functionsowing to the simple nature of the constituent matrices. This leads to a discussion of vanishingbut non-analytic contributions at t = 0, such as t k log( t ). By going to the continuous κ -basis, wealso verify the correct linear behaviour in q := e − t close to zero or t → ∞ , that matches with theoscillator construction. In order to probe for hints of non-analyticity in the complex q plane, we usethe oscillator expression (2.41) to obtain the coefficients of the general matrix element R nm ( q ) till q using the NCAlgebra [20] package. We move onto construct the associated Pad´e and Borel-Pad´eapproximants and perform various consistency checks.Quite a lot of work (see [8, 10] and references therein) has been done to understand the one-loopstructure of the theory since the work in Refs. [5–7]. An analysis was also done using open-closedstring field theory of Zwiebach [21] in [7] where it was shown that it naturally incorporates the shiftin the closed string background—just like in gauge field theories. In this regard, we must mentionthe somewhat recent work of Sachs et al. [10] where the quantum (in)consistency of OSFT has beenprecisely characterized in the language of QOCHA (quantum open-closed homotopy algebras).3e must also mention in passing another more recent gauge choice called the Schnabl gauge,where tree amplitudes (and loop amplitudes to some extent) simplify immensely; both the kineticterm and the interaction terms become more tractable in this conformal frame. This gauge wasoriginally chosen while constructing the non-perturbative tachyon vacuum solution of OSFT interms of surface states called wedge states [22]. Additionally, at the one-loop level new interestinggeometrical structures arise [23] which may help with computations in the more physical opensuperstring field theories where the tachyon would be projected out, but the gauge structure ismuch more intricate.We however continue to choose the Siegel gauge since the computational techniques are morereadily available in this frame. Due to the severe divergences from the closed string tachyon (andthe absence of winding states, etc. See the discussion by Okawa in [3, § § §
2, we first review some essential aspects ofperturbative OSFT and the Moyal representation. This is followed by a description of some knownresults from the oscillator analysis of the tadpole and a summary of our notations for quick reference.In §
3, we apply the Feynman rules in Moyal space to the tadpole and present algebraic expressions forthe integrand, with focus on the ghost sector. We shall be quite explicit throughout the discussionssince we are also seeking to clarify a minor mismatch with the oscillator construction, and becausemost of the operations are elementary block matrix multiplications or Gaussian integrations. Next,we analyse the squeezed state matrix R ( t ) in § §
5. Finally, we close by makingsome comments in relation to our results and directions for future work in § In this section, we review some essential aspects of open string field theory that provides contextfor the subsequent discussions and also in order to set the notations. For the general structure ofthe theory, we follow closely the very excellent lectures by Zwiebach and Taylor [3]. See also [3, 9]for modern developments and [2, 5, 25] for classic treatments of the subject. We shall then reviewthe Moyal representation of the ∗ product [12, 14, 18] using which most of the calculations in thispaper are done. Next, we recall some results [7] from the closely related oscillator formalism, wherealternate expressions can be written down for the physical quantities we study, and which we seekto improve upon. We close this section by collecting together some oft used notations and slightmodifications from prior conventions. 4 .1 Gauge choice and quantization String field theories are spacetime formulations for interacting strings that are similar in spirit tothe quantum field theories. Two essential requirements demanded from such theories are that a) thekinetic term should lead to the correct physical states , and b) the interacting action must reproducethe S-matrix elements of the Polyakov first quantized string theory by providing a single cover ofthe associated moduli space. A very useful toy model to study is the open string field theory forbosonic strings and where these statements have been rigorously proven [24].
Basic ingredients of OSFT
Open string field theory is a second-quantized formulation of bosonic open string theory that hasas its dynamical variable the classical string field
Φ, which may be represented as an element of thestate space of a matter-ghost boundary conformal field theory (BCFT): | Φ (cid:105) ∈ H BCFT = H matter ⊗ H ghost , (2.1)and contains a component field for every state in the first quantized string Fock space. An elegantcovariant formulation of this theory has been given by Witten with the following classical action: S cl [Φ] = − (cid:104) Φ , Q B Φ (cid:105) bpz − g o (cid:104) Φ , Φ ∗ Φ (cid:105) bpz , (2.2)which has the general structure of a Chern-Simons theory. It employs the BRST quantizationprocedure which ensures that the underlying worldsheet theory is physically equivalent to the onein covariant quantization. The string field may also be thought of as being valued in a gradedalgebra A which is chosen as the space of string functionals of the embedding coordinates (matter)and the reparametrization ghost field arising from fixing the worldsheet metric to conformal gauge( γ ab ∼ δ ab ), i.e. A = { Φ[ X µ ( σ ); c ( σ )] } , (2.3)where σ ∈ [0 , π ] denotes the canonical worldsheet parameter of the open string. We shall befocussing on the ghost sector primarily and hence discuss it in more detail in § c = 26.The basic ingredients of the above action are the first-quantized BRST operator Q B , the BPZinner product (cid:104) . , . (cid:105) bpz (or the (cid:82) operation), and an associative but non-commutative ∗ productbetween the string fields subject to the following “Witten axioms”: Grading:
The string fields are subject to a Z grading for the ghost number, G Φ and Z for Grass-mannality. The c ghost and the b anti-ghost are assigned ghost number charges of +1 and − | Φ (cid:105) ∈ H BCFT at ghost number+1 and is also Grassmann odd.
Differential:
The BRST operator Q B = (cid:72) dz πi j B ( z ) defines a map Q B : Λ n (cid:55)→ Λ n +1 , i.e. it’s adegree one operator under the grading. It is nilpotent: Q B ≡
0, and satisfies the derivationproperty: Q B (Φ ∗ Φ ) = ( Q B Φ ) ∗ Φ + ( − ) G Φ1 Φ ∗ ( Q B Φ ) . Associativity:
The binary ∗ product is assumed to satisfy: (Φ ∗ Φ ) ∗ Φ = Φ ∗ (Φ ∗ Φ ). See [1] for a precise treatment of these algebraic structures. In recent formulations that have proven useful, thiswould define a differential graded algebra (DGA), which encodes the maps [10]. The requirement of associativity maybe relaxed to obtain a homotopy associative algebra or a cyclic A ∞ structure [9, 10]. PZ inner product:
This is an invariant, bilinear form of ghost number − (cid:82) operation it induces a map (cid:82) : A → C that respects thefollowing relations: (cid:82) Q B Φ = 0, (cid:82)
Φ = 0 if G Φ (cid:54) = +3, and cyclicity: (cid:90) Φ ∗ Φ = ( − ) G Φ1 G Φ2 (cid:90) Φ ∗ Φ . These axioms uniquely determine the action by the requirement of extending the gauge symmetryfrom the free theory to the interacting case.This field theory reproduces a single covering of the moduli space of Riemann surfaces gener-ated by the underlying matter-ghost boundary conformal field theory (BCFT). Hence, all on-shellscattering amplitudes are guaranteed to be generated through a Feynman diagrammatic expansion.It also encodes rich non-perturbative string physics even at the classical level, as has been shownin the study of tachyon condensation [3, 22] and the computation of gauge invariant observables,called Ellwood invariants [8], for example.Next, let us turn towards the ∗ product which is one of the central aspects of Witten’s OSFT. The ∗ product operation The interaction between open strings is implemented by using the ∗ product which endows thestate space H BCFT with the structure of a non-commutative algebra [4]. For the matter functionals,this can be imagined as by imposing delta function overlap between the two halves of each string:the right half of the first string matches with the left half of the second string, which requires thefollowing connection conditions: X ( r ) ( σ ) − X ( r − ( π − σ ) = 0 , P ( r ) ( σ ) + P ( r − ( π − σ ) = 0 , (2.4)for the matter sector and in the ghost sector: c ± ( r ) ( σ ) + c ± ( r − ( π − σ ) = 0 , b ± ( r ) ( σ ) − b ± ( r − ( π − σ ) = 0 , (2.5)where now the parameter σ is restricted to 0 ≤ σ ≤ π/ r = 1 , ,
3. See [26] (and referencestherein) for a careful treatment of the ghost sector and of the general N -string vertex case.It is worth mentioning that in concrete calculations, the delta function overlap above is imple-mented by evaluating correlation functions of the BCFT on canonical domains such as the upperhalf plane (UHP) where the Neumann functions may be constructed explicitly. In particular, thethree half-discs corresponding to the three open string worldsheets can be glued together consistentlyusing conformal maps discussed in [3] to obtain the 3-vertex.A wealth of information has been gained about the structure of the theory using powerful Rie-mann surface theory employing elegant conformal mapping techniques. To appreciate how non-trivial the construction of the interacting SFTs is, even for the bosonic open string is, it is necessaryand instructive to understand the geometry of the conformal frame dictated by the underlyingworldsheet theory. However in this work which focusses on the algebraic approach, it suffices to re-mark that since the conformal frame has a somewhat complicated geometry, it introduces non-trivialconformal factors and branch-cut structure in both the matter and the ghost sectors. This makesexplicit study of the string diagrams highly non-trivial in general, especially for loop amplitudesrequiring constructions involving higher genus Riemann surfaces [6]. Siegel gauge
From the resemblance of Witten type OSFT to the Chern-Simons action and p-forms, one can inferthat the classical action in (2.2) is invariant under the following gauge transformation, once theWitten axioms are satisfied: δ Λ Φ = Q B Λ + Φ ∗ Λ − Λ ∗ Φ , (2.6)6here Λ is a ghost number zero, Grassmann even string field. Conversely, the cubic action is theunique action allowed by extending the linear gauge symmetry ( δ Λ Φ = Q B Λ) to the non-linear level.Because of this huge gauge symmetry, we must first fix a gauge before deriving the Feynmanrules of this theory. A venerable gauge choice is the
Siegel gauge where the kinetic term (cid:104) Φ , Q B Φ (cid:105) simplifies drastically. This is obtained by dictating that the string field satisfies: b | Φ (cid:105) = 0 , (2.7)where b is the anti-ghost zero mode. Then we can rewrite Φ as Φ = b c Φ by virtue of the anti-commutation relation { b , c } = 1. Now, the kinetic term can be rewritten in terms of the totalmatter + ghost Virasoro zero mode: ˆ L = ˆ L X + ˆ L gh (2.8)by making use of the relation { Q B , b } = L as S kin = (cid:104) Φ | ˆ c ( ˆ L − | Φ (cid:105) (2.9)where we revert to the first quantized operator language for convenience. Now, one may express thepropagator in terms of a Schwinger parameter as: α (cid:48) b ( L ) − = α (cid:48) b (cid:90) ∞ dt e − tL , (2.10)where we assume that the integral exists. We can interpret the action of the operator e − tL as tocreate a rectangular worldsheet strip of length t and width π , the canonical range for σ . The cubicterm representing the ∗ product now results in a Riemann surface or string configuration constructedout of three such rectangular strips, which is flat everywhere, except for a curvature singularity atthe common joining point. The external states in a given interaction can now be placed as vertexoperators on the appropriate semi-infinite strips to evaluate the correlators [25]. The operator formalism [26] in terms of explicit matter-ghost oscillators, ˆ α µn , ˆ b n , ˆ c n for a given BCFT,provides another concrete realization of the Witten type overlap relations (in addition to the onebased on worldsheet path integrals above). The correlation functions on the canonical domains arenow expressed in terms of the nine Neumann matrices, which are infinite matrices derived from the Neumann functions for the corresponding domain. These come with state space and mode numberlabels. Since these are quite challenging to handle analytically, the interactions were difficult toanalyze in this language for hand-calculations.In [12] a basis for the open string degrees of freedom was studied by Bars which diagonalizes theinteraction vertex, and makes the connection to non-commutative geometry as originally proposedby Witten rather manifest. The ∗ product was implemented as the Moyal product in the phasespace of even string modes. This could also explain the spectroscopy of the Neumann matricesstudied in [17]. These algebraic transformations correspond to diagonalizing the reparametrizationoperator K (see § σ = π/ z = + i (in the canonicalhalf-disc coordinates) as may be expected from the geometric picture. This leads to a reduction inthe effective number of Neumann matrices. We start with the Heisenberg algebra h N generated by the N pairs of phase space operators X i , P j and a central element C =: iθ , satisfying the canonical commutation relations:[ X i , P j ] = C δ ij , [ C, X i ] = 0 = [ C, P j ] , (2.11) This can be accomplished by a gauge transformation, at least at the linear level [2, 3]. N + 1 dimensional Lie algebra. It is also an associative algebra as may be seen fromthe Jacobi identity. A very useful construction out of this is its universal enveloping algebra , whichis the Weyl algebra A N ∼ = U ( h N ). Its elements are the formal polynomials in X i and P j modulo thecanonical commutation relations.Let us denote the generators of h N by T i . Then, a natural basis for A N is the collection of alldistinct Weyl-ordered formal homogeneous polynomials T i . . . T i k + permutations , (2.12)which makes it isomorphic to the symmetric algebra (cid:12) ( h N ). This naturally leads one to consideran association with variable t i (that would become the Moyal coordinates x µ n , p µ n later) and anidentification with the polynomial algebra K [ t i ] with elements P ( (cid:126)t ) = (cid:88) k =0 Π i ...i k t i . . . t i k (2.13)with symmetric coefficients Π i ...i k valued in the field K (which will be taken as C for OSFT).The non-commutativity of U ( h N ) means that the product of two Weyl ordered polynomialswould require further reordering. This induces a deformation of the usual commutative productin K [ t i ] and results in a ∗ algebra. The Lie bracket in h N (or the algebra g in general) uniquelyfixes this product and using the BCH formula, an explicit representation in terms of a bidifferential operator may be obtained (see [18, § ∗ algebrawhich for the ghost sector of OSFT is given by (2.22) or (2.26) for brevity.The generalization to OSFT requires an infinite number of modes (to realize the Virasoro alge-bra that guarantees its consistency) and hence we are essentially considering U ( h ∞ ). For physicallyinteresting string configurations, one also needs to enlarge from the space of polynomials to expo-nential functions ( § We discuss the discrete Moyal formalism, extensively developed in [13, 14, 16, 28] by Bars et al.first. Consider the open string field as a functional of the X µ ( σ ) , c ( σ ) degrees of freedom. This maybe made explicit by going to the oscillator representation in terms of the constituent Fourier modes x e , x o , c e , c o and the zero modes: x = π (cid:82) dσ X µ ( σ ) for matter and c for the c -ghost. In the Siegelgauge that we choose for perturbation theory, we can consistently drop the pieces proportional to c . Now, the discrete Moyal map is obtained by first taking a half-Fourier transform with respectto “half” of the degrees of freedom to convert the string field Φ( x, c ) defined in coordinate space toMoyal space A (¯ x, ξ, ξ gh ). Then the Moyal star product is applied on the string fields valued in thephase space doublets ξ = ( x e , p e ) of “even” string modes. The maps between the even (e) and odd(o) modded subspaces are implemented by the matrices: T : H o (cid:55)→ H e , R : H e (cid:55)→ H o (2.14)where the modes (for Neumann boundary conditions) are obtained from: X ( σ ) = x + √ (cid:88) n ∈ Z + x n cos nσ, P ( σ ) = 1 π p + √ (cid:88) n ∈ Z + p n cos nσ (2.15)The ∗ product then becomes diagonal after this change of variables and additionally, the product islocal in the midpoint coordinate ¯ x (and the ξ variable corresponding to the b dependence). We will We closely follow [18, §
2] in this discussion to motivate the Moyal product. T that arise naturally in this transformation tophase space variables. These matrices will be crucial for the evaluation of string diagrams attemptedin this paper.Although the Moyal map employs infinite linear combinations in string mode space and hence isdefined formally, it captures several aspects of the physics OSFT including subtle contributions fromthe midpoint [16, 28]. It provides a concrete realization of the split-string picture [29] while givingone prescription for treating the midpoint anomalies by providing a consistent truncation [13, 14,28]. For the reduced star product [26] in Siegel gauge, the ghost Witten vertex is equivalent to thediscrete Moyal basis star representation. It is one of the aims of this paper to test the applicabilityof this basis at the one-loop level.The b anti-ghost, which is analogous to the embedding coordinate, satisfies overlapping conditionsand the c ghost, which is similar to the momenta, satisfies anti-overlapping conditions as expressedearlier in (2.5). Hence, we can expect some slight asymmetry between the two sets (see (C.2)) ofodd Moyal coordinates ( x o , p o ) and ( y o , q o ) we provide in Appendix C on the bc system. Since weare mostly interested in the ghost contributions in this paper, we have only illustrated the generalidea in the matter sector (which was developed first historically, see [13]) before focussing on thetreatment of the ghosts. Some relevant matter contribution would be presented in Appendix A. Thecontinuous κ basis would be briefly reviewed in § §
2] and [12] concerning this basis.
To go from the string field | Φ (cid:105) defined in Fock space of ( b, c ) ghosts to Moyal space, one performs aFourier transform over half the number of degrees of freedom:ˆ A ( ξ , x o , y o , p o , q o )= (cid:90) d ¯ c e − ξ ¯ c A (¯ c, x o , − p o /θ (cid:48) , y o , − q o /θ (cid:48) )= 2 − N (1 + w (cid:62) w ) − (cid:90) dc N (cid:89) e> ( − idx e dy e ) e − ξ c + ξ w (cid:62) y e + θ (cid:48) p o S (cid:62) x e + θ (cid:48) q o Ry e Φ( c , x n , y n ) , (2.16)where ξ is a fermionic object encoding the zero mode dependence, and θ (cid:48) is the common non-commutativity parameter in ghost space. This operation may also be implemented by taking aninner product with a bra (cid:104) ξ, ξ gh , ξ | defining the Moyal basis. The matrix S will be defined below.Restricting to the ghost sector in Siegel gauge, we roughly identify: A ( ξ gh ) := (cid:104) ξ gh | Φ (cid:105)∼ (cid:90) dx bo Φ[ c, b ] , (2.17)by isolating the zero mode dependence as ˆ A ( ξ , ξ gh ) = ξ A ( ξ gh ).We find that it is more convenient to work with objects having even labels instead of the odd parity elements that appear naturally in the ghost sector. We emphasize that these are not theoriginal even degrees of freedom but special (infinite) linear combinations: x ce = T eo y o , p ce = R (cid:62) eo q o , x be = κ − e S eo x o , and p be = κ e S eo p o , (2.18)9here T eo = 4 π (cid:90) π dσ cos eσ cos oσ = 4 o i o − e +1 π ( e − o ) , and its inverse (2.19a) R oe = 4 π (cid:90) π dσ cos oσ (cid:16) cos eσ − cos eπ (cid:17) = 4 e i o − e +1 πo ( e − o ) , and (2.19b) S eo = 4 π (cid:90) π dσ sin eσ sin oσ = 4 i o − e +1 eπ ( e − o ) , (2.19c)with mixed parity labels, in the open string limit N → ∞ . These matrices satisfy the relations: T R = e , R T = o , S S (cid:62) = e , S (cid:62) S = o , (2.20)along with a few more useful relations that we collect below in § § (cid:62) refers to the matrix transposewhich differs from the (¯) notation used in [14]. The infinite vectors w, v are given by: w e = √ i − e +2 , v o = 2 √ i o − πo = 1 √ T o . (2.21)After this preparation, the ∗ product among string fields valued in Moyal space is implemented bythe bidifferential operator( A ∗ B )( x be , p be , x ce , p ce ) = A exp (cid:34) θ (cid:48) (cid:32) ←− ∂∂x be −→ ∂∂p be + ←− ∂∂x ce −→ ∂∂p ce + ←− ∂∂p be −→ ∂∂x be + ←− ∂∂p ce −→ ∂∂x ce (cid:33)(cid:35) B (2.22)where ←− ∂ and −→ ∂ are respectively the left right and left fermionic derivatives obeying the standardanti-commutation rules, and θ (cid:48) is the non-commutativity parameters for ghosts.As mentioned earlier, the ghost part of the string fields may be more succinctly obtained bytaking the inner product with the oscillator bra (cid:104) ξ gh | defining the Moyal basis: (cid:104) ξ , ξ gh | = − − N (cid:16) w (cid:62) w (cid:17) − (cid:104) Ω | ˆ c − e − ξ (ˆ c −√ w (cid:62) ˆ c e ) e − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh , (2.23)and we have the vectors λ gh = (cid:18) √ R (cid:62) ˆ b o − √ κ − e ˆ b e + 2 κ − e wξ (cid:19) , λ gh = (cid:18) √ R (cid:62) κ o ˆ c o √ i ˆ c e (cid:19) . (2.24)Here we have transformed to the even basis the expression given in [14] by utilizing some simplealgebraic relations satisfied by the relevant matrices. Next, we define the off-block diagonal matrix σ , labelled by even mode integers: σ := − θ σ ⊗ e , (2.25)where σ is the second Pauli matrix and we have chosen the non-commutativity parameter θ (cid:48) = θ =+1 for convenience, by a choice of units. Excluding the matter sector, we can now write the Moyalstar product between two fields as: (cid:0) A ∗ B (cid:1) [ ξ gh ] = A exp (cid:32) ←− ∂∂ξ gh Σ −→ ∂∂ξ gh (cid:33) B ; where now Σ := − iε ⊗ σ. (2.26)10he trace operation associated with the Fock space inner product (cid:104) Φ | ⊗ · · · ⊗ n (cid:104) Φ n | V n (cid:105) ∼ Tr (cid:16) ˆ A ∗ · · · ∗ ˆ A n (cid:17) (2.28)is then represented as integration over Moyal (phase) space with the appropriate measure:Tr := det σ (cid:48) | det(2 πσ ) | d/ (cid:90) ( dξ ) ( dξ gh ) (2.29)where we have restored θ (cid:48) for generality, by defining σ (cid:48) := θ (cid:48) σ ⊗ e . Metric in ghost space
We can now combine the ghost (non-zero) modes into the two doublet vectors: ξ = (cid:18) x be − p ce (cid:19) and ξ = (cid:18) x ce p be (cid:19) (2.30)which we denote together again by ξ gh . Under an SO (4) rotation to the new basis, ξ gh = x be p be x ce p ce → ξ = (cid:20) x be − p ce (cid:21) , ξ = (cid:20) x ce + p be (cid:21) the block matrices transform as: ε ⊗ α → − iε ⊗ iα, ε ⊗ β → I ⊗ − σ β,I ⊗ α → I ⊗ α, I ⊗ β → − iε ⊗ iσ β, (2.31)where α is block diagonal and β is off block diagonal.This allows one to use the Sp (2) metric + iε ab (with ε = − − ε ) in the ( b, c ) ghost phasespace suggested in [13] and makes the SU (1 ,
1) symmetry manifest. The presentation also becomescleaner due to the similarity of the algebraic expressions with the matter sector. Notice that wehave the canonical ∗ anti-commutator in the ghost Moyal plane: { ξ ni , ξ mj } ∗ = − iε nm σ ij (2.32)Hence, it is consistent to impose an − iε ⊗ tensor product factor while defining dot products. In allfermionic bilinears and quadratic terms, this metric factor would be understood to be present.2.2.4 Monoid subalgebra and regularization
A very interesting feature of the discrete Moyal basis is the consistent regularization developedin [28] involving a cutoff prescription in the number of string modes 2 N defining the phase spacedoublet. It allows for a deformation of the spectrum from the frequencies valued in the non-negativeintegers, to a sufficiently reasonable set of frequencies κ n . The finite versions of the N × N matrices T, R, S and N × w e , v o from (2.19a) are in general dependent on all the frequencies κ n , n = 1 , . . . , N . These are solved for by requiring that the following relations are satisfied: R = κ − o T (cid:62) κ e , R = T (cid:62) + vw (cid:62) , v = T (cid:62) w, w = R (cid:62) v (2.33) Although we do not use the zero mode dependence, it is instructive to mention the structure here in the normal-ization using the odd modes (cid:90) dξ Tr (cid:18) ˆ A ( ξ , ξ ) † ∗ (cid:18) ∂∂ξ − θ (cid:48) v (cid:62) ∂∂q (cid:19) ˆ A ( ξ , ξ ) (cid:19) = (cid:104) Φ | ˆ c | Φ (cid:105) = 1 . (2.27) It is somewhat interesting to compare this to the spectrum of the so called fractal strings [31]. preservation of associativity while taking double sums. It essentiallyremoves the null elements of the algebra by hand and hence is topologically different from the stringfield algebra, even in the open string limit. It would therefore be interesting to study this structureon its own and because it correctly captures certain aspects of perturbative OSFT as shown in [13]and as we shall see in the following.The regularization also leads to a (Moyal) star subalgebra constructed out of finite numberof modes . The elements of this subalgebra are string configurations corresponding to quadraticexponentials—which are the analogues of the Hermite polynomials in the functional formalism—but now defined in Moyal space: A N ,M,λ ( ξ ) := N e − ξ (cid:62) Mξ − ξ (cid:62) λ . (2.34)Here, the string field is parametrized by complex (anti-)symmetric matrices M , a complex vector λ and the normalization factor N , which is independent of the ξ . These form a monoid or a semi-groupstructure i.e it is closed under the ∗ product, is associative, and has the unit element (the number1). It is in general non-commutative and may not have an inverse element, although the genericelements do have inverses. Thus, being just short of forming a group due to the lack of an inverse,it is a monoid or a semi-group containing many interesting string fields.In particular, the perturbative vacuum state for the ghost sector (in the Siegel gauge) belongsto this class and is given by the monoid element:ˆ A gh ( ξ gh ) = ξ N gh exp (cid:104) − ξ gh (cid:62) M gh ξ gh (cid:105) , (2.35)where the matrix M gh has the block diagonal form in the purely even basis: M gh = − (cid:20) R (cid:62) κ o R
00 2 κ − e (cid:21) , λ gh = 0 , (2.36)and N gh is a normalization factor.The subalgebra is a helpful structure for evaluating string diagrams, to which we turn next. Therules would be provided later in (3.7) while illustrating the tadpole computation in § In order to study string diagrams using this formalism, we require the gauge fixed action written inMoyal space: S GF = − (cid:90) d d ¯ x Tr (cid:18) α (cid:48) A (¯ x, ξ ) ∗ ( L − A (¯ x, ξ ) + g o A (¯ x, ξ ) ∗ A (¯ x, ξ ) ∗ A (¯ x, ξ ) (cid:19) , (2.37)where A (¯ x, ξ ) contains only the non-zero ghost modes and the full string field has the explicit zero-mode dependence ˆ A (¯ x, ξ , ξ ) = ξ A (¯ x, ξ ). We remark that this form of the action is also applicablefor the finite N truncations. In the full open string field theory, all star subalgebras necessarily contain an infinite number of modes forconsistency with the Witten axioms. Here we are only considering the deformed theory. We can relegate the subtletiesof the closure of sub-algebras in string field theory by working at a finite value of N , which is somewhat similar to leveltruncation, and hence amounts to imposing a UV cut-off. Although this regularization cannot realize the Virasoroalgebra and breaks the gauge invariance, it does preserve the non-linear Gross-Jevicki matrix identities [26] (see also(2.44) below) satisfied by the infinite Neumann matrices. This is because the fundamental matrices continue to satisfythe same relations as their infinite N counter-parts (whenever they are regular) even after the deformation. corresponding to the identity state | ˜ I (cid:105) under the reduced star product studied in [26]. ξ space. Due to the interplay between kinetic term and the interaction term in OSFT, the propagatorbecomes complicated in ξ space and involves a potential term. The external states A i ( ξ ) thatcorrespond to the operator insertions on the semi-infinite strips are joined together using the ∗ product.The intermediate string fields are propagated using the operators q L i (see (2.10)) and the finaltrace operation (Gaussian integration over ξ, ξ gh ) implements the inner product. Here the variables q i = e − t i encode the modular parameters of the intermediate strips t i . See [14, 15] for more detailsand examples.In case of diagrams with loops, one also needs to perform a state sum; if we consider the contribu-tion only from the ordinary ghosts, we can insert a (normalized) Fourier basis e + iξ (cid:62) η + ip ¯ x e − ξ gh (cid:62) η gh .A ( dη ) integration at the end then implements the state sum. For example, the tadpole diagramthat we will be focussing on in § t i . We note that for thepurpose of numerical calculations, it’s also useful to consider the Feynman rules in the Fourier basisgiven in [14, 15]. In the oscillator construction [26], the Fock space of open (bosonic) string fields is constructed byacting with the creation operators α µ − k , b − n , c − m on the vacuum | ˆΩ (cid:105) . The star product is thenimplemented by using n -vertices belonging to the tensor product of the dual spaces H ( i ) ∗ . Inparticular, we have the three-vertex (cid:104) V | and the two-vertex (cid:104) V | whose explicit structure encodesthe Witten-style overlapping conditions (see [3, 26] and references therein). The 3-string vertex fixesall the interactions that may arise in the theory. For the purpose of this paper, we provide only therelevant ghost part [26, 32] appearing in the combined vertex: (cid:104) V | = X (cid:104) V | ⊗ gh (cid:104) V | , gh (cid:104) V | ∼ (cid:104) Ω | exp( − E gh ) , (2.38)where E gh is a quadratic form coupling the ghosts involving the ghost Neumann matrices X rsnm : E gh = (cid:88) r,s =1 ∞ (cid:88) n =1 m =0 c ( r ) n X rsnm b ( s ) m . (2.39)Furthermore only the coefficient matrices for the non-zero modes (in the Siegel gauge) would concernus. These are the ghost Neumann matrices denoted by X rsnm , with r, s ∈ { , , } and by their sym-metry and cyclicity properties, we can restrict to X = X (0) , X = X (+) and X = X ( − ) . Theyare algebraic valued and can be obtained efficiently from CFT using contour integral representations[3, 26].The one-loop tadpole can be represented as a ket (or more properly as a bra) which involves anexponential purely quadratic in the creation operators. These special states then belong to the classof squeezed states in the Hilbert space. |T (cid:105) = (cid:90) ∞ dt e t det(1 − S (cid:101) X )( Q det(1 − S (cid:101) V )) exp (cid:18) − a † M a † − c † R b † (cid:19) ˆ c | ˆΩ (cid:105) . (2.40) These have to be understood in the form of a distribution due to the singular normalization involved and thevanishing quadratic term in the exponents. t dependence in Q and the infinite matrices M , R , (cid:101) X , and (cid:101) V are understood. The relevantinner product involving reflector (cid:104) V | , | V (cid:105) , ˆ L , etc. is presented in (3.63).We quote the following form for R ( t ) derived in [7, §
4] using squeezed state methods presentedin [11]: R ( t ) = X + (cid:2) ˆ X (0 , t ) ˆ X (0 , t ) (cid:3) − S (cid:101) X S (cid:20) ˆ X ( t, X ( t, (cid:21) (2.41)The “hatted” matrices are simply the Neumann matrices dressed with the t dependent propagatorfactors of the following form: ˆ X i k j l nm ( t k , t l ) := e − nt k / X i k j l nm e − mt l / . (2.42)In terms of these, the infinite matrix (cid:101) X is given by (cid:101) X ( t ) = (cid:20) ˆ X ( t, t ) ˆ X ( t, t )ˆ X ( t, t ) ˆ X ( t, t ) (cid:21) , (2.43)and S = ⊗ C , where again C nm = ( − n δ nm is the twist matrix, that arises from the specificoverlap conditions imposed by the Witten type vertex in the matter and ghost sectors. The abovematrices become 2 L × L dimensional in an oscillator level truncation, which roughly correspondsto using 4 N × N dimensional matrices in the discrete Moyal representation for finite N . Expansion around t = 0As observed in [7], the infinite matrix R ( t ) cannot be reliably expanded around the point t = 0(or q := e − t = 1) that we are interested in. This is because an intermediate matrix to be inverted, − S (cid:101) X (0), for the expansion point becomes singular due to a subset of the Gross-Jevicki non-linearrelations satisfied by the unhatted matrices M , ± := − CX , ± in the ghost sector: M + M + + M − = , M + M − = M − M , (2.44a) M + M + M − = , M M + + M + M − + M − M + = 0 , (2.44b) M ± − M ± = M M ∓ . (2.44c)These are mutually commuting matrices and in the limit t →
0, when we have − S (cid:101) X | t =0 = (cid:20) − M − −M −M − M + (cid:21) , (2.45)this allows us to express the determinant in terms of the constituent blocks by the usual formulafor 2 × − S (cid:101) X ) | t =0 = det( + M − M + − M − − M + − M )= det (cid:0) M − M + M − M + (cid:1) = det( ) = 0 , (2.46)which makes the Taylor series ill-defined. This fact is also carefully pointed out in [7, App B]. Theauthors study these expressions numerically and comment on why a level truncated analysis woulddiffer from the correct numerical behaviour which matches with a BCFT based expansion (3.61) asthe level is increased. Since the identities only hold in the infinite L limit, the problem does notarise at finite level, which effectively acts as a UV cutoff for t = 0.Thus, the order of limits t → L → ∞ do not commute and subsequently theinfinite level result gives a factor of − n for the linear term instead of − n as confirmed by numericalstudies at finite level. As we shall see in § t = 0 even for finite N thus altering the UV behaviour.14 .4 Summary of notations used Here we collect some of the notations and conventions that will be used in the rest of the paper.
Phase space basis vectors ξ, ξ gh : The string field A (¯ x, ξ, ξ gh ) is valued in the non-commutativephase space ξ µi = ( x µ , . . . .x µ N , p µ , . . . p µ N ) , ξ ghi = ( ξ n , ξ n ) labelled by even integers. Thedoublet structure ( x, p ) would be understood in the following which for ghosts is in (2.30).The zero mode dependence is factored out in Siegel gauge through ˆ A = ξ A ( ξ gh ). We shallsuppress the Lorentz indices unless required.The integration or the BPZ inner product is mapped to the trace in this phase space. Also, weshall denote ( dξ ) , ( dη ), etc. for integration over the Moyal space modes d d ξ . . . d d ξ N , d d η . . . d d η N ,and suppress the measure factors of 12 πi unless necessary. Constant matrices:
The spectrum is denoted by a 2 N × N diagonal matrix κ , which in the paritybasis is κ = Diag { κ e , κ o } and the labels e, o refer to the even and odd integer mode numbers.In the open string limit N → ∞ , we shall set κ n = 2 n and κ n − = 2 n −
1, corresponding tothe perturbative spectrum. A useful matrix in the ghost sector is:˜ κ gh = (cid:20) R (cid:62) κ o T (cid:62) κ e (cid:21) (2.47)The linear transformations to go to the discrete diagonal basis requires the use of certain(constant) infinite matrices whose elements are simple functions of the mode labels e and o .In the regulated theory, these have their finite dimensional analogues which in general dependon the frequency matrices κ e and κ o . The infinite N limit of the matrices is sufficient to seetheir fall off behaviour at large mode numbers in infinite sums: T eo = 4 π o i o − e +1 e − o , R oe = 4 π e i o − e +1 o ( e − o ) , w e = i − e , v o = 2 √ π i o − o . (2.48)these satisfy the relations presented in [14] of which we mainly use: T R = e , R T = o , R = κ − o T (cid:62) κ e , T T (cid:62) = e − ww (cid:62) w (cid:62) w (2.49) Monoid elements:
We shall be primarily using the monoid subalgebra § A ( ξ ) = N e − ξ (cid:62) Mξ − ξ (cid:62) λ . For the perturbative ghost vacuum ˆ A gh = ξ A gh , where A gh has parameters N gh = 2 − N (1 + w (cid:62) w ) and M = M gh (2.36). For external states built on the perturbative vacuum, it issufficient to consider a generating functional with M X = M X and M gh = − iε ⊗ M gh with ageneral λ X , λ gh and construct states as polynomials ℘ ( ξ, ξ gh ) . We shall assume that the interchanging functional operations as usually done in QFT can beperformed here as well, although it does not seem that straightforward.
Normalization factor:
The Witten type vertex and the Moyal vertex are related by a (divergent)factor. By using the regularized theory, it leads to a renormalization of the bare coupling g tothe physical g T when considering the D25 brane reference BCFT. The two vertices are relatedas: (cid:104) Φ | Φ ∗ Φ (cid:105) = µ − Tr [ A ∗ A ∗ A ] (2.50) The η, ξ here are (unfortunately) unrelated to the η ( z ) , ξ ( z ) conformal fields defined by FMS [33] and in the recentdevelopments in superstring field theories. A i ( ξ ) := (cid:104) ξ | Φ i (cid:105) and µ = − N ( d − (1 + w (cid:62) w ) − d − (cid:16) det(3 + tt (cid:62) ) (cid:17) − d (cid:16) det(1 + 3 tt (cid:62) ) (cid:17) . (2.51) Modular parameters:
We use the variable t for the worldsheet lengths so as to match the usualconvention for the nome q = e πiτ . This requires that τ (cid:55)→ it/ π . q = e − t , q = e − t , q = e − t , etc. (2.52)Some simple matrix functions that would be convenient for writing down integrands can thenbe defined in terms of q and the mode label n as f i ( n ; q ) ( i = 1 , , ,
4) to be given in (3.16)and some auxiliary functions h i ( n ; q ) and g i ( n ; q ) in (4.1).We shall be quite explicit in the following, since we are seeking to resolve the minor mismatch inthe oscillator construction and since many of the steps are simple block matrix multiplications orGaussian integrations. The reader can skip these intermediate steps and essentially consider the finalform of the expressions if desired. We shall also retain the “gh” superscript although it is usuallyclear from the context when the quantities refer to the ghost contribution. When the superscript isnot used, it refers to the matter sector which we shall sometimes denote by an “X” superscript.In the next section, we shall apply the Feynman rules in Moyal space (described in § In this section, we write down the one-loop tadpole integrand in the Moyal representation whilefocussing on the ghost contribution. This can then serve as a starting expression for examining thenon-analyticities in the ghost sector, as a function of the modular parameter t . We wish to obtain an expression for the one-loop contribution to the tadpole graph in bosonic openstring theory. Since this is an intrinsically off-shell quantity, we need to work in the framework of astring field theory and we choose the Witten type OSFT reviewed in the previous section. The string A e ( ξ ) t ππ Figure 1: The open string tadpole diagram at a given modular parameter t for an external state A e ( ξ ) in Moyal space. The width of each strip is fixed to π and the curvature singularities aresuppressed.diagram for this process is depicted in Fig.1 where an open string state at zero momentum ( p e = 0)in a D25-brane background appears from vacuum, splits into two open strings and then annihilate16ach other, just like in QFT. It is parametrized by a single modular parameter associated with thelength of the internal propagator. The corresponding integrand at a fixed modular parameter t , maybe obtained by identifying two legs of an off-shell 3-point diagram and integrating over a completeset of (normalized) quantum states e + iξ (cid:62) η + ip ¯ x e − ξ gh (cid:62) η gh as described in § A e ( ξ ) asfollows: I e ( t ) = − g o (cid:90) d d ¯ x d d p (2 π ) d ( dη )(2 π ) dN ( dη gh ) Tr (cid:104) A e ∗ (cid:16) e − iξ (cid:62) η + ξ gh (cid:62) η gh e − ip · ¯ x (cid:17) ∗ (cid:16) e − t ( L − (cid:16) e iξ (cid:62) η − ξ gh (cid:62) η gh e ip · ¯ x (cid:17)(cid:17)(cid:105) (3.1)where Tr denoted ξ integrations and L = L X + L gh is the total propagator. Here, we have set l s = √ α (cid:48) = 1. Additionally, in the discrete formalism we may be allowed to rescale themodes appearing in the matter and ghost degrees of freedom in order to set the non-commutativityparameters θ = 1 = θ (cid:48) .The ghost number saturation condition for the Witten vertex dictates a total of +3 ghost numbercharge at each vertex. Since we restrict to off-shell states of ghost number 1, this then leads to theprojection onto ghost number (3 − / − ξ which is always understood to be present.The expressions for the ghost sector contributions are naturally simpler compared to the mattersector due to the absence of the ghost zero mode c (in the Feynman-Siegel gauge). Additionally,the ghost contribution is in a sense universal. Hence, we restrict to pure ghost external states inthis section and consider the matter contribution later in App A. The fields we consider would nowbe of the form ℘ ( ξ gh ) A gh ( ξ gh ), where ℘ ( ξ gh ) represents a polynomial in ξ gh and A gh is the vacuummonoid defined in (2.36): A gh = N exp (cid:104) − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh (cid:105) (3.2)These integrands may therefore be obtained by evaluating from a generating functional W ( λ gh , t )dependent on an element valued in the monoid subalgebra A ( ξ gh ) = N exp (cid:104) − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh (cid:105) , (3.3)differentiating with respect to this parameter λ gh appropriately and then setting it to zero at theend of a calculation: ℘ ( ξ gh ) A gh = (cid:32) ℘ (cid:32) − (cid:126)∂∂λ gh (cid:33) A (cid:33) (cid:12)(cid:12)(cid:12) λ gh =0 , (3.4)as done in usual quantum field theory calculations. Hence, it would be sufficient to analyse the classof monoids of the form A ( ξ gh ). Furthermore, we restrict A e ( ξ gh ) to be in the SU (1 ,
1) singlet sector[14, 19] of twist even pure ghost excitations, since the tadpole state is a twist even singlet.
Method of evaluation
Interchanging the order of integration (between η and ξ ) in (3.1) and using associativity of the ∗ product allows us to obtain various formally equivalent expressions:(a) A ∗ A → A [ η, ξ ] → A A ( t ) → Tr → (cid:82) dη (b) A ∗ A ( t ) → A [ η, ξ, t ] → (cid:82) dη → A A (cid:48) [ ξ, t ] → Tr(c) A ∗ A ( t ) → A [ η, ξ, t ] → A A → Tr → (cid:82) dη ,17d) A ( t ) ∗ A → A [ η, ξ, t ] → A A → Tr → (cid:82) dη , and(e) A ( t ) ∗ A → A [ η, ξ, t ] → A A → (cid:82) dη → Tr,where the last two are possible due to the assumed cyclicity of the trace.We choose the first sequence due to its relative simplicity. The second one allows us to identifythe Fock space state by integrating A (cid:48) ( ξ ) with | ξ (cid:105) but it involves a somewhat complicated inversenested inside another inverse which makes direct evaluations difficult. It does lead to the correctbehaviour near t = 0 as we shall mention in § N regularization from § N , one can ensure uniform convergence of the integrand asa function of t . Perhaps this could justify some of the algebraic manipulations we use, but in generalone cannot avoid subtleties associated with order of limits, namely the non-analyticities from closedstring physics may be extracted only in the open string limit. We shall return to this point in thelater sections. After this preparation, let us list the three monoid elements appearing in the amplitude along withtheir parameters: A gh = N exp (cid:104) − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh (cid:105) , M = M gh , λ = λ gh , N = N gh , (3.5a) A gh = e + ξ gh (cid:62) η gh , M = 0 , λ = − η gh , N = 1 , (3.5b) A gh = e − ξ gh (cid:62) η gh , M = 0 , λ = + η gh , N = 1 . (3.5c)Here we recall that M gh is a symmetric matrix but the metric in ghost space is set to be + iε ab (with ε = − − ε ) and hence the full structure of the matrix for the quadratic term is of the form − iε ⊗ M gh . This makes the combination an anti-symmetric matrix, as required for anti-commutingdegrees of freedom. Additionally, we have suppressed a metric factor in the linear term ξ gh (cid:62) λ gh ,whose explicit form is ξ gh (cid:62) ( − iε ⊗ N ) λ gh .To commence evaluation, we first take the ∗ product of A and A to obtain: A N ,M ,λ := A N ,M ,λ ∗ A N ,M ,λ (3.6)This can be written down by applying the monoid algebra relations given in Ref.[14] and mentionedbriefly in in § A ( ξ ) and A ( ξ ) from the class ofshifted Gaussians (quadratic exponentials with a linear term), the string field obtained through the ∗ operation is parametrized by: m = ( m + m m )(1 + m m ) − + ( m − m m )(1 + m m ) − , (3.7a) λ = (1 − m )(1 + m m ) − λ + (1 + m )(1 + m m ) − λ , (3.7b) N = N N det(1 + m m ) exp (cid:104) + 14 λ gh (cid:62) α σK αβ λ ghβ (cid:105) where , (3.7c) K αβ = (cid:20) ( m + m − ) − (1 + m m ) − − (1 + m m ) − ( m + m − ) − (cid:21) , m i := M i σ. (3.7d)Applying this rule to the two string fields for our case in (3.6) immediately leads to the parameters: A ( ξ gh ) := N exp( − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ ) , where M = M gh , λ = − (1 − m gh ) η gh + λ gh , N = N exp (cid:18) η gh (cid:62) σm gh η gh − λ gh (cid:62) ση gh (cid:19) . (3.8)18here we have used K = 0 , K = 1 = − K and K = m gh and once again the ghost spacemetric is implicit.Next, we need the t evolved monoid element A ( ξ gh , η gh , t ), for which we use the action of L gh on a general monoid element N e − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh . Unfortunately, the Virasoro operator L gh isno longer diagonal in this basis: L gh = Tr˜ κ gh − (cid:18) ∂∂ξ gh (cid:19) (cid:62) M gh − ˜ κ gh (cid:18) ∂∂ξ gh (cid:19) + ξ gh (cid:62) ˜ κ gh M gh ξ gh (3.9)unlike the oscillator case: The simplicity in the interaction term has made the kinetic term com-plicated. Hence L X + gh has a non-trivial action on the string fields, which can however be writtendown in closed form. This leads to the following transformation rules [14] in terms of hyperbolicfunctions of the “spectral matrix” ˜ κ gh (2.47): A ( t ) := e − tL gh A N gh ,M gh ,λ gh ( ξ gh ) = N ( t ) exp (cid:16) − ξ gh (cid:62) M gh ( t ) ξ gh − ξ gh (cid:62) λ gh ( t ) (cid:17) , where (3.10a) M gh ( t ) = (cid:20) sinh t ˜ κ gh + (cid:16) sinh t ˜ κ gh + M gh M gh − cosh t ˜ κ gh (cid:17) − (cid:21) sech t ˜ κ gh M gh , (3.10b) λ gh ( t ) = (cid:104) cosh t ˜ κ gh + M gh M gh − sinh t ˜ κ gh (cid:105) − λ gh , (3.10c) N gh ( t ) = N gh exp (cid:104) + 14 λ gh (cid:62) ( M gh + coth t ˜ κ gh ) − λ gh (cid:105) × det (cid:20)
12 (1 + M gh M gh − ) + 12 (1 − M gh M gh − ) e − t ˜ κ gh (cid:21) , (3.10d)and a very similar expression in the matter sector, except for the extra dependence on the zero modemomentum p µ and the vector w . Notice that the correct boundary conditions for t = 0 and t = ∞ are taken into account in the above rules.Now, applying this transformation on (3.5c), for which the matrix of parameters M gh vanishes, wereadily obtain the string field: A ( t ) = N ( t ) exp (cid:16) − ξ gh (cid:62) M ( t ) ξ gh − ξ gh (cid:62) λ ( t ) (cid:17) , with parameters M ( t ) = tanh t ˜ κ gh M gh , λ ( t ) = +sech( t ˜ κ gh ) η gh , N ( t ) = 2 − N N (cid:89) n =1 (1 + e − tn ) exp (cid:18) η gh (cid:62) M gh − tanh t ˜ κ gh η gh (cid:19) . (3.11)We can now use the property that the remaining ∗ product between A and A ( t ) may be droppedas total derivative pieces contribute only to boundary terms under a trace ( ξ integration with appro-priate measure factors inserted). We therefore define a new string field configuration A A ( t ) =: A gh ( t ), under the ordinary (local) product in function space, with parameters: M ( t ) = M + M ( t ) ,λ ( t ) = λ + λ ( t ) , N ( t ) = N · N ( t ) . (3.12)Hence, the trace in (3.1) simply results inTr[ A ( t )] = N det(2 M ( t )) exp (cid:20) + 14 λ (cid:62) M − λ (cid:21) =: C η exp (cid:104) − η gh (cid:62) Q η η gh + λ gh (cid:62) L (cid:62) η η gh (cid:105) . (3.13)19n order to perform the remaining Gaussian integration over η gh , we have separated the quadratic,linear and zero degree terms in η gh by collecting the contributions from N and the argument ofthe exponential in the first line of (3.13) above. In terms of the matrices that are used in the Moyalrepresentation § Q η ( t ) = − (cid:104) σm gh + σm gh − tanh t ˜ κ gh + ( m gh (cid:62) + sech t ˜ κ gh (cid:62) − σm gh − (1 + tanh t ˜ κ gh ) − ( m gh + sech t ˜ κ gh − (cid:105) , (3.14a) L (cid:62) η ( t ) = − σ (cid:104) m gh − (1 + tanh t ˜ κ gh ) − (1 − m gh − sech t ˜ κ gh )) (cid:105) , (3.14b) C η ( t ) = N det(2 M ) exp (cid:20) + 14 λ gh (cid:62) M − λ gh (cid:21) = det( M ) exp (cid:20) + 14 λ gh (cid:62) M − (1 + tanh t ˜ κ gh ) − λ gh (cid:21) . (3.14c)where we have used the subscript η to specify the variable in the quadratic form, a convention weshall be following from now onwards .At this point it is convenient to introduce the (Euclidean) nome q = e +2 πiτ = e − t and thefunctions: f ( n ; q ) = (1 − q n ) , f ( n ; q ) = 1 + q n ,f ( n ; q ) = 1 − q n , f ( n ; q ) = (1 − q n )(3 − q n ) = f + 2 f f , (3.16)in order to convert the hyperbolic functions to exponentials for typographical simplicity. We canthen rewrite the coefficient matrices obtained above in terms of the matrix functions f i (˜ κ gh ; q ). Thesehave block diagonal structure but contain non-diagonal matrices in the upper block. Additionally,they do not commute with matrices such as m gh and M gh . However, using matrix relations such as˜ κ gh (cid:62) = ( M gh ) − ˜ κ gh M gh , m gh (cid:62) = − σM gh , and σ (cid:20) α α (cid:21) σ = (cid:20) α α (cid:21) (3.17)for block diagonal matrices, one can simplify the above expressions for η gh coefficients as Q η ( q ) = − (cid:18) σM gh σ + M gh − f ( q ) f ( q ) (cid:19) − (cid:18) M gh − f ( q ) f ( q ) − σf ( q ) M gh σ + σf ( q ) − f ( q ) (cid:62) σ (cid:19) = − (cid:0) σf ( q ) M gh σ + M − f ( q ) + σf ( q ) − f ( q ) (cid:62) σ (cid:1) , (3.18) L η ( q ) = 14 σf ( q ) − f ( q ) (cid:62) M gh − , (3.19)where we have dropped one argument of f i ( q ; ˜ κ gh ) as shall be done in other places as well fortypographical simplicity. Here we point out that the + ve sign in the exponential factor in the first line of (3.13) is different from the usual − ve sign for Grassmannian Gaussian integral, since the antisymmetric metric factor ε adjoining λ produces anextra -ve sign upon taking a transpose. Explicitly, we have the following signs:( − iε ) (cid:62) ( − iε ) − ( − iε ) = − ( − iε ) . (3.15)Since we insist on using an − iε metric in ghost space, there is the extra − ve sign which makes the exponential partin the Gaussian integral identical to the matter sector. of theform (1 + tanh t ˜ κ gh ) − = e − t ˜ κ gh cosh t ˜ κ gh = 12 f , (3.20a)1 − sech t ˜ κ gh = f f , (3.20b) e − t ˜ κ gh (cid:0) cosh t ˜ κ gh + sech t ˜ κ gh − (cid:1) = f f . (3.20c)Additionally, we can obtain the half-angle relations by noting that f ( n ; q ) = f ( n ; √ q ) f ( n ; √ q ).Finally, we perform the integration over η gh (3.13) to obtain a purely quadratic functional dependenceon λ gh in the exponential of the form + 14 λ gh (cid:62) F λ gh , where the matrix F in the ghost sector can bewritten as F ( t ) = M − ( t ) + L (cid:62) η Q − η L η . (3.21)Here, the first term M − arises from the η gh independent overall factor C η ( t ) in (3.14c) definedabove.Collecting all the factors together, the ghost contribution to the generating functional W [ λ, λ gh , t ]is given by: W [ λ gh , t ] = (1 + w (cid:62) w ) det(2 Q ghη ) exp (cid:20) + 14 λ gh (cid:62) F λ gh (cid:21) . (3.22)We shall include the matter sector contribution from App A, which is obtained through a very similarcomputation—with the only difference being the integration over the zero mode momenta p µ alongthe Neumann directions, and the use of a different set of constant matrices for defining the monoidelements. The matter contribution to the generating functional serves to provide a consistency checkfor our analytical expressions. Only the determinant factors need be included in numerical checkswhen considering overlap with the perturbative vacuum state | ˆΩ (cid:105) = ˆ c | Ω (cid:105) . And for purely ghostexcitations, we use this scalar piece for the matter sector—it contributes to the measure factor anddoes not affect the structure of the R nm ( q ) factors in (2.40), that we are primarily interested in.Finally, the total matter+ghost generating functional has the structure: W [ λ X , λ gh , t ] = (cid:16) ww (cid:62) (cid:17) d +28 e t det(2 Q η ) | det(2 Q ψ ) | d/ exp (cid:20) (cid:16) λ X (cid:62) F X λ X + λ gh (cid:62) F gh λ gh (cid:17)(cid:21) (3.23)where X denotes the matter part from the embedding coordinates X µ ( z ), have combined the con-jugate variables η X and p into a single “vector” ψ := (cid:18) η X p (cid:19) and denoted the matter coefficient matrix Q with the subscript ψ . Next, we can consider the block structure of the matrices Q η , L η , and F gh . To this end, we recallthat the matrices M gh and ˜ κ gh (given in § M gh = − (cid:20) R (cid:62) κ o R
00 4 κ − e (cid:21) , ˜ κ gh = (cid:20) R (cid:62) κ o T (cid:62) κ e (cid:21) . (3.24) Yet another useful relation is f + f = 2 f f .
21e remark that the M gh above is given by i × M gh as compared to the one given in [14] whereas thematrix ˜ κ gh remains the same. Then the 2 N × N coefficient matrices have the explicit constituentblock structure: Q η ( q ) = + 14 (cid:20) κ − e f ( κ e ) + T κ − o f ( κ o ) T (cid:62) − i [ f ( κ e ) − T f ( κ o ) R ] − i (cid:2) f ( κ e ) − R (cid:62) f ( κ o ) T (cid:62) (cid:3) (cid:2) κ e f ( κ e ) + R (cid:62) κ o f ( κ o ) R (cid:3) (cid:21) , and L η ( q ) = 12 (cid:20) T κ − o f ( κ o ) T (cid:62) i f ( κ e ) − i R (cid:62) f ( κ o ) T (cid:62) κ e f ( κ e ) (cid:21) , (3.25)where again the blocks are labelled by half-phase space degrees of freedom ( x c,be , p c,be ).Let us observe that the infinite sums over the odd integers κ o in all the four blocks of Q η divergebadly for t < f , f , and f are unbounded as κ o increases. Hence, these matrixelements are not analytic in a neighbourhood of 0. Only the t → + limit is well-defined for whichthe matrix Q η vanishes due to the zeroes of the functions f , f and f at that point (as we shalldiscuss below). Strictly speaking, this prevents the expansion we seek involving Q − η . However,the matrix L ( t ) also vanishes at t = 0 due to the zeroes in f and f . Hence, the combination L ( t ) (cid:62) Q ( t ) − L ( t ) in F ( t )—which does involve infinite sums—can be taken to vanish at t = 0 for thepurpose of this work. This behaviour signals that the expansion we obtain may be asymptotic andnot a convergent expansion, owing to this non C ∞ nature.Additionally, we notice that in the open string limit N → ∞ , the order of the pole from thecombined determinant factors, Q η and Q ψ in (3.23), becomes infinite as well. This is consistentwith our expectations of an essential singularity at t = 0 associated with the Shapiro-Thorn closedstring tachyon state in (1.1).In general, due to the relatively simple structure of the T matrix, we can expect combinationsof the generalized hypergeometric functions, J F J − , to arise from the infinite sums in Q η . Thenon-analyticity in Q η matrix elements would then be a log branch cut. The non-diagonal terms,with n (cid:54) = m are of the form: Q xx n, m = ( − m + n π ( m − n ) (cid:26) q (cid:18) q (cid:18) Φ (cid:18) q , , − m (cid:19) − Φ (cid:18) q , , − n (cid:19) + Φ (cid:18) q , , m + 12 (cid:19) − Φ (cid:18) q , , n + 12 (cid:19)(cid:19) − (cid:18) q , , − m (cid:19) + 4Φ (cid:18) q , , − n (cid:19) − (cid:18) q , , m + 12 (cid:19) +4Φ (cid:18) q , , n + 12 (cid:19)(cid:19) − ψ (0) (cid:18) m + 12 (cid:19) − ψ (0) (cid:18) − m (cid:19) +3 ψ (0) (cid:18) n + 12 (cid:19) + 3 ψ (0) (cid:18) − n (cid:19)(cid:27) , (3.26a) Q xp n, m = − i ( − n + m mq π n ( m − n ) (cid:26) − nq Φ (cid:18) q , , − m (cid:19) + mq Φ (cid:18) q , , − n (cid:19) + 2 n Φ (cid:18) q , , − m (cid:19) − m Φ (cid:18) q , , − n (cid:19) + nq Φ (cid:18) q , , m + 12 (cid:19) − mq Φ (cid:18) q , , n + 12 (cid:19) − n Φ (cid:18) q , , m + 12 (cid:19) + 2 m Φ (cid:18) q , , n + 12 (cid:19)(cid:27) , (3.26b) Q pp n, m = ( − n + m π ( m − n ) (cid:110) m (cid:16) H − n − + H n − (cid:17) − n (cid:16) H − m − + H m − (cid:17) + q m (cid:18) Φ (cid:18) q , , − n (cid:19) + Φ (cid:18) q , , n + 12 (cid:19) − q n (cid:18) Φ (cid:18) q , , − m (cid:19) + Φ (cid:18) q , , m + 12 (cid:19)(cid:19)(cid:19) +4 (cid:0) n − m (cid:1) tanh − (cid:0) q (cid:1) + 4 log(2)( m − n ) (cid:111) (3.26c) while the diagonal matrix elements are given by: Q xx n, n = 18 π n (cid:26) q (cid:18) q Φ (cid:18) q , , − n (cid:19) − (cid:18) q , , − n (cid:19) − q Φ (cid:18) q , , n + 12 (cid:19) + 4Φ (cid:18) q , , n + 12 (cid:19)(cid:19) π (cid:0) q n − (cid:1) + 3 ψ (1) (cid:18) − n (cid:19) − ψ (1) (cid:18) n + 12 (cid:19)(cid:27) , (3.27a) Q xp n, n = i π n (cid:26) q (cid:18) − q Φ (cid:18) q , , − n (cid:19) + nq Φ (cid:18) q , , − n (cid:19) + 2Φ (cid:18) q , , − n (cid:19) − n Φ (cid:18) q , , − n (cid:19) + q Φ (cid:18) q , , n + 12 (cid:19) + nq Φ (cid:18) q , , n + 12 (cid:19) − (cid:18) q , , n + 12 (cid:19) − n Φ (cid:18) q , , n + 12 (cid:19)(cid:19) − π nq n (cid:0) q n − (cid:1)(cid:27) , (3.27b) Q pp n, n = 18 π (cid:26) H − n − + 2 H n − + 2 q Φ (cid:18) q , , − n (cid:19) − nq Φ (cid:18) q , , − n (cid:19) +2 q Φ (cid:18) q , , n + 12 (cid:19) + nq Φ (cid:18) q , , n + 12 (cid:19) + π n (cid:0) − q n + q n + 3 (cid:1) + nψ (1) (cid:18) − n (cid:19) − nψ (1) (cid:18) n + 12 (cid:19) − − (cid:0) q (cid:1) + 8 log(2) (cid:27) . (3.27c) In this case, the J F J − functions get further expressed in terms of Lerch transcendents Φ( z, s, a ),a generalization of the zeta and the polylog functions, defined classically [34] by the infinite seriesrepresentation: Φ( z, s, a ) = ∞ (cid:88) n =0 z n ( n + a ) s . (3.28)In all of the above, the Φ functions with the argument (cid:60) ( a ) < .We can now examine some series expansions to notice that these are functions having a leadinglogarithmic branch cut t log t for the blocks Q xx and Q pp , and t log( t ) for the blocks Q xp = Q px (cid:62) : Q pp = 16 t − t π + 512 t t (cid:18) t )3 π − π + 1024 (cid:19) + O (cid:0) t (cid:1) , Q xp = − it (18 log( t ) + 19 + 6 log(2))9 π − it (360 log( t ) + 151 + 24 log(2))9 π + O (cid:0) t (cid:1) . (3.30) Since the three functions f , , ( t ) have first order zeroes at t = 0 (or q = 1), and because the singlesum over the odd frequencies κ o still retain the same order of zero for both finite and infinite N , wecan factor out this zero. Hence, in the open-string limit, corresponding to N → ∞ , we can simply divide out by t in order to expand the inverse. The physically correct order of operations would beto consider the expansion only in the open string limit. However, one can also attempt an expansionfor the deformed theory defined at finite N , and see where it leads us; since both have similar formalstructure. The oscillator counterpart of this issue with order of limits, namely level L → ∞ followedby t → § by a factor of 2 already at the leading correction in t .Let us therefore factor out the parameter t = − ln q and introduce the two matrices: Z ( q ) := − Q η ( q )ln q , Y ( q ) := − L η ( q )ln q (3.31) This would differ from the representation in terms of the original Lerch functions, which take the formΦ ∗ ( z, s, a ) = ∞ (cid:88) n =0 z n [( n + a ) ] s/ . (3.29)for (cid:60) ( a ) <
0, where we omit any term with n + a = 0. We do however expect to miss some of the very interesting non-analyticities of the form e − jπ /t = e + jπ q in ouranalysis. Although the oscillator and Moyal representations are formally isomorphic, there are subtle differences due tothe special nature of the Witten type vertex. See [16, 28] for a careful discussion of these matters and for a detailedanalysis of midpoint issues.
23n order to rewrite the matrix F ( q ) in (3.21) as below: F ( q ) = 12 ( M gh ) − f ( q ) − ln q Y ( q ) (cid:62) Z ( q ) − Y ( q ) . (3.32)This form will turn out to be convenient when we study the behaviour of the matrix R ( t ) in thelimit t → + directly in the modular parameter t later in § F has component blockswhich would be labelled as F xx , F xp = F px (cid:62) and F pp in terms of the phase space doublet indices( x, p ) as usual.The matrix Z ( q ) as it appears above is bounded at t = 0 and hence would still be amenable to anexpansion. However, the resulting expression for F need not be analytic because the matrix inverse Z ( t ) − allows for infinite sums that alter the pole-zero structure. Furthermore, there are doubleinfinite sums involved when this is sandwiched between Y (cid:62) and Y . Perhaps these non-analyticitiesmay be relatable to the closed string states arising in this degeneration limit geometrically. In orderto simplify the analysis, we shall restrict to the case when the Y s contribute only diagonal matrices—corresponding to even parity elements—thus eliminating some of the multiple summations. We hopeto look at the other cases in more detail when occasion offers itself. In this work, we are primarily interested in the analytic behaviour of the squeezed state matrix R ( t ) or equivalently F ( t ) in the limit t → + , but as a check on the correctness of our expressions,we shall study the determinant factor numerically in App A using similar methods as in [7]. Thedeterminant part corresponds to the overlap with the perturbative vacuum state, i.e | A e (cid:105) = | ˆΩ (cid:105) :the open string tachyon at zero momentum, and has interesting divergence structure of its own.However, as encountered in [7], it is awkward to study this factor analytically due to the essentialsingularity at t = 0.The full matrix element contributing to the ghost sector does not lend itself to an expansionbecause in general each of the matrices whose determinant would be required would appear as apower series starting at degree 0 (constant term). As the minimal degree does not decrease orincrease along a row or a column, this form of the determinant proves unwieldy for a systematicexpansion. We therefore do not perform a series based analysis of the determinant using the diagonalbasis in this work and instead focus on the finite factor from the R matrix: (cid:104) ˆΩ | ˆ c n ˆ b m |T ( t ) (cid:105) ∼ R nm ( t ) × det ( · · · ; t ) (3.33)Additionally, as part of a series of papers on off-shell conformal field theory (see [6] and referencestherein), the N -tachyon scattering case has been studied in great detail by Samuel et al. andaddresses these questions much more directly using advanced Riemann surface theory upto the one-loop level. In this approach, the measure factors corresponding to the matter + ghost determinantsare evaluated in terms of line integrals involving rational combinations of elliptic functions and theirderivatives. It may be possible to extend some of their results to the overlap with a general Fockspace state other than the tachyon case considered there. In the Siegel gauge, the ghost contribution to the tadpole state can be expressed in terms of Fockspace kets and Moyal fields as: |T ( t ) (cid:105) = (cid:90) dξ gh T ( ξ gh , t ) | ξ gh (cid:105) . (3.34)Comparing to (3.1), we have the Moyal string field T ( ξ gh , t ) ∼ (cid:90) ( dη gh ) (cid:104) e + ξ (cid:62) η gh ∗ ( e − tL gh e − ξ (cid:62) η gh ) (cid:105) , (3.35)24here again we have left the overall sign unfixed. We transform the expression for (cid:104) ξ gh | in the oddbasis given in [14] to the basis labelled by even integers, that we use, and write this as a bra: (cid:104) ξ gh | = − − N (cid:16) w (cid:62) w (cid:17) − (cid:104) Ω | ˆ c − e − ξ (ˆ c −√ w (cid:62) ˆ c e ) e − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh , (3.36)where we have the vectors λ gh = (cid:18) √ R (cid:62) ˆ b o − √ κ − e ˆ b e + 2 κ − e wξ (cid:19) , λ gh = (cid:18) √ R (cid:62) κ o ˆ c o √ i ˆ c e (cid:19) , (3.37)and M gh is the matrix defining the perturbative ghost vacuum A gh : M gh = −
12 Diag { R (cid:62) κ o R, κ − e } . We remind the reader of the metric convention we have been using—where the − iε factor is implicit—and hence ξ gh (cid:62) M gh ξ gh = − iξ (cid:62) M gh ξ as well as ξ gh (cid:62) λ gh = − i ( ξ (cid:62) λ − ξ (cid:62) λ ).To probe the structure of the state |T ( t ) (cid:105) , one usually finds its overlap with various Fock spacebasis states (cid:104) ϕ | . Hence, we must consider the corresponding overlap amplitudes in Moyal space andthen transform back to Fock space.In order to convert the amplitude written in Moyal space, (3.1) to the one in terms of Fock spacestates, we need to construct the appropriate perturbative string fields A e ( ξ ). To this end, we givethe corresponding expressions in the oscillator formalism: (cid:104) A e |T ( t ) (cid:105) ∼ (cid:90) dξ (cid:104) A e | ξ gh (cid:105)(cid:104) ξ gh |T ( t ) (cid:105) , (3.38)where we have denoted the external state by | A e (cid:105) and introduced a complete set of states (cid:104) ξ gh | —theappropriately normalized bra defining the Moyal basis in ghost space to be given below in (3.36).We recall that we can restrict to the SU (1 ,
1) symmetric [19] combination of pure ghost externalstates, since the tadpole state is a singlet under this symmetry. In particular, the matrix R nm defining the quadratic form in the exponential of the squeezed state satisfies: m R nm = n R mn , (3.39)i.e R κ is a symmetric matrix. This does not demand the full SU (1 ,
1) but can be achieved byrestricting to the discrete Z subgroup. The ghost sector matrix R ( t )The ˆ β c,be,o , ˆ β Xe,o oscillators (described in App C) can now be directly used to construct the perturbativestring fields A e ( ξ, ξ gh ) that correspond to the matrix elements R nm ( t ) (and M nm ( t ) in the mattersector) when written in terms of Fock space states. The pure ghost fields would be of the form ℘ ( ξ gh ) A gh ( ξ gh )where ℘ ( ξ gh ) is an appropriately normalized polynomial, which would be the analogue of Her-mite polynomials acting on Gaussians in a representation in terms of position space functionalsΦ[ X µ ( σ ) , c ( σ )].In terms of these, the relevant matrix elements get mapped to the following Moyal polynomialswith ghost bilinear pieces: R ee (cid:48) ← − δ ee (cid:48) + 8 iκ e (cid:48) p ce p be (cid:48) , (3.40a) R oo (cid:48) ← δ oo (cid:48) + 2 i ( κ o Rx b x c (cid:62) R (cid:62) ) oo (cid:48) , (3.40b) i R eo ← − i ( p c x c (cid:62) R (cid:62) ) eo , (3.40c) i R oe ← +4 i ( κ o Rx b p b (cid:62) ) oe . (3.40d) Upto a t independent normalization factor to which we return in § R eo , R oe terms vanish identically which reflects the twist symmetry of the tadpole state |T (cid:105) .This can also be seen numerically as we have verified. We remark in passing that we can also obtainthe matrix elements for the matter part by using the oscillators given in [13] as: M ee (cid:48) ← − (cid:0) κ e δ ee (cid:48) − κ e κ e (cid:48) x e x e (cid:48) (cid:1) (3.41a) i M eo = i M oe ← (cid:0) κ e x e ( p (cid:62) T ) o (cid:1) (3.41b) M oo (cid:48) ← (cid:0) κ o δ oo (cid:48) − p (cid:62) T ) o ( p (cid:62) T ) o (cid:48) (cid:1) (3.41c)which may be useful for future applications.Now that we know the required form of the polynomials, we can proceed to construct themstarting from the generating string field A ( ξ gh , λ gh ) given in (3.3) using ℘ ( ξ gh ) A = (cid:32) ℘ (cid:32) − (cid:126)∂∂λ gh (cid:33) A (cid:33) (cid:12)(cid:12)(cid:12) λ gh =0 (3.42)while taking into account the implicit − iε ⊗ N metric factors everywhere, including the linearterm. Explicitly, we make the replacements: ℘ (cid:16) x c , p c , x b , p b (cid:17) (cid:55)→ ℘ (cid:18) + i ∂∂λ x b , − i ∂∂λ p b , − i ∂∂λ x c , + i ∂∂λ p c (cid:19) . (3.43)Once we have the matrix F ( t ) defining the quadratic form in λ gh in the exponential of the generatingfunctional W ( λ gh , t ) for the integrand (3.22), we can plug it in the above map which produces theFock space amplitudes from the ones in Moyal space. Then we can rewrite the matrix element R nm ( t )corresponding to (cid:104) ˆΩ | ˆ c m ˆ b n |T ( t ) (cid:105) (or equivalently the perturbative monoid element p b n p c m A gh for thepurely even parity case, etc.) as follows: R n, m = − (cid:18) δ nm + 42 m F pp n, m (cid:19) R n − , m − = δ nm + (2 n − R F xx R (cid:62) ) n − , m − , R n, m − = − i ( R F xp ) (cid:62) n, m − , R n − , m = − i ( κ o R F xp ) n − , m . (3.44a)(3.44b)(3.44c)(3.44d)where the upper indices on F refer to the N × N blocks in the 2 N × N matrix F ( t ) belongingto the phase space representation used, namely “momenta” p c p b , “position” x c x b and the mixedcases. The negative sign in the first equation (and implicit in the following) is due to the particularway the ghost zero mode ξ is incorporated in the Moyal basis. This gives a normalization constant( − µ − ) ( § F , (3.32) , we notice that the matrix elements in the purely momentum sector, F pp are particularly simple since the N × N block matrices in Y that contribute to theproduct are all diagonal matrices. Hence, the infinite summations are sidestepped. By using theabove map, we find that these correspond to the purely even parity elements of the R matrix. In § T , R matrices(in the infinite N limit).Because of the twist symmetry of the Witten type vertex (cid:104) V | and the reflector (cid:104) (cid:101) V | , we havevanishing of the mixed parity elements R eo = 0 = R oe . This requires that block F xp = 0, which In the above, we have inserted extra factors of i in the mixed parity cases to make the string fields real. in Q − η , L (cid:62) xαη ( Q − η ) αβ L βpη = . (3.45)We can now express the relation (3.44) as: R = − C + σ (cid:20) κ o R
00 1 (cid:21) F (cid:20) R (cid:62) − κ − e (cid:21) σ = − C + C + (cid:20) q κ e − q κ o (cid:21) + σ (cid:20) κ o R
00 1 (cid:21) L (cid:62) η Q − η L η (cid:20) R (cid:62) − κ − e (cid:21) σ = (cid:20) q κ e − q κ o (cid:21) + 14 σ (cid:20) f ( κ o ) T (cid:62) − i κ o f ( κ o ) R i f ( κ e ) κ e f ( κ e ) (cid:21) Q − η (cid:20) T κ − o f ( κ o ) − iκ − e f ( κ e ) − i R (cid:62) f ( κ o ) − f ( κ e ) (cid:21) . (3.46)Here, we have inserted the σ matrices simply to interchange the two blocks on the diagonal in orderto match our conventions for the parity basis. We have written the above to show that the the L η matrices do not result in two more infinite sums—but only one extra infinite sum—which getssimplified by using the T R = e , RT = o relations after the matrix inverse is expanded as a formaloperator series as we do in § One of the interesting results from our analysis is that our starting expressions correctly reproducethe linear order behaviour of the matrices R nm ( t ) and M nm ( t ) that appear in the definition of theone-loop tadpole state in (2.40) as expected from BCFT. The oscillator and the Moyal formalismare formally isomorphic but this is one of the instances where the subtleties in the definition ofthe propagator and level truncation result in different forms. It is difficult to say where exactlythe isomorphism breaks down but it may be attributable to the level truncation which breaks thegauge symmetry of OSFT and the peculiar nature of the Virasoro zero mode operator ˆ L in theMoyal basis[16, § that the difference for the linear correction term from the twomethods is only a factor of 2. Verification of the linear behaviourZeroth Order
For t = 0, the matrix F becomes simply F | t =0 = 12 M gh − f (˜ κ gh ; q ) | q =1 = − (cid:20) T κ − o T (cid:62) κ e (cid:21) × (cid:20) R (cid:62) (1 + 1) T (cid:62)
00 2 (cid:21) = − (cid:20) T κ − o T (cid:62) κ e (cid:21) , (3.47)by quickly noting that f ( n ; 1) = 2 , T R = e and R T = o . This when substituted into (3.44) givesthe t independent piece to be R nm | t =0 = ( − n δ nm = C nm (3.48)and by a similar short calculation, we can show that M nm | t =0 = ( − nm δ nm = C nm (3.49) Since Q η is symmetric and ( M gh ) − f ( q ) is already block diagonal, it suffices to consider only three independentblocks. Due to non-associativity, a factor of 2 issue arises also in the computation of the closed string tachyon mass [35]through the Ellwood-Hashimoto-Itzhaki-Zwiebach invariant. C is the twist matrix which is crucial in defining thereflector vertex (cid:104) ˜ V | and arises from BPZ conjugation and the Witten style overlapping conditions.These precisely correspond to the closed string tachyon state (1.1) which dominates due to thedivergence structure arising from the determinant factor near t = 0.Here, we have assumed that there are no extra poles from the infinite summations in Y (cid:62) ( t ) Z ( t ) − Y ( t )that cancels the single power of t multiplying it. This will certainly be true for R n, m associatedwith the diagonal blocks in Y ( t ) but can also be seen to hold for R n − , m − by examining theblock structure in (3.46). But more importantly, we can take this as the correct prescription sinceit matches with the BCFT prediction for the structure of |T ( t ) (cid:105) ! First Order
Interchanging the order of summation over κ o (odd integers) and the non-negative integers definingthe exponentials e − t of f i ( n ; t ), appearing in the various blocks in Q η and L η given in (3.25), wecan expand them to the lowest order in the parameter t : Q η = t (cid:20) + T T (cid:62) ) κ e (cid:21) + O ( t )= t (cid:34) − ww (cid:62) w (cid:62) w κ e (cid:35) + O ( t ) , (3.50) L η = t (cid:20) iκ e − iR (cid:62) κ o T (cid:62) (cid:21) + O ( t ) , (3.51)where we have used the relations: T T (cid:62) = − ww (cid:62) w (cid:62) w , R = κ − o T (cid:62) κ e , (3.52)and the off-block diagonal elements in Q η do not contribute since f ( n ; t ) starts at O ( t ). Thequantity w (cid:62) w diverges linearly as O ( N ) and expressions involving it should be treated with care toavoid inconsistencies. Hence, we shall keep the O (1 /N ) term and argue when it may be dropped.The matrix V := − ww (cid:62) w (cid:62) w (3.53)appearing in the first block of Q η above can be readily inverted using a Taylor series in 1 / ¯ ww < N ≥
1. We make the following ansatz involving a function µ ( z ): V − n, m = δ nm + µ ( w (cid:62) w ) w n ( w (cid:62) ) m (3.54)and require VV − = = V − V to find W := V − = + ww (cid:62) w (cid:62) w , (3.55)which may then be verified by a direct substitution. Then we find that Q − η = 1 t (cid:20) W 00 κ − e (cid:21) + finite + subleading , (3.56)showing that: L (cid:62) η Q − η L η = − t (cid:20) T κ o Rκ − e R (cid:62) κ o T (cid:62) κ e W κ e (cid:21) + O ( t )= − t (cid:20) T T (cid:62) κ e W κ e (cid:21) + O ( t )= − t (cid:34) − ww (cid:62) w (cid:62) w (cid:16) κ e + κ e ww (cid:62) κ e w (cid:62) w (cid:17) (cid:35) + O ( t ) . (3.57)28ow we can consider the open string limit for the second block since there are no divergent termsin this expansion, while we retain the T T (cid:62) form for the first block. Isolating the linear term from ( M gh ) − f ( q ), we obtain + t (cid:20) T T (cid:62) κ e (cid:21) This when substituted into (3.32) leads to: F = − (cid:20) T κ − o T (cid:62) − t T T (cid:62) κ e − t κ e (cid:21) + O ( t ) . (3.58)Consequently, we can readily write down the squeezed state matrix R nm to this order using (3.44): R n, m = − δ nm − m × −
12 2 nδ nm − t m ×
14 4 n δ nm = δ nm − nt δ nm + O ( t ) , (3.59) R n − , m − = δ nm + (2 n − (cid:104) R (cid:16) − T κ − o T (cid:62) + tT T (cid:62) (cid:17) R (cid:62) (cid:105) n − , m − = − δ nm + (2 n − t δ nm + O ( t ) . (3.60)The mixed parity cases R n, m − vanishes identically as we have argued before. This enables us toexpress the general matrix element as: R (Moy) nm = C nm − n C nm t + O ( t ) (3.61)As shown in [7], the linear correction in t is completely generated from the conformal transformationof the external Fock space state, and is determined from a BCFT analysis of the conformal mapdone near t = 0.It precisely coincides with the above form, whereas a Taylor expansion based on the oscillatorexpressions gives a linear coefficient off by a factor of 2: R (osc) nm = C nm − n C nm t + O ( t ) . (3.62)As explained carefully in [7], the two limits involving the level (size of the matrices) and the modularparameter, L → ∞ and t → § t = 0. Here we have used the finite N versions of the matrices (A.22) whichensure that the star algebra relations are satisfied. For N = 84 (requiring inversion of 168 × t varying from 10 − to 16 × − in steps of 10 − , we obtain the linear fit given inTable 1. We emphasize that the higher order terms starting at t are the ones that really encode anyeffects of the Shapiro-Thorn massless closed-string states. Unfortunately, our algebraic approachonly allows to successively approximate these coefficients (as we do in § The various orders for evaluating the overlap mentioned in § A ∗ A → A ( η, ξ ) → (cid:90) dη → A A (cid:48) ( ξ ) → Tr corresponds29 ( lin )11 − (0 . − . × t ) R ( lin )22 +(0 . − . × t ) R ( lin )33 − (0 . − . × t ) R ( lin )44 +(0 . − . × t )Table 1: Linear behaviour of the matrix elements R nm near t = 0 based on numerical evaluationof 168 ×
168 size matrices. The fit reinforces the agreement between the Moyal and the BCFTpredictions for the structure of R ( t ).to the manner in which the amplitude would be evaluated in the oscillator method (see § |T (cid:105) = − g T K (cid:90) ∞ dt , (cid:104) (cid:101) V | b (2)0 e − t ( L (1)0 + L (2)0 ) | V (cid:105) , , . (3.63)and the amplitude is obtained by taking the inner product with an external state (cid:104) A e | . Here thesuperscripts refer to the string Hilbert spaces in the first quantized formalism.The corresponding matrix F ( t ) defining the quadratic form in λ gh towards the generating func-tional, W ( λ gh , t ) in this particular order of evaluation is then F ( t ) = ( M (cid:48) ) − , with M (cid:48) = 2 f ( q ) M gh + (cid:18) f f + f f m gh (cid:19) M − f f + σ f f M gh σ − σ q ˜ κ gh f ( q ) + 2 (cid:32) q ˜ κ gh f (cid:33) (cid:62) σ − (cid:18) f f + f f m gh (cid:19) (cid:62) (3.64)A quick inspection of the above structure reveals that similar to the earlier evaluation order, thematrix to be inverted in M (cid:48) vanishes at t = 0. Collecting the linear order terms after some simplealgebra results in an identical expression for the linear correction term, namely: R nm = C nm − nC nm t + O ( t ) , (3.65)and hence we conclude that the order of limits problem does not arise in this order of evaluationeither. However, further expansions are made awkward by the somewhat complicated form of theabove expression, which requires two matrix inverse operations nested one inside the other.Hence, from the above exercice we can infer that constructing the tadpole state out of the Fourierbasis and then combining their contribution to the overlap amplitude (by the η gh integration as wehave done earlier) would be preferred over considering the overlap with the state itself, which maybe somewhat counter-intuitive. Of course, just the linear order behaviour does not fix a prescriptionuniquely or prove the correctness of these expressions. Nonetheless, this is an encouraging resultshowing the subtleties in the map between Moyal space and Fock space.It would have been more interesting if associativity was indeed violated in this calculation, whichcould display the similarity to the oscillator inner product directly. Hence, we have not been ableto clarify the order of limits issue completely. Fourier Basis
We must remark that the issue encountered in the oscillator basis also arises if one attempts toexpand the amplitude in Fourier space defined by the conjugate variable η gh . The Feynman rules in30ourier space were studied and given in detail in Refs. [14, 15] by Bars et al. The propagator andvertex take of the form:∆( η gh , η (cid:48) gh , t ) ∼ exp (cid:104) η gh (cid:62) F gh η gh + η (cid:48) gh (cid:62) F gh ( t ) η (cid:48) gh (cid:62) − η (cid:48) gh (cid:62) G gh ( t ) η gh (cid:105) , and (3.66)Tr (cid:16) e − ξ gh (cid:62) η gh ∗ · · · ∗ e − ξ gh (cid:62) η ghn (cid:17) ∼ exp − (cid:88) i 0, the matrix to be inverted becomessingular simply due to the linear dependence of the blocks Q ( t ) ∼ t (cid:20) (cid:21) ⊗ M gh − (3.69)where the first block has vanishing determinant. Hence this form cannot be used as the startingpoint for a systematic series expansion around t = 0. However, we remark that for numericalpurposes, the Fourier basis provides quicker analytic expressions since the ∗ products are alreadytaken care of. The disadvantage is numerical instability due to using much bigger sized matrices ascompared to the ξ basis.Thus in summary, we have demonstrated in this section that the expected behaviour from BCFT iscorrectly reproduced by the Moyal expressions in ξ space. For showing the validity of the relations,we have used the map (C.9) from th oscillators as operators in Fock space to differential operatorsin Moyal space. One really interesting aspect is the non-analyticity of these matrix elements alreadyseen at the quadratic stage: the higher order terms come with factors of log( t ) as we shall showlater in § e − π n/t . In this section, we wish to study the behaviour of the matrix element factor R nm —defining thesqueezed state in the (ghost) exponential factor of the integrand as appearing in (2.40)—near t =0 + using expansions in various basis functions. Naturally, one can find an absolutely convergentexpansion in the nome q := e − t for | q | < 1, corresponding to open string degrees of freedom. Becauseof the essential singularity at t = 0 coming from the massive closed string states, the expansion inother basis functions such as { t s , ln t } or equivalently { ( − ln q ) s , ln( − ln q ) } would not be a convergentexpansion, but could at best be an asymptotic expansion. This is consistent with our understandingof the quantum inconsistency of bosonic OSFT (or any open bosonic string theory) at the loop level.In the following, we simply explore the utility of the Moyal formulation to directly learn aboutthe structure of the integrand as a function of the parameter t . The expressions we obtained in § t = 0 and we found that it reproduces the correctzeroth and linear order coefficients, which is somewhat non-trivial. Furthermore, we can develop aseries expansion involving special functions to successively approximate the true analytic form for R nm ( t ) by our method. 31 .1 Even parity matrix elements near t → + In order to perform an expansion in t , let us introduce the following auxiliary functions derived fromthe functions f , f , f and f employed earlier (3.16) h i ( n ; t ) := f i ( n ; t ) t , g i ( n ; t ) := h i ( n ; t ) − h i ( n ; 0) = f i ( n ; t ) t − f (1) i (0) , giving explicitly h ( n ; t ) = (1 − e − nt ) t , h ( n ; t ) = 1 + e − nt t ,h ( n ; t ) = 1 − e − nt t , h ( n ; t ) = 3 − e − nt + e − nt t , and g ( n ; t ) = (1 − e − tn ) t , g ( n ; t ) = 1 + e − nt t + 2 n,g ( n ; t ) = 1 − e − nt t − n, g ( n ; t ) = 3 − e − nt + e − nt t − n. (4.1)These can be thought of as certain basis functions with a well-defined asymptotic behaviour. Also,we notice that | g i ( n ; t ) | < n for i = 1 , , 4; a boundedness property which we will use later.We remark at this juncture that the functions tanh t and sech t which appear in the original formof the block matrices (3.14) may be Taylor expanded in terms of the Bernoulli numbers B n and theEuler numbers E n around the point t = 0. However, reorganizing the multiple sums and productsfollowed by applying any identities involving them quickly becomes challenging. Therefore, wecontinue to use the much more straightforward (and uniform) representation in terms of exponentialfunctions in our analysis.Moving on, we illustrate this expansion scheme for the case of even parity matrix elements, R n, m , for convenience. Since its expression involves the inverse of a matrix function, which isdifficult to obtain analytically (at least in the discrete diagonal basis), we employ a formal series torepresent the inverse . After this step, one can find expansions around the point t = 0 althoughthe sub-matrices do lead to more terms without any apparent patterns for resummations. To thisend, we split the the matrix to be inverted Z ( t ), which appeared in (3.31) as follows: Z ( t ) = Z + δ Z ( t ) , so that we may write Z ( t ) − = (1 + Z − δ Z ( t )) − Z − := (1 + M ( t )) − Z − = ∞ (cid:88) s =0 ( − s M s ( t ) × Z − . (4.2)where we have defined a matrix function M ( t ) := Z − δ Z ( t ). Here we recall that Z must be definedas the limit lim t → + Q η ( t ) t but note that the matrix Q η ( t ) in (3.25) is not analytic at t = 0 due to theinsufficient fall-off behaviours of the T n, m − matrix elements as n, m increases T ∞ n, m − = ( − ) n + m m − π (4 n − (2 m − ) (4.3)which behave like 12 m − m .We also remind the reader of the block matrix forms from (3.55) Z = (cid:20) V 00 κ e (cid:21) , V := − ww (cid:62) w (cid:62) w , and W := V − = + ww (cid:62) w (cid:62) w , (4.4) This is justified because the domains of analyticity of the two maps overlap. Z − = (cid:20) W κ − e (cid:21) . (4.5)The matrix δ Z ( t ) is then expressed in terms of functions g , g and g in a form very similar to Z ( t ): δ Z ( t ) = (cid:20) κ − e g ( κ e ) + T κ − o g ( κ o ) T (cid:62) − i ( g ( κ e ) − T g ( κ o ) R ) − i (cid:0) g ( κ e ) − R (cid:62) g ( κ o ) T (cid:62) (cid:1) (cid:0) κ e g ( κ e ) + R (cid:62) κ o g ( κ o ) R (cid:1) (cid:21) ⇒ M ( t ) = (cid:20) W (cid:0) κ − e g ( κ e ) + T κ − o g ( κ o ) T (cid:62) (cid:1) − i W (cid:0) g ( κ e ) − T κ − o g ( κ o ) T (cid:62) (cid:1) − i (cid:0) κ − e g ( κ e ) − T κ − o g ( κ o ) T (cid:62) (cid:1) ( κ − e g ( κ e ) + T κ − o g ( κ o ) T (cid:62) κ e ) (cid:21) , (4.6)where we remind the reader that each matrix element is in general an infinite sum — owing tothe matrix products — and we have used the relation R = κ − o T (cid:62) κ e for rewriting the structureusing only the T and κ matrices. However, in taking powers of the matrix M symbolically it ismore helpful to keep the matrix R since then we can readily apply relations such as R T = o and T R = e in order to reduce the number of terms.Note that although the individual blocks in M ( t ) have at least a first order zero at t = 0, theproducts of these blocks still retain only a first order zero due to the higher order poles arising fromthe infinite sums. This is because we are interchanging the order of summation in double sums whichare not absolutely convergent. Therefore all the higher matrix powers ( M ( t )) s continue to contributeto the t term in R nm ( t ) in our expansion scheme and consequently these coefficients cannot beobtained exactly by the above series. This drawback is again due to the infinite dimensional natureof the problem.Because of the logarithmic branch points, the terms for various s are not analytic, althoughthey vanish at t = 0 as remarked above. However, we expect that the contributions fall off withincreasing values of s (as seen from the tractable s = 0 , , t → + . The matrix Y ( q ) presented in (3.31) is nowwritten in terms of the functions h ( n ; q ) and h ( n ; q ) as follows: Y ( q ) = (cid:20) T κ − o h ( κ o ) T (cid:62) i h ( κ e ) − i R (cid:62) h ( κ o ) T (cid:62) κ e h ( κ e ) (cid:21) . (4.7)We shall now try to investigate the effect of working with a finite size truncation for the matrices vsdirectly using the infinite N versions of the expressions. Because the functions involved in the infinitesums satisfy the boundedness property: | g i ( n ; t ) | < n for i = 1 , , 4, we find that on examining thestructure of the matrix powers M s , the contribution from the extra term ww (cid:62) w (cid:62) w in W remainssubleading and always goes as N − p for some p ≥ 1. Since we only work with the partial sums fordefining the series representation of the inverse, i.e s = 0 , , . . . , S , say, these terms do not add upto give extra finite N corrections. Consequently, we can drop these O (1 /N p ) extra terms from ourcalculations and effectively set W = to do the relevant infinite sums over odd/even integers. Inother words, we have made a choice of order of limits that allows us to use the infinite N expressionsconsistently.Now, in order to study these partial sums using a series representation in t , we shall now commenton their analytic structure. As the functions arising from the infinite sums over the odd/even parityindices are uniformly convergent only for (cid:60) ( t ) > 0, term by term differentiation is not justified.Such mathematical niceties would have existed if we kept N < ∞ but then one misses the nicenon-analytic behaviour expected in the quantum theory which signals the inconsistency attributedto closed string states. In the following, we therefore work directly in the open string limit. Inaddition, as we are expecting only an asymptotic expansion due to physical reasons, it may bepossible to justify sending N → ∞ at this stage of the calculation for practical reasons.33n concrete terms, the above procedure would result in an expansion of the form: R nm ( t ) = ∞ (cid:88) r =0 (Λ r + log( t ) ˜Λ r ) t r (4.8)where the coefficients Λ r and ˜Λ r receive contributions from the partial sums over s and the log( t )piece will be shown to result from the non-analytic behaviour of the special functions that arise.There are more non-analytic terms than the simple log( t ) dependence that can be admitted (see(4.23)) since we do not have absolute convergence and hence the individual coefficients Λ r and ˜Λ r may not all exist. We are at this point only looking for hints of non-analytic behaviour and cannotrigorously account for any missing subleading terms.After these digressions, let us return to the series expansion at hand. In the following we illustratethe general procedure and also display some coefficients that contribute to the final matrix elements.We shall denote the expansion for R ( t ) in terms of the matrix products by a sequence of functions R ( t ) = ∞ (cid:88) s =0 R ( s ) ( t ) , (4.9)where we have chosen R (0) to match the linear order expansion we derived in § not furnish anasymptotic basis as can be seen from the basic criteria for the gauge functions φ n +1 ( z ) = o ( φ n ( z )) , (cid:18) as 1 z → + ∞ (cid:19) (4.10)not being satisfied by these. Except the first two terms, the rest all contribute starting at O ( t ) andconsequently these provide only an asymptotic approximation to the true function.The simplest block to look at is the purely even block R n, m given by (3.44) which we providehere again: R n, m = − (cid:18) δ n, m + 42 m F pp n, m (cid:19) (4.11)This is because it involves Z − sandwiched between Y xp and Y pp which are diagonal matrices andhence is easy to keep track of in a power series expansion. In block matrix form, the matrix F pp from (3.32) corresponding to the even parity elements is given by: F pp = − κ e f ( κ e ) + t ( Y (cid:62) Z − Y ) pp , where we can expand( Y (cid:62) Z − Y ) pp = Y (cid:62) px ( Z − ) xx Y xp + Y (cid:62) px ( Z − ) xp Y pp + Y (cid:62) pp ( Z − ) px Y xp + Y (cid:62) pp ( Z − ) pp Y pp = − h ( κ e )( Z − ) xx h ( κ e ) + i h ( κ e )( Z − ) xp κ e h ( κ e )+ i κ e h ( κ e )( Z − ) px h ( κ e ) + 116 κ e h ( κ e )( Z − ) pp κ e h ( κ e ) (4.12)The matrix powers M , M , . . . required for implementing this procedure requires some block matrixmultiplications. These can be performed using the NCAlgebra package [20] and recursively applyingthe relations satisfied by the T, R matrices such as T R = e , R T = o , T (cid:62) T = o − vv (cid:62) , etc.using the “NCReplace” series of commands .We are not at this point able to explicitly resum the series and demonstrate that this convergesbut it is still instructive to look at the functional behaviour of each of these contributing termsseparately. The sth power would give 2 × s − terms. Each such term is a product of s elements from the matrix M which in turn have sub-structure. .2 Illustrations for geometric series We have therefore obtained a few lower order terms by this method when the expressions reduce toa sum of terms with infinite sum over a single index (the odd integers). At higher values of s , thereare many terms which still involve only a single infinite sum but the few remaining terms involvingdouble and triple sums (over both even and odd integers) lead to computational problems.In the following, the integer in the superscript corresponds to the power s in the series expansionfor the inverse. s = 0 term The first term in the expansion corresponding to s = 0 is given by: R (0)2 n, m = − δ nm − m (cid:20) − 14 2 nf (2 n ; t ) δ nm + t (cid:0) h (2 n ; t ) − h (2 n ; t ) (cid:1) δ nm (cid:21) = (cid:18) e − nt ( nt − nt + e − nt nt + e − nt nt (cid:19) δ nm = (cid:18) − nt + 6 n t − n t O (cid:0) t (cid:1)(cid:19) δ nm (4.13)which contributes to the leading behaviour in the even sector, namely, C nm − nC nm t + O ( t ). Thecoefficients increase rapidly initially but then decrease as expected due to the factorial suppression. s = 1 term For s = 1, the infinite sums arising from the matrix products over the odd integers can be performedusing Mathematica . R (1)2 n, m = − m × − t (cid:104) Y (cid:62) M Z − Y (cid:105) pp (4.14)Since the matrices T n, m − and R n − , m have a relatively simple structure expressible in terms ofintegers, we expect these to be in general in terms of hypergeometric functions J F J − with argumentsof the form q k ; k ∈ Z + . For the diagonal matrix elements, we obtain: R (1)2 n, n = (cid:0) q n − (cid:1) π n log ( q ) (cid:26) − q n +2 Φ (cid:18) q , , − n (cid:19) + 2 nq n +2 Φ (cid:18) q , , − n (cid:19) + nq n +2 Φ (cid:18) q , , − n (cid:19) + 4 q (cid:0) q n − (cid:1) Φ (cid:18) q , , − n (cid:19) − nq (cid:0) q n − (cid:1) Φ (cid:18) q , , − n (cid:19) + 4 q Φ (cid:18) q , , − n (cid:19) − nq Φ (cid:18) q , , − n (cid:19) + 4 q Φ (cid:18) q , , n + 12 (cid:19) + 4 nq Φ (cid:18) q , , n + 12 (cid:19) − q n +2 Φ (cid:18) q , , n + 12 (cid:19) − nq n +2 Φ (cid:18) q , , n + 12 (cid:19) + 4 q n +2 Φ (cid:18) q , , n + 12 (cid:19) +3 nq n +2 Φ (cid:18) q , , n + 12 (cid:19) − q n +1 Φ (cid:18) q , , n + 12 (cid:19) − nq n +1 Φ (cid:18) q , , n + 12 (cid:19) − nq Φ (cid:18) q , , n + 12 (cid:19) − π n q n log( q ) − π nq n − γq n + 14 π nq n + 4 γq n +8 log(2) (cid:0) q n − (cid:1) + 2 (cid:0) q n − (cid:1) ψ (0) (cid:18) − n (cid:19) (2 n log( q ) + 1) For the s = 1 case, which is very similar to the original matrix Z ( q )((3.26a), (3.27a)), these functions reduce tothe Lerch transcendent representations and the appropriate analytic continuations—see (3.28). (cid:0) q n − (cid:1) ψ (0) (cid:18) n + 12 (cid:19) (2 n log( q ) − − nψ (1) (cid:18) − n (cid:19) (cid:0) − q n − q n + q n + 4 q n (4 n log( q ) + 1) + 2 (cid:1) + nψ (1) (cid:18) n + 12 (cid:19) (cid:0) q n − q n − q n + q n + 8 n (cid:0) q n + 1 (cid:1) log( q ) + 2 (cid:1) +16 q n tanh − (cid:0) q (cid:1) − q n tanh − (cid:0) q (cid:1) + 2 π n − − (cid:0) q (cid:1) + 4 γ (cid:27) . (4.15) The above expression would simplify for particular integer values n . A very useful series rep-resentation for understanding these special functions is given by Erd´elyi [34], which is valid for | log( z ) | < π , s = 2 , , . . . and a (cid:54) = 0 , − , − , . . . Φ[ z, s, a ] = z − a ∞ (cid:88) k =0 k (cid:54) = s − ζ ( s − k, a ) log k ( z ) k ! + [ ψ ( s ) − ψ ( a ) − log( − log( z ))] log s − ( z )( s − (4.16)where ζ ( s, a ) = Φ(1 , s, a ) is the Hurwitz zeta function. Substituting this into the Mathematica output would give us the log( t ) dependence we wanted (as z = e − t in our case). The resultingexpression can be truncated at a finite k to obtain an expansion in t and log t as for instance: R (1)22 = log t (cid:20) t π − t π + O (cid:0) t (cid:1)(cid:21) + (cid:34) t + 323 (cid:18) π − (cid:19) t − t (cid:0) − π + 4 log(2) (cid:1) π + 4 t (cid:0) − π + 240 log(2) (cid:1) π + O (cid:0) t (cid:1)(cid:35) . (4.17)For (cid:60) ( a ) > 0, the two functions defined by Lerch Transcendents and Hurwitz Lerch transcendentcoincide and one can simply replace the former with the latter. This is useful since Mathematica isable to expand Hurwitz Lerch functions with arguments e t arguments near t = 0. This is anotherway to obtain the series expansions, although it is sightly less computationally efficient.Similarly, the non-diagonal elements ( n (cid:54) = m ) are expressed as: R (1)2 n, m = ( − ) n + m π m ( n − m ) log ( q ) (cid:40) (cid:0) q m − (cid:1) (cid:0) q n − (cid:1) nm (cid:20) − m q Φ (cid:18) q , , − n (cid:19) + n q Φ (cid:18) q , , − m (cid:19) − m q Φ (cid:18) q , , n + 12 (cid:19) + n q Φ (cid:18) q , , m + 12 (cid:19) − m nψ (0) (cid:18) − n (cid:19) log( q ) + 4 m nψ (0) (cid:18) n + 12 (cid:19) log( q ) − m ψ (0) (cid:18) − n (cid:19) − m ψ (0) (cid:18) n + 12 (cid:19) + 4 m tanh − (cid:0) q (cid:1) + 2 m ψ (0) (cid:18) (cid:19) + 4 mn ψ (0) (cid:18) − m (cid:19) log( q ) − mn ψ (0) (cid:18) m + 12 (cid:19) log( q ) + n ψ (0) (cid:18) − m (cid:19) + n ψ (0) (cid:18) m + 12 (cid:19) − n tanh − (cid:0) q (cid:1) − n ψ (0) (cid:18) (cid:19)(cid:21) + (cid:0) q m − (cid:1) (cid:0) q n − (cid:1) nm (cid:0) n (cid:0) q m + 1 (cid:1) (cid:0) q n − (cid:1) + m (cid:0) q m − (cid:1) (cid:0) q n + 1 (cid:1)(cid:1) × (cid:20) nq Φ (cid:18) q , , − m (cid:19) − mq Φ (cid:18) q , , − n (cid:19) − nq Φ (cid:18) q , , − m (cid:19) + mq Φ (cid:18) q , , − n (cid:19) − nq Φ (cid:18) q , , m + 12 (cid:19) + 2 mq Φ (cid:18) q , , n + 12 (cid:19) + nq Φ (cid:18) q , , m + 12 (cid:19) − mq Φ (cid:18) q , , n + 12 (cid:19) + nψ (0) (cid:18) − m (cid:19) − nψ (0) (cid:18) m + 12 (cid:19) mψ (0) (cid:18) − n (cid:19) + mψ (0) (cid:18) n + 12 (cid:19)(cid:21) + (cid:0) q m − (cid:1) (cid:0) q n − (cid:1) (cid:20) − q Φ (cid:18) q , , − m (cid:19) + q Φ (cid:18) q , , − m (cid:19) + 4 q Φ (cid:18) q , , − n (cid:19) − q Φ (cid:18) q , , − n (cid:19) − q Φ (cid:18) q , , m + 12 (cid:19) + q Φ (cid:18) q , , m + 12 (cid:19) + 4 q Φ (cid:18) q , , n + 12 (cid:19) − q Φ (cid:18) q , , n + 12 (cid:19) − mψ (0) (cid:18) − m (cid:19) log( q ) + 4 mψ (0) (cid:18) m + 12 (cid:19) log( q ) − ψ (0) (cid:18) − m (cid:19) − ψ (0) (cid:18) m + 12 (cid:19) +4 nψ (0) (cid:18) − n (cid:19) log( q ) − nψ (0) (cid:18) n + 12 (cid:19) log( q ) + 3 ψ (0) (cid:18) − n (cid:19) + 3 ψ (0) (cid:18) n + 12 (cid:19)(cid:21)(cid:27) (4.18) By construction, these non-diagonal elements all satisfy the SU (1 , 1) condition [19] R nm m = R mn n (4.19) order by order in s . For specific values of n, m , the above expressions do simplify, for instance: R (1)24 = R (1)42 π q log ( q ) (cid:110) ( q − ( q + 1) (cid:0) q + 1 (cid:1) (cid:16) q + 1) (cid:0) q + 4 q + 1 (cid:1) (cid:0) q + 1 (cid:1) tanh − q +( q − q (cid:0) q + 9 q + 34 q + 65 q + 78 q + 65 q + 34 q + 9 q + 3 (cid:1)(cid:1) − (cid:0) q − (cid:1) (cid:0) q + 4 q + 1 (cid:1) tanh − q (cid:111) . (4.20) However, we find that after Φ( z, , a ) simplifies, the log t terms from those terms are absent in aseries expansion for the non-diagonal elements. For example, we find interestingly enough that: R (1)24 = − t log(2) π + 48 t log(2) π + t (31 − 512 log(2))3 π + O (cid:0) t (cid:1) R (1)46 = − t log(2) π + 160 t log(2) π − t (1408 log(2) − π + O (cid:0) t (cid:1) R (1)26 = 8 t log(2) π − t log(2) π + t (928 log(2) − π + O (cid:0) t (cid:1) R (1)28 = − t log(2) π + 80 t log(2) π + t (103 − π + O (cid:0) t (cid:1) , etc. (4.21)The log branch cuts from arctanh terms have cancelled after the sum over the block matrix in-dices ( x, p ). The individual infinite sums from M all diverge badly for t < log-free expansion at this order in s . s = 2 term We are able to construct the matrix elements R (2)2 n, m for a general n, m although they are a longercombination of special functions, namely products of Hurwitz Lerch transcendents and Lerch tran-scendents which are not particularly illuminating. Hence, we only provide the series expansions forcertain matrix elements to show the general numerical structure: R (2)22 = 2 t (44 log(2) − 27 log(3))3 π + t (cid:18) − 16 log (2) π − 304 log(2)3 π + 72 log(3) π (cid:19) + t (cid:18) − 323 + 7115 π + 64 log (2) π + 9416 log(2)45 π − 894 log(3)5 π (cid:19) + O (cid:0) t (cid:1) , (4.22a) R (2)24 = 2 t (27 log(3) − 44 log(2))3 π + t (cid:18) 16 log (2) π − 108 log(3) π + 76 log(4) π (cid:19) + t (cid:18) − π − 96 log (2) π + 411 log(3) π − π (cid:19) + O (cid:0) t (cid:1) . (4.22b)37 num44 ℛ ( ) ℛ ( ) ℛ ( ) ℛ ( ⊕ ⊕ ) (a) ℛ ( ) ℛ ( ) ℛ ( ⊕ ) - - - - (b) Figure 2: The individual contributions from the various matrix powers s = 0 , , R ( t ) and (b) R ( t ). The numerical estimate for N = 64 isalso plotted for the R case and is seen to closely follow the analytic sum. For R , the fit is notquite good since it starts only at the quadratic order and more terms would be required to accountfor the small but comparable contributions.Next, we can combine the contributions from these three terms and analyse how well they approxi-mate the behaviour by comparing to a numerical evaluation of the same as we do in Fig. 2. However,from the open-closed correspondence we expect the above expansion in terms of the t r and log( t )basis to be incomplete. The crucial point is that one cannot dictate that the summation over s andthe Taylor series expansions over r above must commute. Hence, the summation over s can lead tothe subleading terms from closed string states of the form: R nm ( t ) = ∞ (cid:88) k =0 ∞ (cid:88) j =0 ∞ (cid:88) s ,s =0 c k | j | s s (log s ( t − )) s t j e − π kt (4.23)where c k | j | s s are some specific (real) coefficients and we have suppressed the mode labels n, m forsimplicity.By adding the contributions from the higher matrix powers in s , one may obtain a subset of theabove coefficients to higher accuracy, the ones corresponding to the k = 0 level in this expansion.The exponentially small parts from k ≥ on-shell closed string states and it would be interesting to recover someinformation about those states.In summary, we have provided a formal procedure for successively approximating the coefficientsin an expansion near t = 0. We do not claim to the efficiency or numerical control resulting fromthis method. We must also acknowledge that this procedure does not extend in practice beyondthe lowest orders due to some of the double sums (involving generalized hypergeometric functions)that arise from the block matrix multiplications. Because of the non-analytic behaviour—which hasits physical origins in the worldsheet picture—it is intrinsically difficult to identify the divergencesor the subleading terms in the algebraic method we have used. Nonetheless, we hope that it hasaugmented the knowledge levels on the algebraic structure of OSFT at the quantum level.In the next subsection, we shall study the behaviour of these matrix elements near the otherlimit of the modular parameter, that is, t → + ∞ by using an expansion in the variable q = e − t near q = 0 + . Since the oscillator based expressions are much more suited for this kind of an expansion,we only check till the linear order term for consistency. We will find that our expressions correctlyreproduce the numbers that can be generated from the oscillator based expansion. This physical input from the CFT picture can be taken into account explicitly by the formalism of Hardy fields employed in real asymptotics, which allows to amalgamate many “exponential scales”. See chapter V, App. 1 of [36]and chapters 3 , .3 Expansions near t = ∞ in the continuous κ -basis In order to obtain the expansion for R nm in the large t (or small q ) limit, as a power series in q = e − t , one goes to the continuous κ -basis [13, 16, 17] that we discuss later in § j = 0 representation of SL (2 , R ) associated with the discrete basisdefined in terms of mode number labels, that we have been using. We choose it to convert some ofthe infinite sums to integrals for the purpose of numerical evaluation of the series coefficients. Incertain cases, one can correctly guess the exact algebraic numbers by using the RootApproximant command in Mathematica if they stabilize as the “WorkingPrecision” is increased. q To commence the evaluation, let us define a matrix S in terms of purely the even frequencies κ e as S := diag { κ / e , κ − / e } . Then we can express the matrices Q η and L η that contribute to R ( q ) as: Q η = 14 S − (cid:20) f + tf t (cid:62) − i ( f − tf r ) − i ( f − r (cid:62) f t (cid:62) ) ( f + r (cid:62) f r ) (cid:21) S − =: 14 S − Q κ S − , L η = 12 S − (cid:20) tf t (cid:62) i f − i r (cid:62) f t (cid:62) f (cid:21) S − =: 12 S − L κ S − , (4.24) where we have introduced the block matrices Q κ and L κ after absorbing the numerical factors.Here, we have employed the matrix t := √ κ e T √ κ o which is the operator tanh π ˆ Q and r is the formal inverse of the matrix t , i.e. r := √ κ o R √ κ e .Then we can rewrite the matrix which appears in the matrix F ( q ) (3.32) as: L (cid:62) η Q − η L η = S − L κ (cid:62) ( Q κ ) − L κ S − = S − (cid:20) tf t (cid:62) − i tf r i f f (cid:21) (cid:20) f + tf t (cid:62) − i ( f − tf r ) − i ( f − r (cid:62) f t (cid:62) ) ( f + r (cid:62) f r ) (cid:21) − (cid:20) tf t (cid:62) i f − i r (cid:62) f t (cid:62) f (cid:21) S − (4.25) in block matrix form.To obtain the inverse, once again we perform a geometric series expansion by separating thedegree zero term through Q κ ( q ) =: Q κ + δ Q κ ( q ) . Since as q → + , we have f ( q ) → − , f ( q ) → − and f ( q ) → − , ( Q κ ) − = (cid:20) Λ 00 4Ω (cid:21) , (4.26)where we define the infinite matrices:Λ := 11 + 3 tt (cid:62) , Ω := 13 + r (cid:62) r = 13 (1 − Λ) . (4.27)Next, we insert the following binomial inverse series in order to obtain F ( q ) which in terms of thesenew matrices become: F ( q ) = 12 M gh − f ( q ) + ∞ (cid:88) s =0 ( − s S − L κ (cid:62) (cid:104) ( Q κ ) − δ Q κ (cid:105) s · ( Q κ ) − L κ S − . (4.28)We shall denote the expansion as: F ( q ) = ∞ (cid:88) s =0 ( − s F κ | s ( q ) , where we set (4.29) F κ | ( q ) = 12 M gh − f ( q ) + S − L κ (cid:62) ( q )( Q κ ) − L κ ( q ) S − . (4.30) As we do not use the parameter t = − log q in this section, we hope the repeated use of the symbol wouldn’t giverise to any ambiguities. F pp n, m , which contains theexplicit term:( L κ (cid:62) ( Q κ ) − L κ ) pp = L κ (cid:62) px ( Q κ ) − xx L κxp + L κ (cid:62) px ( Q κ ) − xp L κpp + L κ (cid:62) pp ( Q κ ) − px L κxp + L κ (cid:62) pp ( Q κ ) − pp L κpp , (4.31)where we have used the doublet indices x, p as subscripts for typographical convenience.In the following, we simply restrict to the general structure of the lowest order s = 0 termsince we are primarily interested in certain consistency checks in the q → + limit. The oscillatorexpansion is much better suited for expansions near this limit and hence we return to that methodin § q and q and verify that they match withthe exact results from the oscillator expressions.Upon inserting the constituent block matrices, the momentum block F κpp | in (4.30) has the struc-ture: F κpp | n, m ( q ) = − · nf (2 n ) δ nm + 2 √ nm · 14 ( f (2 n ) f (2 m )Ω n, m − f (2 n ) f (2 m )Λ n, m )= − n δ nm + √ nm − Λ) n, m + √ nm (cid:20) − n, m q n − n, m q m + (cid:18) Λ n, m + Ω n, m − (cid:114) nm (cid:19) q n +(Ω + Λ) n, m q m + 4Ω n, m q n +2 m − n, m q n +4 m − n, m q n +2 m + (Ω − Λ) n, m q n +4 m (cid:21) , (4.32)without any summations over repeated indices. The next term in the expansion becomes moretedious but starts contributing at q (due to the infinite summations over the odd index).To this end, it is worthwhile to note that the constant part of the matrix L κ is of the form: L κ (0) = (cid:20) tt (cid:62) i − i (cid:21) (4.33)This allows us to collect the coefficient of q from( L κ (cid:62) ( Q κ ) − L κ ) (0) pp = − κ e − √ κ e − tt (cid:62) tt (cid:62) √ κ e (4.34)which when substituted into the expression for R n, m in terms of F n, m (3.44): R (0)2 n, m = − (cid:104) − − √ κ e − tt (cid:62) tt (cid:62) √ κ e (cid:105) n, m = (cid:114) nm (cid:20) − tt (cid:62) tt (cid:62) (cid:21) n, m , (4.35)which is precisely the even parity elements of the Neumann matrix X n, m in (2.41) as was derivedin the oscillator formalism.Similarly, the coefficient of q is given by the matrix: R (1)2 n, m = − √ nm (cid:20) (Λ t ) n, (Λ t ) m, + 12 (cid:16) (Λ t ) n, (Ω r (cid:62) ) m, + ( n ↔ m ) (cid:17)(cid:21) × − m (4.36)by considering the s = 1 power and noticing that the q n , q n terms do not contribute at this orderfor any n . Now, one can show that Ω r (cid:62) = Λ t ; hence the above reduces to: R (1)2 n, m = 8 (cid:114) nm (Λ t ) n, (Λ t ) m, . (4.37)40 .3.2 Numerical evaluation in the continuous κ -basis The matrix elements of the rational functions involving the t matrix such asΛ t = 11 + 3 tt (cid:62) t = t 11 + 3 t (cid:62) t can be obtained by numerical integration by going to the continuous Moyal basis , known as the κ -basis. The κ basis diagonalizes the operator K = ( L + L − ) of SL (2 , R ) [13, 16, 17]: K | κ (cid:105) = κ | κ (cid:105) (4.38)which commutes with the vertex, and is useful for performing analytic and numerical calculations.The t matrix is diagonalized in the infinite N limit to give the eigenvalues: t κ = tanh( πκ/ t n, m − = √ n T n − , m − √ m − (cid:90) ∞−∞ dκ v n ( κ ) tanh( πκ/ v m − ( κ ) (4.39)where we start with defining the overlap functions v n ( κ ) = (cid:104) κ | n (cid:105) = y n ( κ ) √ n (cid:113) κ sinh πκ (4.40)which are a class of polynomials that arise naturally in the continuous basis and are analogous tothe Hermite polynomials for the number operator. These are orthogonal with respect to the weightfunction w ( κ ) = (cid:18) κ sinh πκ (cid:19) − (4.41)A generating functional for these polynomials is given by: (cid:88) n ∈ Z + z n n y n ( κ ) = 1 κ (1 − e − κ arctan z ) = f κ ( z ) (4.42)and they satisfy the recurrence relation: y n +1 ( κ ) + y n − ( κ ) = − κn y n ( κ ) , (4.43)among many other relations listed in [16]. Setting y ( κ ) = 0 , y ( κ ) = 1, leads to the polynomials: y ( κ ) = 1 , y ( κ ) = − κ,y ( κ ) = 12 κ − , y ( κ ) = − κ + 43 κ,y ( κ ) = 124 κ − κ + 1 , y ( κ ) = − κ + 13 κ − κ, (4.44)and so on and so forth. The notational conflict in using κ for the continuous basis and for the spectral matrix would be restricted to thissubsubsection. The following properties are taken from App A of [16]. t matrices such as (cid:16) F ( tt (cid:62) ) (cid:17) n, m = (cid:90) ∞−∞ dκ v n ( κ ) F (cid:16) tanh (cid:16) πκ (cid:17)(cid:17) v m ( κ ) , (4.45a) (cid:16) tF ( t (cid:62) t ) (cid:17) n, m − = (cid:90) ∞−∞ dκ v n ( κ ) tanh (cid:16) πκ (cid:17) F (cid:16) tanh (cid:16) πκ (cid:17)(cid:17) v m − ( κ ) , etc. (4.45b)using which we have evaluated (4.37) upto a WorkingPrecision of 16 in Mathematica . The resultingnumbers for some matrix elements are listed in Table 2. Now, by using the oscillator based expansion R Moy | nm . − . . − . − . . − . . . − . . − . − . . − . . R nm ( q )using the continuous κ basis for the Moyal ∗ .in (2.41) and (4.57), we obtain the linear coefficient to be in terms of the ghost Neumann matrices: R (1) | osc n, m = − (cid:0) X n, X , m + X n, X , m (cid:1) . (4.46)These rational numbers are tabulated in Table 3 and found to be the stabilizing value as theWorkingPrecision for the numerical integrations above is increased. Indeed, we may also express R osc | nm − − − − − − − − Table 3: Exact linear coefficients in a few even parity matrix elements R nm ( q ) obtained using theoscillator method in terms of Neumann coefficients.the Neumann matrices X ( ± ) in terms of the matrixˆ m ∗ := (cid:18) − S − T (cid:62) (cid:19) (4.47)defined in [14] to analytically prove that both expressions for the linear term coincide. This expansioncan thus result in interesting relations between the Neumann matrices and matrices arising from theMoyal structure which may be established by using the canonical way of expressing all the Neumannmatrices in terms of the matrix t and the frequency matrices κ e and κ o [14].Regarding studying the determinant factor in the integrand using a q expansion(see also 3.1.3),which is common for all matrix elements, we find that the lowest power of each matrix element do ot decrease along a row or a column which is required for a systematic expansion. Essentially, onecannot separate the degree zero piece as there is no nice way to express det(1 + A − B ) in terms ofdet A − B .Although one can include the higher powers (( Q κ ) − δ Q κ ) s to obtain the exact coefficients fora q series, this would necessitate many more numerical integrations arising from collecting powerstogether and results in numerical uncertainties. The oscillator basis on the other hand furnishes theexact coefficients since the Neumann matrices are known exactly from CFT. We therefore simplycontend ourselves with the zeroth order and the linear coefficient using the κ basis and comparewith the oscillator based expansion. This serves as a consistency check on the correctness of ourexpressions in the t → + ∞ limit. Hence for the purpose of constructing a q -series, we employthe oscillator based expressions in (2.41) expressed in terms of Neumann matrices in the followingsubsection. This can then be used to search for some hints of the non-analyticities expected fromthe underlying geometrical picture. Let us pause for a moment and do a quick check on the overall determinant factors near the t → ∞ or q → p = 1 = − m for the“lightest” tachyon state and one off-shell tachyon state with p = 0, and a tachyon propagator with t → ∞ . The off-shell 3-tachyon amplitude has been known[6, 26, 32] to be of the form: g ( k i ) = g T K × K − ( k + k + k ) , (4.48)where we recall that g T is the on-shell 3-tachyon coupling (by definition) and K = 3 √ 34 . Hence, weexpect the leading asymptotics to be: g w (cid:62) w ) d +22 (2 π ) d ( N +1 / q − det(2 Q ghη ) | det(2 Q Xη ) | d/ (2 Q p ) − d/ → g T K × K − q − ( − q ) − d/ (2 π ) d ( N +1 / (4.49)as q → 0, when d = 26. The 2 π factors arise from the η X and p integrations and the manner inwhich the basis states e iξ (cid:62) η are normalized. We have also set l s = √ Q p arising from the momentum integration is dominated by − log q and hence werequire: lim t →∞ g w (cid:62) w ) d +22 det(2 Q ghη ) | det(2 Q Xη ) | d/ = g T K (4.50)The determinant factors involve the block matrices (3.18), (A.11): Q ghη = + 14 (cid:20) κ − e f ( κ e ) + T κ − o f ( κ o ) T (cid:62) − i [ f ( κ e ) − T f ( κ o ) R ] − i (cid:2) f ( κ e ) − R (cid:62) f ( κ o ) T (cid:62) (cid:3) (cid:2) κ e f ( κ e ) + R (cid:62) κ o f ( κ o ) R (cid:3) (cid:21) , Q Xη = + 12 (cid:20) κ − e f + T κ − o f ( κ o ) T (cid:62) − i ( f ( κ e ) − T f ( κ o ) R ) − i ( f ( κ e ) − T f ( κ o ) R ) (cid:62) (cid:0) κ e f + R (cid:62) κ o f R (cid:1) (cid:21) (4.51) In the q → f → , f → f → det(2) − N det(1 + 3 tt (cid:62) ) det (cid:16) r (cid:62) r (cid:17) det(3 + tt (cid:62) ) d/ det (cid:16) r (cid:62) r (cid:17) d/ = 2 N ( d − det(1 + 3 tt (cid:62) ) det(3 + r (cid:62) r )det(3 + tt (cid:62) ) d/ det (1 + 3 r (cid:62) r ) d/ = 2 N ( d − det(1 + 3 tt (cid:62) ) det(3 + tt (cid:62) ) − d det( tt (cid:62) ) d − , (4.52) This may be read off from open string partition function. r := t − = κ − o t (cid:62) κ e and we have substituted r (cid:62) r = ( tt (cid:62) ) − . Multiplying with the remainingfactors and using det( tt (cid:62) ) = (1 + w (cid:62) w ) − / , we have: g w (cid:62) w ) d +22 × N ( d − det(1 + 3 tt (cid:62) ) det(3 + tt (cid:62) ) d det( tt (cid:62) ) d − = 2 N ( d − g w (cid:62) w ) − d + det(1 + 3 tt (cid:62) ) det(3 + tt (cid:62) ) d , = − µ g , (4.53)where µ is the normalization factor that relates the interaction term in the Moyal and the oscillatorformalisms( § N → ∞ . In terms of µ , the couplings are related as g T = − µ × g K − and hence the N dependence is removed. The LHS now becomes: g T / K which is off from the expected result of g T K by a factor of 16 K = 9 / 32 = 0 . q using the oscillator expression In this work, we have been mainly interested in the behaviour of the finite matrix elements as t → + . This corresponds to looking at the q → − limit, and hence may also be indirectly inferredfrom a series expansion near q = 0 (the t = + ∞ limit) due to the expected non-analyticities.Physically, one would expect that the t evolved string field becomes ill-defined when (cid:60) ( t ) < 0; thepropagator would result in divergent sums while acting on a string field for t < 0. Thus, intuitivelywe would expect the matrix elements to be uniformly convergent for | q | < | q | = 1 which obstructs an analytic continuation beyondthe unit disc in the q plane.We therefore proceed to directly use the oscillator expression given in [7] to probe the q → − limit. The matrix elements R nm ( q ), can be given a systematic expansion in powers of q as follows.The matrix whose powers are taken in the geometric series expansion has a minimal degree 1.Therefore, the matrix powers start contributing only from higher and higher powers onwards as theinfinite sums in the matrix products would not alter the order of the zeroes. This allows us to obtainthe exact coefficients by adding up the contribution from a finite number of matrix powers. A q-series expansion By a theorem of Sierpi´nski (see [38, § z = 1) but diverges at every other point. In our particular case, wewould have a series with radius of convergence 1, that converges at q = 1 to either +1 or − § 4] is naturally suited for systematically finding a q series expansion for R nm ( q ) since the propagator is simple in this basis. Again, the ghost sectoris relatively simpler as compared to the matter sector due to the absence of the momentum zeromode.As the hatted matrices (2.42) appearing in (2.41) for R ( q ) do not seem to satisfy any niceidentities unlike the M , ± matrices, we resort to a geometric series for studying the matrix inverse( − S (cid:101) X ) − . Inserting this formal expansion into (2.41), we have: R ( t ) = X + (cid:2) ˆ X (0 , t ) ˆ X (0 , t ) (cid:3) ∞ (cid:88) s =0 ( S (cid:101) X ) s S (cid:20) ˆ X ( t, X ( t, (cid:21) (4.54)and let us introduce the infinite matrices R ( s ) nm by rewriting: R nm ( q ) = ∞ (cid:88) s =0 R ( s ) nm ( q ) (4.55)44n terms of the variable q .At the risk of further over-complicating the notation, let us also introduce a constant matrix X as follows: X := (cid:20) X X X X (cid:21) , (4.56)which is essentially the S (cid:101) X ( t ) matrix stripped off the t dependent propagator pieces and the C matrices. The C matrices and the q n/ factors from the propagator effectively make the contributionfrom the s th power term into: R ( s ) nm ( q ) = δ s, X nm + ∞ (cid:88) p = s +1 ( − p q p (cid:88) | (cid:126)µ | = p (cid:104) X n,µ (cid:0) X . . . X (cid:124) (cid:123)(cid:122) (cid:125) s terms (cid:1) | µ µ s +1 X µ s +1 m + X n,µ (cid:0) X . . . X (cid:124) (cid:123)(cid:122) (cid:125) s terms (cid:1) | µ µ s +1 X µ s +1 m + X n,µ (cid:0) X . . . X (cid:124) (cid:123)(cid:122) (cid:125) s terms (cid:1) | µ µ s +1 X µ s +1 m + X n,µ (cid:0) X . . . X (cid:124) (cid:123)(cid:122) (cid:125) s terms (cid:1) | µ µ s +1 X µ s +1 m (cid:105) , (4.57)where we are only summing over the set of integer partitions of the power p into s + 1 terms: | (cid:126)µ | = µ + . . . + µ s +1 = p, and its permutations. For performing these block matrix computations we have again used the NCAlgebra package [20] which among its many powerful features handles block matrices in asomewhat more reliable and easier manner as compared to Mathematica ’s built-in functions. Forinstance, the block matrix powers which grow exponentially with the degree can be quickly evaluatedas formal expressions using the “NCDot”/“MM” (MatrixMultiply) command. These can then befed into a “module” for inserting the X , ± exact values. Essentially, the output of the NCAlgebra commands are used to construct lists and we apply the transformation rules on them to convertthem to the coefficients.For low values of s , one can use the “Permutations” and “IntegerPartitions” commands in Math-ematica to insert the appropriate indices and perform the (constrained) summations . Again, thisbecomes computationally challenging since the number of terms in each block grows exponentiallywith s as 2 s − and we had to contend ourselves with s ≤ 17 truncation due to time and energyconstraints.To obtain till the q coefficient exactly, one needs to include the s = 0 , . . . , 17 contributions(the s = 18 terms start only at q ). Once we have an expansion in terms of exact coefficients, wecan find the corresponding diagonal or near diagonal Pad´e approximant ( n ≈ m ) and look at itspole-zero structure in the complex q plane as we do in App B. This is a useful exercice in general,when the available data is limited due to a multitude of reasons.We have obtained the coefficients till the q term for a general matrix element R nm symbolically.For particular values of n, m , the expansions can then be readily obtained. We provide a few elements I would like to thank the UC San Diego Mathematics department for making available this package using whichparts of the computations in this work were performed. One can also employ “If” conditionals to do these summations by brute-force for low enough s . The routineneeds to check r (cid:88) p =1 p (cid:88) s =1 ( p − s + 1) s If conditionals and also perform multiplication and addition for the size of thePermutations of Integer Partitions to obtain the first r + 1 coefficients exactly. This number grows very quickly. R ( q ) = − − q − · q + 2 · · q − · q + 2 · q + · · · + 2 · · · · q + 2 · · q + O ( q ) ≈ − . − . q − . q + 0 . q − . q + 0 . q − . q − . q + 0 . q + 0 . q − . q − . q + 0 . q + 0 . q − . q − . q − . q + 0 . q + 0 . q + O ( q ) , (4.58) R ( q ) = 193 + 2 · q + 2 · q − · q + 2 · · q − · · q + · · · + 2 · · q + O ( q ) ≈ . . q + 0 . q − . q + 0 . q − . q + 0 . q + 0 . q − . q + 0 . q + 0 . q − . q − . q − . q + 0 . q + 0 . q − . q − . q + 0 . q + O ( q ) , (4.59)and for two non-diagonal elements, we have: R ( q ) = − · − · · q + 2 · · q + 2 · · · q + 2 · · · q − · · · + 2 · · · · · q + O ( q ) ≈ − . − . q + 0 . q + 0 . q + 0 . q − . q − . q − . q − . q + 0 . q − . q − . q + 0 . q + 0 . q − . q − . q − . q + 0 . q + 0 . q + O ( q ) , and for (4.60) R ( q ) = 2 · · + 2 · · q − · · q − · · · · q − · · · q − · · · − · · · · · q + O ( q ) ≈ . . q − . q − . q − . q − . q − . q + 0 . q + 0 . q + 0 . q − . q + 0 . q + 0 . q − . q − . q + 0 . q + 0 . q + 0 . q − . q + O ( q ) . (4.61)It is interesting to note that the coefficients are all nice rational numbers given that the Neumannmatrices are only algebraic valued. We observe that there is a (rather slow) non-monotonic fall-offof the coefficients. However, we can see from Table 4 below that for low n, m , they still approximatethe function near q = 1. We expect these Taylor series expansions to correspond to certain specialcombinations of elliptic functions. As it is difficult to identify the form of the function from theseries—and it varies for each matrix element—we tried to look up the numbers in the OEIS .Although we haven’t found any match so far, it may be possible that one can express these in terms It offers a feature to check rational sequences by searching for the numerator sequence and denominator sequenceseparately. series n, m (1) 1 2 3 41 0 . . . − . . . − . . . − . . . − . . . . R n, m evaluated at q = 1 or t = 0 usingthe oscillator based expansion till q . The diagonal elements are all consistent with being +1 withthe off-diagonal ones vanishing, since the twist matrix C nm = ( − ) n δ nm reduces to + δ nm in the evensector. In the odd sector, we have checked that there is consistency with − δ nm as well.of rational expressions of elliptic functions and their derivatives, line integrals, etc. Once oneobtains the expression in terms of elliptic functions, one can convert them to Jacobi Θ functionsand then apply the Jacobi imaginary transform to obtain the closed string contributions explicitly,similar to [6].Furthermore, it is interesting to compare this expansion to the one we obtained in § t variable. We find that they do all follow eachother sufficiently closely near the t → + region (which maps to the q → − region) as can be seenfrom some sample matrix elements plotted in Fig.3a; in the non-diagonal case there is a numericaldifference since the true function is expected to vanish as q → − , but notice that the scales differ. ℛ ( ⊕ ⊕ ) R (a) ℛ ( ⊕ ⊕ ) R - - - - - (b) Figure 3: A comparison of the behaviour of the matrix element R n, m near t = 0 obtained using thefirst three terms ( s = 0 , , 2) in the Moyal basis (green) and using the first 19 terms (till q = e − t )in the oscillator basis (orange, dashed) plotted for (a) R ( t ) and (b) R ( t ). The two furnish verysimilar values for the diagonal case but differ for the non-diagonal case, which was expected giventhe vanishing behaviour near t = 0. In this section, we write down the ghost sector expressions for the corrections to open string prop-agator at the one-loop level ( N = 2 , g = 1). This corresponds to the self-energy diagram in QFTsand in case of the bosonic theory, the diagrams are similar to the φ theory. We will begin by This is expected the case as the Schottky double is a torus and elliptic functions are the natural doubly periodicfunctions should appear in any physical quantity[6, 7]. The extended nature of the world-sheet however also allows for “twisting” the internal propagators [39]. t and t each ranging from 0 to ∞ and the analytic structure becomes much more intricate (andinteresting) consequently. On ghost number assignments Recall that for one-loop diagrams, the perturbative quantization procedure dictates that statesof all ghost number, G i ∈ Z , must propagate in the loop subject to the ghost number saturationcondition for the corresponding genus by the Riemann-Roch theorem. These are the so calledspacetime “ghost strings” which are different from the ordinary reparametrization bc ghosts [25] onthe worldsheet.Applying this rule for the one-loop 2-point function, we require that the vertex operators forthe two states | Φ (cid:105) and | Φ (cid:105) corresponding to the two propagators of “length” t = − ln q and t = − ln q carry the ghost charges: G = 3 − − G = 2 − G , (5.1)when both external lines are connected to the loop but only G = 1 = G when only a single lineis connected to the loop as in . This results from the requirement of total ghost number +3 for theWitten type vertex. For the first case, the condition requires that both states be of either even orodd ghost number which is true also for the Schnabl gauge analysis[23].While considering the two diagrams of the first type, we will account for only the contributionfrom the ghost number +1 quantum states in this work. However, while constructing the quantumeffective action it is essential that we remove this restriction. Hence, our analysis would necessarilybe limited in its physical validity. The remaining two are not one particle irreducible and have thetadpole as a subgraph. Hence they share some of the structures. At the end, all the four diagramsshould be added with equal weight (= +1) in order to match with the first quantized results on-shell[39]. Covering of moduli space As expounded in [6, § 5] by Samuel et al., the moduli space is covered by four string diagrams asdepicted in Fig.4, of which one is non-planar and the rest three are planar. Of these three planarcases, two have the one loop tadpole as a subdiagram and hence has zero momentum transfer. Withthe appropriate change of variables, these are guaranteed to have the same form of the integrand.These diagrams smoothly cross-over as the modular parameters are varied in order to provide a singlecovering of the moduli space (Ref.[6] clearly demonstrates this). Additionally, the ghost factors areno longer trivial as in the tree level cases.We see that the last two diagrams differ by the way two legs of the off-shell four-point functionare glued together to form a loop. The Witten vertex is cyclic but not permutation symmetric andhence we find these inequivalent diagrams for obtaining a single covering of moduli space as requiredby consistency with the Polyakov amplitudes on-shell. In the following, we consider only pure ghostexternal states for convenience. It was shown that in addition to the physical poles corresponding toan intermediate particle going on-shell, there are also unphysical poles in the off-shell amplitude[6].Hence these diagrams contribute to very interesting off shell structure. To be precise, what we call “one-loop” here would correspond to the lowest order O ( (cid:126) / ) correction if the relationbetween the open string and the closed string coupling are taken into account and hence would actually be “half-loop”( √ (cid:126) ) level, as per standard Polchinski conventions. Planar is used in the sense of Feynman graphs; the string diagrams are still non-planar due to the unique structureof the Witten type vertex. A t t (a) A A t t (b) t t A A (c) t t A A (d) Figure 4: The four diagrams contributing to one-loop 2-point function. Diagrams (a) and (b) maybe considered to arise from the s channel and the last two: (c) and (d), from the t channel. I ( s )12 | The non-planar contribution to the open string propagator is given by: A ( s )12 | = (cid:90) ∞ dt (cid:90) ∞ dt I ( s )12 | ( t , t ) = (cid:90) dq q (cid:90) dq q I ( s )12 | ( q , q ) , (5.2)where the legs 2 and 3 are identified, and the labels 2 and 3 are therefore redundant. The integrand(in the ghost sector) can be expressed as below: I ( s )12 | ( q , q ) = (cid:90) dη gh Tr (cid:104) A ( q , η gh , ξ gh , λ gh ) ∗ A ( ξ gh , λ gh ) ∗ A ( q , η gh , ξ gh ) (cid:105) , (5.3)and we have explicitly indicated the arguments for clarity. Here we have defined the monoids: A ( q , η gh , ξ gh , λ gh ) = q L gh (cid:104) A ( ξ, λ gh ) ∗ e + ξ gh (cid:62) η gh (cid:105) A ( q , η gh , ξ gh ) = q L gh (cid:104) e − ξ gh (cid:62) η gh (cid:105) , (5.4)in terms of the simpler elements: A ( ξ gh , λ gh ) = N e − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh ,A ( η gh , ξ gh ) = e + ξ gh (cid:62) η gh ,A ( η gh , ξ gh ) = e − ξ gh (cid:62) η gh ,A ( ξ gh , λ gh ) = N e − ξ gh (cid:62) M gh ξ gh − ξ gh (cid:62) λ gh . (5.5)49here as before, λ gh , λ gh are the sources which may be used to insert the specific asymptotic stringstates at the end of the calculations. Here again, we choose to remove the first ∗ product thatappears between A and A while evaluating the trace under the assumption of associativity. Weapply the sub-algebra rules for the monoid and the propagator rules to write down the parametersfor the resulting string fields below: A gh ( q ) : M ( q ) = M gh , λ gh ( t ) = q ˜ κ gh ( − (1 − m gh ) η gh + λ gh ) , N ( q ) = C (12) η exp (cid:104) − η gh (cid:62) Q (12) η η gh + λ gh (cid:62) L (12) (cid:62) η η gh (cid:105) , (5.6)where the coefficient matrices appearing in the normalization factor N c are given by: Q (12) η = − σm gh − 18 (1 − m gh (cid:62) ) M gh − f ( q )(1 − m gh ) (5.7a) L (12) (cid:62) η = − (cid:20) σ + 12 M gh − f ( q ) (cid:21) (5.7b) C (12) η = N exp (cid:20) λ gh (cid:62) M − f ( q ) λ gh (cid:21) , (5.7c)and for the monoid A gh ( q ): M ( q ) = f ( q ) f ( q ) M gh , λ gh ( q ) = 2 q ˜ κ gh f ( q ) η gh N ( q ) = det (cid:20) f ( q ) (cid:21) exp (cid:20) + 14 η gh (cid:62) M − f ( q ) f ( q ) η gh (cid:21) . (5.8)Now, let us proceed to evaluate the string field resulting from taking A ∗ A ( q ) =: A ( q ). Theparameters for the resulting expression are the following: m ( q ) = (cid:20) m gh + f ( q ) f ( q ) ( m gh ) (cid:21) (cid:20) f ( q ) f ( q ) ( m gh ) (cid:21) − + (cid:20) f ( q ) f ( q ) m gh − m gh f ( q ) f ( q ) m gh (cid:21) (cid:34) m gh f ( q ) f ( q ) m gh (cid:35) − , (5.9a) λ ( q ) = 2[1 − m gh ] (cid:20) f ( q ) f ( q ) ( m gh ) (cid:21) − q ˜ κ gh f ( q ) η gh + (cid:20) f ( q ) f ( q ) m gh (cid:21) (cid:20) m gh f ( q ) f ( q ) m gh (cid:21) − λ gh , (5.9b) N = N N ( q ) det (cid:20) f ( q ) f ( q ) ( m gh ) (cid:21) exp (cid:20) + 14 λ (cid:62) α σK αβ λ β (cid:21) , (5.9c)where in the last expression, we must substitute: K = (cid:18) m gh + ( m gh ) − f ( q ) f ( q ) (cid:19) − , K = (cid:18) f ( q ) f ( q ) ( m gh ) (cid:19) − ,K = − (cid:18) m gh f ( q ) f ( q ) m gh (cid:19) − , and K = (cid:18) f ( q ) f ( q ) m gh + ( m gh ) − (cid:19) − . (5.10)Combining the two string fields by taking the ordinary product of functions and taking the ξ gh trace,we are left with C (12 | η (cid:90) ( dη ) exp (cid:104) − η (cid:62) Q (12 | η η + L (12 | (cid:62) η η (cid:105) (5.11)50here the λ , dependences are implicit. Thus, the final contribution from the ghost sector becomesthe following: det (cid:0) Q (12 | η (cid:1) exp (cid:20) + 14 L (12 | (cid:62) η Q (12 | − η L (12 | η (cid:21) . (5.12)Here, the argument of the exponential mixes the components of the “vector” (cid:126)λ = (cid:32) λ gh λ gh (cid:33) and for a general one-loop n -point function, we obtain an n component vector. This is similar inspirit to working in Fourier space but here we only work with 2 N × N matrices. In certain cases,we can use these formal expressions for numerical calculations; the advantage of this representationis the straightforward application of the transformation rules, although they involve several inversesof infinite matrices and intermediate matrix multiplications. One may observe that the planar amplitude with both external states on the same boundary of theannulus, A ( s )12 | , comes with a relative positive sign with respect to the amplitude above. Hence, thecombined integrand can be written as I ( s )12 | + I ( s )12 | = (cid:90) dη gh Tr (cid:104) A ( q , η gh , λ gh ) (cid:110) A ( λ gh ) , A ( q , η gh ) (cid:111) ∗ (cid:105) . (5.13)This is special for the 2-point function since for general diagrams, the permutation non-invariance ofthe Witten vertex requires that we treat such diagrams, with lines on different boundary components,as contributing to separate amplitudes in general (colour ordering). The anti-commutator structurein the amplitude allows for taking advantage of the partial twist symmetry of these monoid elements.Here, we simply remark that we can write:ˆ A ∗ ˆ A = ( − ) Ω (cid:16) Ω( ˆ A ) ∗ Ω( ˆ A ) (cid:17) , (5.14)where we have included the ghost zero modes ξ i in the form of ˆ A i = − ξ ( i )0 A i for clarity. This leadsto some partial simplifications and we hope to report in this direction in the future.As mentioned earlier, the two remaining planar graphs have the one-loop tadpole as a subdiagramand are related by interchange of the external states labelled 1 and 4. We consider the integrand I ( t )41 | ( q , q ) = (cid:90) dη gh Tr (cid:104) A ( q , λ gh , λ gh ) ∗ A ( η gh ) ∗ A ( q , η gh ) (cid:105) . (5.15)which one can think of as being obtained by identifying the 2 and 3 legs of a t channel diagram (seeFig. 4). One can again write down formal expressions for the parameters in the integrand in termsof lightcone like variables, although we are unable to simplify them for further analysis at this point. In this work, we have primarily focussed on the finite contributions from the squeezed state matrixelements R nm ( t ) characterizing the tadpole state in the ghost sector of OSFT, and looked for hintsof non-analyticity as a function of the modular parameter t . Using the Moyal representation ofthe star product, we were able to write down formal expressions for the generating functionals forcorrelators. Since all integrals in this formalism are of the Gaussian kind, we obtained these in51erms of determinants and inverses of infinite matrices, which is one of the main difficulties withthese methods.Due to the partial analytic control we have over the infinite matrices, we were able to study thebehaviour of R nm ( t ) near the two boundaries of moduli space by employing expansions in t andin q = e − t for the matrix inverse—although conformal techniques become awkward in this basis.In particular, we were able to demonstrate the utility of the formalism by correctly capturing thelinear order behaviour ( § t = 0 which matches with BCFT prediction (3.61). However, weare now able to see this purely from the OSFT perspective. In the oscillator representation, thisexpansion becomes ill-defined and produces results that differed by a factor of 2. In the processof identifying the zeroth and linear order coefficients, we have thus uncovered a subtle differencebetween the Moyal and the oscillator methods, owing to the Fourier transform (2.16) and theresulting somewhat peculiar form of the propagator (3.9).Ideally, one would like to see the signatures of the closed string states by generating an expansioninvolving the closed string variable ˆ q := e π / ln q , starting from a closed form expression in the q variables and doing the Jacobi imaginary transform. This way one could recover the off-shell physicsassociated with the closed string spectra. Unfortunately, the algebraic approach we employ in thiswork is not tailored for this endeavour and hence we have studied the effects of closed string physicsonly indirectly.Nonetheless, we have performed consistency checks of our analytic expressions by examiningvarious limiting regimes of interest and found general agreement with the oscillator and BCFTresults. Beyond the linear order, we are able to successively approximate the matrix elements of R nm ( t ). However, the algebra becomes quite unwieldy as may be expected from the fact thatthe aforementioned infinite matrices are constructed out of non-commuting blocks. We have alsoemployed the oscillator expression (2.41) to generate a series in q till the 18 th degree (for general n, m ) and used it to analyse hints of non-analyticity. We however, refrain from making any claimspertaining to the margin of errors or the efficiency yet, since these are much less clear.To summarize, the present work makes a modest attempt at answering perturbative questions inOSFT using the Moyal formalism and complements the CFT and oscillator investigations. Due tothe strong divergences from the closed string tachyon, the full amplitude is unphysical but still servesas a useful probe of the structure of this very special string field theory. Recently, more physicalsuperstring field theories have been fully constructed which can describe the Ramond sector [1, 9, 40,41]. The work in [42] has correctly reproduced the 4-point amplitude involving spacetime fermions.It would be of utmost interest to study quantization of this theory from which the tachyon isprojected out.One promising avenue would be extending the recent progress made in the direction of partialgauge fixing [43]. This still remains somewhat mysterious and a better understanding of the gaugealgebra at the quantum level may also shed more light on how closed string degrees of freedom areencoded in open superstring field theories. Acknowledgements I wish to thank Itzhak Bars, Loriano Bonora, Ted Erler, Yuji Okawa, and Martin Schnabl for helpfulconversations and discussions. I am grateful to the participants and organizers of SFT 2018, HRI,Allahabad , for providing a kind and inspiring environment during which part of this work was done.I also thank the UCSD Mathematics Department for making available the NCAlgebra package usingwhich parts of the calculations in this paper were performed.52 Determinant factors In order to compute the divergent part of the integrand, we must include the matter contributionas well. Here we present this computation; we will use this combined result in the following to lookat the convergence properties of the finite N regularization for the simplest loop amplitude. Matter sector Gaussian integrals Similar to the ghost sector, we evaluate the matter sector integrand by performing the state sumover matter degrees of freedom by choosing a Fourier basis: e − iξ (cid:62) η e ip ¯ x . Here, however, the presenceof the matter zero mode results in additional terms which were absent in the ghost sector by virtueof the Feynman-Siegel gauge.The matter contribution to the integrand is given by: I Xe ( q ) = (cid:90) d d ¯ x (cid:90) d d p (2 π ) d ( dη )(2 π ) dN Tr (cid:104) A e ∗ e − iξ (cid:62) η − ip ¯ x ∗ ( q ˆ L X − e iξ (cid:62) η + ip ¯ x ) (cid:105) (A.1)Once again, we choose the monoid elements appropriate for excitations on the perturbative vacuum: A ( ξ, λ ) = N X e − ξ (cid:62) M X ξ − ξ (cid:62) λ ,A ( ξ, η, p ) = e − iξ (cid:62) η e − ip ¯ x ,A ( ξ, η, p ) = e + iξ (cid:62) η e + ip ¯ x , (A.2)where we have assumed Lorentz symmetry over all the matter indices in ξ µ . Note the extra factorsof i as compared to the ghost sector and the loop momentum p µ . The monoid A ( ξ ), which servesas a generating functional, is chosen to have zero momentum since this is the one-point function forthe D 25 brane case. We recall for convenience that the matrix M X and the normalization factorused in defining the matter vacuum are given by M X = (cid:20) κ e T κ − o T (cid:62) (cid:21) , N X = det(4 σM X ) d/ = 2 Nd (1 + w (cid:62) w ) d/ . (A.3)As before, we sequentially apply the monoid algebra rules for doing the string products. The rulesin the matter sector are identical to the ghost sector with the choice of basis we are using, includingthe signs in the exponentials from Gaussian integrations. In [13] to obtain the parameters for themonoids A := A ∗ A : A ( ξ, p ): M = M X , λ = (1 − m X )( iη ) + λ, N = N exp (cid:18) − η (cid:62) σm η + i λ (cid:62) ση (cid:19) , p = − p. (A.4)For the propagator rules[15], there are extra terms from the momentum p : M ( t ) = (cid:2) sinh t ˜ κ + (sinh t ˜ κ + M X M − cosh ( t ˜ κ )) − (cid:3) (cosh ( t ˜ κ )) − M X , (A.5a) λ ( t ) = (cid:104) (cosh ( t ˜ κ ) + M M X − sinh t ˜ κ ) − ( λ + iwp ) (cid:105) − iwp, (A.5b) N ( t ) = N e − p t exp (cid:2) ( λ + ipw ) (cid:62) ( M + coth t ˜ κM X ) − ( λ + iwp ) (cid:3) det (cid:16) (1 + M M X − ) + (1 − M M X − ) e − t ˜ κ (cid:17) d/ . (A.5c)Applying this to A in (A.2) and rewriting the hyperbolic functions in terms of the functions f i (˜ κ ; q ),we have the parameters for 53 ( ξ, p, q ): M ( q ) = f ( q ) f ( q ) M , λ ( q ) = 2 q ˜ κ f ( q ) ( − iη + iwp ) − iwp, N ( q ) = 2 dN q p det( f ( q )) d/ exp (cid:20) − 14 ( η − wp ) (cid:62) M − f ( q ) f ( q ) ( η − wp ) (cid:21) , p = + p. (A.6)As before one may now remove the remaining ∗ product in the trace and simply set A A ( q ) =: A ( q ) with parameters: M ( q ) = 2 f ( q ) M ,λ ( q ) = i (cid:18) f ( q ) f ( q ) − m (cid:19) η − i f ( q ) f ( q ) wp + λ, N ( q ) = N N ( q ) , p = 0 . (A.7)The trace operation is simply a functional Gaussian integral and producesTr[ A ( q )] = N ( q )det(2 M ( q ) σ ) d/ exp (cid:18) λ (cid:62) M − λ (cid:19) := C η exp (cid:104) − η (cid:62) Q η η + L (cid:62) η η (cid:105) , (A.8)where we suppress the q dependence for typographical simplicity. Collecting the η dependence fromthe various factors, the coefficient matrices which appear in the quadratic exponential above are thefollowing Q η = Q η | + Q η | + Q η | , with Q η | = 14 σm , Q η | = 14 M − f ( q ) f ( q ) , Q η | = 18 (cid:18) f f − m (cid:19) (cid:62) M − f (cid:18) f f − m (cid:19) , L η = L η | + L η | + L η | , with L (cid:62) η | = i λ (cid:62) σ, L (cid:62) η | = p w (cid:62) M − f f , L (cid:62) η | = i (cid:18) λ − i f ( q ) f ( q ) wp (cid:19) (cid:62) M − f ( q ) (cid:18) f ( q ) f ( q ) − m (cid:19) , and C η = C η | · C η | · C η | , with C η | = N , C η | = 2 dN q p det( f ( q )) d/ exp (cid:20) − p w (cid:62) M − f f w (cid:21) , C η | = det (cid:18) M − f ( q ) (cid:19) d/ exp (cid:34) (cid:18) λ − i f f wp (cid:19) (cid:62) M − f (cid:18) λ − i f f wp (cid:19)(cid:35) . (A.9)We can rewrite the above expressions for the symmetric matrix Q η after some matrix algebra as: Q Xη = 18 (cid:104) M − f ( q ) + σf ( q ) M σ + σf ( q ) − f ( q ) (cid:62) σ (cid:105) (A.10)where we remind the reader that f ( q ) := f + 2 f f .Combining the four terms, we have the block matrix form: Q Xη ( q ) = 12 (cid:20) κ − e f + T κ − o f ( q ; κ o ) T (cid:62) − i ( f ( κ e ) − T f ( κ o ) R ) − i ( f ( κ e ) − T f ( κ o ) R ) (cid:62) (cid:0) κ e f + R (cid:62) κ o f R (cid:1) (cid:21) with the appropriate factors of 2 π and i s inserted in the measure. η gives1(2 π ) dN C η (det 2 Q η ) − d/ exp (cid:20) L (cid:62) η Q − η L η (cid:21) (A.11)Now rewriting L (cid:62) η = i λ (cid:62) α (cid:62) + p w (cid:62) β (cid:62) , for compactness using a little algebra in terms of α (cid:62) ( q ) := 12 (cid:16) M − f + f (cid:62) σ (cid:17) (A.12a) β (cid:62) ( q ) := 12 (cid:16) M − f − f (cid:62) σ (cid:17) (A.12b)the argument of the exponential factor involving Q − η becomes:116 (cid:110) − λ (cid:62) α (cid:62) Q − η αλ + p w (cid:62) β (cid:62) Q − η βw + 2 ipw (cid:62) β (cid:62) Q − η αλ (cid:111) where the Lorentz contraction with λ is understood. Then identifying the quadratic and linearpieces in the centre of mass momentum p (conjugate to the matter zero mode) as follows: Q p = − ln q + 18 w (cid:62) M − f w − w (cid:62) β (cid:62) Q − η βw (A.13a) L p = − i w (cid:62) M − f λ + i w (cid:62) β (cid:62) Q − η αλ (A.13b)we can finally perform the integration over p , to yield the matter contribution to the generatingfunctional: W X ( t, λ ) = (1 + w (cid:62) w ) d/ (4 π ) d ( N +1 / | det( Q η ) Q p | d/ exp (cid:104) − λ (cid:62) Q Xλ λ (cid:105) where, (A.14) Q Xλ = 14 (cid:40) α (cid:62) Q − η α − M X − f ( q ) + 116 α (cid:62) Q − η βww (cid:62) β (cid:62) Q − η α Q p (cid:41) , (A.15)where d = 26 is required for having c = 0 for the BCFT. Combined integration over η, p Another equivalent form that would be more suitable for numerical calculations of the determinantfactor is obtained by performing the integration over η and p after combining them into a single(2 N + 1) × ψ : ψ := (cid:18) ηp (cid:19) (A.16)Now, we can trade using the inverse Q − η , which is numerically extensive, in favour of working witha larger size matrix while evaluating determinants. In terms of ψ , we can write the expressionobtained by taking the trace over ξ (A.8) asTr( A ( q )) = Γ ψ exp (cid:104) − ψ (cid:62) Q ψ ψ + L (cid:62) ψ ψ (cid:105) , (A.17)where the matrix Q ψ would now be (2 N + 1) × (2 N + 1) dimensional and can be written as: Q ψ = (cid:20) A BB (cid:62) C (cid:21) , where A = Q Xη , above in (A.11) , B = 18 ( M − f + σf ) w, a 2 N × , C = − ln q + 18 w (cid:62) M − f w, a scalar (A.18)55nd the (2 N + 1) × L ψ would simply be of the form: L ψ = i (cid:20) (cid:0) M − f − σf (cid:1) λ − w (cid:62) M − f λ (cid:21) = i (cid:20) (cid:0) M − f − σf (cid:1) − w (cid:62) M − f (cid:21) (cid:20) λ (cid:21) =: ˆ L ψ (cid:20) λ (cid:21) . (A.19)The remaining factor Γ ψ is then given by: C ψ = (1 + w (cid:62) w ) d/ exp (cid:20) λ (cid:62) M − λ (cid:21) (A.20)Now, performing the Gaussian integration over ψ as (cid:90) ( dψ )(2 π ) (2 N +1) d C ψ exp (cid:104) − ψ (cid:62) Q ψ ψ + L (cid:62) η ψ (cid:105) = (1 + w (cid:62) w ) d/ (4 π ) d ( N +1 / det( Q ψ ) − d/ × exp (cid:26) λ (cid:62) (cid:20)(cid:16) ˆ L ψ Q − ψ ˆ L ψ (cid:17) N × N + 12 M − f ( q ) (cid:21) λ (cid:27) . (A.21)Here, the first term in the exponential (cid:16) ˆ L ψ Q − ψ ˆ L ψ (cid:17) N × N denotes the first 2 N × N square block inthe (2 N + 1) × (2 N + 1) matrix ˆ L ψ Q − ψ ˆ L ψ which turns out to be the non-zero entry in that matrix.We use only the λ independent factors while numerically computing the matter contribution to theintegrand next. Some Numerical Results In analogy with the analysis done in the oscillator formalism [7, § 4] to study the determinant factor(the scalar part) to the one-loop tadpole, we plot log S ( t ) vs 1 /t for various values of the finitecut-off N . This is an interesting exercice as both the oscillator and the Moyal representations havetheir advantages and disadvantages. In the oscillator case, we have the exact Neumann matricesand the analytical expression for the matter-ghost determinant involves lesser number of inversesand matrix multiplications (which makes a numerical analysis more reliable). However, the leveltruncated Neumann matrices do not satisfy the Gross-Jevicki non-linear identities and so wouldn’tbe fully internally consistent. For the Moyal representation, the finite N deformation is in a senseconsistent since the matrices satisfy identical algebraic relations as the open-string ( N → ∞ ) limitwhenever they do not lead to associativity anomalies. But associativity anomalies are essential toobtain the correct closed string physics in OSFT and hence we examine the convergence rate in theMoyal formalism as well.We use the finite N versions of the matrices and vectors given in [13] for our analysis: T eo = w e v o κ o κ e − κ o , R oe = w e v o κ e κ e − κ o ,w e = i − e (cid:81) o (cid:48) | κ e /κ o (cid:48) − | (cid:81) e (cid:48) (cid:54) = e | κ e /κ e (cid:48) − | , v o = i o − (cid:81) e (cid:48) | − κ o /κ e (cid:48) | (cid:81) o (cid:48) (cid:54) = o | − κ o /κ o (cid:48) | . (A.22)Now, in addition to the matter-ghost contribution we have, we need to insert the extra factorswhich relate the Witten vertex and the Moyal star. Hence, while doing the numerical analysis wehave multiplied by the additional factor − µ − K , where K = √ and µ is given in (2.51). Wefind that the results are comparable although the convergence rate is not as good. This was to beexpected as because of the substructure of the Neumann matrices, more number of inverses andmatrix multiplications are required which also affects the convergence rate. However, the relationbetween the size of the matrices L and 2 N in the two regularizations is not direct as in the latter, thematrix identities are satisfied even when N < ∞ . Therefore, there is no analogue of the U V cut-offseen in the level truncation approach as the determinant is still singular for finite N as t → ∞ .56 ●● ● ● ● ● ● ● ● ● ●●●●●●●●●■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▼▼▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼○○○ ○ ○ ○ ○ ○ ○ ○ ○ ○□□□ □ □ □ □ □ □ □ □ □◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ π t ● ■ ◆ ▲ ▼ ○ □ ◇ t [ ℐ vac ( t )] Figure 5: The log of the overlap amplitude with the perturbative vacuum plotted against 1 /t forvarious values of the matrix size N . The green line is the expected infinite N behaviour with slope2 π . We see that the result steadily approaches this line as N increases. B Pad´e approximants for R nm ( q ) In this appendix, we shall try to infer the analytic properties of the functions represented by theexpansions from § q plane by considerations of their Pad´e approximants . These are mero-morphic functions of the expansion parameter that have identical Taylor series coefficients till thefinite data generated for an unknown function (by using various computational techniques).Specifically, the r/s Pad´e approximant till order N is a rational function, constructed as thequotient of two polynomials of degree r, s respectively such that r + s = N and: P rs ( z ) := A r ( z ) B s ( z ) = a + a z + · · · + a r z r b z + · · · + b s z s = p + p z + · · · + p N z N + R N ( z ) , (B.1)where the expansion coefficients p i , ( i = 0 , · · · , N ) coincide with the series expansion at hand.Generally, the diagonal/symmetric case r = s ≈ (cid:98) N/ (cid:99) captures the zeros and poles of the unknownfunction more accurately and provides the fastest convergence to the true function as N increases.One can estimate if the poles so obtained are spurious or not by roughly checking how much theyoverlap with the zeros in the complex z plane as the value of N increases. An accumulation ofnon-spurious poles could signal an essential singularity or a branch cut [44].We have constructed the Pad´e approximants for a few matrix elements in Table 5 to demonstratetheir utility and to show that these provide a better approximation compared to relying on the Taylorseries as can be seen by comparing to Table 4 above. In order to look for hints of non-analyticity, k R P R P R P R P − . . − . . − . . − . − . − . . . . − . . − . − . P kk evaluated at q = 1 for various matrix elements. The values areconsistent with what one expects for the diagonal and non-diagonal elements, namely ( − ) n and 0respectively, although the convergence as k increases is not uniform.we can study the poles and zeros of these rational functions as we do in Fig. 6 for two purely oddparity matrix elements. As can be observed from the plots, the poles do not appear to accumulate57ear the unit circle (or near q = +1 for that matter) at this order, and a few of them even seem tobe somewhat spurious since they overlap a nearby zero. But notice that the poles are still consistentwith being outside the unit disc. Let us therefore consider the absolute values of the corresponding × × ××× ×× ×× ∅∅∅ ∅∅ ∅∅ ∅∅ × poles ∅ zeros - - ( q )- - ( q ) (a) × ×× ×× ×× ×× ∅∅ ∅∅∅ ∅∅ ∅∅ × poles ∅ zeros - - ( q )- - ( q ) (b) Figure 6: The zeros and poles in the complex q plane of the 9 / R ( q ) and (b) R ( q ) obtained using the oscillator expansions based on the exactNeumann matrices.residues at these poles to ascertain the relative strength of the poles. We divide out by the constantterms in these expansions—which is always X nm as can be seen immediately from (2.41)—in orderto provide the numbers more intuitive. Furthermore, it is useful to consider the absolute values ofthe location of the poles to see if they are indeed approaching the boundary of the unit disc. Wehave performed these checks for several matrix elements and have presented the data in Table 6corresponding to the R case above. Because the off-diagonal functions vanish at the point t = 0,we may expect to see stronger signals for these. Once again, we remark that with the current limiteddata there is not a robust behaviour that may be claimed to hold and also that the residues may notbe representative of the (non)analytic structure due to possible rapid oscillations. To get a better q i | q i | Rescaled residue at q i − . . . − . − . i . . − . . i . . − . − . i . . − . . i . . . − . i . . . . i . . . − . i . . . . i . . / R , their absolute values and thecorresponding residues. For being more useful, we have rescaled all the residues with the constantterm X = ≈ . of the absolute value (rescaled by X nm ) of these approximants in the complex q plane as in Fig. 7 below. Next, we have at our disposal (a) (b) Figure 7: A plot displaying the absolute value and phase for the 9 / R and (b) R in the complex q plane. The “spikes” correspond to the location of the (simple)polesand the strength of the residues can be visually estimated by noticing how fast these are diverging.The phases are indicated using colours such that positive real numbers are assigned red, negativereal numbers are assigned cyan, and the hue varies linearly. All the numbers for absolute values arerescaled by the constant piece X nm .another approximation scheme which is known to work better for low values of N : the Borel-Pad´eapproximation. In this method, one combines Pad´e approximants with the Borel transform by firsttaking the Borel transform of the truncated series, then finding its Pad´e approximant and finallydoing the inverse Borel transform. The Borel transform of the truncated power series in q isobtained by replacing each coefficient p k by p k /k !, i.e: N (cid:88) k =0 p k q k → N (cid:88) k =0 p k k ! q k (B.2)whose Pad´e approximant P rs ( q ) Borel can be obtained in a similar manner as above.The final step is to perform the inverse Borel transformation that involves an integration alongthe positive real axis: ˜ P rs ( q ) = (cid:90) ∞ dt e − t P rs ( tq ) Borel . (B.3)The interesting case is when the integrand has poles on the positive real axis which can correspondto ambiguities from subleading terms, not ordinarily seen in a power series expansion. Hence, wehave analysed the pole structure of P rs ( q ) Borel but have found that although there appears to bepoles at certain positive values of q , these are not stable as the order r/s is varied. For R , forinstance, we have in Table 7 but for many other matrix elements we have checked, this behaviour ismuch less clear as the imaginary parts are not stable.However, we have evaluated the above integral for q = 1 and have found the expected result of C nm = ( − ) n δ nm to good enough accuracy. We have also examined the (scaled) residues of P rs ( q ) Borel towards this line of analysis. In short, the essential singularity expected for q = +1 and branch cuts The code for generating this plot was taken from a Mathematica Stack Exchange page. See the discussion in [45, § r, s } Pole of ( P rs ) Borel on R + { , } none { , } . { , } . { , } . P rs ) Borel , as r, s is varied for R ( q ).due to log( − log q ) do not show up conclusively at this order, indicating the need for much higherorder coefficients or some other underlying features of the functions. C The bc system and ˆ β oscillators In the BRST formulation, the worldsheet ghosts are introduced as part of the gauge-fixing procedureanalogous to the Faddeev-Popov ghosts in gauge field theories. In the first quantized theory, theworldsheet ghost and the anti-ghost are denoted by c ( z ) and b ( z ) respectively . These are anti-commuting fields with conformal weights − τ of the underlying worldsheet theory to 0, we can have themode expansion for these fields as follows: b ±± ( σ ) = (cid:88) n ∈ Z ˆ b n e ± inσ = π c ( σ ) ∓ ib ( σ ) , c ± ( σ ) = (cid:88) n ∈ Z ˆ c n e ± inσ = c ( σ ) ± iπ b ( σ ) . (C.1) Moyal coordinates Analogous to the matter sector, we can have “positions” and “momenta” linear combinations [14, § x n , ˆ p n , ˆ y n and ˆ q n as follows:ˆ x n = i √ b n − ˆ b − n ) , ˆ p n = i √ c n − ˆ c − n ) , ˆ y n = 1 √ c n + ˆ c − n ) , ˆ q n = 1 √ b n + ˆ b n ) (C.2)so that we may write: b ( σ ) = i √ (cid:88) n ∈ Z + ˆ x n sin nσ, π b ( σ ) = − i √ (cid:88) n ∈ Z + ˆ p n sin nσ,c ( σ ) = ˆ c + √ (cid:88) n ∈ Z + ˆ y n cos nσ, π c ( σ ) = ˆ b + √ (cid:88) n ∈ Z + q n cos nσ. (C.3)Schematically, we may represent [16] this as: b → x, c → y ⊕ c π b → p, π c → q ⊕ b (C.4)After choosing the Siegel gauge, we take the physical string field to be dependent only on the c mode. Here we understand that the b factor has been explicitly “factored” out. By virtue of thecanonical (anti-)commutation relations { ˆ c n , ˆ b m } = δ n + m, , n, m ∈ Z , We follow Polchinski conventions [46] for the bc ghost CFT. 60e have the corresponding structure: { ˆ x n , ˆ p m } = δ nm , { ˆ y n , ˆ q m } = δ nm , but now n, m ∈ Z + . (C.5)At this point, it is essential to introduce the SL (2 , R )/conformal vacuum and the associated ghostvacua constructed out of it. The conformal vacuum | Ω (cid:105) is the vacuum invariant under the globalconformal group generated by the L , ± Virasoro generators. Because of the two ghost zero modesˆ c and ˆ b , we can have the two fold degenerate vacua |±(cid:105) on top of this: |−(cid:105) = ˆ c | Ω (cid:105) , | + (cid:105) = ˆ c ˆ c | Ω (cid:105) , (C.6)at ghost numbers +1 and +2 respectively. One has the freedom to work with either of these twovacua and henceforth we define states by using the |−(cid:105) vacuum, conventionally denoted as | ˆΩ (cid:105) orsometimes | ˆ0 (cid:105) .From the underlying BCFT based on worldsheet path integrals, we require three ghost insertionsto account for the conformal killing vectors (CKVs) for the disc (tree level) amplitudes. In the Fockspace language, this translates to the additional normalization condition on the vacua: (cid:104) + |−(cid:105) = 1 ⇐⇒ (cid:104) ˆ c − ˆ c ˆ c (cid:105) = 1 . (C.7)Thus, in every non-vanishing inner product, it is assumed that the ghost number requirement issaturated to +3 in this form . ˆ β oscillators The relation between the matrix elements F nm corresponding to the half-phase space degrees offreedom ξ gh and the usual Fock space matrix elements R nm ( t ) can be obtained by using the formof the ˆ c n , ˆ b n oscillators in the diagonal basis. To this end, we must employ the action of the linearmaps between the two bases on these operators. The Moyal images of the Fock space states can beobtained by acting on the vacuum monoid with the so-called ˆ β oscillators:ˆ c n (cid:55)→ ˆ β cn , ˆ b n (cid:55)→ ˆ β bn , where ˆ β O A ( ξ ) := (cid:104) ξ | ˆ O| ψ (cid:105) . (C.8)These are thus simply the counterparts for ˆ c n , ˆ b n and the usual ˆ α oscillators (in the matter sector)used in bosonic string theory and may be expressed either as differential operators or phase space fields with left and right ∗ action on the string field in the ξ basis. We choose the differentialoperator representation in our discussion that follows.In [14], the oscillators are given for the odd parity degrees of freedom x o , p o , y o and q o , that canalso be used to represent the bc ghost system in the Moyal language. We have applied the canonicaltransformation that takes the odd basis to the even basis ( § β ce := 1 √ (cid:34) − iθ (cid:48) sgn( e ) κ − e p b | e | + θ (cid:48) ∂∂p c | e | (cid:35) , ˆ β co := 1 √ (cid:20) R | o | e x ce − i sgn( o ) S (cid:62)| o | e κ − e ∂∂x be (cid:21) , ˆ β be := 1 √ (cid:34) θ (cid:48) p c | e | − i sgn( e ) θ (cid:48) κ e ∂∂p b | e | (cid:35) , ˆ β bo := 1 √ (cid:20) − i sgn( o ) S (cid:62)| o | e κ e x be + T (cid:62)| o | e ∂∂x ce (cid:21) , (C.9)where we have restored the non-commutativity parameter θ (cid:48) for the ghost sector and the summationsover repeated indices are restricted to only the positively modded variables. We remind the readerthat the matrix S eo arises naturally while defining the Moyal product in the ghost sector and simplyequals S eo = κ e T eo κ − o . It also satisfies SS (cid:62) = e , S (cid:62) S = o , which can be proven from theproperties of the T and R matrices. We thus set the total spacetime volume to 1 through this normalization, which may be accomplished by a toroidalcompactification of all 26 bosonic coordinates, including the timelike direction. In general, for Dp branes, the tangen-tial/longitudinal directions may be compactified. The ghost number assignments are understood to be for the vertex operators as per modern conventions. We only give the ghost parts without the zero-mode contribution since once we have chosen the Siegel gauge, onlythis form would be relevant to the discussion that follows. The twisted ghost butterfly case It is an interesting exercice to consider the overlap with the twisted ghost butterfly state instead ofthe perturbative vacuum. This is one of the simplest star algebra projectors and is defined by thefollowing state in Fock space: | ψ B (cid:105) = exp (cid:20) − L (cid:48)− (cid:21) | Ω (cid:48) (cid:105) , (D.1)where the prime refers to the twisted ghost conformal field theory studied by Gaiotto, Rastelli, Sen,and Zwiebach (GRSZ) [47](see also [14, 48]). Because the (total) stress tensor on the canonical stripcoordinate w is twisted as: T (cid:48) ( w ) = T ( w ) − ∂j g ( w ) , (D.2)where j g = c b is th ghost number current in the original CFT, (and similarly for the anti-holomorphiccomponent), the new Virasoro operator above is given by L (cid:48)− = L − − j − . (D.3)In Moyal space, this state is represented by the twist even and SU (1 , 1) symmetric string fieldˆ A (cid:48) B = ξ N B e − ξ gh (cid:62) M B ξ gh , (D.4)where N B = 2 − N , M B = − (cid:20) κ e κ − e (cid:21) . (D.5)We remark that this string field satisfies β be ∗ ˆ A (cid:48) B = β ce ∗ ˆ A (cid:48) B = ˆ A (cid:48) B ∗ β b − e = ˆ A (cid:48) B ∗ β c − e = 0 , ∀ e > , (D.6)where now the β b,ce are fields in Moyal space instead of differential operators: β be = p c | e | − i e ) κ e x b | e | , β ce = 12 x c | e | − i sgn( e ) κ − e p b | e | . (D.7)Moving on, let us write M B =: χM gh , where we introduce the matrix χ = (cid:20) Γ (cid:62) 00 1 (cid:21) , with Γ := T κ − o T (cid:62) κ e . (D.8)We mention that the matrix elements of Γ (cid:62) can be evaluated exactly in the infinite N limit and aregiven by:Γ (cid:62) n, m = ( − ) n + m +1 nπ ( n − m ) (cid:20) ψ (cid:18) 12 + n (cid:19) + ψ (cid:18) − n (cid:19) − ψ (cid:18) 12 + m (cid:19) − ψ (cid:18) − m (cid:19)(cid:21) . (D.9)Then, we have M B ( M gh ) − = χ . Now, let us consider the monoid defined byˆ A (cid:48) B ∗ e ξ gh (cid:62) η gh · ( q L e − ξ gh (cid:62) η gh ) . This has the parameters: M = M B + f ( q ) f ( q ) M gh = (cid:20) χ + f ( q ) f ( q ) (cid:21) M gh ,λ = (cid:20) m B − f ( q ) f ( q ) (cid:21) η = (cid:20) χm gh − f ( q ) f ( q ) (cid:21) η, and N = 4 − N det( f ( q )) exp (cid:20) η gh (cid:62) (cid:18) ( M gh ) − f ( q ) f ( q ) + σχM gh σ (cid:19) η gh (cid:21) . (D.10)62s for the perturbative vacuum state, we can next take the trace over ξ gh and then perform theGaussian integral over η gh . This time, we set λ gh = 0 and have the non-vanishing coefficient matricesas follows: Q Bη = 14 (cid:20) ( M gh ) − f ( q ) f ( q ) + σχ (cid:21) (cid:20) χ + f ( q ) f ( q ) (cid:21) − (cid:20) f ( q ) f ( q ) − χM gh σ (cid:21) + 14 σχM gh σ + 14 ( M gh ) − f ( q ) f ( q ) , (D.11a) C Bη = 2 − N (1 + w (cid:62) w ) / det ( f ( q ) χ + f ( q )) . (D.11b)Finally, the Gaussian integration results in C Bη det(2 Q Bη ) which may be looked at numerically. 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