JHEP01(2008)018 hep012008018.pdf January 8, 2008 November 9, 2007 December 12, 2007 chnabl gauge and Published: Received: strips of variable width to Accepted: ize over intermediate on-shell r expanded local frames. We nge symmetries. Interestingly, agrams produce Riemann sur- iversity, uires two Schwinger parameters achyons. The computation involves of Technology, http://jhep.sissa.it/archive/papers/jhep012008018 /j Published by Institute of Physics Publishing for SISSA String Field Theory, Bosonic Strings, D-branes. We give a careful definition of the open string propagator in S [email protected] [email protected] C.N. Yang Institute for Theoretical Physics, Stony Brook Un Stony Brook, NY 11794,E-mail: U.S.A. Center for Theoretical Physics, MassachussettsCambridge, Institute MA 02139, U.S.A. E-mail: SISSA 2008 c Leonardo Rastelli Barton Zwiebach Abstract: ° present its worldsheet interpretation.and The contains propagator the req BRSTthe operator. left It and builds toevaluate surfaces the explicitly the right by four-point of gluing amplitude off-shell ofa off-shell states subtle t with boundary contracted term, o crucialthe to familiar enforce on-shell the physics correctfaces emerges excha more even than though once. stringstates. Off-shell, di the amplitudes do not factor Keywords: The off-shell Veneziano amplitude in Schnabl gauge JHEP01(2008)018 4 6 9 4 1 23 27 31 35 36 37 12 17 34 35 38 39 ings. variant open string field ted Veneziano four-point ] make sense off-shell, their t. These studies had useful s learned that: closed string amplitudes gave hroughout the history of string tring field theory [6]. Off-shell hyons would have a sensible off- hell extensions. It was clear that – 1 – ⋆ /L ⋆ ⋆ B /L ⋆ B/L B , and B/L of off-shell amplitudes would emerge from a field theory of str set Although the amplitudes of light-cone string field theory [2 2.1 Deriving the2.2 propagator CFT representation2.3 of CFT representation of 4.1 Boundary term 4.2 Proof4.3 of exchange symmetry The propagator revisited A.1 Basic properties A.2 Reordering formulas A.3 Wedge states 3.2 Second diagram 3.1 First diagram 1. Introduction Off-shell amplitudes have been the subjecttheory. of much interest It t wasamplitude [1] suspected for from the on-shell theshell extension. scattering beginning of Even open that three-point vertices stringa the would tac consistent have celebra off-s 2. The propagator, Contents 1. Introduction properties are unusual.theory With [3], the the development studyapplications. of of a off-shell In Lorentz amplitudes fact,contraints co began [4, the 5] in expected that earnes propertiesamplitudes helped were of in studied the off-shell mostly construction using of the closed Siegel s gauge. It wa 5. Discussion A. Notation and algebraic identities B. Correlators on the cylinder C. Proof of the rearrangement formula 4. Boundary term and exchange symmetry 22 3. The off-shell Veneziano amplitude: first two diagrams 11 JHEP01(2008)018 either, because the xamined in the useful of local coordinates at e amplitude. ties, but not all gauges s in Schnabl gauge, the ell vertices. ng states. oring coordinates at the ic approach. We use the de. Its off-shell version in ctures, each surfaced pro- he tachyon vacuum string eld theory action are sym- uld reproduce the familiar f the light-cone momentum in the computation of [32]. emann surfaces. As it turns map from the string diagram s with base the moduli spaces ut do not actually satisfy the nalytic solutions [10 – 13] use uge the Veneziano amplitude 3 uli space of Riemann surfaces: times not independent. 1 because the string diagrams ude is fairly complicated as it agrams at factorization develop at if cut, result in two allowed he amplitude and emphasize at formation by a regular marginal ons. heir inverses. Further analysis and t split the diagram into two separate pieces. ymmetric, full symmetry arises by summing he vertices are, by construction, fully symmetric. : near poles, all of which must arise from the – 2 – The second property implies that the string diagrams for 1 factorization which provide the two off-shell factors. 2 propagator, the amplitude is a product of the relevant off-sh of Riemann surfaces andthe fibers punctures spanning where the possible thepunctures, choices scattering each states Riemann are surface contributes inserted. only Ign once to th Siegel gauge provides amplitudes that obey the above proper The first property arises because the vertices in the string fi The simplest amplitude to consider is the Veneziano amplitu The off-shell Veneziano amplitude in Schnabl gauge was first e In this paper we begin a detailed study of off-shell amplitude all of which cannot be the same. They do not satisfy property 3 In open string field theory the vertices are only cyclically s There are also factorizations inThere which are cutting a some line further does small no disagreements in our computati 3. The amplitudes satisfy 2. The amplitudes are integrals over sections of fiber bundle 1. The amplitudes have permutation symmetry among scatteri 1 2 3 + over inequivalent orderings. In closed string field theory t a given amplitude give a construction of the appropriate mod they produce all surfaces ofduced fixed only once. genus and The fixed thirdinfinitely property number arises long of because pun strips string di (orsubdiagrams tubes, in closed string theory) th do. Off-shell light-cone amplitudesbreak do the symmetry not among satisfy states property p by assigning to them values o metric and so is the propagator. gauge in which itfield was [7]. possible The to string field obtainoperator that an represents is a analytic another finite marginal form solution de the for in wedge Schnabl t states gauge [14 – [8, 18]gauge 9]. that condition. are Other natural Recent a in related Schnabl work gauge, appears b in [19]–[28]. Schnabl gauge is the centralwas first topic discussed in by this Giddings paper. [29],to who the In found upper-half Siegel the plane conformal ga andon-shell then result. showed that The therequires closed-form amplitude constraints wo expression that involve for ellipticapplications the functions were and amplit discussed t in [30, 31]. paper by Fuji, Nakayama, andmore geometrical Suzuki conformal [32], field who theoryeach used interpretation step an of the algebra t integrationsout, over the we moduli have found of the that relevant a Ri subtle boundary term is missing Schwinger parameters associated with propagators are some JHEP01(2008)018 is ↔ ro- ⋆ ) (1.3) (1.1) (1.2) 2 /L cancels ,p formally 1 ⋆ p L By looking that 5 denote, respec- In Siegel gauge P L 4 uli space and each or ry is restored if we and a symmetric limit and etry under ( ⋆ mplitude. es include finite integrals and another to represent B is phenomenon explicitly ary. In Schnabl gauge the finition for the propagator he string amplitude. er frame. m that arises when 1 btleties, property 1 holds. /L .1) where the Schwinger rep- ult that violates the manifest that makes this overcounting 1 and Λ o amplitude, where we find a shows that the boundary term xchange. Hence a symmetrized ll amplitudes that are not built Q simplifies the amplitude consid- in appendix A. o used this propagator. : . ⋆ ⋆ Q . L . ⋆ ⋆ ⋆ Λ ⋆ ⋆ and naively assume that this L B − L B ⋆ e L B L L B Q − = 2 1 L B Ψ=0 is – 3 – } L B B -channel contribution to the off-shell four-point ⋆ s →∞ ? lim ⋆ P ≡ = is the BRST operator, and Λ Q,B is moved both to the left and to the right) requires no { − P Q Q change the moduli; the string diagrams produce Riemann uses one Schwinger parameter, exactly the number needed to p do not 0 to give /L 0 P b in ⋆ /L denotes BPZ conjugation, ) is violated. We then demonstrate that the requisite symmet ⋆ 4 . With two parameters for each internal line we overcount mod As first stated in [7], and as we review in section 2, a propagat Property 2 does not hold. In Schnabl gauge off-shell amplitud In computations one may use ⋆ The presentation of B. Zwiebach in Strings 2007 (Madrid) als Our conventions are the same as in [14, 23, 9] and are reviewed ,p 5 4 3 /L p ( We explain how this term arisesresentation of from the a propagator regulated uses version two of independent (1 cutoffs Λ amplitude following from (1.2) we find that the exchange symm include on the right-hand side of (1.2) an extra boundary ter exchange symmetry between incoming and outgoing states of t This term is needed because the naive computation gives a res each propagator at the factorization properties of the tively, the antighost and the Virasoro zero modes in the sliv duce the moduli spacepropagator of uses surfaces two with Schwinger punctures parameters, on one to the represent bound is used to take themprovides an to inverse infinity. for the We BRST checkthat operator. that arises this Further from analysis concrete (1.3) de isversion actually of antisymmetric the under propagator the (where e represented by a Schwinger parameter with a cutoff Λ inverts the BRST operator in the gauge Here This is the propagator used in [32] to compute the Veneziano a over parameters that boundary terms. We conclude that, after proper care of all su 1 coordinate for the redundant directionerably. of The integration possible that existence of a set of consistent off-she surfaces more than once. This is illustrated in the Venezian with the 1 surface is produced anfor infinite the off-shell number Veneziano of amplitude. times. It is We the exhibit presence of th compatible with the familiar on-shell result. JHEP01(2008)018 . . 1 B − (2.1) (2.2) 6= = tighost ⋆ } = 0 and B ⋆ 2 ,C C ⋆ sliver and is a B breaks into two { H off-shell Veneziano , for example. While We then discuss the 0 c such that d one line on-shell. It is, ith this prescription the = erm arises from a suitably C e factorization on on-shell to write and evaluate than al formulae for the off-shell is not BPZ even, C nd nontrivial feature of the ated CFT interpretation for have the expected symmetry B moduli in order to simplify the tores the exchange symmetry. Of course, the string diagrams s have been relegated to three chyon. We find that the residue 3 we compute the two diagrams at lowest level, by looking at the strip separating identical looking § 5 we discuss our results and some § is the subspace of states that satisfy . In CB . 0 S = . /L ′ 0 b = 0. The full state space ⋆ 4 we identity the culprit in a subtle boundary 2 § 0. The operator , where – 4 – Ψ = 0 ′ B B /L S ≥ ⋆ BC, P n ). In B 4 = and ,p P S 3 p ( , and . From BPZ conjugation rules it follows that acting on star products of off-shell states. Our result here ↔ C is not relevant, but one could take ) oscillators, − 2 C squares to zero, B/L n 6= B/L 2 ,p b 1 ⋆ B p . These formulas are the analogs of the simple strip-plus-an C ∗ /L ⋆ B , in general denotes the antighost zero mode in the conformal frame of the 0 = 1. The form of c the product of two Witten vertices with two lines off-shell an } B − This paper begins in section 2 with an explanation for (1.1). We were somewhat surprised that property 3 does not hold – the = not B,C ⋆ 0 is a generalization of the on-shell result in [9]. We find a rel the action of instead, the symmetrized product ofdo not two suggest different off-shell vertices. factorization sincevertices. there is no long CFT interpretation of complementary vector subspaces representation of the Siegel gauge propagator that correspond (1.2). We rewrite theresults. amplitudes in As terms would of bethe expected, corresponding the ones formulas are intachyons, much we Siegel note simpler gauge. that the amplitude Byunder computed the examination so exchange of far ( fails th to the gauge condition (2.1). To show this consider an operator term that happens not to vanish.regulated We propagator explain and how the demonstrate boundaryWe that t revisit its the addition formal res propagator arguments provides of an inverse section for 2.1Veneziano the amplitude and BRST are operator. show given that in Theinteresting (4.54) fin w open and (4.59). questions. In appendices. A collection of useful formula 2. The propagator, 2.1 Deriving the propagator An open string field Ψ is in Schnabl gauge if it satisfies Here { linear combination of by integration over moduli space is probably the most novel a Schnabl gauge. amplitude does not exhibit factorization.pole This that is arises checked, when theis intermediate line is an on-shell ta Then introduce the orthogonal projectors c Clearly, the operator JHEP01(2008)018 0 J is P /L . In B . As ′ (2.4) (2.5) (2.8) (2.6) (2.3) (2.9) (2.7) (2.10) K P Ψ = = 1. We in = Ψ: ′ P ⋆ 0 P P b C 0 c . + = P (2) BC . ⋆ 0 QP . |P b , the left-most ⋆ ⋆ 0 = ⋆ has an inverse 1 12 c P R 0 − ⋆ obtained by restricting ⋆ h = L = (2) (2) = 0, and P . = K to cancel the QBC K ⋆ ⋆ ⋆ L B B...B ⋆ ⋆ ⋆ = P L B ′ P ⋆ ⋆ (2) (2) B 0 = (2) (2) on both sides gives P c L B = (1) (1) 0 L B , ng fields of the form ⋆ P ⋆ b P L = B i ′ ) , and (1) (2) (1) Ψ = 0 is not invertible. One introduces a L B b Q 0 K (2) QBC . Assuming QBC . 0 0 , cannot be simplified. The propagator ⋆ 0 L BC . ⋆ . c . c Q 0 , P P Ψ (1) (2) L (2) ⋆ B ′ ⋆ c B K ⋆ h − = 0 = ⋆ ⋆ L Q ( 0 B c B 0 2 1 P L B | Q | | ′ ⋆ b L L C P 12 12 12 = Q C = − L B = , 2 i≡ R R R − ′ 0 0 QBC L h h h B = – 5 – c c Ψ ⋆ , the action of = 0 0 = . We must have = = = J b B b PK ⋆ 0 ⋆ . It is very important to note that the propagator ⋆ ⋆ QP QP P, P ⋆ P ≡ L P = ⋆ C ⋆ 0 P (1) B/L (1) = c QBC P = 0) and the right-most Ψ = P , P ⋆ L B 2 = K −P = P = Ψ gives B P h ⋆ (1) B 0 K 1 2 C c KP Q = 0. We also note that 0 0 0 equals the projector to the gauge slice. This is easily done ⋆ ⋆ b = L L (1) B (1) i Qb BP 0 = L B , except for the fact that Ψ PK b Q | . ′ 0 K P c 12 L B P , namely R , QP − P h = 6= Ψ = 0) one has Ψ 0 ⋆ = = = b because cannot be equal to K P P h S 2 1 such that P (1) PK PK P |P 12 H→ R : h P In Schnabl gauge the kinetic term is To find the propagator one considers the kinetic operator must give is BPZ even: In Siegel gauge first noticed in [7], the propagator takes the form one can almost invert Indeed, one readily verifies that This is reasonable: when solving and both left- and right-hand sides are clearly in the gauge. Indeed, as desired As opposed to the situation in Siegel gauge, this the kinetic term to string fields in the gauge slice, i.e. stri propagator there to leave us in the gauge ( P Schnabl gauge, In Siegel gauge ( have One can similarly check that We note that P JHEP01(2008)018 al hat can and per- = 0) . For 1 A (2.15) (2.11) (2.13) (2.12) (2.14) ξ wedge, A ( A L . Given t A A L +1 L i t t and defined . A , ) L A k 2 t e state A A ) L t + L and L ∗ − i 1 L A − ( t L T t , − W ( t e tation of this action will be ∗ W ∗ e. In this subsection we find ® 1 . A definition of the propagator is . . . 1) to the unit-width 0, with the operator 1) (0) , ∗ . The local insertion then occurs − − A , which holds for arbitrary TL t ≥ 2 1 2 ) ( t have been somewhat formal — we L − 1). All together we have a wedge of 2 A ( ) 1 2 t A ξ . 1 2 z = ∗ A associated to the state − P 1 ◦ W L TL t α − ) t z t dT e ( t ] ∗ − Im( . For arbitrary Fock space states 2 1 ∗ W e A ∞ A , tan + L (1+ 1 . 0 /L L 2 1 = 2 1 L − t Z 2 2 π ) t . In the following subsection with study the f T ∗ A A ≤ W − + L L ) – 6 – ( e 1) )= t (0) )= z 2 − B/L − αL φ ξ ∗ ∗ t ( (1 ( A − 1 ◦ 1 1 e 2 f e Re( ∗ − f t A ∗ 1 W ­ = L ≤ 1 t W A z 2 1 ≡ ( A = ∗ t t i is built starting at dt − ] 1 t B/L TL TL ] t 1 A A − − ] ), so we have [ 0 A L t [ A Z t φ , dT e dT e t . In the CFT language the test state is inserted on the canonic . dt h t between every consecutive pair of states )= (1 + ’s with impunity. As we will see later, there are subtleties t enter in the propagator (2.8), we must understand how these o ∞ ∞ 1 P k 2 1 1 0 0 0 ∗ − Z Z Z /L = t = ⋆ /L = = W 1 2 ...A ⋆ P )= ∗ 2 . B 1)+1= ⋆ 2 has an interesting geometrical picture, shown in figure 1. Th is obtained by gluing a wedge of width A = 0 by means of t A t 1, this wedge state removes a piece of surface. − 1) + ∗ ] ] /L t z ∗ ⋆ and 1 A A ( − ≤ 1 1 2 B A t t ’s against 1 ( ( A · ( 2 1 L 1 ≤ L B/L 1 L + The manipulations leading to the derivation of It will be convenient to introduce the state [ Let us begin by considering the action of 1 we have 2 1 2 by multiple string fields this generalizes to mapped to cancel We simply write width 2 Here we have used (A.7), (A.10), and unit wedge and the wedge [ The state [ at There is a wedge The state [ be visualized as the unit wedge followed by the gluing of another wedge of width that 0 invalidate such cancellations unlessused. a suitably regulated 2.2 CFT representation of Since ators act on states andused their to star obtain products. thethe The string CFT CFT diagrams represen representation associated of with the this operator gaug A action of JHEP01(2008)018 L t (2.18) (2.19) (2.20) (2.21) (2.17) (2.16) conformal . The local . t ) t − inserted using a (1 1 2 A A . ◦ W f r . ∗ with t ◦ ] + . Since we began with t t k t z ◦ W A [ ) t and to the right, provide t ∗ )= , . z (1+ ) ( ) 1 2 ξ t . . . r inserted at the ) (see figure 1). The effect of ( t T − T ∗ φ t (1 ] A = ◦ 2 1 2 ) (1 + A A W f [ 2 1 ◦ ∗ with ◦ with ∗ , a wedge state t t f t = ] ] t t ] ◦ ◦ , A z A 1 [ t 1 ) ∗ A ξ − ◦ [ ( ) f 1 t – 7 – ∗ f − − ) ◦ t f (1 = ( r − 1 2 ◦ lands at T L (1 strip. f 1 2 − W A t t ◦ L ) W )= = ) t ξ t ξ . It follows that ( ( t A dt φ f tz insertion is thus a wedge of width (1+ L is just to scale the local coordinate by a factor of 1 L t 1 2 t 0 + L A → T . As shown in [14] Z t r z A )= )= k is a wedge state of width ξ (0) = ( r A t ] f L t ...A A boundary midpoint and the local patch filling the wedge. [ ◦ ∗ is that of a conformal map, so we can determine what is the full ) 2 t A A (1+ The geometrical picture for the state [ ∗ 1 2 1 f denotes the map A t ( 1 L It follows from (2.13) that As we can see, the effect of a wedge of unitprecisely width, the the missing extra piece wedge of state surface. factors All to in the all, lef where on the state Figure 1: coordinate domain for the local coordinate that covers the width map applied to the state We can use this to rewrite (2.12) as where Indeed, as constructed, the state JHEP01(2008)018 . ) t . (2.24) (2.26) (2.25) (2.22) (2.23) − i (1 2 1 2 . Note the 2 W BA ). ∗ t BA L t i − t 2 ∗ ] or 2 A 1 1 (1 , each with a local . L t − 2 1 . 2 t ). Shown is one term t 2 BA i BA A [ ∗ W 2 A ) L 1 ∗ t t ∗ ∗ . − t ) − the result in (2.11). The t ] 1 1 BA 2 + 1 L (1 A A L W 2 1 ( t A B L of width s of size . [ + L t t L BA B 1 ∗ ] W 1 − L B 2 A 1 ∗ A t 2 A ∗ 2 − B t [ 1) t ∗ ] A 1) ( 1 A 2 ∗ 1 L − ∗ W − L t t A − A ] t t [ 1 ∗ 1 ∗ + ( ∗ +1 − + L 1 + ( A 1 t W t L 1 assuming that the states are in Schnabl ] 2 t − B A t 2 − W L t A ∗ + t +1 A W t L [ 1 1 ] W t + 2 ∗ A A ∗ + L A ∗ A [ 1 t 2 1) 1) ] 1 B L ∗ 1 without – 8 – A t − − − ∗ ) t , L t BA ∗ t 1 L B − W 1 BA + ( + ( t ∗ A [ (1 − ∗ L t 1 2 1 h 1 h − t ∗ W 1], to the sides of [ t , ) insertion but rather local insertions of + L dt dt t t W , we get BA [0 ) 1 1 − B α dt W L tB 0 0 + L t ], ∈ (1 1 Z Z W 2 1 − 0 t B h + L . . . Z +1 W = − t 1 dt )= +1 2 t A αB 1 B dt 1 M ( 0 A − A 1) 1 Z ), with ∗ 0 t ≡ − 1) = Z 1 + − − α A + M ( (1 W ) = ( 2 1 ) L B 2 + L ) = ( A 2 B ∗ A A wedge state in the integral representation (2.26) of 1 − ∗ A 1 B ( A L B ( Now we proceed with the action of L B In terms of states of type [ Rearranging we get Figure 2: The geometrical picture is that of a sequence of glued wedges insertion, flanked from the left and from the right with wedge only, the other two terms have no gauge. We do this on the product of two states, beginning with structure of terms requires the evaluation of Using ( The calculation is then straightforward wedges of width JHEP01(2008)018 , ∗ L k and /L A − t (2.28) (2.34) (2.29) (2.32) (2.33) (2.27) (2.31) (2.30) ] TL + R e A . L 1), 1 . produces 1 . + R k − ≤ = ∗ L s A 1) t 1) L + L − . W /L s − s L ( ∗ n T k 1 2 e k + R φ ( − ) A L A ) W − + R ⋆ 1) L + R e L A ∗ L s − L ∗ L − s s ∗ − ] ( s L 1 ( k L − ( + A . . . . − T e A match seamlessly. More- ) s L T e [ . ) ∗ L ∗ e 2 anking the result from the s ) ∗ W 2 − t ∗ − A ences. First, 1 in figure 2. This expression ∗ + − A ing that . . . (1 L ∗··· . . . (1 ( e ∗ ...... 2 1 2 TL T ∗ 2 and ∗ e ∗ φ s − s ds ) 2 W ] A 2 1 induces contractions ( + R + R 2 L ∗ A A A ∞ L L s A ) . t L 1 [ ] dT e − 1) /L + R s 1 Z + R L ∗ − L ∗ − ( L ∞ s T s 1 − = T ] BA e 0 1) [ L e ( − 1 W ( Z s − A − ∗ ∗ s T A ∗ e [ ( ) e = W 1 t TL − ∗ ∗ ∗ φ A ∗ − e e A ⋆ 1 1 L – 9 – 1 + 1) (1 ⋆ ∗ s A A 2 1 L − A TL 1 ) L s ⋆ ∗ TL ( − T s A W TL acting on a single state: e 2 − 1 1 e e ∗ L TL − ⋆ − ⋆ t 1 s + dt s − W (1 L − + ) are added to the left and to the right of the [ 1 ) )= /L s /L dT e W t L s 0 T ⋆ n ) ds e s Z W φ dT e − ∞ B s dT e − ∞ − ds 0 1 = s (1 ∞ (1 ds Z (1 Z ∞ ∞ 0 e 2 1 0 1 A Z ∞ ∗··· = , one to the left and one to the right of the sequence of states. Z Z 1 s 2 L 1 B ds dT e Z )= A φ 1). In the language of overlap states − k = = s ⋆ ∞ ∞ ) = ∗ 1 A 1 0 k ≥ L A A 1 W )= Z Z ∗ A s ∗ ⋆ k φ 1 1 ( ∗ L L = = A ⋆ . . . ∗ ∗ TL . . . 2 − . . . ∗ e ∗ A 2 2 ∗ A A 1 ∗ ∗ A 1 1 ( A A ⋆ wedges, as shown in the figure. For one state only we have ( ( 1 L 2 ⋆ ⋆ ] 1 1 L L A [ The exponential can be broken up using (A.9): The off-shell states have beenleft expanded and and from there the are right. wedges fl The first line on the right-hand side of (2.26) is represented makes it manifest that the local coordinate patches of It is interesting to compare with (2.12). There are two differ 2.3 CFT representation of Therefore we have: we get With this and (A.11) we find To generalize we recall the second identity in (A.7), and not two extra factors of over, extra strips of width We begin with the evaluation of 1 Second, the range of integration is different. While 1 induces expansion ( JHEP01(2008)018 + L . B 1 − − (2.36) (2.35) (2.37) (2.39) s (2.38) (2.40) ⋆ W B ∗ ¸ i 1) action (A.6) 2 − . 1) . s ⋆ ( . ¸ − ) . 1 2 s 1 BA 1 B ( 1 1) L − − 2 1 i W − − s s s 1 s s + L ( W − ∗ W W 1 2 1 s W . ) B 1 ∗ 1) − ∗ 1) + 1 L + L s − ∗ W W s − 2 ] − s − s B s + s B L ∗ 2 s ( ] W ( A 1 2 W ∗ 2 s 2 1 B A 1 L on wedge states, as given ] ∗ − W [ ∗ A s 2 − ∗ A 2 W [ ⋆ + L + equire the evaluation of L s W 1 + L ∗ 2 L ∗ A ∗ e 3. B B ∗ B s A W 1 B ( A L BA s s s ∗ [ ] ] ∗ L − s ] ∗ L ∗ and the resulting term combines 2 1) 1 s s 2 s − ∗ s ∗ 2 A ] 1 A 1 A A s A 1 W ∗ − ⋆ [ [ L 1 ] A [ A − L 1 1 A − ∗ s s ∗ L B s ∗ ∗ [ s ( 1) ss s − A 1 s A s s [ ] W A and in the second we use the A ] ∗ 1) − ∗ W 1 A 1 1) ∗ − W 1) 1 + L L 1) 1) s and B A s ( − − ∗ [ + ( − − B 1) − s 1 2 BA s ⋆ [ 1 2 − ∗ − ( ∗ ∗ s 1 2 W B W A 1 A ( ∗ 1 + R + ( 2 + ( 1) 1 2 + ( ∗ L − L W A s – 10 – A B s − s 1) 1 1 1 s L 1 W + − ( ∗ − W ∗ s A 1 1 1 − s − = 1 2 s ( ( s 1 s L A A A 1 ⋆ 2 1 ∗ s + − L W − W W W s 1) 1) 1) 1 B s W B + ∗ ∗ L ∗ − ∗ − − − W · s W 1 s B ds A − 2 A ∗ − s W L A s · ⋆ L + ( + ( + ( ds action requires a few formulas. Given the ∞ 1 s BA s + 1 ) B ) W L 1 1 ⋆ ∞ ds Z + s L 1 A A + L we get B BA B ∞ Z − B L B 1 1 1) 1) − s )= Z − − h − − − = 2 s h + R ′ ⋆ A ∗ W ds B + ( + ( B 1 ∗ M ) = + L ( ∞ 2 − 1 1 s = ( A ≡ Z sB A ′ ( ′ W ∗ ⋆ ⋆ = 1 M M s )= L B 1 ds A 2 ( − ∞ s A ⋆ ⋆ 1 ∗ in the first term can be moved to act on L W Z B ⋆ 1 + R − B A ( B The calculation of the ⋆ ⋆ L B In the overlap notation, we have The on star products, one needs the action of action on wedge states, In the first factor we replace with the second term: We can now begin the calculation of in (A.25), (A.26). Given the structure of terms in (2.33) we r Since The second line on the right-hand side is illustrated in figur JHEP01(2008)018 . (3.4) (3.2) (3.1) (3.3) s ] is not 2 ) A ). One’s ′ [ 4 but rather (2 ∗ Ψ B s F ] , 1 3 + A (Ψ (the “second di- (2) ) ↔ ′ F ) (2 2 + F ), flank [ Ψ erms have no to the right, , ∞ + (1) 1 , , P F ussed at length in the next [1 E . (2) ) tices with a propagator. For E ∈ 4 -channel contribution is given F ) . s s Ψ 4 E , , ∗ Ψ ) ). Two similar terms terms in (2.40) ) 4 3 E ′ 2 Q ) Ψ (2 Ψ A 4 ∗ ∗ ∗ 3 Q F Q . Ψ 1), with ( 1 3 ⋆ ∗ ⋆ ⋆ ⋆ + (Ψ A − 3 L B ( L ⋆ (Ψ B ⋆ s ⋆ ⋆ ( (2) L B L B L (Ψ B L 1 2 B P F , L L B B , − 2 + 2 , , – 11 – Ψ L B Ψ 2 2 (1) ∗ ∗ Ψ Ψ 1 ? F = 1 ∗ ∗ 34 – in contradiction with the symmetry of the start- Ψ ? Ψ 1 1 = D P D ↔ Ψ Ψ (the “first diagram”) and s = D D F s ≡− ≡ ≡ (1) F by moving the BRST operator in ) . Wedges of width F ′ s 2 (2) (1) (2 F is symmetric under the exchange of (Ψ F F F BA s F or 1 ordered as 1234 along the boundary, the i BA A term in the representation (2.40) of insertion at the positions shown in gray. The two remaining t is BPZ even, B P local insertions of Figure 3: 3. The off-shell Veneziano amplitude: firstThe two four diagrams point amplitude isarbitrary obtained states by Ψ joining two cubic ver have the symmetric under the exchange 12 In this section we evaluate by Then agram”) for external off-shell tachyons. We shall find that section. ing point (3.1). To the rescue will come a boundary term, disc Since with first instinct is to process JHEP01(2008)018 . t (3.8) (3.6) (3.7) (3.5) (3.11) (3.10) . 2 i . . p t . E 3+ ) t t C X Thus we find − 3+ E C . , and noting that ) (1 − t ) 2 1 E B i . = z i W ( i (1) 0 u is given by | ∗ c (1 + 1 Ψ ar on wedges of width t 4 + c ] 1 (1) 0 4 t (0) t part. It is convenient to − cV c i c i F [Ψ + B ∆ V 1) ¢ s t 1) π ∗B − 2 = 0 punctures 3 and 4 collide. For (1) ( t ¡ − t ] , 3 c . We have 3 i 2 shown in figure 4. Note that states i ) cV 2) t 3 [Ψ − ) (3.9) p , ∗ ( ξ (0) c i = (∆ ( ) 2 + 1]. For t 0 1 to rewrite the ghost correlator as i | , B 1 − i − 1 cV t p D [0 c (1 Ψ ( i | 2 1 1) ∈ = V tan − t − , the first diagram t 2 π ( = = – 12 – 1 B , W 3+ t +3+ i , Q C i u 1 ] z i cV E = i 2 Ψ , D ) | , with ∼ t z 2 Ψ | ) [Ψ 4 (1) z 3 ) is represented by the insertion of ∗ 1) ∆ p F + L 1 − ] (1 + − 3 + 1 B c dt i ∆ 2 − B p − [Ψ ( 2 D t by − (1) 1 = (∆ i c 0 dt i = Z i 1 t i 0 Ψ (0) | ∆ Z , c L − 2 − 4 ) is represented by the local insertion of 1) 2 ´ t = p − ] 2 π i ( c ³ + (1) D 1 − F p The first string diagram ( = are matter primary operators of dimension ∆ − i V (1) = F As usual, we will use the Mandelstam variables: s = 1 the punctures are uniformly spaced on the boundary. The conformal map of Ψ t The string diagram is the cylinder of total widthWe 3 consider + general external states of the form where Figure 4: 1 and 2 appear on wedges of unit width, while states 3 and 4 appe implies that [Ψ 3.1 First diagram Using (2.26), recalling that ( the external states are all annihilated by The CFT correlator factorizesuse into cyclicity a and matter the part identification times a ghos JHEP01(2008)018 2 . to ) P 2 3 p + (3.16) (3.14) (3.12) (3.13) (3.17) (3.18) (3.15) 2 4 p + i < j ) s 2 i ( ′ · p α 2 i P p − , respectively. . with + 2 i . The relevant 2 2 te the modulus P i i´ ∞ p j ′ p t t ij α P + . θ P π 2 1 − π ′ p 3 2 14 α 2 3+ to 3+ + ´ θ s − h h t i 1, and 4 ´ + , t P t π sin (2 sin ariable ′ 3+ ³ π information is the angular m ) α i f the trigonometric identity ³ 2 i 3+ = tan t − p ) 4 4 2 4 π vertex operators: go to 0 p P ´³ 2 i i´ 3 3+ + + . p t 2 3 where t h P p , z 2 i i ( dt to a more relevant variable — the π ′ p +2 . z ′ 1, is defined as the coordinate of s sin α 13 X 2 3+ t θ 2 α θ (2 − ³ and h 2 ′ . 2 ≤ located with clockwise ordering on the . δ ) , α t ) = t 2 2 i D 1 λ 4 sin 1 i p ) i´ 0 P ³ , P π + 1 − t ∆ ) ≤ Z on the boundary of the disk P 1 2 3 = tan (3 + π p + ´ w w P ) = tan t iθ 2 i 3 + 3 , denoting the coordinate on the disk, a map to e 3+ p i 4 2 4 iθ 1 = (2 π p +2 – 13 – , and e h X t s w i 3 = + = p − s , z (2 ′ = X 3+ ( ′ are mapped to sin ′ w , P C z α , with 0 ³ α ³ 12 i = α 2 2 e w E λ δ ) θ ) ( − t P 2 i 1+ t D p z , P = ) 3+ 1 i´ i i´ C π P (1+ t P V t E + X (2 π 2 2 · π 2 i = tan p 3 4 2 p 3+ (1) + 3+ 2 ip 2 1 c h e P h we find p ′ + α x (1) sin s (0) sin , z − 3 c X ³ + ³ 4 · t · · 3 ´ 1) (2 ip ′ = 0 2 π e − α 1 ( ³ − z c (0) = 1, the punctures 4 π X i´ ) = 4sin 2) · w t 2 x = − ip ( π of the four-punctured disk. Let us then review how to calcula at e c (1) 3+ 1 1) λ B sin(3 h F P − D ( − X sin · x 1 ³ It is useful to make a change of variables from · ip of a disk with four punctures e D the upper-half plane is denote the positive angle of rotation that is needed to go fro configuration is shown in figure 5. With Our disk is presentedseparations as between a the circular punctures. unit disk and We introduce the relevant the angle v after a map to the upper-half plane in which boundary of the disk. The modulus One can readily verify that for points This is immediately evaluated using (B.9). Making also use o 3sin λ Assembling our results back into (3.10) we have To evaluate the matter correlator, we specialize to tachyon Using (B.11) we find that modulus Placing JHEP01(2008)018 is λ (3.21) (3.22) (3.20) (3.19) he total is defined as the , at the modulus i < j λ 1 − 1 , with · etric identities we find the . le between two punctures is sion of punctures 1 and 2. ij 1 4 ) ) θ 4 4 z z = 0) = 0, since then punctures = es the final result t i − − ( . t 2 3 . λ ¢ ¢ z z π ¤ ( ( t 34 24 2 2 3 2 3+ θ θ z z πt h ¡ ¡ 3+ ¤ 2 £ t = 4. The angle π , . sin sin 2 ) ) 3+ j cos 3 sin 4 4 ¢ ¢ , £ P ¤ z z 2 2 t 13 12 2 2 , π θ θ to − − 3+ sin ¡ ¡ – 14 – i 2 3 , £ =1 P z z λ i λ sin sin )( )( sin 4 , 3 2 i − z z − P 2, which corresponds to the configuration of equally = = 1 / 3 − − λ 1 1 λ = z z ( ( i t = π λ 3+ = 1) = 1 h t 2 ( λ times the ratio of the separation between the punctures and t 4sin π is then λ A disk with four punctures We apply (3.20) to the first string diagram (figure 4) and find th circumference. the given by 3 and 4We collide, also which check is that conformally equivalent to the colli spaced punctures on the boundary of the disk. Using trigonom alternative useful formulas We readily check that this result correctly implies that Figure 5: To apply this result tosimply cylinder given diagrams by we 2 note that the ang (lowest) clockwise rotation angle that takes The modulus A short calculation using the values indicated in (3.18) giv JHEP01(2008)018 (3.29) (3.28) (3.27) (3.30) (3.25) (3.26) (3.23) (3.24) 1] and , [0 ) 2 i p ∈ P λ + t . +2 s 34 2 (2 χ ′ − , α ¸ t 2 ′ ) − ) α 12 t i´ 2 3 λ . ′ χ − p t ) ) α 2 i + − λ π λ λ p − , 2 4 factor of one. The first line , 2 ) p , 2 . (1 3+ − λ − / 2 4 + course, one must still add the 2 =1 t ) , 1 s ) h i X / ( 2 4 ) − ′ 1 · (1 2 ′ λ (1 p ) λ / α 2 sin α χ 34 + 1 dλ λ − − (1 − 2 3 ) χ ³ − s 2 p − 1 ′ ¸¸ λ · ¸ ( − ′ 4 3 − (1 i´ t α ) t λ s α t ¡ 4 3 2 i − ′ − · 2 p − π α ¡ = π = 4 4 3 / 2 2 . χ − 3 3 t 3+ ¡ P / 3+ 2 i 1 1 3 dt − · + dλ λ we have ¸¸ p ) h π 2 2 − − i 2 √ t p 2 t ) ) λ t / dλ λ sin P , χ 3 λ λ + 1 π sin ′ 2 π ) 2 1 t − 0 2 / α that vanish on-shell: ³ 2 j p 2 − − 3+ Z 1 – 15 – → − p 3+ + π 0 2 (1 χ 4 · s ´ h t Z − i λ λ 3+ ´ + + , as defined by (3.22). An alternative expression is ) p t t · ´ 2 i λ λ sin i = (1 = (1 = (2 2 p )sin and p ′ π / 2 t ( i i i and small − ′ α 1 ) X t t t 3+ t ) ij α − ³ λ 2 4 π π ³ λ (1 χ X δ π π 2 i 3 2 − ³ (3 + p D 3+ 3+ 3+ − i´ dλ δ ) 2 2 1 (3 + t h h h 2 P π D · (1 / ′ ) 4 π ≡ π 1 · α 2 sin sin sin π π 0 3+ − ij · Z 4 · h χ -channel diagram to obtain the region of integration ´ ´ = (2 t i 2 = (2 π as a function of sin p (1) t ³ ³ · · (1) F X F ³ δ D ) π as integration variable the full amplitude (3.15) becomes =(2 λ (1) F In (3.27) one views where we have defined variables Note that on-shell thethen second gives line us in the the familiarcontribution above on-shell from result Veneziano the gives amplitude. a Of as well as the Jacobian Making use of the identities a calculation gives the relatively simple result which also implies that for small Using JHEP01(2008)018 - t e (3.32) (3.31) (3.33) (3.34) (3.35) . + 1 = 0. 41 s ′ χ , . 1]? Though α ¸ , ) match at the 34 41 . This has the χ χ λ [0 2) but requires 1 , . ¸ ¸ / ) ) (1) − Ψ g diagram arises t ∈ 34 34 λ λ (1 = 1. χ his string diagram: F ole at ( ( χ 1 λ (1 t ¸ → 2 2 ¸ h 2 3 h h 4 2 h π π · · √ · 2 · and 3 χ χ χ 2] to 12 ¸ ¸ · ¸ -channel diagram can we / χ ) ) ) χ (1) t s 1 ¸ λ λ factor of the Witten vertex. and Ψ fferent off-shell factor. This ¸ λ , ( ( F 3 ctures. Since we have shown 3 4 1 1 the second line in (3.29) does [0 4 − √ h h 4 (understanding that subscripts Ψ √ 3 · · ∈ 3 , it is clear that the contributions (1 1]. It is an important consistency · 2 2 · 1 → , +1 · λ − − · h t s ,j 3 [0 ′ ′ ¸ · ¸ 2 corresponds to α α 2 +1 / ∈ i − − -channel contribution and introducing , Ψ − ) ) s + 1 3 χ t λ + 1 1 ′ λ λ 1 s = 1 α Ψ ′ s ′ → − − − α λ ) α → λ ij − (1 (1 − 2 2 2 χ − − − – 16 – ¶· s t ¶· i , Ψ ′ ′ i (1 p 2 α α p 2 − − Ψ − -channel contribution (3.29) as s s ′ X X α → µ dλ λ dλ λ µ − δ 1 near zero and use (3.24) to obtain δ 2 2 D / / D ) 1 1 λ 4, as can be seen from the top equation in (3.26). The value ) 0 0 π dλ λ / and replacing π Z Z 2 (2 1 / t (2 ¶ ¶ 1 i i = 3 Z p p cover the modular region ≃ to denote that this is the λ . ≃ ¶ 2. This matching occurs for arbitrary value of is not BPZ symmetric. and i / s that can be easily read from (3.29). The contribution from th (1) t p X X s pole F 2 pole | µ µ | = 1 , we find →∞ h δ δ B/L X (1) λ t D D (1) λ µ ) ) and F δ − F π π and D 1 ) (1) s 1 π -channel answer is obtained by noting that the correspondin F h → = (2 = (2 t λ (1) (1) 4 corresponds to the maximum modulus that can be attained in t s t = (2 / 2) = 1. Happily, this is the case because F F / The We can use (3.29) to obtain the form of the amplitude near the p We can now ask: To include the contribution from the (1) t (1 = 3 F 2 from (3.1) by the replacements Ψ this would work on-shell, itnot does make sense not beyond work off-shell. In fact, λ For this we simply expand for the modulus for then the contribution from other cyclic orderings of the pun that this first diagram gives thefrom correct the on-shell remaining amplitude diagrams should vanish on-shell. The external states 1 andStates 2 3 appear and with 4, thehappens which expected collide because off-shell in the string diagram, carry a di simply extend the region of integration in (3.29) from effect of exchanging or just are defined mod 4). We rewrite the functions adding the subscript channel would be boundary point Letting condition on the off-shell amplitude that the integrands of Together, h JHEP01(2008)018 . L B ⋆ ⋆ L (3.37) (3.38) (3.40) (3.41) (3.36) (3.39) B − in (3.15). operators. ′ ⋆ (2) B . F on were over the À . ) and 4 À and ) ⋆ Ψ 4 L Q (2) Ψ . ∗ (3.4) to the second dia- F ∗ , } 3 rrangement formula (3.38). 3 1 ilar treatment in [32]. ather lengthy to evaluate. À , ⋆ Ψ ) (Ψ ≥ L 4 B Q 1 L 2 τ ⋆ ( 1 Ψ τ aluate τ L − L ∗ − 2 1 − τ e T 3 Be − − Ψ ⋆ + e L es. 1 , Be Q L 2 τ ( 1 τ ) − , Be T ⋆ 2 ⋆ − ) − e L Ψ B , e on the two star products. The resulting 2 ⋆ 0 ∗ , Ψ ⋆ on star products is somewhat simpler than B B e 1 ) 2 ∗ ≥ 2 2 /L 1 dτ 2 (Ψ ⋆ Ψ 1 dT L B/L – 17 – ,τ B ∗ (Ψ 1 2 τ dτ 0 1 L 2 − dT τ M ≥ (Ψ − Z ∞ Be ⋆ 1 ⋆ 0 is defined by τ ¿ − L | B Z Be 2 , the right-hand side of (3.38) would be equal to 2 ¿ ¿ = 2 = M τ ,τ 2 operators are moved to the right of the dτ − ⋆ ⋆ ⋆ ⋆ 1 1 τ dτ L B B L = B 1 dτ and L B L B dτ (2) M 1 and τ Z F M L M ≡ { Z − , it is advantageous to reorder = = ⋆ ) ′ /L (2) ⋆ (2 is shown in the left part of figure 6. Note that if the integrati F B F M Left: region of integration on the right-hand side of the rea With the use of the rearrangement formula the contributions Since we have seen that the action of Using BPZ conjugation, we could write In appendix C we prove the elegant rearrangement formula Right: The same region of integration in triangular variabl 3.2 Second diagram The formulas developed so far are in principle sufficient to ev in such a way that the Figure 6: where the region of integration gram can be written as and proceed by computing the action of expression is the sumInstead, of we ten are inequivalent going terms to and use would a be simpler r methodthe inspired action by of a sim The region whole range of positive JHEP01(2008)018 (3.45) (3.46) (3.47) (3.44) (3.42) (3.48) (3.43) , . , ¶ 2, and using g 2 À § ) A ) + 2 1 t 1 2 t + − t − g 1 (1 1 , (1 + 2 1 t A , 2 1 } 1 ) t 1 W µ 2 W t ∗ m − = ≥ ∗ rst contribution in (3.42) 1 A t (1 1 2 ] t t 2 1 ] ors, 4 . , 1) 4 ℓ ℓ , l + W C [Ψ C der are given by − 1 i 1 [Ψ i ∗ t , ) ∗ ) 3 + L 2 4 4 ℓ 1 t + r C t r ] B , t ] ( (∆ i ( 2 2 3 ) c 1 4 ∗ t 2 4 V B Ψ [Ψ 1 r − ) t ) ≤ + ( ] 4 + L 3 3 c 3 2 r r 3 2 ) t B ( BQ ( 3 Ψ +∆ [ 3 ∗ 3 r = ≤ ∗ V Q ( 2 [ ∆ ) c ∂cc t 4 ) 0 ] 2 ) ) 1 ∗ ¶ t 1 r 2 2 ) 1 ( in figure 10. r r , − t 1 2 π – 18 – ( ( t [Ψ 1 2 , r 2 (1 c c V t − 1 2 ∗ 2 ) B B µ ≤ t , we find 1 (1 ) ) ) i + 2 r 2 1 2 W 1 1 1 t ( + t 1 − r r 1 cV 1 1 t − 2 t ( ( W 2 3 V c c ≤ (1 − + = 2 1 on the star products by the techniques of +∆ 0 i = 1 | ≡ h ≡ h ≡ h W 3 ∆ τL 2 g 1 g 2 ¶ One of the terms in the second string diagram. ¿ m − ,t 2 A A 2 A t 1 π , r t Be 2 dt 2 t 1 µ dt + 2 1 2 c , we find dt c M M ≡ { 2 1 Figure 7: Z τ = dt − 2 = e c M = Z (2) 2 , r − t F 1 2 , 1 = τ = − (2) 1 e Taking states of the from Ψ r F = 1 Here the insertion points and the circumference of the cylin Evaluating the action of where we have defined the following matter and ghost correlat t where the region of integration is the triangular region shown on the right partis of shown figure as 6. a cylinder The string of diagram width for 2 + the fi JHEP01(2008)018 . , so ¶ g 2 g 2 (3.53) (3.50) (3.49) (3.52) (3.55) (3.51) (3.54) A A . ) + = , 2 p g 1 g 3 1 ¶ t A A g 3 P ) 2 2 , + ¶ µ A p t ) . + m 1 + 2 1 +2 A p 1 ¶ s g 1 γt 1 2 ) + 2) (3.56) + 2)] (2 t s γt 2) 1 A ′ − ( 1 ) ′ α 1 − t 2 4 µ ¯ ¯ γt sin( α − ( p ) γt − m 4 2 4 γ + ¯ ¯ − p 2 3 ) A sin( p ) 2 . ( 1 + 1)) ℓ ′ + sin( 1) +∆ C α γt − 1 2 3 i 1 ) γt 3 t ) , t ) p − 1 ( l fied by using cyclicity and the 1 ( ℓ ′ 4 γ C sin( − (∆ γt . 2 ¯ ¯ ¢ α i + γt cos( ( ) ) ( ) ¤ · − (∆ 1 l 1 2 2 4 µ . . Evaluating the matter correlator 4 ) sin( r − 2 2 4 ¯ ¯ ( − 2 )) γt ) X + γt cos( p · c 2 i t − 2 2 1 +∆ ) p p 4 1 + p µ 3 ) ′ + 1)) 4 2 3 r 2 ¢ + γt + α ∆ p r ( p P ) sin( 1 2 1 e ( +∆ c 1 ′ t 2 π + p ¶ 3 t c ) ( ( s α P − ′ 1 ( 4 ∆ γ γt ′ = B t + α ¶ r π ) ( ) t α ) 2 ( + 2) sin( i ¶ 1 2 2 3 π 2 2+ c − (2 1 V 1 r ′ ) µ ¯ ¯ 2 t γt t ) + cos( ( π µ ) 3 α – 19 – γt 2 2 − = 1 ( . Explicit computation reveals that r − 2 1 π ℓ ( ¯ ¯ 3 µ γt ℓ C c ∂cc π γt sin( ) + cos( γ γ i ) ) 2 )+sin ¶ ¡ 1 +∆ ) 2 2 1 − p 1 2 4 ≡ r r 2 γt r ( ( sin( sin ∆ + 1)) [( γt dt ¯ ¯ ¯ ¯ (cos( ( γ c c ( 1 ¶ 1 · · · X +∆ 2 t + 1)) 2 1 hB hB ( dt cos( t µ ∂cc π 1 ∆ ¡ γ 2 δ t sin B c ( = = M ¶ ) D µ γ 1 − Z 2 ) 3 g 1 g 2 πγ 2 t · r π ) π 2 ( A A ) sin( γ dt c = (2 ( γ 1 ) µ 2 ) sin( g 2 2 − dt 2 r γ A ( sin( sin c dt c = £ M + 2 1 B Z sin( ) ) 1 ′ g 1 1 dt 1 1 πγ πγ 2 − t (2 r A 1 c ( M F + − c = πγ Z ) + ′ = = − ≡ h (2 g 1 g 2 (2) g 3 = F A A ) F A ′ + (2 F (2) F An entirely analogous calculation gives It is convenient to introduce the shorthand As before, the evaluation of the ghost correlators is simpli periodic identification to write Combining terms we find We now specialize to the case of tachyons Using the formulas of appendix B, a calculation gives through (B.11) and collecting all the terms we finally find where that JHEP01(2008)018 4, / (3.57) (3.58) (3.59) = 1 λ 2. / . The curves for the string = 1 c λ M λ . Figure 8 shows 2 t and 1.0 . 1 1 t 2 corresponds to parameter / − ult quoted in [32], as can be f integration 1 2 t t = 1 1. Shown is also the diagonal 2 0.8 2 + . t ≤ 1 . t 2 = = 1 corresponds to ) 2 1 2 = ,t t t 1 ) + 1)) γt t γ 0.6 1 unit square. The triangle above the diagonal = ( + 1)) t 2 ≤ there 2 1 ( 2 1 t t t γ , ) sin( ( ( sin 1 1 2 γ t ( , T γt 2 – 20 – sin varies as a function of 1 4 intersect the diagonal twice. The curves with 0.4 / sin λ sin( − 1 = on the 2 2 t t λ λ = 0. The point = 2 + 1 − λ < λ t 1 1 0.2 t = for the second string diagram (see figure 6). on the unit square 0 c M there = 0 and 1 λ 0.0 T 2 t 0.6 0.4 0.2 0.0 1.0 0.8 Curves of constant modulus 4 are above the diagonal, and the point / = 0 are the lines 1 λ diagram. Using (3.20) we find λ> To understand the result we express it in terms of the modular Figure 8: A small computation then gives is the region of integration This is in agreement (upchecked to using an the overall change minus of sign) variables with the res It is interesting now to appreciate how the curves of constant that defines the upper right triangle as the relevant region o of so in fact, all curves of constant JHEP01(2008)018 λ ), 2 is a ρ, t u (3.62) (3.60) (3.63) (3.64) (3.61) ( → . ) 0 there is a 2 ¶ produces the is, it is easiest ,t 34 2 the curves of . 1 → 1 χ / χ t 1 2 u γt . 2 ¶ ¶ t that depend on 2 − 1 d ¶ ≤ t / du ′ 1 u 1 1 γt α λ ) µ γt − does not contribute to γt λ · ) ≤ sin λ − 12 4 cos ariables ( χ are equal to zero. In fact, / 2 − ¯ ¯ ¯ ¯ − ation is restricted over the (1 tion. A little thought shows n of figure 8. Indeed, − 2 − 2 1 t t 2 π r 1 ′ (1 2 1 st arise because the integrand γt γt α 2 µ : 1 tion domain. or +1=0. As γt γt − · − u ) s sin s γt 1 ′ cos ′ λ t 34 ¯ ¯ ¯ ¯ α sin µ α χ · − ¯ ¯ ¯ ¯ − 2 and values of ) µ χ 1 / 1 ) (1 ¶ 1 1 . 1 γt . 2 γt 34 2 / 1 γt 1 χ γt − 1 ≤ s 1 ) + 1) sin ′ dλdu λ γt 2 1 π − ¯ ¯ ¯ ¯ λ λ t α 1 + 1) γt · t 1 − Z + cos 1 ( ¶ t − sin t ≤ λ p 12 γ 2 + cos ( – 21 – ¶ χ 2 4 ≡ (1 γ 2 = ¯ ¯ ¯ ¯ p γt 4 2 4 each curve of constant modulus has two pieces 2 π ρ dt X u γt sin / 1 2 µ γt µ 1 sin X · dt δ (cos γt µ is that its derivative with respect to D (cos 3 ≤ sin c δ ) M ¯ ¯ ¯ ¯ γ 2 u Z D π λ · · ) · dt π 1 ≤ (2 dt = (2 we can rewrite the amplitude as 1 − π ) ′ λ = (2 = ) . Remarkably the full Jacobian gives F ′ u 1 du (2 1 + γt F γt dλ sin (2) + + 1, it is a regular function at this kinematic point. To see th F − s ′ (2) 1 α F γt is equal to zero the integrand is regular for 1 t The diagram studied in this subsection (the second diagram) Using the modulus inside the integration domain and one piece outside of it. Fo constant modulus are entirely contained inside the integra the pole at The amplitude then collapses to a relatively simple form: To simplify the answer further we introduce the variable — the details of which have been anticipated in our discussio parameter for the curvesupper of right constant triangle. modulus, and For the 0 integr to start with thegoes representation to (3.56). infinity somewhere Any over singularitythat the mu compact this domain can of only integra happen at the corners where either The region of integration here includes 0 where when The motivation for introducing candidate singularity. To explore it, we make the change of v factor cos JHEP01(2008)018 , ¶ (4.5) (4.4) (4.6) (4.1) (4.2) (4.3) gives ⋆ (3.65) L tion as ⋆ 2 T L ⋆ − Λ e − 2 e d dT can give a con- − 1, the amplitude ⋆ . µ − L L ⋆ À 1 ) = Λ . T . 4 − s s − ′ ⋆ ′ e Ψ L α α d. It gives an additional Λ 2 ∗ . , the factor esult consistent with the B Be T L ⋆ 3 ave the expected symmetry s still behaves as in (3.32). 2 − L . Let us repeat this manipu- e ⋆ (Ψ dT . Under BPZ conjugation the ⋆ Λ 2 he propagator: ⋆ → ∞ . Λ − lues of L 0 2 e ⋆ dT ⋆ t Z Λ + 1 = 0 means that near that pole ⋆ Λ 2 1 →∞ − s Λ B ′ , L e 0 dt ⋆ dT α ⋆ Z L 1 ⋆ Λ − . 1 ⋆ Λ 0 Λ , T B 0 ≡ Λ Z L Λ ⋆ − Z B ⋆ L Λ Q dρ ⋆ Λ 1 + P Be 1 + Λ L 0 1 B Q L = Z ⋆ Q dT – 22 – ⋆ L , ⋆ = 0, corresponding to the exchange of on-shell 2 ≡ = ⋆ Λ ⋆ Λ L T ⋆ ΛΛ ⋆ 2 0 1 B s − L ′ T ). The error can be traced to the naive manipula- Z P e dt ). 4 ΛΛ α Λ − , ⋆ 1 4 B e 2 P L ,p B dt 1 ,p 3 Ψ L 3 p − ∗ 1 c ( p M dT T ( 1 Z = Λ − Ψ 0 ⋆ ↔ ↔ Z ¿ ) Be ) ΛΛ 2 2 2 ≡ P ,p ,p ≡− dT Λ 1 ⋆ 1 1 , arising from the Jacobian, makes the integrand a finite func p ⋆ Λ L p 2 0 , we have t B ⋆ Z ΛΛ 1 get interchanged, L F ⋆ dT = Λ } 0 ⋆ Z − Q,B = { 0 and this establishes our claim. While there is no pole for ⋆ The failure of (3.63) to contribute to the pole at → ΛΛ 2 P tribution. Naively one would argue that in the limit Λ We then have t The last term iscontribution the to boundary the four term point that function: was previously droppe which yields The extra factor of It is useful to understand intuitively why the boundary term where Ultimately we are interested in taking the limit Λ cutoffs Λ and Λ Using under consideration has a pole for massless states. There are also poles at positive integer va the full amplitude obtainedWe from conclude the that the first full and amplitudeunder second computed the so diagram exchange far ( does not h tion (3.2), where a boundarylation term more was carefully. inadvertently We dropped introduce a regulated version of t 4. Boundary term and exchange symmetry The calculation ofexpected the symmetry previous ( section has failed to give a r JHEP01(2008)018 ↔ ) (4.7) (4.8) (4.9) 2 2, we § ,p 1 p , À limits can be , ´ ⋆ 1 2 ⋆ ⋆ tractions that go + ⋆ Λ e the prescription t B L limit first, we have − ⋆ 2 Q ts Λ the techniques of W B L expands the surface by a → ∞ ∗ restored by the boundary ⋆ barring calculational errors ⋆ = L tion to amplitudes. It is the ts ⋆ ⋆ ys property (ii). We confirm ] nd is also true for the second iously, property (i) also holds. Λ 4 is symmetric under ( , do not depend on the precise ies exists. We also re-examine ive calculation already gave the − ) ΛΛ ′ e [Ψ surfaces. Had we taken the limit B P (2 defined above provides an inverse ∗ F F . ⋆ P . As anticipated above, we shall see , ⋆ + ts ⋆ + ⋆ ] Λ ) 3 ′ e B ΛΛ ΛΛ (2) (2 [Ψ F ≡ P F F ∗ ⋆ ¶ 1 2 has an unambiguous limiting shape for large + . If one takes the Λ + t and →∞ Λ lim , s – 23 – (2) c Λ − − M non-degenerate Λ ⋆ e F 2 (1) e ts sums over all surfaces obtained by contraction with F →∞ + ≡ Λ ⋆ W B L Λ ³ s (1) + lim F ,B 2 = →∞ Ψ lim ⋆ s ∗ Λ as a suitable limit of F 1 to infinity, in particular the order of the Λ and Λ Ψ B , the factor ⋆ →∞ lim ⋆ ¿ F Λ Λ e µ t dt 2 1 that invalidates this argument. While 1 s 1 Λ Z B L P ≡ − ). = 4 vanishes for external on-shell states; . We define ⋆ depends on the prescription used to take this limit. We choos , which induces a very large expansion followed by a set of con ,p ⋆ ⋆ 3 B B ΛΛ B L p 0 first we would have indeed found only degenerate surfaces. ⋆ F ( F F Λ The terms previously computed, − → e (i) ⋆ (ii) the total amplitude L B interplay with rise to degenerate Riemann surfaces and no regular contribu down to zero size. This is a large set of scales that go from one down to about to the kinetic operator in some appropriate sense. 4.1 Boundary term We now turn to an explicit evaluation of the boundary term. By scale factor of order Using (4.3) we see thatthis this leads prescription to must a BPZ givethis even a propagator, fact four-point so in amplitude theterm that rest (4.6) obe with of the the limitsIt section: taken is according the to non-trivial (4.7). exchange thatwork symmetry Less a of obv is prescription section with 2.1 the and confirm right that propert the propagator where we have set find safely interchanged. This isdiagram, obvious because for the the integration first region diagram, a that by requiring that way one takes Λ and Λ Λ Λ and Λ Property (i) is necessary sinceright for result. on-shell We tachyons claim the that na the correct prescription is JHEP01(2008)018 . ℓ . C ¾ . (4.11) (4.12) (4.13) (4.16) (4.14) (4.15) Bi i ) is kept. 1 2 4 r + À ¾ ( ⋆ 1 2 i ∞ x c s 1 2 ) + . − 3 À = 2 x + 2 x r 1 2 1 2 ⋆ ( x s x W c + + W t ) − 2 x x . + L 2 + − L 2 x r ⋆ B W 2 ( B i ts W c , ∗ ∗ 1 2 ∗ ) ∗ x 1 m x W + x À ] ] r 1 2 ] x ⋆ 4 2 1 + x 4 L ] A ( 4 =3+3 s + 4 c g + t B tion (4.7). The first term − [Ψ 1 2 [Ψ 2 x [Ψ A − 2 x ∗ [Ψ , ∗ + + ⋆ L 2 − h ∗ surface with t t 2 ⋆ W ∗ ts x ⋆ W − ℓ B are x iagram of figure 9. Restricting − ] ) to the integral – the first term ts s ] C ∗ x x, ℓ 3 ] 2 4 ⋆ ∗ ] ∗ i 3 = (4.10) 2 W 4 p x ) 3 ts x ] = x 4 ∗ + ] [Ψ ´ ] are in Schnabl gauge: 4 [Ψ 2 3 r 4 [Ψ 3 2 1 [Ψ x W ⋆ ∗ p ( 4 ∗ ( . ∗ c + [Ψ + L ts ′ ∗ 1 2 [Ψ t ] [Ψ 1 2 α ⋆ B x ∗ 4 ⋆ B + + − =2+2 L ) ∗ + x ts 2 x ⋆ x ts 1 ] 3 ∗ ⋆ 4 2 B ] ] 1 2 x [Ψ 4 3 r s ts 3 4 1 2 ( and Ψ 1+ ∗ + + − L − c 2 i ,W + 2 x [Ψ W 3 x 2 ) x [Ψ [Ψ p B 2 3 π ] ⋆ 2 x ℓ first: then both the lower and upper limits ∗ ∗ s 3 + L ∗ ∗ C r Ψ P x , r i 2 1 ( ⋆ ,W − = ⋆ ′ B ⋆ ,W ) ∗ c [Ψ 2 2 x – 24 – ts 2 α 4 + ts ) ts ] ∗ t ℓ 1 r ] π ] ∗ Ψ 1 4 ( ¶ Ψ 3 3 − 1 2 r Ψ 1 2 ∗ ,W X 2 ⋆ ( π → ∞ ∗ · = 2 + [Ψ 2 c [Ψ ¿ + 1 =2+ [Ψ 4 ts ′ 1 2 x h µ ⋆ ⋆ ∗ + L Ψ 4 x ip 3 ∗ γ Ψ s . On the other hand, the second term in (4.7) gives a s Ψ W e ⋆ + ∗ h B 1 2 ) 4 − ¿ B ,W ts +Ψ dx 3 ℓ 2 1 x ] ∗ 2 + r 3 3 C ¶ F t 3 ( , r i +Ψ Ψ 1 2 Ψ Ψ ∞ ⋆ Ψ − ) 3 h 1 X 0 s · ,W 4 + ⋆ [Ψ ∗ Ψ 1) 1) t 2 3 Z 2 , assuming that Ψ r ts ¶ ∗ 1 = 1 ( − ip − − − 1) 2 1 Ψ ⋆ B c ⋆ e 1 1 2 2 Ψ ) 2 W s 1 ) − ∗ ts 3 3 2 + = + ( + ( µ t r r 1 )( Ψ − ( 4 ( W ⋆ − B c Ψ ) , r ½ ¿ X 1 s ⋆ 1) · h ⋆ 2 B F +Ψ 2 ts s − ) 3 3 − ( ip 2 ½µ Ψ Ψ = 0 dx ⋆ e W − r ) (1 ( ³ 1 1 1) 1) ts t dx c ∞ r r (1 ) ( 0 ⋆ − − B t − + 1 s ⋆ Z s X s r · ( 1 1 2 Z + ( + ( c ip e = = h −h ⋆ B = = F B ΛΛ g m F A A We believe that this singularin surface (4.7) gives does no not contribution contribute to regular contribution: of integration in (4.11) go to infinity, and only the singular We also define We now remove thein regulators (4.7) using the is symmetrized an prescrip instruction to send We evaluate the action of where the matter and ghost correlators are defined by Back in (4.8) and changing the integration variable to Here the insertion points and circumference of the cylinder The first term in thisto amplitude the is case illustrated of in tachyons, the we string find d JHEP01(2008)018 x . 1], , 34 χ [0 (4.18) (4.17) (4.20) (4.19) ¶ ∈ x ′ x x γ ′ γ in (4.12). sin B µ F 12 e χ )) ) ) . ¶ x 2 i x ′ p ] is covered once as γ ′ − 4 1 P γ , + (2 + (1 ) sin ′ + 2)) t [0 ′ 2 4 γ p γ (2 µ x ( ′ ∈ 1 ( + · 4 α ′ . 2 3 2 λ p γ − sin ) p ( sin + x 2 2) s ( P / ′ 2 ′ − . 3 ) α sin α x − √ ∞ (1 3 4 ) ′ 4+ ( 1 ) x γ − ′ 2 i = p γ ¶ (1 + λ sin P )) 2 i 8 2 π p + x − ) sin( t · x P 1 dx ′ ) +2 2 2 s α ′ – 25 – p (2 + , the integral (4.20) goes into minus the same ∞ (2 ′ γ 0 ′ + (1 + γ 2 1 α Z /x , 3 p ´ goes from 1 to 1 4 )) p ) π + 81 s − ). The modular region x x 4 sin( x ( 2 i ′ ′ x p → 3 ( γ − α X 2 + 2)) λ − P √ ³ x ) = ′ (2 + ′ x δ ′ α µ ( g γ ′ γ D ) sin( 2 ¶ ) = ′ γ ) A . Thus we can restrict the integration region to − ( 2 γ π π t 2 /x ′ sin( 34 sin( µ α · · χ (1 − sin sin( ´ λ ) = (2 p λ ↔ = m − λ X A 12 ³ χ (1 δ 2 The string diagram for the first term in the boundary amplitud D − ) s ′ π α − (2 λ · − = Using (B.11), the matter correlator is Figure 9: B F goes from 0 to 1, and once again as From (B.9) we find the ghost correlator expression with One readily finds that Under the change of variables The modular parameter for this geometry is Assembling partial results, JHEP01(2008)018 3 . 4 √ . ¸ 3 ¸ 34 2 π (4.21) (4.27) (4.28) (4.23) (4.26) (4.25) (4.22) (4.24) χ 34 χ ¶ ∼ ¶ 3 ′ 3 4 γ ′ χ √ γ 4 3 √ . ¶ sin 3 2 µ / )) 34 ) µ 1 χ 12 x x ) . : χ 12 ¶ λ ¸ χ 34 − ¶ 3 34 χ ¶ − 2 π 4 = 0. . (2 + √ χ (1 ′ 2 π ′ 3 µ ¸ ¶ (1 γ ↔ B γ ymmetry, let us extract ′ µ ( . µ 2 34 π 4 + ection we prove that the γ F 0, we have ′ try, ) χ 12 12 + 4 µ sin γ χ χ 34 ¶ ∼ · sin ta. ′ sin χ factorize into the product of e, 34 ,p ¶ e 2 γ x χ 3 ′ ¶ 2 . 3 µ − p n holds in Siegel gauge, where ) γ ¶ t 2 π ( 4 12 ′ √ x )) not sin χ ) 2 π χ α 3 µ 1 x = 0 and − x − µ ↔ ¶ µ µ 12 ) ¶ 0. As ) 12 x · − 34 χ 12 λ ′ (1 + 2 χ )) χ χ x ¸ ¶ γ ∼ ′ (2 + x (1 − ¶ ′ 8 ,p π ¶ 3 ′ γ = 1 x γ x 3 γ 4 ′ sin ( √ p (1 + 1 x 4 1 4 dx γ 3 √ 12 ′ 2 s µ we use (2 + ′ 1 3 sin γ χ − ′ µ 0 0 or s sin α λ − γ ′ sin Z − 2 α ·µ − · 4 ∼ ) µ 34 − to 2 1 – 26 – · 1 − 2 x χ sin( λ 2 i · − · x ´ under ( ¶ p dx 3 is the conformal factor that arises in inserting a p ¸ ¸ 2 34 x ′ dλ λ P √ χ 4 1 ′ (1 + x γ / ′ − α µ ¶ 1 X + 1 4 π 2 i + 1 γ 1 0 2 1 ³ ¶ p x B s ′ ′ s sin ′ Z − ′ δ 2 x π = α γ t ′ α ′ ´ µ α D γ µ α 4) p ) − 12 − / dλ − sin ´ π · χ ) · 3 p ↔ −F µ ¶ λ ´ (2 X ´ √ ′ p 12 ³ p B γ − ′ χ X δ F ≃ γ ³ ¶ D (1 X ′ sin X ) δ 2 ³ γ ³ ′ π D − pole δ γ δ ) | s ·µ (2 ′ D π sin D 2 1 α ) ) · − + 1 = 0, which arises for − π (2 π ·µ λ s factor s = ′ (2 − · · (2 F α B = ≃ F ≃ B F pole pole | | As a first check that the boundary term restores the exchange s Though symmetric, the amplitude near the pole does s B F F two off-shell vertices. A factorized answer would read the pole at To change variables of integration from tachyon vertex operator on the Witten vertex. Factorizatio So the exchange symmetrysymmetry holds is near exactly the obeyed pole. for In arbitrary the values of next the subs momen provided we antisymmetrize the integrand under the exchang Indeed, the coefficient (3 so we get: Combining this with (3.32), we find that for the full amplitud In particular, when the momenta are on-shell The amplitude then becomes The boundary term is antisymmetric under the exchange symme JHEP01(2008)018 34 (4.31) (4.30) (4.32) (4.29) χ du du discussed dependent, 12 K χ . Performing λ ¯ ¯ ¯ ¯ χ . 2 ∞ ¶ 2 γt 2 ) in the last term / γt 1 ¶¾ λ ) sin 12 λ ¯ ¯ ¯ ¯ − are also e the total amplitude . The pole arises for χ ll vertices attached to ¯ ¯ ¯ ¯ 4 − T du u / x ′ . x ue. There are other ways (1 γ (1 Z ′ ¸ being θ 2 π γ terpretation of the Schnabl 34 − does not vanish on-shell. As nly prescription leading to a sin χ µ auge propagator is a strip of , ¯ ¯ ¯ ¯ · 34 ¶ ght exchange symmetry. χ ′ B 34 2 the kinetic operator ¸ χ γ does not agree on shell with the ′ − . It follows that the total ampli- 0, and it encodes the vanishing of F ¯ ¯ ¯ ¯ ) t ′ γ ′ s 34 λ α γ + e ′ sin F χ ) − γ − . ′ µ> λ ) µ sin (2 2 λ ↔ 12 ¯ ¯ ¯ ¯ (1 ΛΛ − χ F t t − ′ P − 12 ¶ · α + χ χ x (1 )). The function 34 − ′ ¸ ) 2 x χ x γ →∞ ¤ (2) ′ ¯ ¯ ¯ ¯ λ − )=1 for t Λ γ s x – 27 – F ′ µ ′ − sin ( ? α π x = lim γ ′ θ + − µ 3+ γ (1 £ (3(1 + 2 P − 0, sin (1) − ¯ ¯ ¯ ¯ · s π/ sin F 4. The limits of integration for · ′ dλ λ are given in (3.29), (3.63) and (4.23), respectively. Col- 12 4 / α t χ / µ< = 1 = ¯ ¯ ¯ ¯ 1 − B ′ ′ 0 π s integration in (4.11) would range from 1 to γ F γ Z ′ 3+ F γ x ´ λ> dλ λ p sin 2 1 ¯ ¯ ¯ ¯ is symmetric under ½ · 0 , and χ ) and Z ) B )=0 for X ¸ ′ 2 ¢ ¶µ 2 µ t ³ (2 p / ( λ δ 1 θ + F and length equal to the Schwinger parameter ) − F D − 1 λ ) X t + B π 4 1 π ¡ e − δ F µ (2 (2) D θ (1 ) − (2 + F 2 π 2 1 π , = · π/ − · computed with the prescription (4.29) is symmetric just lik (1) B ; where the string diagram manifestly splits into two off-she = s e = (2 F F e γ F s A natural question is whether the prescription (4.7) is uniq computed with the prescription (4.7). However, F → ∞ s the boundary integrand for in section 2.1 whilefour-point (4.7) amplitude does. that is It correct appears on-shell that and (4.7)4.2 has is the Proof the ri of o exchange symmetry Our final result for the off-shell Veneziano amplitude is where tude The difference it has a transparent geometric interpretation. The Siegel g canonical width is the step-function Here F standard Veneziano formula, since thewe boundary will discuss term in (4.30) section 4.3, (4.29) does not really invert lecting our results we have as we will discuss shortly. T each side of apropagator long does propagator. not suggest By off-shell contrast, factorization. the geometric in to achieve a BPZ symmetric propagator, perhaps the simplest Using this prescription, the the same steps as above, we would arrive at the boundary term JHEP01(2008)018 } . . . { under (4.34) (4.33) (4.35) (4.40) (4.38) (4.36) (4.37) (4.39) s F . . . Consider a ¶ f 12 12 u χ i on moduli space, ¶ ¶ χ ¯ ¯ ¯ ¯ u 34 12 = x 34 χ χ ′ χ f ) ) x γ ′ u u σ σ curve as just functions γ ( ( λ locally u u − tal derivative. We recall sin . to ¯ ¯ ¯ ¯ 12 2 d 34 i ¶ . dσ χ 34 χ i 2 u ) ) χ γt 12 u ¯ ¯ ¯ ¯ ¶ 34 σ σ . Given the parameterization χ ′ γt χ ( f 12 = 12 du ) γ 1 ′ sin − χ f u χ t du σ γ u ¸ : u , 1]: (1 ) sin v 34 diagram (figure 10). The curve is − ) , ¯ ¯ ¯ ¯ u λ )= χ )= = ¯ ¯ ¯ ¯ σ 2 are reflections of one another: 2 [0 . (1 ( σ d − 2 − (1) = λ ¶ 2 dσ ,t ) ,t ¶ u σ ∈ t 2 u 1 1 − γt σ 34 12 t t 34 (1 − + χ σ χ − ( , on the constant γt ¶ χ = t ¯ ¯ ¯ ¯ ) − (1 v sin · 34 34 34 2 2 x σ and ′ ¯ ¯ ¯ ¯ ( χ χ χ (0) (1 x ) ) ) γ − i ′ u u u − σ σ σ , t , t u and γ in the ( ( ( 12 d 34 ) ) , v – 28 – dσ sin χ u u u u χ 34 σ σ and goes from ¯ ¯ ¯ ¯ λ 1 χ )= ¸ and at 1 ( ( 34 12 d d 2 1 ) 12 du 1 2 σ (1) χ dσ dσ χ γt t t σ t ( χ ) du λ ) u (0) = ¯ ¯ ¯ ¯ γt v 12 12 are invariant under this exchange and the momenta σ σ ′ u = , we are effectively testing the symmetry of ( χ χ − 12 sin λ − γ , we have , as given in (4.32), is symmetric under the exchange ′ ) ) ↔ t v )= χ 1 − 2 s 34 ¯ ¯ ¯ ¯ γ t σ σ t σ (1 34 χ 1 F 2 integration. For this only the terms inside the braces − sin χ t t (1 )= − − ¯ ¯ ¯ ¯ 2 − ↔ λ u γt 2 34 and (1) χ 1 µ· γt µ (1 (1 ,t (1 and ) t s u χ 1 1 u u ¶µ sin σ d t t ¸ ¯ ¯ ¯ ¯ 12 λ dσ ( ( µ µ 12 as ¤ χ . We will show that the symmetry holds µ u u t χ − v dσ dσ dσ π t 34 d (0) 3+ 1 4 1 1 1 dσ π χ du £ u 0 0 0 3+ ↔ µ 12 Z Z Z and introduce a companion variable µ Z χ ↔ u θ ). Since sin ) − 4 − − − u · − − σ 12 ). We demand that ( ,p = = = = χ σ v 3 ( p µ v ( 0 = only through dσ ), ↔ σ f s ( i u ) F u u 2 Z The symmetry property would be established if We now demonstrate that : ,p σ − 1 p that is, before performing the the exchange need to be looked at. ( enter of We begin by showing thatthe the definition middle of term is the integral of a to Through these relations we can now view Consider one of the curves of constant The middle term in (4.33) is and the fact that invariant under the reflection parameterization of this curve with a parameter with the condition that the points at JHEP01(2008)018 4 / 1 = 1 (4.46) (4.45) (4.49) (4.48) (4.47) (4.43) (4.44) (4.41) (4.42) 2 . t λ < = 1 and ¶ + 2 12 t 1 : χ t ¯ t ¯ ¯ ¯ ¯ 12 χ i x . For ′ = u f x γ ′ 2 . u 34 γ χ f ,t curves that do not 12 sin u to χ ¯ ¯ ¯ ¯ . It is then sufficient λ ¸ i = 1 − 34 34 ¤ u χ 1 t ¯ ¯ ¯ ¯ χ 34 t ′ , π t χ i 3+ γ cond diagram. The point π ′ . . . u corresponds to 12 ↔ £ 3+ ulation in the first diagram γ χ i . + 12 f − u sin u χ f 12 sin u 12 . u ¯ ¯ ¯ ¯ u χ es from 34 · χ Z ies are satisfied: ¤ ¯ ¯ ¯ ¯ χ f − ¯ t + 34 u x + π ¯ t χ ′ . . 34 3+ . x ¶ ¸ π γ − ′ χ £ ¤ 3+ ¤ ¤ ¯ ¯ ¯ ¯ 12 . . . ¤ γ 12 t ¯ t t terms t ¯ t 34 χ χ − x f π sin ′ t sin π χ πt ¸ i u 3+ πt u 3+ ¯ ¯ ¯ ¯ 3+ x 3+ γ ¤ ¸ ) ′ u πt £ £ £ ¯ t 3+ 34 £ ) 34 = λ Z γ π χ + χ . The point λ 2 i ¯ ¯ ¯ ¯ 3+ sin extra sin ¯ sin t u sin − − sin ′ λ ¯ ¯ ¯ ¯ £ t ¤ − · λ γ 2 ′ ¯ t + 12 = = (1 , u γ π χ = = (1 t 2 ¯ ¯ ¯ ¯ 3+ 34 sin ¤ – 29 – sin t ′ t ) · £ ¯ t χ − . . . ¯ ¯ ¯ ¯ ¯ t 34 γ · ′ λ u ¯ t π χ f 4, in which case the constant ¯ t χ − 3+ γ + sin + u / ¸ 12 π ¸ − £ 3+ λ u sin 1 ) χ + 34 ¤ 34 ¯ ¯ ¯ ¯ Z = λ u t χ χ − (1 sin µ ¸ π t 4 we get the t − u − λ − 3+ ) λ / π λ> = − £ 12 3+ λ 1 χ + (1 . . . f 12 u − t sin λ u χ − − · − · i u u λ< (1 χ u t ¸ 34 Z χ + ¤ 0= 0= t u − µ· π t χ 3+ + = 1 and some value π ¸ £ 3+ 1 ¤ 34 t t χ − are the points on the curve that are also on the diagonal sin π t u for the curve in question is given by (3.58), using 3+ · π + £ 12 3+ λ χ + u sin < u − · − u (see figure 10). Hence, 0 = 0 = ¯ t Consider the first relation. It follows from the modulus calc corresponds to = i The second term on the first relation is associated with the se 1 Then that In the last equality we assumed The modulus u intersect the diagonal and the integration region indeed go the integral is really of the form The above identities are antisymmetric under the exchange t where to show that: It now follows that (4.33) holds if the following two identit (figure 10). Therefore, for JHEP01(2008)018 = 1, 2 (4.52) (4.51) (4.50) (4.53) t + 1 t 2 (see figure first equality / 1 > 1 . ¯ t . + 2 u ¯ t γ = = corresponds to modulus is the function of 2 ¯ t x 2 − ′ γ . It then follows from (4.48) ¯ t t u 45) also holds. This completes γ , such that 2 sin . = t , γ ¯ t . 1 = ¯ t ¤ γ ) and x ′ x + 1)) 1 x = +2) γ ′ t 1 x ¯ t ( γ 1 x 1 ( 3(1+ π sin 3 π we have £ for 1 ( 2 2 2 1+ λ ¯ t , since it lies on the same curve) is (3.59), sin 3 π + and sin u – 30 – 3 4 = 4 3 and , ′ = 1 (or γ − ¯ t = λ u − λ u − = → 1 − 2, as required. Therefore: 1 ¯ t / 1 1 1 γ ¯ t Auxiliary diagram for the proof of symmetry. γ x < sin 1 1 ¯ t 1+ = ′ = γ ′ 1 γ ¯ t 3, and some values sin Figure 10: has the requisite exchange symmetry. s π/ F 1, we have coincides with the right hand side of (4.47). This proves the = γ 12 χ i u end of section 3.1, the tachyons. It is naturalpropagator, now with to its revisit regulation theoperator and formal symmetrized on computatio the limits, gauge pr slice. We ask if the propagator the second diagram nor the third one contribute for before, The off-shell continuity of the integrand for combined JHEP01(2008)018 (4.62) (4.64) (4.63) (4.61) (4.65) . The indeed ⋆ , which e gets a ⋆ will give i 0 B | L ⋆ 1 Λ c L − ⋆ + e Λ = 0. It is truly − QBC , e BC . ¢ BC . for large Λ i ⋆ ⋆ ⋆ plays a key role in 0 QBL | L L ⋆ B ⋆ ⋆ ′ 1 0 L ⋆ ≃ − c Λ Λ ⋆ BC . ∼ ψ L ⋆ Λ + − ⋆ − ⋆ — for the symmetrized ′ L 0 e e Λ − L ⋆ e ⋆ ψ − Λ e the term Λ 1 − Λ e dition. It may be that the L ot exist because they could e gauge slice. Nor there are − ¢ BLL − 1 at spacetime background. In BC s additional subtleties. With e o construct exactly marginal ¡ L ge condition or a shortcoming Λ vial state in the gauge slice. If Λ = Λ f (4.60) for the computation of ple. 1 Be QB B − L = 0. Acting on arbitrary states, L i e that satisfies the gauge condition, 0 we cannot even obtain (4.60). We + | ′ 0 = − 1 Bη P c ψ QB 1 + ¡ BC + BC BC . ¢ − ⋆ ⋆ ⋆ ) goes to infinity. Interestingly, if we take QBC L L L ⋆ = ⋆ ⋆ ⋆ ⋆ LBL BC ⋆ Λ Λ Λ ⋆ Λ ⋆ L = 0 and − − − L L = e e ⋆ BC Λ i L Λ ⋆ Λ Q e B 1 – 32 – = 0. Using also 0 Λ L L − | L ⋆ Λ e − 1 BQη B Λ trivial term in (4.63). Comparing with (4.60) we c e L Λ B − L + )= L + − Q = BQη ⋆ Be − η L Λ and this gives a regular surface. We thus see that, apart ⋆ L Λ L QBC + ⋆ BQBL − QBC BC L e = B Λ Λ − ⋆ L B e η L L − C } L ∼ = = − Λ ¡ ⋆ − , is more delicate. It vanishes in our prescription, since on K ⋆ L e ⋆ ⋆ B,Q ⋆ ⋆ Λ Λ { L of something which is in the gauge. The state B − − ⋆ L ΛΛ (see (C.5)), we get e Λ = 1 Q Q P L − ¡ ⋆ Λ e Λ B − L = L e Λ BQη exact term, we get the expected answer — just = 0. Let us now see if K − = ⋆ e Q : K ΛΛ η then Bη ⋆ P ⋆ ΛΛ Let us now examine the extra To derive (4.60) we begin with If we do not use the symmetrized prescription for P prescription and not from the prescription (4.29), for exam from the identify Λ = Λ singular surface as soon as one parameter (Λ or Λ last term, the extra term on thea right-hand gauge side condition is fixes thereforebe a the truly added gauge tri to completely any suchappearance physical states of state cann the while extra preserving terma indicates the unitary that gauge matter Schnabl con CFT, gauge Siegelsuch ha states gauge in has Siegel gauge no at trivial zeroSchnabl momentum states gauge, in on however, the there th standard is fl a BRST trivial state satisfies the gauge condition: Expanding the first two terms one gets In the above operator equation we find a singular surface — thus no contribution. The same holds for making it impossible to fix the gauge completely. The state Clearly Consider the terms inside the parentheses. As Λ becomes larg will not attempt here tostring discuss amplitudes, the nor possible if it implicationsof reflects o the a presently shortcoming defined of propagator. the gau holds for large Λ and rewrite this as trivial since it is the tachyon vacuum solution andsolutions features for in general the operators difficulties in t Schnabl gauge [9]. JHEP01(2008)018 , ⋆ Λ e ng to limits (4.68) (4.66) (4.67) (4.69) = . ⋆ ¸ s ¸ ⋆ ⋆ →∞ ⋆ B ⋆ Λ ⋆ B ¶ and L 1 m, these terms L Λ Λ e − = 0 in (4.67) and − ⋆ ⋆ and Λ e = 1 s /s B s , ⋆ − µ first boundary term, s L ¸ ⋆ )] + ⋆ hat makes the exchange ⋆ 1 →∞ ⋆ Λ ⋆ ording to (4.7). In fact, ⋆ Λ B vanishes, as desired. We − B L large g of the propagator. With re complicated amplitudes. B 1 x L hese terms give a vanishing ⋆ is fixed to a specific value), 1 Λ ( e exactly equal and opposite. dary term, s Λ nce that our definition of the B s 1 erage each term over the two L x . · erse of the kinetic term. B L + · ⋆ ) to taking , the Λ Q Q 1 x L B to the right as in (4.4), we sym- ] · − − x − ⋆ . x − Q L ⋆ s ⋆ ) 1 ⋆ ) [( ⋆ ⋆ Λ ⋆ in (4.67) we have a singular expression 1 ⋆ Λ x e + B in both cases and the two terms cancel L B ) B to the right and the second group is the L − 1 x x · ∞ + − ( ∞ − Q L 1 2 + ( e ) x = · s 1 = 0 in (4.68) we have a singular expression Q [( L + + − e ⋆ ) /s ⋆ , x ¸ ⋆ Λ ⋆ 1 x /s ( xL ⋆ – 33 – B s + Q ⋆ L − − − L L s x e e ⋆ Λ ) ( Λ B ⋆ e 1 L s − x x e is from 0 to − dx dx − + 1 Λ ( ⋆ x ⋆ s s Λ s s xL B x L B s 1 0 − − L e e Z Z · − . If we take is moved symmetrically to the left and the right the boundary 2 1 ∼ = = ∞ x Q x Q dx dx = ⋆ ⋆ ⋆ ⋆ = ⋆ ⋆ Λ ⋆ ⋆ Λ s ∞ ⋆ ⋆ s s B s s B L L /s ΛΛ Z Z L ⋆ Λ P Λ s B to the left. There are now two boundary terms, which we can bri L = ∼ = − e ⋆ Q − L ⋆ Λ Λ and, similarly, if we take B L − e = 0 and 0. When the propagator is inserted in a generic string diagra · in (4.68). The range of Λ 2 1 /s B L ⋆ → → ∞ ∞ = s x x ⋆ = In summary, when the Before concluding this section we discuss another rewritin ΛΛ /s P ⋆ with a common form by a few applications of the CBH formula. For the In the last line we have dropped terms that are subleading for each other out. correspond to codimension one loci in moduli space (since the benefit of hindsightsymmetry we more can transparent. write themetrize Instead over propagator of the in two commuting a ways of form commuting t With the symmetrized prescription the final right-hand side thus confirm the structurepropagator predicted is in consistent (4.60). with This its is fundamental role evide as an inv terms cancel and the propagator can be written as The first group of terms is the result of moving s where the surface degenerates.contribution. It The seems two safe remaining to possibilities assume correspond that t Apart from the integration ranges,According the two to boundary the terms ar prescriptionvalues (4.7), we are instructed to av result of moving with irrespective of how the limits are taken. For the second boun with the usual understandingwhen (4.69) that is inserted the in cutoffs a tree are level removed four-point amplitude acc commute. The precise definition (4.7) may be necessary for mo JHEP01(2008)018 k and s: , one λ /L ameter erators 1 liar [33, 6]. Given s over moduli space, resence of the BRST alue of the Schwinger + 1 exterior derivative that is written without ese boundaries. Thus, k ular contribution. In the to the two boundaries of , namely, from collapsed e string diagrams map all A better understanding of nfinite values of Schwinger a string diagram (diagram es an inverse, following the an those needed to produce mes singular — it produces s, boundary terms will arise ulus annuli are produced. ine parallel to the boundaries on-shell but helped restore the sk. The spine thus collapses to ample, propagators and states , BRST action gives degree ’s act on external states vanish i undary gave the on-shell ampli- tudes. Q ns for open string loop diagrams. : the line formed by the set of all that acts on an external state. Our spine Q ’s in order to get them to act on the external ⋆ – 34 – /L ⋆ B ’s or . acts on the states, and the degree u labeled by external states Ψ Q B/L is more subtle. Previously it was thought that the only rel- n ’s arising from the various propagators results in two effect Ψ Q Q ··· 1 Ψ where n Ψ ··· i Ψ form Ω acting on external states and, in addition to the modular par Q ’s set Schwinger parameters to limit values. ’s end up acting on external states and, k ··· Q 1 Q Q and the need for two Schwinger parameters to represent the op Ψ . Since string amplitudes are integrals of differential form i n Q ’s are moved across P Ψ , Q ··· , . The BRST action on differential forms on moduli space is fami For any given string amplitude the diagrams in which We have obtained a symmetrized form (4.69) of the propagator In Siegel gauge, an open string diagram has a The second effect of 1 ⋆ Ψ 2. Some 1. Some Ω /L 1 operator 5. Discussion The key new features of the Schnabl gauge propagator are the p the action of the various forms Ω some degree d on-shell. These diagrams havethe more moduli Schwinger parameters space, th one extra parameter for each example made thistwo) clear: with a the four-point amplitude includes extra parameter of integration evant boundaries arisepropagators. when In Schwinger the parameters four-stringtude go amplitude (diagram this to one). familiar zero bo parameter We — have naively found a thatfour-string degenerate amplitude a surface it boundary — gave a can at boundaryoff-shell also term infinite exchange that give symmetry. v vanished a We reg doparameters not can know contribute if on-shell boundaries in at general i string ampli boundary terms. In aas string the diagram with various propagator states. We have not discussed the rules needed to deal with th work remains to bethe done propagator to might fully arise understandpreliminary by tree discussion further amplitudes. at analysis the of beginning how of it section defin 4.3. string midpoints on all thethat propagators — divides on the each strip, stripopen the in string l two midpoints equal to infinity. parts.are wedges In and In the all disk string Schnabl midpoints picture,a gauge are for point. at th ex the It center appears of theIn that di this a gives planar interesting one-loop complicatio the amplitude annulus the diagram. spine Ifthe is closed the a string spine curve degeneration. collapses homotopic It the is annulus not beco clear how finite mod JHEP01(2008)018 the (A.1) (A.2) (A.3) (A.4) (A.5) ξ . arctan 2 π his paper. LR wishes or Investigator Award. ds rather new kind of s canonically associated by the U.S. DOE grant ≡ rfaces [34, 35] — through he National Science Foun- ) artial support during the expressed in this material ws of the National Science ξ . ( hese facts. seful algebraic identities. We . . grams. We now see that in + diagrams may have significant . R + + L ⋆ rence) and Scuola Normale Su- B ring physics in open string field B B , f B + ) ξ ± + ( L ]= ]= b B B + + L ) ) ξ ξ ( = = ( ′ f f B,L B,L ± + [ [ πi dξ 2 , B , B – 35 – , , ⋆ I + R + + L L ≡ L L ± + L + L ]= ]= L + + L = , B ) = ± ξ ( L L,L L,L + [ [ T L ) ) ξ ξ denote the left and the right part of an operator. We have ( ( are the zero modes of the stress tensor and of the antighost in ′ f f R B πi dξ indicates BPZ conjugation. We define 2 and and ⋆ I L L ≡ L It used to be thought that covariant open string field theory i The superscript Two basic commutation relations are with a certain way toStrebel decompose the quadratic moduli space differentials ofSchnabl and Riemann gauge su minimal (and areastring presumably string diagrams. in dia A good any geometricalimplications projector understanding for of closed gauge) these string one fieldtheory. theory fin and for closed st Acknowledgments We thank Yuji Okawato for thank useful the comments Galileo onperiore Institute a for (Pisa) draft Theoretical for Physics versioncompletion (Flo of their of t great this hospitality work.DE-FC02-94ER40818. and The The work the work of L.R. INFN ofdation is B.Z. for supported Grant in No. is p part PHY-0354776 supported by t andAny in by opinions, part the DOE findings, Outstandingare and Juni those conclusions of or theFoundation. recommendations authors and do not necessarily reflect theA. vie Notation and algebraic identities In this appendix we collectrefer our to main [14, definitions 23, and 9] several u for a moreA.1 detailed Basic exposition properties of many ofThe t operators The subscripts From (A.3) we deduce conformal frame of the sliver, JHEP01(2008)018 : . y , n n φ (A.9) (A.8) (A.7) (A.6) φ ∗ and (A.15) (A.10) (A.14) (A.12) (A.11) (A.13) ∗ x ...... ∗ . Algebraic ∗ y . m , m n φ n ) , , φ φ ]= ) ) φ + L n n + L + L ∗ φ φ B L ) . assmann-odd string B x,y ∗ − + L − ⋆ . . . + R + L L − ⋆ L L ∗ − L ( . . . B L T B y . ( ∗ ( m ( = = − T φ ∗ ∗ m ) − , . φ ]= ) + L ) ) ...e . . . ses, the relevant Lie algebra is + L are all derivations of the star . . . L + L + L . L, y L, y L ∗ ∗ ...e L ∗ joint) representation of L x,y + − L − , 2 1 − , ∗ 1 = = ! L ⋆ ⋆ φ φ B ( φ ⋆ − 1 denotes its Grassmann property: 2 ) L k on products of string fields: T TL L ( φ + L = 0. One then verifies that k − φ 1 1 1 1 ( L − ) TL T − − ⋆ φ L ( 1 2 e e + L ∗ − − ⋆ − − 1 − − ∗ y B =1 L e e + + L L ⋆ − à k m B =1 , − when [ , x + + L L L k m . . . B B = ( L , x P . . . B = ( B B T T T ∗ P TL , T ∗ 1) , T T − − − 1 yx ⋆ TL 1) − − e e e e e 1 − φ e e e L , and ( αx − φ ∗ and commutation relation [ , y ( e , + R – 36 – = = = = ∗ + L + L 1 =2 L n and L y =2 L 1 n y B ) φ =2 n + + + + ! =2 X L L L L n m X 1) 1) φ m ⋆ T y X m X m − B B B B − − − + + ) ) ⋆ e T α 0 1 0 1 TL TL + = + n and e n + L + L e − TL ( n − − ( B n φ à L L φ TL e e x − β α (1 − φ φ , − − xy ∗ − e ∗ e e e = ⋆ ∗ L + L e ∗ , ( L x ( x L = = )= )= = T ...... T . . . n n . . . − ) ) ∗ = ∗ − − e φ φ βy ∗ + L + R ∗ ) e ) 2 ⋆ L L + ∗ ∗ 1 ) ) 1 x 1 L − − 1 φ φ αx ⋆ L L ⋆ , e ( ( ...... Lφ B Bφ L + L T T ∗ ∗ e − L 2 2 e φ φ − ∗ ∗ ) = ( ) = ( ) = ( ) = ( L n n n n 1 1 φ φ φ φ φ φ ( ( ∗ ∗ ∗ ∗ ⋆ TL TL ...... − − e ∗ ∗ ∗ ∗ e 1 1 1 1 φ φ φ φ ( ( ( ( ⋆ ⋆ L B L B The operators algebra. This implies a simple action of Recurrent special cases are: As two by two matrices, Other identities that can be proved by similar methods are: Here and elsewhere, a string field in the exponent of it is 0 modfield. 2 We for also a have Grassmann-even string field and 1 mod 2 for a Gr A.2 Reordering formulas We often need to reordertwo-dimensional, exponentials with of generators operators. In all ca identities are most easily derived by using the explicit (ad JHEP01(2008)018 , + L elian (A.21) (A.25) (A.24) (A.28) (A.30) (A.26) (A.23) (A.29) (A.22) (A.16) (A.27) (A.19) (A.20) (A.17) (A.18) , . ℓ . = 0 α α W i W ] (0) ⋆ , the semi-infinite cylinder φ B . α . ◦ } 1) W , , f ge states are: edge states are: , . . h − = 0 α α , } + 1 1 α = ℓ α W W , |Ii − 1)+1 α + + L L + L α W . + − , ) + ( . B B + α α +1) z . W αL + ) α ) ( R β α UHP z B + − T ( α T L i α α W + B ∼ e − T T − α + L e e e e ∼ − − (0) αB = W , z + 1) W W W W φ + L αB − , z 0 α = ◦ B = = = = |Ii 0 = = (1+ = = (1 β f ≥ + annihilate the identity and commute with α α α α coincides with the identity of the star algebra; – 37 – ≥ L ◦ α α α α z = ( = [( W 2 α W W W W + z R ∗ ) ) α α α W W W ⋆ − ) ) ⋆ Im + L + L B α e TL 2 BW | W W ≡ I + + L L Im L L TL B ) − | ] z − W − − 1+ − = 0 e B B + R ⋆ ⋆ L z e i ( + ◦ L L L L W − − ( α T ≡ { 1 B T ⋆ 1) − − ℓ ≡ { − B W − e | ( C B f e + L − 0, are surface states defined by their overlap with a generic α ( for the semi-infinite cylinder of circumference L and α ℓ W ( ≥ C + R i ≡ h + ( α L , L , φ α − . The state α coincides with the SL(2) vacuum. The wedge states obey the ab + 1, W + L i W +1 + 1) α h 0 L α | C α [( = = , 1 φ α W W hence they annihilate all wedge states, The derivations Other important conservation laws are We also use the notation the state of circumference Some identities for the action of Virasoro exponentials on w Some identities for the action of antighost operators on wed In the last equality we have used the definition of the surface Fock state A.3 Wedge states The wedge states Clearly algebra In the operator formalism, the wedge states can be written as JHEP01(2008)018 . ) , ) 23 (B.6) (B.1) (B.2) (B.7) (B.4) (B.8) (B.3) (B.9) (B.5) r s ( (B.10) 13 s r∂c 12 [19] defined s − 4 ) B r r a sum of small . ( − c j p · 24 i p s ′ . , )= α 14 r B 2 ( )) s ¯ ¯ ) 3 , r , counterclockwise. The 12 ij r s follows [19]: ( ℓ eft boundary downwards. r s s 2 ∂cc C ¯ ¯ 3 ) À can be moved across a ghost r − = z ∂cc . ( , i