From the octagon to the SFT vertex — gluing and multiple wrapping

Journal of High Energy Physics, Jun 2017

We compare various ways of decomposing and decompactifying the string field theory vertex and analyze the relations between them. We formulate axioms for the octagon and show how it can be glued to reproduce the decompactified pp-wave SFT vertex which in turn can be glued to recover the exact finite volume pp-wave Neumann coefficients. The gluing is performed by resumming multiple wrapping corrections. We observe important nontrivial contributions at the multiple wrapping level which are crucial for obtaining the exact results.

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From the octagon to the SFT vertex — gluing and multiple wrapping

Received: April the octagon to the SFT vertex | gluing and Open Access 0 1 c The Authors. 0 1 0 ul. Lojasiewicza 11 , 30-348 Krakow , Poland 1 Konkoly-Thege Miklos u. 29-33, 1121 Budapest , Hungary 2 Institute of Physics, Jagiellonian University 3 MTA Lendulet Holographic QFT Group, Wigner Research Centre for Physics We compare various ways of decomposing and decompactifying the string eld theory vertex and analyze the relations between them. We formulate axioms for the octagon and show how it can be glued to reproduce the decompacti ed pp-wave SFT vertex which in turn can be glued to recover the exact nite volume pp-wave Neumann coe cients. The gluing is performed by resumming multiple wrapping corrections. We observe important nontrivial contributions at the multiple wrapping level which are crucial for obtaining the exact results. vertex; AdS-CFT Correspondence; Integrable Field Theories - gluing and 1 Introduction 2 Cutting pants into DSFT vertex, octagon 2.1 2.2 2.3 The decompacti ed SFT vertex Naive resummation of the octagon 3 The structure of multiple wrapping corrections Mirror channel compacti cation | cluster expansion 3.2 The structure of the multiple wrappings 4 Resumming the octagon 5 Resumming the string vertex 6 Conclusions A Large volume expansion of the plane-wave DSFT vertex B DSFT vertex axioms from octagon axioms 1 Introduction The AdS/CFT correspondence relates string theories on anti de Sitter backgrounds to conformal gauge theories on the boundary of these spaces [1]. The energies of string states correspond to the scaling dimensions of local gauge invariant operators which determine the space time dependence of the conformal 2- and 3-point functions completely. In order to build all higher point correlation functions of the CFT one needs to determine the 3-point couplings, which is in the focus of recent research. String theories on many AdS backgrounds are integrable [2{5] and this miraculous in nite symmetry is the one which enables us to solve the quantum string theory dual to the strongly coupled gauge theory.1 In the prototypical example the type IIB superstring theory on the AdS5 S5 background is dual to the maximally supersymmetric 4D gauge theory. Integrability shows up in the planar limit and interpolates between the weak and strong coupling sides. The spectrum of string theory, i.e. the scaling dimensions of local gaugeinvariant operators are mapped to the nite volume spectrum of the integrable theory, which has been determined by adapting nite size techniques such as Thermodynamic Bethe Ansatz [7{9] (consequently developed into a NLIE [10] and the quantum spectral 1For a review see the collection in ref. [6]. curve [11, 12]). Further important observables such as 3-point correlation functions or nonplanar corrections to the dilatation operator are related to string interactions. A generic approach to the string eld theory vertex was introduced in [13] which can be understood as a sort of nite volume form factor of non-local operator insertions in the integrable worldsheet theory. There is actually one case when the 3-point function corresponds to a form factor of a local operator insertion. In the case of heavy-heavy-light operators the string worldsheet degenerates into a cylinder and the SFT vertex is nothing but a nite volume form factor, see [14{16]. Another approach through cutting the string worldsheet corresponding to a 3-point correlation function into two hexagons was introduced in [17], see also [18{23] for further developments. The string eld theory vertex describes a process in which a big string splits into two smaller ones. In light-cone gauge xed string sigma models on AdS5 S5 and some similar backgrounds, the string worldsheet theory is integrable and the conserved J-charge serves as the volume, so that the size2 of the incoming string exactly equals the sum of the sizes of the two outgoing strings. Initial and nal states are characterized as multiparticle states of the worldsheet theory on the respective cylinders and we are interested in the asymptotic time evolution amplitudes, which can be essentially described as nite volume form factors of a non-local operator insertion representing the emission of the third string. In order to be able to obtain functional equations for these quantities we suggested in [13] to analyze the decompacti cation limit, in which the incoming and one outgoing volume are sent to in nity, such that their di erence is kept xed. We called this quantity the decompacti ed string eld theory (DSFT) vertex or decompacti ed Neumann coe cient. We formulated axioms for such form factors, which depend explicitly on the size of the small string, and determined the relevant solutions in the free boson (plane-wave limit) theory. Taking a natural Ansatz for the two particle form factors we separated the kinematical and the dynamical part of the amplitude and determined the kinematical Neumann coe cient in the AdS/CFT case [24], too. These solutions automatically contain all wrapping corrections in the remaining nite size string, which makes it very di cult to calculate them explicitly in the interacting case, especially for more than two particles. It is then natural to send the remaining volume to in nity and calculate the so obtained octagon amplitudes. One can go even further and introduce another cut between the front and back sheets leading to two hexagons, which were introduced and explicitly calculated in [17]. Since we are eventually interested in the string eld theory vertex, we have to understand how to glue back the cut pieces. This paper is an attempt going into this direction. Clearly, gluing two hexagons together we should recover the octagon amplitude.3 Gluing two edges of the octagon we get the DSFT vertex, while gluing the remaining two edges we would obtain the nite volume SFT vertex, which would be the ultimate goal for the interacting theory. The study of various observables in integrable quantum eld theories in in a natural way can be decomposed into a number of stages. Firstly, the problem posed in in nite volume typically yields a set of axioms or functional equations for the observable 2In this paper we will use terms size and volume interchangeably to mean the circumference of the cylinder on which the worldsheet QFT of the string is de ned. 3We will not consider this case, however, in the current paper. in question which often can be solved explicitly. The key property of the in nite volume formulation is the existence of analyticity and crossing relations which allow typically for formulating functional equations [25{28]. Secondly one considers the same problem in a nite volume neglecting exponential corrections of order e mL. In this case the answers are mostly known like for the energy levels, generic form factors4 and diagonal form factors [29, 30]. However, some of these answers are still conjectural and are not known in various interesting cases. Thirdly, one should incorporate the exponential corrections of order e mL, which are often termed as wrapping corrections as they have the physical interpretation of a virtual particle wrapping around a noncontractible cycle. The key example here are the Luscher corrections for the mass of a single particle [31] and their multiparticle generalization [32]. Once one wants to incorporate multiple wrapping corrections, the situation becomes much more complicated however in some cases this can be done [33]. In the case of the spectrum of the theory on a cylinder, fortunately one does not need to go through the latter computations as there exists a Thermodynamic Bethe Ansatz formulation which at once resums automatically all multiple wrapping corrections and provides an exact nite volume answer [34]. Unfortunately for other observables like the string interaction vertex we do not have this technique at our disposal and we may hope that understanding the structure of multiple wrapping corrections will shed some light on an ultimate TBA like formulation. This is another motivation for the present work. In fact one of the new results of the present paper is an integral representation for the exact pp-wave Neumann coe cient which involves a measure factor reminiscent of various TBA formulas. In [17], a formula for gluing two hexagons was proposed: insert a complete basis of particles on the mirror edge5 and sum over them This is in fact a rather formal expression as the observable in question is divergent. Also we allowed for a generic measure factor. It will indeed turn out that the measure factor is nontrivial for multiple wrapping. In this paper we analyze the multiple wrapping terms for a massive free boson theory which corresponds to the relevant quantities being evaluated for the pp-wave geometry. The outline of this paper is as follows. In section 2 we will review the decompacti ed SFT vertex axioms as well as introduce the axioms for the octagon. We will also deduce the measure factor by requiring that gluing the octagon through (1.1) reproduces the exact pp-wave decompacti ed SFT vertex. Then in section 3, we will revisit the gluing procedure (1.1) from the point of view of cluster expansion (or equivalently compacti cation in the mirror channel) and isolate the key ingredients which are necessary for obtaining the nite volume answer for a generic observable. We will also illustrate this structure with the well known relativistic examples of ground state energy and LeClair-Mussardo formula for the nite volume 1-point expectation value [36]. In the following two sections we will 4By this we mean form factors with no coinciding rapidities in any channel. 5In the relativistic case this would correspond to inserting particles in the channel with space and time interchanged i.e. with rapidities + i =2. show that one can provide natural choices for these ingredients which enable us to glue the octagon into the decompacti ed SFT vertex and then glue the decompacti ed SFT vertex into the exact nite volume pp-wave vertex.6 We close the paper with a discussion and two appendices, one of which contains the derivation of the integral representation of the pp-wave Neumann coe cient, and the other a discussion of the relation between octagon axioms and decompacti ed SFT vertex axioms in the context of the gluing formula. Cutting pants into DSFT vertex, octagon The string eld theory vertex describes the amplitude of the process in which a big string (#3) splits into two smaller ones (#1 and #2), see left of gure 1. In light-cone gauge xed string sigma models the conserved J-charge serves as the volume, which adds up in the states and the asymptotic time evolution amplitudes can be understood as nite volume form factors of a non-local operator insertion. In calculating these quantities we go to the decompacti cation limit, in which the volumes J3 and J2 are sent to in nity, such that their di erence J3 leading to in nite volume form factors, see the middle of gure 1. These form factors automatically contain all wrapping corrections in the remaining nite size string, which makes very di cult to calculate them explicitly. It is then natural to send the remaining volume to in nity and calculate the so obtained octagon amplitudes. See the right of gure 1 for the geometry. Since eventually we are interested in the string eld theory vertex we have to glue back the cut edges. Gluing two edges of the octagon we get the decompacti ed SFT vertex, while gluing the remaining two edges we obtain the seeked for SFT vertex. The decompacti ed SFT vertex In our previous paper [13] we formulated the axioms of the DSFT vertex, which we also called the generalized Neumann coe cients. Here for simplicity we quote the axioms for a relativistic theory with a single type of particle. For initial particles living on string #2 with rapidities i they read as follows: The exchange axiom is NL( 1; : : : ; i; i+1; : : : ; n) = S( i i+1)NL( 1; : : : ; i+1; i : : : ; n) 6Here we are concerned with just the bosonic case so we do not consider issues related with the prefactor. The periodicity axiom explicitly includes the volume of the small emitted string: NL( 1 + 2i ; 2; : : : ; n) = eip1LNL( 2; : : : ; n; 1) The kinematical singularity axiom, which relates form factors with di erent particle numbers, takes the form: iRes 0= NL( 0+i ; ; 1; : : : ; n) = @1 which reads as7 NL( 1; 2) = where the functions dL( ) involve all order wrapping terms. They are given explicitly in terms of deformed Gamma functions which have a rather nontransparent de nition. The above formula exactly coincides with the decompacti cation limit of the pp-wave Neumann coe cient [35]. Remarkably enough, there exists a very compact and transparent integral formula for dL( ) which we derive in appendix A. It takes the form k( ) = The multiparticle solutions can be xed from the kinematical residue equation and have NL( 1; : : : ; n) = pairings (i;j) pairs Thus we sum for all possible pairings of the rapidities f ig and take the product for the pairs of the 2-particle expressions. Clearly this form is compatible with Wick theorem in the free boson theory. From the decompacti ed string vertex one can go in two opposite directions. Either one can glue together the remaining two mirror edges (the dashed lines between #2 and #3 and between #2 and #3' in gure 2) thus obtaining the nite size SFT vertex, which is really the ultimate goal of this program, or one can go in the opposite direction and send the remaining volume L to in nity thus obtaining the octagon. In the case of the free massive boson (the pp-wave) the exact nite size SFT vertex Neumann coe cient (up to an overall normalization) can be expressed very compactly as NLL12 ( 1; 2) = NL1 ( 1; 2) In section 5 we will describe how this form can be obtained by gluing together the decompacti ed SFT vertex. 7Here we choose the NL(;) = 1 normalization. amplitude is on the right. The glued mirror edges between #31 and #3'1 are indicated by a dotted nite size string represented by the circle serves as a non-local operator insertion in the Now, however, we will concentrate on the octagon which appears when we send the remaining volume, L, to in nity. E ectively, this limit not only sends the volume of string #1 to in nity but also cuts the space of string #3 into two disconnected pieces, which we denote by #3 and #3'. They are connected by crossing through string #1 on one side and through string #2 on the other. This suggests the octagon description as shown of gure 2. Let us formulate the functional relations for this quantity. The octagon following axioms: The octagon amplitude, when particles with rapidities i are in string #2, satisfy the The exchange axiom relates particles on the same kinematical edge to each other thus is not changed compared to the DSFT vertex axioms: O( 1; : : : ; i; i+1; : : : ; n) = S( i i+1)O( 1; : : : ; i+1; i; : : : ; n) In the periodicity properties we have to cross a particle from domain #2 to #3 rst, then from #3 to #1, then from #1 back to #3' and nally to #2 leading to a 4i O( 1 + 4i ; 2; : : : ; n) = O( 2; : : : ; n; 1) In the kinematical singularity axiom particles in domain #2 can feel particles in domain #3 only by crossing with i , (and not by crossing with i ), thus we have iRes 0= O( 0 + i ; ; 1; : : : ; n) = O( 1; : : : ; n) i.e no S-matrix factors appear, which make their determination easier. Particles on di erent edges of the octagon can be obtained by analytical continuation, what we describe in detail in the appendix B. summing up for all multi-particle mirror states, represented by empty circles. Physical particles are represented by solid circles. The two particle octagon solution for the free boson theory is O( 1; 2) = O( 1; : : : ; n) = pairings (i;j) pairs The multiparticle solutions can be xed from the kinematical singularity axiom and take Our main problem now is to understand how to obtain the DSFT vertex with string #1 nite size L, by gluing together the two mirror edges between #1 and #3 and between #1 and #3' (see gure 2). Naive resummation of the octagon A very formal de nition of gluing two mirror edges was proposed in [17]. We demonstrate this idea on the example how the DSFT could be obtained from the resummation of octagons. The idea of the gluing is to interpret the cutting as a resolution of the identity n! i=1 where ju1; : : : ; uni denotes an in nite volume mirror state living between the spaces #3 and #1, while hun; : : : ; u1j is its dual mirror space living between the space #1 and #3'. In formulas it means for a two particle DSFT vertex that NL( 1; 2) = 1(u)O( 1; 2; u+; u )e LE(u) + 1Z 1 du1Z 1 du2 2(u1; u2)O( 1; 2; u1+; u2+; u2 ; u1 )e L(E(u1)+E(u2)) + : : : = u i32 . Graphically it can be represented as on gure 3. Since the mirror particle-anti-particle pairs come on the opposite edges of the octagon the amplitude is singular due to the kinematical singularity axioms. However, it is very natural to normalize the amplitude by the \in nite" empty glued octagon: norm = 1 + 1(u)O(u+; u )e LE(u) + 1 Z 1 du1 Z 1 du2 which exactly su ers from the same divergences. Indeed, the special \free" form of the octagon amplitudes guarantees that the normalization in the denominator removes all the disconnected singular terms and only nite regular expressions remain: NL( 1; 2)= O( 1; 2) + 1(u)Oc( 1; 2; u+; u )e LE(u) + 2(u1; u2)O(u1+; u2+; u2 ; u1 )e L(E(u1)+E(u2)) + : : : 1Z 1 du1Z 1 du2 2(u1; u2)Oc( 1; 2; u1+; u2+; u2 ; u1 )e L(E(u1)+E(u2)) + : : : u− u+ last is disconnected. where O( 1; 2; u1+; u1 ; : : : ; un+; un )c denotes the connected part, i.e. the one which is connected with the following graphical rules: put the rst vertex for 1 and the last for 2, while in between n vertices for each ui. Left side of the ui represents ui , while the right ui+: For each propagator O( 1; uj ) draw an edge from 1 to the right/left of uj . For a propagator O(ujj ; ukk ) leave the j side of vertex uj and arrive at the j side of uj . See gure 4 for the diagrams representing O( 1; 2; u+; u ). The connected part of the multiparticle octagon consists of exactly those graphs which are connected. Actually the sum of these terms are not singular and takes a very special nite form. For one pair of mirror particles we have O ( 1; 2; u+; u ) = O( 1; u+)O( 2; u ) + O( 1; u )O( 2; u+) c = O( 1; 2)(k(u k(u) = which generalize to n particles as O ( 1; 2; u1+; u1 ; : : : ; un+; un ) = O( 1; 2) Y(k(ui c This can be checked by noticing that the connected form factors satisfy the kinematical of the connected form factors. NL( 1; 2) = O( 1; 2)dn( 1)dn( 2) ; dn( ) = eR 11 d2u k(u )e mL cosh u At the leading wrapping order the naive result is correct, indicating that we are missing some relevant contributions from multiple wrappings. Actually the missing terms come one could demand that NL( 1; 2) should satisfy the DSFT vertex axioms. In appendix B we show that it is equivalent to the teleportation requirement, which can be rephrased as that after an analytical continuation i we have Expanding both sides and taking into account that k( i ) = k( ) we can see that the residue terms must sum up to e ipL=dL( ). Evaluating at leading order gives from the double pole term produces a term 12 e 2ipL, which can be canceled choosing 1(u) = 1 + e mL cosh u + O(e 2mL cosh u) Calculating systematically the higher order terms we can nd that which gives the expected results: 1(u) = n(fuig) = Y 1(u)e mL cosh u = log 1 dL( + i ) = (1 1(u) = 1 + O(e mL cosh u) Clearly the relevant nontrivial terms are kinds of \diagonal" contribution associated with multiple wrapping. In order to understand better their role and origin we will now look at the gluing process from the point of view of so-called cluster expansion in relativistic integrable eld theories. Then we will revisit again the gluing of the octagon as well as describe how one can glue the decompacti ed SFT vertex into the nite volume one. The structure of multiple wrapping corrections In this section we rst exhibit explicitly the exactly known observables for a free massive boson which will give insight to the multiple wrappings, starting from the completely standard examples of free energy and going on to the quite intricate formulas for the exact string vertex. We then analyze the general structure of the wrapping corrections. The ground state energy. The ground state energy can be obtained from the large R limit of the torus partition function with the upper/lower signs corresponding to a free boson/fermion. Expanding the above formula in a power series in e mL cosh gives multiple wrapping contributions to the ground Incidentally the exact equation which holds in the interacting case has a very similar E0(L) = cosh log 1 E0(L) = where "( ) is a solution of the relevant TBA equation. The LeClair-Mussardo formula. The nite volume expectation value of a local operator is given by the following formula [36] hOiL = Here Fnc( 1; : : : ; n) is the in nite volume (connected) diagonal form factor of the operator O. Remarkably enough the above formula again generalizes to the interacting case through the simple substitution mL cosh hOiL = NLL12 ( 1; 2) = NL1 ( 1; 2) For completeness let us quote here the nite volume expansions for the decompacti ed SFT vertex as well as the nite size string vertex. It is illuminating to recognize the structural similarity of the multiple wrapping terms appearing in these expressions with the relativistic formulas given above. The decompacti ed string vertex. The formula for the decompacti ed string vertex with two particles on string #2 takes the form: NL( 1; 2) = O( 1; 2)dL( 1)dL( 2) where the logarithm of the function dL( ) is log dL( ) = We note a surprising similarity with the ground state energy formula. nite size string vertex. The formula for the string vertex with all the three strings being of nite size Li has been derived by a direct calculation in [35]. The formulas there can be recast into a simpler form when expressed in terms of rapidities and take the form (again up to an overall normalization): Let us now review the approach of mirror channel compacti cation aka cluster expansion. Let us rst consider the case of the partition function evaluated on a torus of size L where R is very large in order to extract the ground state. As in the derivation of the TBA, it is convenient to perform this calculation in the mirror channel (which is compacti ed to the large size R). Then the partition function is by de nition the summation over all states in the mirror theory weighted with e EL where E is the mirror channel energy: Here for the free boson, mode numbers can coincide hence we have n1 n2. In contrast for an interacting theory or for a free fermion we would have a sharp inequality. In the next step one takes the continuum limit however, and this is the key point, one has to take care of the diagonal terms and rst separate them out Writing all the contributions appearing in (3.9) gives n1=n2 We see that this coincides with the rst terms of the expansion of measure factor log 1 e mL cosh . Let us emphasize that their interpretation is not as are not even part of the spectrum. Provisionally a useful interpretation of these terms is that they represent multiply wrapped single particles. This interpretation seems to provide the correct intuition for the treatment of such terms in all cases considered in this paper. The LeClair-Mussardo formula arises when we insert a local operator into the above expansion. We thus have to evaluate the diagonal expectation values of the type hn1n2j O jn2n1iR for asymptotically large R, i.e. neglecting any wrapping terms in R. In this limit the expectation value can be written8 as a linear combination of appropriate measure factors 8This formula is still conjectural for more than two particles but there is overwhelming evidence that it is correct [30, 37]. (with the only explicit R dependence) and in nite volume diagonal form factors of the local operator O. For the case of a free massive boson we have the following explicit formulas for up to two particles hn1j O jn1iR = hn1n2j O jn2n1iR = hn1n1j O jn1n1iR = 0 + 2 Note the factor of 2 in the diagonal double wrapping term. It is exactly this factor (and diagonal term in (3.11) and e ectively transforms the measure factor 9I.e. neglecting all e mR terms. appearing in the LeClair-Mussardo formula. The structure of the multiple wrappings Looking at the above two examples, we see that the computation of the nite volume observable hX iL can be summarized as regularizing the mirror channel (e.g. by compactifying nite but large volume R), decomposing the summation over a complete basis of states into independent sums of single and multiple wrapped particles with appropriate combinatorial factors, and nally providing an expression for the diagonal nite volume asymptotic9 expectation values, namely Z hX iL = h;j X j;iR +X hn1n2j X jn2n1iR e (En1 +En2 )L (3.20) 1 X D hn1n2n3j X jn3n2n1iR e (En1 +En2 +En3 )L In an interacting or fermionic theory one has to ip some signs as there all quantized mode numbers must be distinct. The key remaining information are the above R-regularized diagonal expectation values. We may expect that they have the following general form The measure factor is the only place with explicit R dependence. For the free boson we expect it to take a simple form ( ; ; R) = Q The second factor in (3.21) should be a quantity de ned in in nite volume associated to the observable X which should follow from some appropriate functional equations. Alternatively we could calculate hX iL instead of Z hX iL. By this we remove many disconnected terms (as Z 1 has the same structure as hX iL.) The modi ed quantity appearing in the expansion is denoted by F Xc (fni gi2 ) where the superscript c indicates that one would have to take just the connected part. Note that care should be taken to de ne these generalized form factors also for the multiply wrapped particles. How to do it in general is by no means obvious. The main result of this paper is to provide the relevant expressions both for the octagon and for the decompacti ed string vertex with two external particles such that summing (3.20) for the octagon yields the decompacti ed string vertex and summing (3.20) for the decompacti ed string vertex gives the exact nite volume string vertex. Before doing so, we summarize the relevant quantities both for the ground state energy and for the LeClair-Mussardo formula. Ground state energy. The ground state energy is related to the torus partition function decomposition degenerates only to one \trivial" term = 1 = Qi REi FX (fni In particular, in our normalization it implies that ( ki)g) = Y REi LeClair-Mussardo formula. In the case of the LeClair-Mussardo formula we analyze the expansion of hOiL instead of Z hOiL, since by this trick we can remove all disconnected terms and the decomposition degenerates only to one term F Xc (fni ( ki)g) = Fnc(fnig) Y ki where Fnc(fnig) is the connected diagonal form factor. Thus the wrapping order appears only as a combinatorial factor. Resumming the octagon the connected contributions (see section 2.3 for their de nition). We propose the following form of the nite volume expectation values which, when inserted into (3.20) will exactly reproduce the decompacti ed string vertex with string #1 being of length L: h;j O 1; 2 j;iR = O( 1; 2) hn1j O 1; 2 jn1iR = O( 1; 2) (k(u1 hn1n2j O 1; 2 jn2n1iR = The key assumption now involves the expectation values when some of the mirror particles wrap multiple times. The exact answer implies that the relevant expectation value does not depend on the wrapping order. E.g. we have = hn1j O 1; 2 jn1iR = O( 1; 2) (k(u1 With the above formulas in place, it is simple to generalize as = u = hfnigj O 1; 2 jfnigiR = O( 1; 2) Y As then one can easily convince oneself (looking e.g. at all terms up to 3rd order or comparing to the free boson ground state energy) that (3.20) can be summed up to give exactly the decompacti ed string vertex = O( 1; 2)dL( 1)dL( 2) Resumming the string vertex Passing from the decompacti ed string vertex with string #1 being of size L1 and strings #2 and #3 being in nite to the nite volume string vertex with all strings having size Li can be done following closely the strategy employed for the octagon. We will again consider a con guration with two particles on string #2. In order to do the in nite volume part of the calculation it is convenient to transport the mirror particles up to string #2. Now the rapidities will have to be shifted by i =2 so in this section we will denote Since we are dealing with a free theory the decompacti ed vertex with multiple particles will be obtained by Wick contractions but now with pairing performed with NL1( 1; 2) = NL1 ( 1; 2) = O( 1; 2)dL1 ( 1)dL1 ( 2) So we see that the nontrivial part of the computation is almost exactly the same as for the octagon (up to the rede nition of u here) and the L1-dependent factors will appear only as an overall product for all particles entering the amplitude. It is clear that we thus get the following expressions: j;iR = NL11 ( 1; 2) hn1j NL11; 2 jn1iR = hn1n2j NL11; 2 jn2n1iR = NL11 ( 1; 2) (k(u1 The product of the dL1 (:) functions can be simpli ed using the functional equations dL1 (u+)dL1 (u ) = 1 In order to see the crucial role of the above remaining u-dependent factor let us consider the expression (3.20) up to the single particle term. We have The 1 particle term thus splits into a di erence of two terms: one with a wrapping factor e mL2 cosh u and the other with the wrapping factor e m(L1+L2) cosh u e mL3 cosh u, where indeed exactly the rst terms in the expansion of It is clear that the two particle term coming from (5.5) will contribute to the exponentiation of the above structure. However the contribution of the doubly wrapped particle is quite subtle and requires some care. In order to motivate our proposal, let us recall that the decompacti ed string vertex axioms introduced in [13] involve an overall monodromy factor eipL1 . Now since we are considering a particle which wraps twice across the vertex, we expect that it would e ectively feel a factor e2ipL1 . Thus it is very natural to expect that the generalization of formula (5.4) to a doubly wrapped particle takes the form NL11 ( 1; 2) (k(u1 Now we can examine the doubly wrapped particle contribution in (3.20): We see that this yields the rst nontrivial double wrapping terms in the expansion of the logarithms in (5.8). In order for this to work it was absolutely crucial that the double wrapped particle feels e ectively the double factor e2ipL1 . It is clear that analogous property should hold for multiple wrapped particles. Repeating the above for higher number of particles we see that we obtain the exact nite volume Neumann coe cient NLL12 ( 1; 2) = NL1 ( 1; 2) dL1 ( 1)dL2 ( 1) dL1 ( 2)dL2 ( 2) The nontrivial topology of the SFT vertex allows for various lines of approach towards determining it exactly. By cutting the vertical edges various number of times and decompactifying one obtains the decompacti ed string vertex of [13], the octagon and two hexagons of [17]. Although the nal goal is the determination of the exact vertex, i.e. with all the three strings being of nite size, the necessity of passing through this intermediate decompacti ed stage is that only then we can formulate functional equations for the relevant quantities which incorporate analyticity and various variations of crossing symmetry. One of the contributions of the present paper was to formulate appropriate axioms for the octagon in the interacting case. Hence a key question is to understand the procedure of gluing back the decompacti ed answers into the nal nite volume result. In [17] a formal expression for gluing back was suggested by a summation over a complete set of mirror particles living on the edge which is being glued. This expression is, however, rather formal as it stands and su ers from divergences. The subtleties arise at the multiple wrapping level which is in general di cult The case of the pp-wave vertex (essentially a free massive boson on the string pants diagram) is a very interesting theoretical laboratory for studying these issues as we have at our disposal exact nite volume answers for the nite size SFT vertex as well as its various decompacti ed variations | the decompacti ed SFT vertex and the octagon. As these expressions are exact and incorporate an in nite set of multiple wrapping corrections we may quantitatively explore the subtleties of the gluing procedure. We argue that the quantitative structure of the gluing procedure may be e ciently understood within the so-called cluster expansion (equivalently compacti cation in the mirror channel). There the main ingredient is the asymptotic large mirror volume expectation value for the observable in question which should decompose into a linear combination of measure factors and appropriate in nite volume quantities. This is a standard way to understand ground state energy and the LeClair-Mussardo formula for one point expectation values in relativistic integrable theories. In the present paper we adopt this framework to the case of the octagon and the decompacti ed SFT vertex. Note, however, that even in the classical case of LeClair and Mussardo there is no proof of the general large mirror volume expectation value formula for more than two particles. In the case of the vertex we also do not provide a proof, however our proposed formulas are very natural from the physical point of view. Also a-posteriori it is very nontrivial that any such formulas exist which reproduce the apparently very complicated nite volume Neumann coe cients. We demonstrated that one can resum the multiple wrapping corrections for the octagon into the exact decompacti ed SFT vertex. This necessitates a nontrivial, but quite natural modi cation of the multiple wrapping measure. We then proceed to interpret this modi cation through the cluster expansion where it turns out to arise from certain diagonal terms. We then show that similarly one can resum the decompacti ed SFT vertex and recover the exact nite volume pp-wave Neumann coe cients. There are numerous further questions to investigate. A key question, and one of the long term motivations of this work, would be to guess some underlying exact TBAlike formulation for the SFT vertex. The integral expression for the pp-wave Neumann coe cient obtained in the present paper is very intriguing in that respect. Also in this paper we did not discuss the hexagons at all. It would be interesting to understand this better, as well as the di erences w.r.t. [17].10 In this paper we focused on the 3-point functions and on the way how they could be described by gluing octagons and the DSFT vertex. The 4-point functions, however, are even more interesting and recently there have been activities using integrable methods in their descriptions [38{41]. It would be very challenging to gure out how two octagons (or their modi cations) could be glued together to describe the four point functions. Actually the geometry of the 4-point function allows for two di erent cuttings into two octagons. Demanding their compatibility might lead to non-trivial constraint on the gluing procedure or on the octagon themselves. Acknowledgments RJ was supported by NCN grant 2012/06/A/ST2/00396 and ZB by a Lendulet and by the NKFIH 116505 Grant. RJ would like to thank the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support during the completion of this work. Large volume expansion of the plane-wave DSFT vertex In this appendix we rewrite the plane-wave DSFT vertex into the form, in which multiple wrapping terms can be easily identi ed. Recall from [13, 35] that the DSFT vertex takes the form11 N ( 1; 2) = O( 1; 2)dL( 1)dL( 2) 10A main di erence between the approach of [17] and the considerations of the present paper is that there the light cone gauge choice is di erent for each of the three strings, while here we concentrate on the conventional light cone SFT vertex picture where we have a single gauge choice, and hence e.g. the total size of the strings is conserved. 11Here we normalized the DSFT vertex as NL(;) = 1. We are interested in the large L expansion of ~ mL . The large L expansion of calculated in [35] and rephrased in [13] as dL( ) = ~ ( ) = where ~exp( ) vanishes exponentially for large L in the following way: @ log ~exp(z) = K0(2 n ) = We can integrate this equation as log ~exp(z) = and the constant of integration is xed from the vanishing large volume limit. By introducing the rapidity variable via z = ~exp( ) = exp This implies that k( ) = log dL( ) = log ~exp( ) = In order to check this expression we rst perform an analytical continuation in + i . In doing so a pole singularity of the kernel k(u ) crosses the integration contour, which contribute to the functional relation giving dL( + i ) = (1 which is required by the kinematical singularity axiom. Continuing further to another singularity crosses the integration contour, which contributes with an opposite residue leading to dL( + 2i ) = e ipL dL( ) = which is the required monodromy property of the function. 1Z 1 du1Z 1 du2 1Z 1 du1Z 1 du2 1Z 1 du1Z 1 du2 DSFT vertex axioms from octagon axioms In this appendix we show how the DSFT vertex axioms could be obtained from the octagon axioms. This will shed also light, how we need to use the octagon amplitude to describe particles in the split #3 and #3' domains. For simplicity we present the ideas for the free theory and for 2 particles only. The generalization for the interacting theory can be easily done at the formal level similarly to eq. (2.14). At a less formal level one has to understand how to regularize the kinematical singularities for the mirror particle-anti-particle pairs. Recall that the DSFT vertex can be written in terms of the connected octagons as NL( 1; 2) = O( 1; 2) + 1(u)Oc( 1; 2; u+; u )e LE(u) + 2(u1; u2)Oc( 1; 2; u1+; u2+; u2 ; u1 )e L(E(u1)+E(u2)) + : : : = u 32i . Let us see now how the DSFT axioms are satis ed. The exchange axiom is trivially reproduced as each term in the expansion has this property. In the free boson theory the connected and the disconnected terms are mapped to each other under the exchange 1 $ 2, thus the connected terms are symmetric themselves. In order to show the kinematical singularity axiom we continue analytically 1 ! 1 + i . As a rst step we continue it into the mirror domain between space #2 and #3: 1 ! 1 + i2 . When it is exactly in the mirror domain it will hit a kinematical singularity of the octagon coming from integrals for mirror particles of type ui . We can avoid this singularity by slightly deforming the contours. However, when we continue the particle's rapidity further to domain #3 we cross the integration contour by a pole singularity. Thus we will have two types of contributions: the direct continuations and the pole contributions. See gure 5 for a graphical representation. The direct term, denoted by Nsum( 1 + i ; 2) is simply Let us explain the notation a bit. In the following we understand by Nsum( 1 + i ; 2) the above sum. So whatever is the argument of Nsum( 1; 2) it means we evaluate the octagon sum at that rapidities and we do not continue it analytically. With this notation the residue term is e ip1LNsum( 2; 1 + 3i ) and as we explained Nsum( 2; 1 + 3i ) now denotes the sum 1(u)O( 2; 1 + 3i ; u+; u )e LE(u) + 2(u1; u2)Oc( 2; ; 1 + 3i ; u1+; u2+; u2 ; u1 )e L(E(u1)+E(u2)) + : : : 1(u)Oc( 1 + i ; 2; u+; u )e LE(u) + 2(u1; u2)Oc( 1 + i ; 2; u1+; u2+; u2 ; u1 )e L(E(u1)+E(u2)) + : : : For the DSFT axioms to be ful lled in the general case it is crucial that the factor e ip1L can be factored out from each term. This can be guaranteed by demanding and assuming that higher order poles do not contribute. The mechanism producing the extra residue term was called teleportation in [17]. This also indicates, how we should split the particles between regions #3 and #3': we should sum for all possible distributions with an additional e ipL factor, whenever a particle is moved to region #3'. Observe that it is crucial that we do not have any contributions from double or higher order integrations as they would spoil the above structure. Actually in the free boson case we know that there is a double pole contribution, which can be compensated by an appropriately chosen measure factor. Thus the existence of higher order poles leads to non-trivial measure factors to guarantee This the equation we should satisfy in the general interacting case. Demanding it for the continued rapidities will give restrictions on the de nition of the connected octagon form factors and the measure. true in the free boson theory as k( + i ) = k( ) and follows from our normalization ires 0= NL( 0 + i ; ) = (1 In order to show the periodicity axiom we need to continue further 1 + i In doing the continuation in each term of the sum Nsum( 1 + i ; 2) we do not expect any teleportation as the 1+ + i = u 1 singularity is regularized in the connected part. 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Zoltan Bajnok, Romuald A. Janik. From the octagon to the SFT vertex — gluing and multiple wrapping, Journal of High Energy Physics, 2017, 1-24, DOI: 10.1007/JHEP06(2017)058