Lattice String Field Theory: The linear dilaton in one dimension
aa r X i v : . [ h e p - t h ] O c t Prepared for submission to JHEP
TAUP-2795-13
Lattice String Field Theory: The linear dilaton in onedimension
Francis Bursa (1) , Michael Kroyter (2) (1)
School of Physics and AstronomyUniversity of GlasgowKelvin Building, University Avenue, GlasgowG12 8QQ United Kingdom (2)
School of Physics and AstronomyThe Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv University, Ramat Aviv, 69978, Israel
E-mail: [email protected] , [email protected] Abstract:
We propose the use of lattice field theory for the study of string field theoryat the non-perturbative quantum level. We identify many potential obstacles and examinepossible resolutions thereof. We then experiment with our approach in the particularlysimple case of a one-dimensional linear dilaton and analyse the results.
Keywords:
String Field Theory, Lattice Gauge Field Theories ontents d ≤ l -level 122.1.1 Moving the non-locality to the quadratic term 172.2 Higher levels 172.3 The reality condition 202.4 Truncation of the action to l = 1 in scheme 4 212.5 Automatization using conservation laws 242.6 The problem with scheme 3 252.7 The action in scheme 1 272.8 Analytical study of the lowest mode 282.9 Adding trivial terms to the action 29 α ′ -independence 343.4 Estimate of statistical errors 34 l max L = 20 364.2.2 Results at L = 10 384.2.3 Including a large number of modes at L = 30 404.3 Changing x min L = 20 424.3.2 Results at L = 10 434.4 Small interval length 444.4.1 L = 6 454.4.2 L = 4 474.4.3 L = 3 .
15 484.5 Comparing different interval lengths 484.6 Relations among different expectation modes h T n i T ( x ) in position space 534.6.2 Correlations 544.7 Varying the dilaton gradient 55– i –.7.1 L = 20 564.7.2 L = 10 564.7.3 Interpretation 574.8 Adding trivial parts to the action 604.9 Scheme 1 614.10 Scheme 2 63 String theory is currently the most promising candidate for the unification of all forces.Unfortunately, it is neither clear what string theory is nor even how to define it. The mostcommon “definition” of string theory found in the literature uses scattering amplitudes thatare obtained from world-sheet perturbation theory. However, this perturbative expansioncannot be considered as defining a theory, since the series obtained is most probably anasymptotic one, i.e. it has a vanishing radius of convergence. This state of affairs isvery similar to the one in field theory, where the Feynman diagrams themselves cannotbe considered as a definition of a theory, but the field theory action, from which they arederived, does define a theory. One could hope that something similar could be achievedfor string theory, which would be defined as a field theory of first quantized strings . Theworld-sheet expressions for the scattering amplitudes would then be derived from the fieldtheory action using standard perturbation theory methods. This approach towards thedefinition of string theory goes under the name of string field theory .Furthermore, string field theory should, at least in principle, be good not only fordefining string theory, but also for studying string theory when the world-sheet tools areless adequate. This is completely analogous to the case of standard field theory, when onecannot rely on the standard perturbative approach at strong coupling, high temperatureor high density. Of course, some of the most interesting questions one can pose relate tosuch regimes. In string theory one could hope that string field theory would be useful forthe study of many questions, among which we can find: • The identification of phases of string theory at large coupling and temperature, thephase transitions of the theory and their type. • The examination of consistency and stability of string theory compactified to differentdimensions and of more general, e.g. non-geometric string theory backgrounds. • The study of solitons, in particular D-branes, time-dependent solutions, and otherclassical objects. See e.g. [1] for an introduction to string theory and the reviews [2–4] for an introduction to string fieldtheory. – 1 –
The study of various quantum effects, such as the scale-dependence of masses andcouplings, which are not protected by supersymmetry. • A particularly ambitious task would be the study of (portions of) the string theorylandscape. A particular example could be the understanding of the landscape thatis related to changes of the open string background. • The study of known string theory dualities and the identification of new ones.While other approaches towards the non-perturbative definition of string theory alsoexist, string field theory is very natural in principle and construction of such theories wasattempted already in the first days of string theory [5, 6]. We now have several suchformulations. Among these formulations, the more promising ones are those in which thetheory is covariant and universal . Such formulations were introduced for the bosonicopen string [7], for the bosonic closed string [8] and to some extent also for the opensuperstring [9–15] and the heterotic string [16–19]. Interesting new ideas regarding closedsuperstring field theory were presented in [20–22].Initially, string field theories were put to the test by demonstrating that they lead tothe same scattering amplitudes as the world-sheet theory, i.e. one attempted to demon-strate that a proper single covering of the relevant moduli space is achieved and that theperturbative expansion for the amplitudes is correctly reproduced [23, 24]. For quite sometime such perturbative studies formed the main focus for research in the field. This state ofaffairs changed following the realization that Sen’s conjectures [25, 26] can be studied usingstring field theory, i.e. that a field theoretical approach is a most adequate one for studyingnon-perturbative classical solutions, in particular solutions describing the condensation ofthe open string tachyon. Following the first attempts to address these questions in thecases of the bosonic string [27, 28] and the superstring [29], the interest of the communitydrifted towards the study of such classical solutions. This led to a large body of work,which culminated with the construction of the first non-trivial analytical solution to stringfield theoretical equations of motion by Schnabl [30] (see also [31–33]). The new tools thatwere developed for the construction of this tachyon vacuum solution were further used forthe construction of other analytical solutions, including the construction of simpler tachyonvacuum solutions [34], similar superstring field theory solutions [35–38] and to solutionsdescribing marginal deformations [39–47], as well as to much further advance in the field.Despite all this progress, a non-perturbative quantum mechanical study of string fieldtheory was never performed. Such a study could be useful for addressing the importantquestion of distinguishing “bare” string field theories from “effective” ones [48]. The latterones being theories that, while capable of reproducing the correct scattering amplitudes,do not make sense as quantum theories at the non-perturbative level, since there is no wayto regularize or re-sum their perturbation series. A related but simpler problem that might “Universality” here refers to the property of having the same functional definition regardless of thebackground. This is “almost” as good as background independence. Universal formulations usually dependon the BRST world-sheet quantization of the string, e.g. in the bosonic case they depend on the bc -ghostsin addition to the usual space-time degrees of freedom. – 2 –lso benefit from such a study is that of the gauge invariance and gauge fixing of string fieldtheories: While it was demonstrated that in a specific gauge Witten’s open bosonic stringfield theory reproduces the correct covering of moduli space, the quantum master equationof this theory, which would ensure gauge invariance at the quantum level, is singular [49].The situation with many other open string field theories seems to be similar. Many otherissues that cannot currently be examined, such as the fate of the closed string tachyon inopen string field theories could also be examined.Another important motivation for such a study is that it could enable us to address thebig challenges that we listed in the beginning of this introduction. Thus, such a study couldbe of use to the general string theory and high energy community, as it would significantlyextend the usefulness of string field theory to the general research in the field. This couldbe particularly important, as string field theoretical research tends much to concentratearound string field theory itself.However, the quantum non-perturbative study of string field theory is an enormousendeavour. In this work we attempt a first small step towards this goal. We consider thesimplest possible string field theory, namely Witten’s bosonic open string field theory andexamine the possibility of studying this theory using lattice field theoretical methods. Ouraim is to provide a proof of concept for a lattice approach to string field theory by identifyingthe many obstacles such an approach would have, suggesting various ways to deal withthese difficulties, and examining these issues “experimentally” by lattice simulations of aparticularly simple setup.The motivation for a lattice study is the complexity of the theories that we are inter-ested in. Even for regular field theories there is not much that can be said analyticallyabout the quantum non-perturbative regime without, e.g. supersymmetry. Furthermore,even in the latter cases there are many aspects of the theory that cannot be addressedanalytically. Given the limitations of analytical studies, it is only natural to consider anumerical approach that is adequate for the study of non-perturbative physics of field the-ories. Among the various possibilities, the lattice approach [50–57] is probably the mostestablished and the most useful one.The choice of Witten’s theory among the various possibilities is also easily motivated,as it is the simplest and the most well understood among the universal string field theories.Unlike the closed string field theories it relies on a single product that can be explicitlyexpressed in terms of known coefficients describing the coupling of various fields. More-over, studying the bosonic theory enables us to avoid the various complications related toproperly choosing picture numbers of string fields, from which superstring field theoriessuffer.As was already mentioned, there are several difficulties with the proposed approach.Let us mention here a couple of obvious ones, and postpone the discussion on the other onesto latter sections. Witten’s theory is cubic, implying that the action is unbounded frombelow. While this is usually attributed to the unphysical nature of many of the componentfields, it is bound to lead to problems for a lattice simulation. We attempt to resolve thisproblem using analytical continuation from a setup, to be defined in what follows, in whichthe cubical terms are purely imaginary. We thus trade the instability by a convergent– 3 –scillatory behaviour.Another complication of Witten’s theory comes from the fact that the bosonic theoryin the critical dimension, as well as in any other dimension above two is presumably not welldefined, due to the presence of the closed string tachyonic instability. Moreover, runninglattice simulations in the critical d = 26 dimensions seems hopeless from a computationalperspective. A possible way to overcome both these problems is to study linear dilatonbackgrounds with d ≤
2. In this paper we focus mainly on the simplest one among all thesemodels, namely the d = 1 case. While the d ≤ String field theory is a second quantization approach to string theory. The classical stringfield is identified with the quantum Fock space of the first quantized (world-sheet) stringtheory. The world-sheet theory is a two dimensional conformal field theory (CFT). Thus,its Fock space is infinite dimensional. Hence, the string field is an infinite sum of regular(component) fields. The string field is assumed to be real. The reality condition for thestring field translates to reality conditions on the component fields, to be described in 2.3.The action of the string field is, S ≡ S + S = − Z (cid:16) α ′ Ψ ⋆ Q Ψ + g o ⋆ Ψ ⋆ Ψ (cid:17) , (1.1)where Ψ is the string field and the star product, the integral, and Q are defined below.The constant α ′ , related to the string tension, defines the string length, a natural lengthscale for the string, l s ≡ √ α ′ , (1.2)– 4 –nd g o is the open string coupling constant. A rescaling of Ψ by g o would result in a globalprefactor of g − o in front of the action. Note, however, that the way we define it here, g o isa dimensionful parameter. Hence, we cannot expect to obtain canonical normalizations forthe component fields both in the way the action is written here and after dividing by g o .Jumping ahead to the equations that will follow upon using level truncation we see that inthe current form of the action, canonical dimension for the scalar tachyon field is obtainedin (2.10). Then, we infer from (2.19) that g o has a mass dimension of − d . Thus, we leavethe action in the form (1.1).In order to make sense out of the action (1.1), the entities that appear in it shouldfirst be defined. The bi-linear star product takes two string fields and gives back a singlestring field. It has the geometric interpretation of gluing the right half of the first stringwith the left half of the second string. Hence, it is a non-commutative, associative product.The introduction of the star product turns the space of string fields into an algebra. Theintegral symbol represents “integration over the space of string fields”. It is performedby gluing of the left half of a single string to its right half, followed by the evaluation ofthe CFT expectation value of the resulting configuration. The kinetic term is producedusing the operator Q , which is the BRST charge of the world-sheet theory. It is given bya contour integral of a current J around the state Ψ in the CFT. An important propertyof Q is that it is an odd derivation with respect to the other two operations, Z Q Ψ = 0 ∀ Ψ , (1.3) Q (Ψ ⋆ Ψ ) = ( Q Ψ ) ⋆ Ψ + ( − ) Ψ Ψ ⋆ ( Q Ψ ) . (1.4)Here ( − ) Ψ represents the parity of the string field Ψ . The physical string field is odd andleads to a minus sign in the definition above, while the string field that plays the role of agauge parameter is even.The fact that the string field includes an infinite amount of component fields is asubtlety that any numerical method should address. A common way to deal with theinfinite amount of fields is to truncate the string field to a finite sum by considering onlyfields whose “level” is below some value l and terms in the action integral, which arebelow some l [58, 59]. This is referred to as a truncation to level ( l , l ). The level of afield is defined to be its conformal weight plus a constant that sets the zero-momentumlowest level state to l = 0. Since the Virasoro operator L , which reads the conformalweight of a state, serves as the (gauge fixed) kinetic term for the string field, the level l is essentially the on-shell mass of the string excitation considered. Hence, level truncationhas the natural physical interpretation of considering only low-mass states.The level is invariant under the action of Q . However, the star product mixes differentlevels: the star product of string fields that were truncated to a given level results inexpressions that are not truncated to this level. Hence, after the evaluation of the action interms of component fields, the action should also be truncated. As l , are sent to infinityone expects to obtain the result of using the full string field. There is no proof that thisshould work and subtleties might well arise. However, in the past it always did work. Theinclusion of a kinetic term implies l ≥ l while the cubicity of the action implies l ≤ l .– 5 –n practice one always works either in the ( l, l ) (which is simpler - has far fewer terms) orthe (more “physical”) ( l, l ) level-truncation.The conformal dimension of a field depends on its momentum. Most past papersconsidered only the zero momentum sector. Those who did consider non-zero momentumeither considered the double limit of a truncation in which the zero-momentum level andthe momentum were considered separately, or took the more physical choice of consideringthe total conformal weight as a single level parameter. In both cases, the only allowedmomentum was along a compactified space-like direction. Hence, this momentum wasquantized and its contribution to the level was always positive, which is sensible from theperspective of level truncation. This is not quite the case that we consider. We do nothave compactified directions and we consider the most general space-time dependence ofthe fields. However, the use of a lattice implies that we have to Wick-rotate the timedirection and to evaluate the action on a finite range of space-time, with some arbitrary(Neumann/Dirichlet) boundary conditions. These two restrictions turn the use of the morephysical level truncation into a sensible choice, which we adopt. Not only would that freeus from considering double limits, but it would also simplify and make more accurate theconsideration of the string-field-theory-inherent non-localities. Thus, we define the totallevel of a string field to be l = l + l p , (1.5)where l is defined by the mass of the specific excitation and l p includes the contributionof the momentum to the conformal weight.Another important issue is that of the space of string fields, a proper definition ofwhich is still lacking. Currently it is not even clear which mathematical concepts areneeded in order to properly define it. However, the general problems with the definitionof this space should not emerge in the context of level truncation. On the other hand, weshould decide whether the space of string fields should be restricted to string fields of agiven (first-quantization) ghost number and whether dependence on the ghost zero modeshould be allowed in its definition.The ghost system that we refer to here is the bc system used to fix the conformal gaugesymmetry on the world-sheet. This symmetry is generated by the energy momentum tensor T ( z ) which is an even object of conformal dimension h = 2. Thus, it is fixed by a system oftwo conjugate odd bosons, b whose conformal dimension is also h = 2 and c with h = − c insertions minus the number of b insertions.These first-quantized ghosts manifest themselves in the second quantized formulation bydeclaring that the string field Ψ is a functional not only of the space-time X µ variables,but also of the bc system. In terms of modes these conformal fields can be expanded as b ( z ) = ∞ X n = −∞ b n z n +2 , c ( z ) = ∞ X n = −∞ c n z n − . (1.6) Our conventions and world-sheet analysis follow Polchinski’s textbook [1]. – 6 –he fact that these fields are canonically conjugate implies [ b n , c m ] = δ n, − m . (1.7)The world-sheet quantum space is defined in terms of vertex operators, which arerestricted to carry ghost number one. The classical string field Ψ generalizes the spaceof vertex operators and should therefore also be restricted to carry ghost number one.However, a proper treatment of the gauge invariance of the classical action (1.1), canmodify this restriction. The gauge transformation that leaves the action invariant is δ Ψ = Q Λ + α ′ g o (cid:0) Ψ ⋆ Λ − Λ ⋆ Ψ (cid:1) . (1.8)A common way to classically fix the gauge is to impose the Siegel gauge, which is a stringfield theoretical extension of the Feynman gauge for the vector component field. This gaugechoice is enforced by requiring that the b -ghost zero mode annihilates the string field, b Ψ = 0 . (1.9)The Fock vacuum is annihilated by b . Hence, the Siegel gauge can also be defined asthe space of states built from the vacuum without using c . A quantum treatment of thegauge symmetry should take into account the fact that the gauge symmetry is redundantand “uses the equations of motion”. The most natural framework for addressing sucha system covariantly is the field-antifield BV formalism. This formalism was applied toWitten’s string field theory in [60–62]. The result is very elegant: prior to gauge fixing theaction should be replaced by a “master action”, which is identical in form to the classicalaction (1.1), except that the string field Ψ should no longer be constrained to carry ghostnumber one . One can gauge fix the master action to obtain the Siegel gauge in thespace of string fields with unrestricted ghost number. A potential subtlety with the masteraction is that according to the general BV formalism it should obey the “quantum masterequation”. However, it is still not clear if this is the case and it is only certain that it obeysthe “classical master equation”.While gauge fixing is necessary in perturbation theory, this is not always the case in alattice approach, in which the infinite gauge orbits are replaced by finite ones. Hence, wecan consider the following four options for our space of string fields, to which we refer inthe following as schemes :1. Classical string fields without gauge fixing, i.e. Ψ carries ghost number one but hasno ghost zero-mode restrictions.2. Classical gauge-fixed string fields. This is probably inconsistent, since there is nojustification for gauge fixing without a proper treatment of the gauge symmetry. In this paper the brackets represent the graded commutator, which for the current case is ananticommutator. The string field should still be an odd object. Hence, the component fields at even ghost numbers haveto be odd. This is also consistent with the general parity assignments of the BV/BRST formalisms. – 7 –. BV string field without gauge fixing. Here, all ghost numbers are considered and thestring field is allowed a c dependence.4. Gauge-fixed BV string field, i.e. Ψ carries all possible ghost numbers but is c -independent.One of the important advantages of the lattice approach to field theory is that it provides aregulator that does not break gauge symmetry. In our case, on the other hand, the latticedoes break the gauge symmetry, since the star product mixes all levels. Hence, one shouldexpect that the gauge fixed schemes, in which the gauge symmetry was already taken careof, would be better behaved. While we will experiment with all the four schemes, we willsee that, as we expect, scheme 4 seems to be the most promising one. d ≤ bc ghost system,which we already described. Other than the ghost system the world-sheet theory dependsalso on a matter system, which can be any CFT, as long as its central charge is c m = 26.This value for the central charge is needed in order to cancel the central charge of theghost system which is c gh = −
26. The vanishing of the total central charge is necessaryfor avoiding conformal anomalies.In this paper, we work with a one dimensional linear dilaton theory. But let us fornow consider the more general case of a linear dilaton in d ≤ . This caseincludes theories with 1 < d <
2, where we identify the dimension d with the central chargeof the non-linear-dilaton part of the matter sector plus one. This identification stems fromthe fact, on which we elaborate below, that the linear dilaton can be realized by a singlenon-homogenous direction, while standard space-time directions correspond to one unit ofcentral charge. The 1 < d < V µ to be defined below, is of the order of the stringscale, m s ≡ l − s = ( α ′ ) − . (1.10)The string scale is usually identified (presumably up to a constant of order one or aroundit) with the Planck mass. Hence, these spaces differ substantially from standard space-times. We will further have to set two other scales, namely the lattice spacing and thelattice size. In order to obtain a proper description of the physics, the first one should be In the two dimensional case, the world-sheet degrees of freedom are the “tachyon” field and the “discretestates”, which are physical only for specific momenta. Scattering amplitudes of both tachyons and discretestates are known [63, 64]. They were considered from the string field theory perspective in [65–67]. Otherrelevant papers include the identification of the ground ring/ W ∞ structure [68], the introduction of ZZ [69]and FZZT D-branes [70, 71], open/closed duality and relation to the supersymmetric theories [72] and thedecay of ZZ-branes [73]. – 8 –ent to zero and the second one should be sent to infinity, in l s units, in order to approachthe continuum limit.In a theory of open strings, the matter fields can be expanded as , X µ ( z, ¯ z ) = x µ − iα ′ p µ log | z | + i r α ′ ∞ X n =1 (cid:16) a µn √ n ( z − n + ¯ z − n ) − a † µn √ n ( z n + ¯ z n ) (cid:17) , (1.11)where µ = 1 , d case and takes a single value and hence can be omitted in the 1 d case. Taking the derivative with respect to z gives i∂X µ ( z ) = α ′ p µ z + r α ′ ∞ X n =1 √ n (cid:16) a µn z − n − + a † µn z n − (cid:17) . (1.12)These fields have the following operator product expansion (OPE) X µ ( z, ¯ z ) X ν ( w, ¯ w ) ∼ − α ′ η µν log | z − w | , (1.13)which translate into the standard commutation relations among the infinite set of creationand annihilation operators, [ a µn , a † νm ] = η µν δ m,n . (1.14)We assume a flat (space-time) metric, η µν = diag(1 , d = 2 , (1.15) η = 1 d = 1 . (1.16)Momentum states are represented as usual using the operators e ik · X . The OPE (1.13)implies that these exponents suffer from short distance singularities and should be normal-ordered. We do not write the normal ordering symbol and assume that all fields that weconsider are implicitly normal ordered. In the standard flat background with constantdilaton, the normal-ordered exponents carry well-defined conformal weights,( h, ¯ h ) = (cid:16) α ′ k , α ′ k (cid:17) . (1.17)Since we consider open strings, our fields are given by insertions on the boundary ofthe CFT theory, which we identify with the real axis, ℑ ( z ) = 0 ⇐⇒ z = ¯ z , (1.18)and the CFT is defined on the upper half plane. One usually extends the CFT to the wholecomplex plane using the doubling trick, according to which z -dependence above the realline is identified with ¯ z -dependence below it. Still, the real line is special, e.g. conformalfields may suffer from extra singularities when approaching it, originating from collisions In the literature it is common to find the variables α µn ≡ p | n | a µn . – 9 –f z and ¯ z . In particular, momentum states should now be normal-ordered according tothe boundary-normal-ordering. Then, they carry a well defined single conformal weight h ( e ik · X ) = α ′ k . (1.19)Another confusion that can arise from the presence of the boundary is the relation betweenthe derivative with respect to z and that with respect to the boundary parameter. We willtry to avoid this issue by always considering only the z variable.In order to fully describe the CFT one has to define the energy-momentum tensor, T ( z ) = T g ( z ) + T m ( z ) , (1.20)where the superscripts stand for “ghost” and “matter”. The ghost part of the energymomentum tensor is fixed, T g ( z ) = − ∂bc ( z ) − b∂c ( z ) , (1.21)but the matter part is theory-dependent. In our case it equals, T m ( z ) = − α ′ ∂X µ ∂X µ + V µ ∂ X µ . (1.22)The second “improvement” term encodes the linear dilaton nature of the background. Thisterm changes the central charge to a new value, c = d + 6 α ′ V . (1.23)In order to obtain the correct value of 26 for the central charge, one has to impose V = 26 − d α ′ . (1.24)In particular, the cases we are interested in are, V µ = (0 , V ) , V = − √ α ′ d = 2 , (1.25) V = − r α ′ d = 1 , (1.26)where the minus sign is just a convention.In a linear-dilaton background the following modifications occur as compared to astandard flat-space background [74, 75]: • The momentum-conservation delta function is modified in order to reflect the break-down of translational invariance, δ d (cid:0) X n k µn (cid:1) −→ δ d (cid:0) X n k µn + iV µ (cid:1) . (1.27)Here, the delta function is formally defined by, δ d (cid:0) X n k µn + iV µ (cid:1) ≡ π ) d Z d d x e − ix · P k n + x · V . (1.28)– 10 – The field X , aligned with the linear dilaton direction, is no longer a (logarithmic)conformal tensor, X µ ( z, ¯ z ) → f ◦ X µ = X µ ( f ( z ) , ¯ f (¯ z )) + α ′ V µ log | f ′ ( z ) | . (1.29)The momentum operators built from X µ remain conformal tensors. However, theirconformal weights change, h ( e ik · X ) = α ′ ( k + ik · V ) . (1.30) • The change in T m induces a change in Q , since the following identity holds in general J ( z ) = cT m ( z ) + bc∂c ( z ) + 32 ∂ c ( z ) , (1.31)where J ( z ) is the BRST current, which should be integrated to obtain Q .The introduction of a linear dilaton introduces complications also from the latticeperspective. Specifically, the fact that the target space is no longer homogenous implies thatit would not be enough to consider the length L of the lattice, as a single length parameter.Instead, we would have to study the dependence on x min and x max , the minimum andmaximum points of the spatial direction X , or alternatively on x min and the lattice interval L ≡ x max − x min . (1.32) In this section we implement level truncation in order to describe various aspects, problemsand resolutions of our program from the string field theory perspective. We evaluate mostexpressions in an arbitrary dimension and concentrate on the d = 1 case at the end.Later on, in section 3, we discuss the computational lattice perspective. Here we start byevaluating the action for l = 0 in 2.1. Next we discuss the way by which higher levels canbe added in 2.2 and explain how to impose the reality condition in 2.3. Then, we describeexplicitly the first case of a higher level, namely l = 1, in 2.4. In order to go to higherlevels one has to define a systematic way for the evaluation of the very many terms thatare present in the action. We sketch a method that can be used for an automatization ofthe evaluation of the action in 2.5 and utilize it in 2.6 for the evaluation of the action ofscheme 3 at l = 1. We further explain there that there is a problem with this action thatprevents us from using scheme 3 together with level truncation, after which we truncatethe scheme 3 action to a scheme 1 action in 2.7. We end this section by an analytical studyof the simplest possible case, that of a single mode, in 2.8, followed by a discussion of onemore potential obstacle towards using our methods, in 2.9. Here, it is assumed that we work in d = 1. For d > x min and x max are the minimum and maximumvalues in the direction of the linear dilaton. – 11 – .1 Truncation to zero l -level We consider classical string fields that are built upon the vacuum of the first quantizedtheory, | Ω i ≡ c (0) | i = c | i , (2.1)where | i is the SL(2) invariant vacuum . The vacuum | Ω i is odd and carries ghost numberone, as is proper for a classical string field. General string field configurations are builtfrom this vacuum by acting on it with the operators b − n with n > c − n with n ≥ a µn with n >
0. Note that the b − n and c − n ghost operators are odd, so each one of themcan act at most once, while the a µn are even and so each one of them can act indefinitely.In order to retain the restriction to ghost number one, which is needed for schemes 1 and2, the number of b and c insertions must be the same. For the other two schemes there isno such restriction. However, if the total number of b and c ghost insertions is odd, therelevant component field must also have odd parity in order not to change the parity ofthe string field.The momentum of the state is quantized when we work with 0 < X < L . AssumingDirichlet boundary conditions, the momentum dependence would come from sin (cid:0) πk · XL (cid:1) ,with k an integer. We can now define the level as the sum of the indices of all the operatorsplus a contribution from the momentum, e.g. the level of c − b − ( a † ) | Ω , p i is l = l + l p ,with l = 3 + 2 + 4 + 4 = 13. However, (1.30) implies that the sine factor is not aneigenvalue of the level operator. Not only that, but the eigenvalues of the two distinctexponents composing the sine are complex. We resolve this problem below.At the lowest (zero) l -level the string field contains only two component fields. We im-pose the reality condition on the string field. This implies that the following two componentfields are real, Ψ = Z d d p (cid:0) T ( p ) e ip · X + T ( p ) e ip · X c (cid:1) | Ω i . (2.2)The first of these fields is the “tachyon” T (not to be confused with the energy-momentumtensor). It carries no c insertion and has ghost number one and so contributes in all fourschemes. The second field T is odd. It carries ghost number two and contains the c zeromode in its definition. Hence, it contributes only in scheme number three. In any case, itcannot contribute to the action without the presence of string fields of ghost number lessthan one. Thus, it does not contribute at all at zero l -level and is set for now to zero.Direct calculation shows that, Q Ψ = Z d d p T ( p ) α ′ (cid:16) p + iV (cid:17) + m ! ∂cce ip · X | Ω i , (2.3)where we defined, m ≡ V − α ′ . (2.4) The conformal symmetry is generated by the Virasoro operators L n , n ∈ Z . The vacuum | i is invariantunder an SL(2) subalgebra of the Virasoro algebra generated by L and L ± . This subalgebra becomesuseful for us already at l = 1 (2.40). The other Virasoro operators are useful at higher levels. – 12 –e see that the constant term inside the parentheses vanishes only for the value of V (1.24)at d = 2. This constant fixes the mass of this mode. Hence, we see that the tachyon becomesmassless exactly in two dimensions, while for d = 1 it is massive.The kinetic term of the tachyon reads, S = − α ′ Z Ψ Q Ψ = − α ′ h ( I ◦ Ψ )(0) Q Ψ (0) i , (2.5)where the expectation value is evaluated in the upper half plane and I is the conformaltransformation, I ( z ) = − z . (2.6)For the evaluation of (2.5) we have to regularize, z → ǫ and continue as in [76]. Theexpectation value factorizes into matter and ghost parts. The ghost part gives, (cid:28) c (cid:0) − ǫ (cid:1) c ′ c (0) (cid:29) = 1 ǫ , (2.7)while for the matter part we have to evaluate (cid:28) e ip · X (cid:0) − ǫ (cid:1) e iq · X (0) (cid:29) = (2 π ) d δ d ( p + q + iV ) (cid:16) ǫ (cid:17) − α ′ p · q . (2.8)Using this result for (2.2) and (2.3) leaves us with integration over the momenta, as wellas with a factor coming from the conformal transformation, (cid:16) dIdz (cid:17) h = (cid:16) ǫ (cid:17) α ′ ( p + ip · V ) − . (2.9)Using the delta function one sees that all ǫ -dependent factors cancel out, regardless of thespecific background and the final result is, S = − Z d d p d d q (2 π ) d δ d ( p + q + iV ) T ( p ) T ( q ) (cid:16) p + iV (cid:17) + m ! = − Z d d p d d q d d x e − ix · ( p + q + iV ) T ( p ) T ( q ) (cid:16) p − q (cid:17) + m ! = − Z d d x e V · x ∇ T ( x ) · ∇ T ( x ) − T ( x ) ∇ T ( x )2 + m T ( x ) ! , (2.10)where we abuse the notation by using the same symbol for the Fourier conjugate fields T ( p ) and T ( x ).We now have to evaluate the cubic term, S = − g o Z Ψ ⋆ Ψ ⋆ Ψ = − g o Z d d p d d q d d k h ( f − ◦ Ψ )( f ◦ Ψ )( f ◦ Ψ ) i . (2.11)The three conformal transformations are obtained by sending the upper half plane to theunit disk using, w = 1 + iz − iz , (2.12)– 13 –hen rescaling w and relocating it to the three points of the “rotated Mercedes-Benz logo”, w → e πin w , (2.13)and finally sending it back to the upper half plane using the inverse of (2.12), z = i − w w . (2.14)The only relevant information about these conformal transformations is, f (0) = 0 , f ± (0) = ±√ , f ′ (0) = 23 , f ′± (0) = 83 . (2.15)Now, the ghost part contributes, D c ( −√ c (0) c ( √ E = 2 · , (2.16)the matter part is, D e ip · X ( −√ e ik · X (0) e iq · X ( √ E = (2 π ) d δ d ( p + k + q + iV )3 − α ′ p · k · − α ′ k · q · − α ′ p · q . (2.17)and the conformal weights contribute, (cid:16) df − dz (cid:17) h − (cid:16) df dz (cid:17) h (cid:16) df dz (cid:17) h = (cid:16) (cid:17) α ′ ( p + ip · V ) − (cid:16) (cid:17) α ′ ( k + ik · V ) − (cid:16) (cid:17) α ′ ( q + iq · V ) − . (2.18)All in all we get, S = − g o Z d d p d d q d d k (2 π ) d δ d ( p + k + q + iV )3 T ( p ) T ( k ) T ( q ) K − α ′ ( p + k + q + V ) = − g o K − α ′ V Z d d x e V · x ˜ T ( x ) . (2.19)Here (as usual) we defined, K ≡ √ , (2.20)and used the delta function in the first equality. In the second equality we moved to x -space, as in (2.10) and defined (as usual) the tilded variables,˜ T ( x ) = K α ′ ∇ T ( x ) . (2.21)Similar relations are to be understood for other tilded variables in what follows. The pre-factor K − α ′ V can be absorbed into a redefinition of the coupling constant. Note also thelinear dilaton factor e V · x , which is common to the quadratic and cubic terms. A rescalingof T by this factor leads to a space-dependent effective coupling, g effo ∼ e − V · x . (2.22)– 14 –his coupling goes from zero to infinity along the linear dilaton direction, which impliesthat the pre-factor g o K − α ′ V can be set to unity by an appropriate choice of the origin.Note, that this effective coupling constant is still dimensionful. It is possible to multiplyit by a proper power of α ′ in order to obtain a dimensionless coupling constant.Let us now perform the advocated field redefinition, T ( x ) = e − V · x τ ( x ) ⇐⇒ T ( p ) = τ (cid:16) p + i V (cid:17) . (2.23)The kinetic term takes now the standard form, S = − Z d d x (cid:0) m τ + ( ∇ τ ) (cid:1) . (2.24)This field redefinition is defined pointwise. Hence, the Jacobian is just a number and canbe ignored. The interaction term transforms under the above field redefinitions into, S = − g o K (cid:0) − α ′ V (cid:1) Z d d x e − V · x ˜ τ ( x ) . (2.25)It is easiest to obtain this expression in momentum space, where one has to use the deltafunction and deform the contour of integration. Note that the resulting interaction termis both space-dependent and non-local. In p -space the spatial-dependence and non-localityreverse their roles.The spatial dependence of the coupling constant implies that we cannot use periodicboundary conditions on the lattice, since it would glue a strong coupling region with a weakcoupling region, which is unphysical. Instead we can use, as mentioned above, Dirichletor Neumann boundary conditions. We choose the former and evaluate the action in a box x µmin < x µ < x µmax , with x µmax − x µmin = L µ . Specifying now to d = 1, we can expand, τ ( x ) = r L ∞ X n =1 τ n sin (cid:16) πn ( x − x min ) L (cid:17) . (2.26)We can now recognize one more advantage of τ over T . We mentioned above that theexpansion of T in terms of sine modes does not involve conformal eigenmodes and itsexpansion in exponents leads to complex eigenvalues. Contrary to that, one can see thatthe expansion in sine modes of τ is well defined and real. Furthermore, we shall see insection 2.3 that working with the τ variable is essential also in order to obey the stringfield reality condition.Another issue that we would like to mention is that of the variational principle. Whilewe do not derive here equations of motion, since we concentrate on the action itself, itcould still be interesting to examine their derivation and their form in the case at hand.String field theory includes an infinite number of derivatives. It is known that there mightbe subtleties with the definition of the variational principle in such theories [77, 78]. Onecould wonder whether our choice of boundary conditions is consistent with a variationalprinciple. The quadratic term is the standard one and so is its variation. The variation of– 15 –he cubic term leads to δS = − g o K (cid:0) − α ′ V (cid:1) Z d d x e − V · x ˜ τ ( x ) δ ˜ τ ( x ) . (2.27)It seems that in order to obtain an equation of motion we need to change the δ ˜ τ ( x ) terminto a δτ ( x ) term, i.e. to integrate by parts the K α ′ ∇ factor acting on δτ ( x ), such that itwould act on the ˜ τ ( x ) factor and on the dilaton factor e − V · x . Such an integration by partswould lead to boundary terms that include the variation of various higher order derivativesof τ ( x ). These expressions should not a-priori vanish. Moreover, setting all these infinitelymany terms to zero could lead to very strict functional restrictions on τ ( x ). However, wecan take another approach. Given the expansion (2.26) we obtain,˜ τ ( x ) = r L ∞ X n =1 K − α ′ (cid:0) πnL (cid:1) τ n sin (cid:16) πn ( x − x min ) L (cid:17) . (2.28)When expressed in this form it seems that ˜ τ ( x ) and δ ˜ τ ( x ) vanish at the boundaries.Nonetheless, this assertion relies on some convergence properties of the expansion, whichmight not be well justified. This issue is related to the discussion in [77, 78] and moregenerally to the problem of properly defining the space of string fields. Attempting toanalyse it would take us too far away. Hence, we do not dwell on these questions further.We can now assign a level to the modes in the expansion above, l ( τ n ) = α ′ (cid:16) πnL (cid:17) . (2.29)We have to include all levels that are smaller than some l <
1. The restriction l < l ≥ l -modes might also contribute . The physical origin of this restriction is that if we decideto probe lower than string-scale size modes, we should also include the higher modes, whichare also of this size.An l that allows for N modes is equivalent to working on an N -site lattice. However,the non-locality and space-dependence of the action simplify if we perform the analysisdirectly in terms of the modes. The free part of the action is now, S = − N X n =1 (cid:16) α ′ + (cid:0) πnL (cid:1) (cid:17) τ n . (2.30)Assuming that we work in the ( l, l ) scheme (that is, if all interaction terms are to beincluded), the interaction term reads, S = − g o K (cid:0) − α ′ V (cid:1) N X n , , =1 K − α ′ (cid:0) πL (cid:1) ( n + n + n ) τ n τ n τ n f n ,n ,n , (2.31) f n ,n ,n ≡ (cid:16) L (cid:17) Z x max x min dxe − V x · (2.32) · sin (cid:16) πn ( x − x min ) L (cid:17) sin (cid:16) πn ( x − x min ) L (cid:17) sin (cid:16) πn ( x − x min ) L (cid:17) . For a Dirichlet expansion these modes actually contribute starting at some l > – 16 –e can substitute − V x = q α ′ x . We left the V dependence for enabling the evaluation ofthis expression with unphysical values of V . We can now evaluate this action on the lattice.Note, that our Wick-rotation was performed in such a way that we have to consider, Z = Z (cid:16) Y n dτ n (cid:17) e S . (2.33) In previous studies of level-truncated string field theory it was suggested to move the non-locality from the cubic term to the quadratic term. The motivation was the simplificationof the (relatively more complicated) interaction term. Furthermore, as it is moved to thequadratic term, the non-locality becomes completely diagonal, which results in somewhatsimplified expressions.At the lowest l level, the action is still given by the sum of (2.24) and (2.25), onlynow the fundamental field, which should be expanded in modes is ˜ τ , while τ is defined interms of it as τ = K α ′ p ˜ τ . (2.34)For simplicity of notations, we drop the tilde from now on, with the understanding thatthe correct variables are used.Expanding in the modes for the tachyon field, the action is modified to S = − N X n =1 K − α ′ (cid:0) πnL (cid:1) (cid:16) α ′ + (cid:0) πnL (cid:1) (cid:17) τ n , (2.35) S = − g o K (cid:0) − α ′ V (cid:1) N X n , , =1 τ n τ n τ n f n ,n ,n , (2.36)where the definition of f n ,n ,n did not change.As higher level fields are added, one can similarly decide whether the better represen-tation is the one in which the untilded fields are the fundamental ones, or the one with thetilded fields. The needed manipulations are completely analogous to what is done here. Evaluating conformal transformations can be tedious, especially at higher levels, where thecoefficient fields are not represented by primary conformal fields. A way to simplify calcu-lations was actually devised even before the CFT formulation of [76]. In this formulationthe action is given by, S = − α ′ h V | | Ψ i | Q Ψ i − g o h V | | Ψ i | Ψ i | Ψ i , (2.37)where the subscripts represent an index of a copy of the Hilbert space. The two-vertex V and three-vertex V live in the spaces H and H respectively. They are squeezed statesand their form for flat background was found in [79–84]. The modification of these worksto a linear dilaton background is relatively simple, due to the similarity with the bosonized– 17 –host sector, studied in these papers. Explicit expressions for d = 2 were given in [66].Both factorize into matter and ghost sectors (the matter sector further factorizes into d independent sectors).The two-vertex is explicitly given by, h V m | = Z d d p d d q h p | h q | δ d ( p + q + iV ) ·· exp (cid:18) ∞ X n =1 ( − n +1 ( a µn ) ( a µn ) ! , (2.38a) h V g | = h Ω | ( c + c ) exp (cid:18) ∞ X n =1 ( − n (cid:16) b n c n + b n c n (cid:17)(cid:19) . (2.38b)The superscripts m and g in these expressions stand for “matter” and “ghost”. Thesuperscripts 1 and 2 over the oscillators represent the two spaces.In order to evaluate the kinetic term we also need to write down the oscillator form of Q , Q = X n ∈ Z c n ( L m − n − δ n, ) + X m,n ∈ Z m − n c m c n b − m − n : , (2.39)where we indicated that the second term is normal ordered. The matter Virasoro oper-ators L mn appearing in (2.39) are obtained from expending the energy momentum tensor T m (1.22). Explicitly, the relevant ones at l = 1 are, L = ∞ X n =1 na † n a n + α ′ ( p + ipV ) , (2.40a) L − = √ α ′ p a † + ∞ X n =1 p n ( n + 1) a † n +1 a n , (2.40b) L = √ α ′ ( p + iV ) a + ∞ X n =1 p n ( n + 1) a † n a n +1 . (2.40c)For the evaluation of the cubic term we need to know the three-vertex, which is un-fortunately more complicated than the two-vertex. h V m | = N Z d d p d d p d d p h p | h p | h p | δ d ( p + p + p + iV ) ·· exp (cid:18) − X r,s =1 (cid:18) ∞ X n,m =1 a rn V rsnm a sm + ∞ X n =1 a rn V rsn p s + 12 p r V rs p s !! , (2.41a) h V g | = h Ω | c c c exp (cid:18) X r,s =1 ∞ X m =0 ∞ X n =1 b rm X rsmn c sn (cid:19) . (2.41b)Here, we suppressed the index µ , on which the oscillators and some of the coefficientsdepend, for clarity. The V rsnm and X rsmn coefficients are independent of the linear dilaton.– 18 –hey are found, e.g. in [3], V rsnm = − √ nm I dw πi I dz πi z m w n f ′ r ( z ) f ′ s ( w ) (cid:0) f r ( z ) − f s ( w ) (cid:1) , (2.42) X rsmn = I dw πi I dz πi z n − w m +2 f ′ s ( z ) f ′ r ( w ) (cid:0) f s ( z ) − f r ( w ) (cid:1) Y I =1 f r ( w ) − f I (0) f s ( z ) − f I (0) , (2.43)where f r are the conformal transformations defining the three-vertex (2.12)-(2.15). Thenormalization factor N and the momentum dependence can be read by comparing to theexpressions obtained for the tachyon using CFT methods. The result is N = K − α ′ V , (2.44) V rs = 2 α ′ log Kδ rs . (2.45)For evaluating V rsn we again compare the expressions obtained using oscillator methodsand CFT methods. The oscillator representation is, h V m | a † n | p i | p i | p i = − X s =1 V sn p s (cid:10) tachyon (cid:11) , (2.46)where the value of the expression without the a † insertion, which equals the expectationvalue for three tachyon interaction is written as (cid:10) tachyon (cid:11) . On the CFT side we obtain, h V m | a † rn | p i | p i | p i = I dz πi r α ′ √ nz n (cid:10) f ◦ ( i∂X )( z ) f ◦ e ip X (0) f ◦ e ip X (0) f ◦ e ip X (0) (cid:11) = I dz πi r α ′ α ′ √ nz n X s =1 f ′ ( z ) p s f ( z ) − f s (0) + iV f ′′ ( z ) f ′ ( z ) !(cid:10) tachyon (cid:11) (2.47)= I dz πiz n r α ′ n X s =1 p s f ′ ( z ) f ( z ) − f s (0) − f ′′ ( z )2 f ′ ( z ) !(cid:10) tachyon (cid:11) . Here, we used the CFT definition of the expression and (1.12) in the first equality. Then,we used the non-tensor transformation rule (1.29) in the second equality and the anomalousmomentum conservation in the last equality. We can now infer, V sn = − I dz πiz n r α ′ n f ′ ( z ) f ( z ) − f s (0) − f ′′ ( z )2 f ′ ( z ) ! . (2.48)Note, that there is no lose of generality from choosing r = 1, due to the cyclicity propertyof the three-vertex, V rs = V ( r + n )( s + n ) ∀ n , (2.49)where the indices are added modulo 3. Also note, that the expression we obtained doesnot agree with the literature even in the limit V → V -independent). The reason is that without a linear dilaton V rsn is onlydefined up to V rsn → V rsn + K sn , for arbitrary constants K sn , due to the non-anomalousmomentum conservation. This freedom is used, e.g. in [85, 86] in order to set P s V rsn = 0.If we do not wish to have a term proportional to iV a † in the definition of the three-vertex,then we have no redefinition freedom and we are forced to use (2.48). One can verify thatthis result indeed makes sense, by noticing that, unlike other expressions for the three-vertex, it is SL(2) invariant. We are almost ready now to address higher levels. The onlyissue that should still be clarified is the form of the reality condition, to which we turnnext. Let us recall the reality condition of the string field. This condition states that the stringfield is left invariant under the combined action of two involutions, Hermitian conjugation
O → O † and BPZ conjugation O → O ♭ . The former is the more familiar one. It sends | i to h | , O n to O − n , where O stands for either a , b or c , while reversing the order ofoperators. It also induces complex conjugation. BPZ conjugation is performed by theaction of the two-vertex h V | (2.38). It also sends | i to h | . However, it does not inducecomplex conjugation, nor does it change the order of operators. It also acts differently onthe various operators, a ♭n = ( − n +1 a − n , c ♭n = ( − n +1 c − n , b ♭n = ( − n b − n . (2.50)The different signs originate from the odd conformal dimension of ∂X ( z ) and c ( z ) versusthe even one of b ( z ). Combining the two involutions leaves us with ( − a even + c even + b odd times the original operators inversely ordered. It is important to note that the coefficientfields also change their order relative to the other expressions. This is important whenthe Grassmann odd coefficient fields of even-ghost-number string fields are considered.The coefficient fields are also complex conjugated. We prefer to work with coefficientfields which are defined to be real. Thus, matching the signs translates into the choice ofputting an extra i factor in front of some of the coefficients. Note, that we do not have toseparate the c factor from the vacuum | Ω i , since it does not contribute a sign and alsocommutes with the rest of the operators, which are Grassmann even when coefficient fieldsare included.We write the action in terms of momentum modes. The rules for settling the realityof the component fields, when applied to the explicit momentum dependence lead to the(almost) standard reality in momentum space,ˆ φ ( p ) = ˆ φ ( − p − iV ) ∗ , (2.51)where ˆ φ is an arbitrary component field and p is the momentum. To get from this expressiona genuine standard reality condition we have to impose the same transformation that weimposed in (2.23), ˆ φ ( p ) = φ (cid:16) p + i V (cid:17) . (2.52)– 20 –ith this definition the reality condition takes the familiar form, φ ( p ) = φ ( − p ) ∗ ⇐⇒ φ ( x ) = φ ( x ) ∗ . (2.53)We would like to work from now on only with the real fields. This can be achieved byredefining Q , the matter two-vertex h V m | (2.38a) and the matter three-vertex h V m | (2.41a)in a way that compensates for the transformation (2.52). The redefinition of | V i is nothingbut the replacement, δ d ( p + q + iV ) → δ d ( p + q ) , (2.54)in (2.38a). The redefined Q is the same as the old one, only with the matter Virasorooperators redefined from (2.40) to the more symmetric form, L = ∞ X n =1 na † n a n + α ′ (cid:16) p + V (cid:17) , (2.55a) L − = √ α ′ (cid:16) p − i V (cid:17) a † + ∞ X n =1 p n ( n + 1) a † n +1 a n , (2.55b) L = √ α ′ (cid:16) p + i V (cid:17) a + ∞ X n =1 p n ( n + 1) a † n a n +1 , (2.55c)and similarly for the other modes. We see that not only the string fields, but also theVirasoro modes obey now the standard reality condition, L † n = L − n . (2.56)It is straightforward to see that with the new definition one recovers (2.24).Transforming the cubic interaction according to (2.52) is nothing but the replacementof p r by p r − iV everywhere in the three vertex. This results in, δ d ( p + p + p + iV ) → δ d (cid:16) p + p + p − iV (cid:17) , (2.57) N → K (cid:0) − α ′ V (cid:1) , (2.58) V sn → V sn − X r =1 V rn . (2.59) l = 1 in scheme 4 We now have all the ingredients needed for defining the l = 1 action, which we evaluatefor a general dimension d . The l = 1 component of the string field can be written in termsof six real component fields,Ψ = Z d d p (cid:18) A µ ( p ) a µ † + B ( p ) b − + iC ( p ) c − + (cid:16) A µ ( p ) a µ † + i B ( p ) b − + C ( p ) c − (cid:17) c (cid:19) e ip · X | Ω i . (2.60)– 21 –f the new six fields, the second line includes the ones which are outside the Siegel gauge.These fields do not contribute to our schemes 2 and 4. The only fields with ghost numberone are the “photon” A and the auxiliary field B . These are the fields that contribute toscheme number 1. Of these, only the photon contributes to scheme 2. Scheme 4 carries allthe fields of the first line, while scheme 3 carries not only all the new fields, but also thefield T , from l = 0, which did not contribute to the action previously. It is important toremember that the fields B , C , T and A are Grassmann odd fields.We now want to evaluate the kinetic term of the new fields. Since we assume that thefields in Ψ are real, we should work with the modified h V | (2.54) and L n (2.55). At thislevel the BRST charge Q is truncated to Q = c ( L m −
1) + c L m − + c − L m − b − c c − c − c b + 2 c − b c . (2.61)Assume for now that we work with scheme 4. Then, we have the even fields T and A andthe odd fields B and C . Note, that due to our treatment of the reality condition, T nowis what we called τ in section 2.1. The form of the BRST charge Q can now be furthersimplified by disregarding all terms that do not include c , Q = c ( L m − − b − c c − c − c b . (2.62)Direct evaluation now gives, S = − Z d d x (cid:16) m T + ( ∇ T ) m A + ( ∂ ν A µ ) i ( m BC + ∇ B · ∇ C ) (cid:17) . (2.63)Here, we used the generalization of (2.4), m l ≡ V l − α ′ , (2.64)for l = 1. Note, that the kinetic term of the vector takes the standard form of a vector inthe Feynman gauge.We now turn to evaluating the cubic terms. Ghost number conservation dictates thatthe only possible interactions include T , AT , A T , A , T BC and
ABC . We have to eval-uate all these terms. To that end we need the coefficients V rs , V rs and X rs . Using (2.48)we obtain, V = 0 , V = r α ′ , V = − r α ′ . (2.65)These values should have been modified according to (2.59). However, they do not change,since they sum up to zero. In fact, this is the case for all odd values of n . For terms atleast quadratic in A we also have to use (2.42) for evaluating, V = V = − , (2.66)while for the terms involving the ghost fields we need (2.43), X = X = − . (2.67)– 22 –he evaluation of the T term is straightforward and leads to the result already ob-tained (2.25), S = − g o N Z d d x ˜ T e − V · x . (2.68)Here and in what follows we leave the x argument (of ˜ T ( x ) ) implicit. Next, we get the T A term, S = − g o h V | Z d d p d d p d d p (cid:0) a † A ( p ) | p , Ω i (cid:1)(cid:0) T ( p ) | p , Ω i (cid:1)(cid:0) T ( p ) | p , Ω i (cid:1) = − g o N Z d d p δ (cid:16) X p i − iV (cid:17) ˜ A ( p ) ˜ T ( p ) ˜ T ( p )( − V p − V p − V p )= 0 , (2.69)where in the last equality we used the fact that we obtain in the integrand an expressionwhich is anti-symmetric with respect to p ↔ p . The T A term, for which we have towrite the space-time indices explicitly, is S = − g o h V | Z d d (cid:0) a µ † A µ ( p ) | p , Ω i (cid:1)(cid:0) a ν † A ν ( p ) | p , Ω i (cid:1)(cid:0) T ( p ) | p , Ω i (cid:1) = − g o N Z d d p δ (cid:16) X p i − iV (cid:17) ˜ A µ ( p ) ˜ A ν ( p ) ˜ T ( p ) (cid:16) − V η µν +( − V p − V p − V p ) µ ( − V p − V p − V p ) ν (cid:17) = − g o N Z d d p δ (cid:16) X p i − iV (cid:17) ˜ A µ ( p ) ˜ A ν ( p ) ˜ T ( p ) · (2.70) · (cid:16) α ′ (cid:0) p µ p ν + p ν p µ − ( p µ p ν + p µ p ν ) (cid:1) + 2 η µν (cid:17) = − g o N Z d d x (cid:16) A µ ˜ A µ ˜ T + α ′ (cid:0) ˜ A µ ˜ A ν ∂ µ ∂ ν ˜ T + ∂ µ ˜ A ν ∂ ν ˜ A µ ˜ T − A ν ∂ ν ˜ A µ ∂ µ ˜ T (cid:1)(cid:17) e − V · x . Then, we evaluate the A term, S = − g o h V | Z d d (cid:0) a † A ( p ) | p , Ω i (cid:1)(cid:0) a † A ( p ) | p , Ω i (cid:1)(cid:0) a † A ( p ) | p , Ω i (cid:1) = − g o N Z d d p δ (cid:16) X p i − iV (cid:17)(cid:16) V X r,s V rs p s + X r,s,t ( V r p r )( V s p s )( V t p t ) (cid:17) ·· ˜ A ( p ) ˜ A ( p ) ˜ A ( p ) (2.71)= 0 . Here, we should have paid attention to the Lorentz indices. The result, however, does notchange by doing so. We can now notice that, in the expressions above, all terms with anodd number of (vector) A fields vanish, as expected. Similarly, the ABC term vanishes.The calculation is the same as in (2.69), except that T should be replaced by Bb − + iCc − .– 23 –ence, we are left with the evaluation of the T BC term, S = − g o i h V | Z d d (cid:0) ˜ T ( p ) | p , Ω i (cid:1)(cid:16)(cid:0) ˜ B ( p ) b − | p , Ω i (cid:1)(cid:0) ˜ C ( p ) c − | p , Ω i (cid:1) + (cid:0) ˜ C ( p ) c − | p , Ω i (cid:1)(cid:0) ˜ B ( p ) b − | p , Ω i (cid:1)(cid:17) (2.72)= − ig o N Z d d x ˜ B ˜ C ˜ T e − V · x The complete action up to l = 1 (for scheme 4) is the sum of S (2.63) and S (2.68),(2.70) and (2.72). So far we considered scheme 4 at l = 1. If we remain at l = 1, but switch to scheme 3, wealready have 8 component fields. This results in over a hundred possible interaction terms.While many of those trivially vanish in light of, e.g. ghost number conservation, manyothers have to be explicitly evaluated. Furthermore, the number of terms grows fast as weincrease the level l , which is essential in order to obtain reliable results. Explicit evaluationof all terms would soon become hopeless. The resolution of this difficulty is to automatethe evaluation of the various coefficients that appear in the action. The quadratic termsare easily calculated. For the evaluation of the cubic terms, an efficient method shouldbe used. As in previous works that used level-truncation, we find that the most efficientmethod for the evaluation of these terms is using conservation laws of the cubic vertex [87].Conservation laws are obtained by evaluating the expectation values of currents inthe geometry of the three-vertex. These currents are built from products of the conformalfields, for which we want to derive the conservation laws, by conformal tensors of functions.The weight of these conformal tensors is properly chosen in order to obtain a current, andthe functions are constrained in order to prevent singularities at any point other than thepunctures, including infinity. Closing such a current around the three punctures leads toa linear combination of modes of the current, while deforming the current to infinity leadsto zero, as long as the functions were properly constrained. Actually, some more terms canbe obtained, both at infinity and around the punctures, if the current is anomalous, as isoften the case (e.g. Virasoro operators in the case of a non-zero central charge c , ghostcurrent, and ∂X in the case of a linear dilaton system). However, these terms are alsoexplicitly derived in [87].Here, we need the conservation laws for the b and c ghosts and for the ∂X (matter)modes. The lowest order conservation laws are, h V | c = h V | (cid:16) √ (cid:0) c − c (cid:1) + . . . (cid:17) , (2.73) h V | c − = h V | (cid:16) (cid:0) c + 8 c + 11 c (cid:1) + . . . (cid:17) , (2.74) h V | b − = h V | (cid:16) √ (cid:0) b − b (cid:1) − (cid:0) b + 8 b + 11 b (cid:1) + . . . (cid:17) , (2.75) h V | a − = h V | (cid:16)r α ′ (cid:0) p − p (cid:1) + 127 (cid:0) a + 16 a − a (cid:1) + . . . (cid:17) , (2.76)– 24 –here again, the superscript refers to the space in which the mode is defined and the el-lipses indicate higher level modes. Note, that the matter conservation law includes themomentum explicitly. In principle, the dilaton slope V could also occur. However, we canalways eliminate it in favour of the momenta using the anomalous momentum conserva-tion (2.57). The result then holds in any dimension. It might differ from the familiar flatspace expressions by terms proportional to p + p + p . Since conservation laws are given in the momentum representation, it is easier to writedown the action in this representation. For now we consider the one-dimensional case at l = 1 in scheme 3. Hence, Lorentz indices, when they appear, can obtain only a singlevalue and are therefore omitted. The quadratic term is given by S (3)2 = − Z dp (cid:16) m + p T ( p ) T ( − p ) + m + p (cid:0) A ( p ) A ( − p ) + 2 iB ( p ) C ( − p ) (cid:1) + B ( p ) (cid:0) B ( − p ) + q α ′ V A ( − p ) (cid:1) α ′ − iV √ α ′ B ( p ) A ( − p ) (cid:17) . (2.77)Here, the first line is the expression that we had before and the second line includes thenew fields. It is seen that all these fields are auxiliary fields, since there are no new kineticterms. Reality of the action is a consequence of the fact that products of even fields carryreal coefficients, while products of odd fields carry imaginary coefficients.For the evaluation of the cubic terms we use the conservation rules, which reduce thegeneral cubic term to that of the elementary tachyon vertex h V | | Ψ i | Ψ i | Ψ i ∝ h V | | Ω , p i | Ω , p i | Ω , p i . (2.78)The conservation laws give the proportionality coefficients, which can be zero and aremomentum-dependent. We already evaluated the fundamental (three tachyon) term, S ( T T T )3 = − g o h V | | Ω i | Ω i | Ω i = − g o N Z dp dp dp δ (cid:16) X j =1 p j − iV (cid:17) K − α ′ P k =1 p k . (2.79)Here, we wrote | Ω i k instead of | p k , Ω i k for short. Also, recall that the coefficient N is givenby (2.58).Even before the use of the conservation laws there are several terms that can bediscarded due to ghost number. The total ghost number of any three coefficient fieldsshould equal three. In our treatment, where we build the states over the ghost numberone | Ω i vacuum, it means that the total ghost number other than that of the vacua shouldequal zero. From the correlation between ghost number and statistics of the componentfields we can also infer that odd fields either do not appear, or appear as a pair, as shouldbe the case for obtaining an even action. That means that we would be able to continue– 25 –ntegrating those fields out, before commencing the simulations. All in all, there are only19 possible terms that have to be evaluated.In the evaluation of S there are six contributions to a generic coefficient, which comefrom the six possible orderings of the three coefficient fields involved. The properties ofthe three-vertex implies that these coefficients can only depend on the cyclic order of thefields. Hence, the term in the action that involves the component fields Ψ Ψ Ψ is givenby, − g o h V | (cid:16) | Ψ i | Ψ i | Ψ i + | Ψ i | Ψ i | Ψ i (cid:17) . (2.80)It turns out that in several cases the two orderings produce expressions that cancel out,after relabeling the three spaces, in particular, due to the momentum dependence of theresult. Another issue, which we have to notice, is that of symmetry factors, i.e. if twocomponent fields are the same, e.g. Ψ = Ψ , the result should be divided by two, while inthe case Ψ = Ψ = Ψ , the result should be divided by six. Even better (computationally)is to divide the result by one and by three respectively, but to evaluate only one of theterms in (2.80), since in these cases there is no issue of different orderings.We are now ready to write down the full expression in terms of (2.79), S (3)3 = − g o N Z dp dp dp δ (cid:16) X j =1 p j − iV (cid:17) K − α ′ P k =1 p k (cid:16) T ( p ) T ( p ) T ( p )+ 89 A ( p ) A ( p ) T ( p ) (cid:0) − α ′ ( p − p )( p − p ) (cid:1) + 16 i T ( p ) B ( p ) C ( p ) − T ( p ) B ( p ) B ( p ) (2.81)+ 16 √ α ′ B ( p ) (cid:0) A ( p ) T ( p ) − A ( p ) T ( p ) (cid:1) ( p − p )+ 32 i B ( p ) B ( p ) T ( p ) (cid:17) . Here, the first two lines are the expression that we had for scheme 4, the third line includesa new bosonic interaction and the last two lines include two new interaction terms involvingodd fields.The path integral (2.33) now contains also integration over the various new modes. Inparticular one expects it to contain integration over the odd variables included, namely, B , C , T and A , Z = Z (cid:16) Y j dT j d T j (cid:17)(cid:16) Y n d B n dB n d C n dC n dA n d A n (cid:17) e S . (2.82)Here and in the rest of the paper T j (denoted τ n above), T j , A n , etc., represent the modesof the various fields. The fields appear in the measure in pairs of an even and an oddfield, with the even ones written first. We need two different indices for the products sincethe number of modes of a given field depends on its l . We would also like, if possible,to integrate out the bosonic auxiliary field B . Since at higher levels it would be quiteimpossible to eliminate all the auxiliary fields, it could be nice to compare the results withand without the elimination of B . The auxiliary bosonic field C does not appear in theaction at all. – 26 –nspecting the action (2.77) and (2.81) we recognize that it suffers from a major prob-lem: A Grassmann integral can be non-zero only if the integrand has a term, which is linearwith respect to all the Grassmann modes. However, a term linear in all the odd fields isabsent in the path integral. Since odd terms enter the various terms in the action eitherquadratically or not at all, the problem of saturating all the modes is that of the regularityof the (bosonic-field-dependent) matrix of coefficients of the terms quadratic with respectto the odd variables in the action. The problem then is that this matrix turns out to besingular.This problem occurs since level truncation does not commute with Grassmann inte-gration. Actually, we faced this problem already at level zero, where we noticed that thefield T , which is present in (2.2) is absent from the action altogether. There, we decided toignore this field temporarily and it indeed enters the action now. However, it is not clearwhich fields should we retain now and which ones should be postponed to the next level.Inspecting the action we see that the field B is present in all the relevant expressions and issaturated in each term by one of the other fermionic fields, namely T , C and A . This is notparticularly surprising, due to the ghost number of the states that these component fieldsmultiply. However, it is not clear which modes should we keep now. The most “natural”choice would be to keep T , since it already “enters too late to the game”. However, onecould object to the idea of adding high T modes before adding the first C modes, since itwould make our cutoff l -dependent instead of l -dependent. Furthermore, since the modesof T and B enter the level truncation at different cut-off values, we would generally havea different number of such modes and it would be impossible to saturate the Grassmannintegral.One could worry that such problems could occur also for scheme 4, which also includesodd modes. This is not the case. The source of the problem we face here is the fact thatthe fields T and A are auxiliary and hence do not have kinetic terms. The kinetic termsprovide regular parts for the matrix. Hence, the integral over the odd fields is regular forscheme 4, except perhaps for some specific values for the bosonic fields.An additional potential difficulty with scheme 3, is that it is likely to inherit fromscheme 1 the problem, to be described in 4.9, of a nearly-massless mode leading to largeinstabilities. In light of all that we do not dwell further into scheme 3. We also would like to check scheme 1, in which we only keep the fields T , A and B .The action is just the truncation of the scheme-3 action to include only these fields. Thequadratic part of the action is S (1)2 = − Z dp (cid:16) m + p T ( p ) T ( − p ) + m + p A ( p ) A ( − p )+ B ( p ) (cid:0) B ( − p ) + q α ′ V A ( − p ) (cid:1) α ′ (cid:17) , (2.83)– 27 –nd the cubic part is S (1)3 = − g o N Z dp dp dp δ (cid:16) X j =1 p j − iV (cid:17) K − α ′ P k =1 p k (cid:16) T ( p ) T ( p ) T ( p )+ 89 A ( p ) A ( p ) T ( p ) (cid:0) − α ′ ( p − p )( p − p ) (cid:1) − T ( p ) B ( p ) B ( p ) (cid:17) . (2.84)The explicit integration of the B field should be much easier in this scheme as comparedto schemes 3 and 4. Before we attempt a numerical study of the case with many modes, we would like toexamine analytically the simplest possibility of retaining a single mode. Hopefully, wecould get some feeling about what should be expected from this simple example. Thelowest lying mode would be the first mode of the tachyon field. Its level depends on thelength L of the range which we consider for X and it is given by T ( x ) = r L sin (cid:16) πn ( x − x min ) L (cid:17) T , (2.85)where T is the only variable in the theory. The action is S = − (cid:16) α ′ + (cid:0) πL (cid:1) (cid:17) T − ˜ g o f K − α ′ (cid:0) πL (cid:1) T , (2.86)where we absorbed some constants into the coupling constant and the single couplingconstant of the theory is found to be f = (cid:16) L (cid:17) Z L − L dxe − V x sin (cid:16) π ( x − L ) L (cid:17) = (cid:16) L (cid:17) π L cosh (cid:0) LV (cid:1) L V + 40 π L V + 144 π . (2.87)Here for simplicity we take the range of integration to be symmetric with respect to theorigin. It is easy to see that, as one should expect, f approaches infinity as L → ∞ .Performing the advocated analytical continuation T → e iπ/ T (see section 3 for details)the action becomes S = − e iπ (cid:16) α ′ + (cid:0) πL (cid:1) (cid:17) T − i ˜ g o f K − α ′ (cid:0) πL (cid:1) T . (2.88)The simplicity of this expression makes it possible to evaluate the partition function ana-lytically. Write, S = − a ( L ) T − ib ( L, V ) T . (2.89)Then, the partition function is given by, Z = Z ∞−∞ dT e S = Z ∞−∞ dT e − aT − ibT = b − Z ∞−∞ dT e − ab − T − iT = 2 πe a b (3 b ) Ai (cid:16) a (3 b ) / (cid:17) , (2.90)– 28 –here Ai is the Airy function. We know that the integral converges, since our a has apositive real part. However, the result is not real, as expected. Nonetheless, when we takethe limit L → ∞ , the partition function approaches a real value. In this limit (setting α ′ = 1) we have (regardless of the value of V ), a → e iπ , b → ∞ , (2.91)where the approach of b to infinity is along the positive real line. The factor in front ofthe Airy function is real and approaches zero as L → ∞ . The Airy function, on the otherhand, is complex. However, it has a real limit,Ai (cid:16) a (3 b ) / (cid:17) → Γ (cid:0) (cid:1) . (2.92)In principle, we were not supposed to expect a real limit here, since we are truncatingto the lowest single mode. Nonetheless, it is encouraging to see that the wild oscillationsconspire to produce a real value already at this stage. Also, we see that reality is really ob-tained only as we take the limit L → ∞ . Thus, comparing finite values does not necessarilymake sense.Using (2.90), we can study the dependence of various expectation values as a functionof a and b . One can obtain different values for these coefficients in many ways, by usingsymmetric as well as non-symmetric limits for x min and x max . The limit b → h S i = − , while h T i approaches zero from the direction of e iπ and h T i approaches zero from the direction e − iπ . Conversely, in the mentioned above limit b → ∞ , we find that h S i = − , while h T i approaches zero from the direction e − iπ and h T i approaches zero from the direction e iπ . We will compare these results to the latticesimulation of section 4 .Another important remark regarding the fact that the normalization factor approacheszero: on the one hand, the normalization factor should be renormalized as we change ourparameters. Thus, from this perspective, there is nothing here to discuss. On the otherhand, keeping only the lowest level amounts to truncating more and more modes as L approaches infinity. This is not a natural limit and we took it only for the purpose ofverifying that we can reproduce on the lattice the analytical expression that we obtainusing it. The natural limit that we would have to consider is taking L to infinity whilekeeping l fixed. This leads to more and more modes, up to infinity at the limit. This is thephysical limit. However, the increase in the parameters, as well as the introduction of anever growing number of modes, imply that a non-trivial renormalization would be needed. Another problem that a lattice simulation in a linear dilaton background faces comes fromthe possible presence in the action of trivial terms. By that we mean the presence in the Note, that in section 4 we use a slightly different convention, in which the factor of e iπ explicitlymultiplies the fields. This leads to different constant phases as compared to what we did here. – 29 –efinition of the cubic interaction of terms that vanish due to the anomalous momentumconservation. Such terms can be added to the definition of the vertex also in the standardcase of a constant dilaton. In any case they take the form of conformal fields inserted atthe three interaction points times a momentum dependent function of the form F ( p , p , p ) = p n p n p n (cid:16) p + p + p − iV (cid:17) n , , ≥ . (2.93)The expression in the brackets is identical to the argument of the momentum conservationdelta function and thus leads to zero contribution of these terms, which can, therefore, beadded to the definition of the interaction at will. In previous works use was made of suchterms in order to simplify the form of the interaction in various contexts, e.g. in [86].While the ambiguity in these terms is usually harmless and could even be useful, in ourcase new complications emerge. The momentum conservation is broken by the introductionof the lattice. Thus, while the introduction of these terms does not change the action beforethe introduction of the lattice, it does influence the results when a lattice is used. A simpleidea for a resolution would be to avoid these terms altogether. However, it is not clear howto distinguish the “genuine” interaction from the trivial terms. The definition of the actionis really ambiguous. This is somewhat similar to the case of a gauge symmetry: there is nocanonical way to gauge fix. The analogue of gauge fixing in our case is the decision of whichis the correct form of the action. However, on the lattice different “gauge fixings” lead todifferent results. One would like to be able to show that as the lattice cutoff is removed,the results tend to the same values regardless of the “gauge choice”. Unfortunately, thisseems to be quite unlikely, since the coefficients of the trivial terms can be arbitrary andmore and more such terms pop up as the level is being increased. One could try to fixthe ambiguity by demanding that the form of the interaction be “as simple as possible”.While this statement makes sense at low levels, it becomes ambiguous at higher levels.Another possibility for a “gauge fixing” is to avoid the appearance of V in the action otherthan in the exponent. While this option does not necessarily lead to the simplest possibleexpressions, as we have already seen in our l = 1 example above, the expressions areunambiguous and are formally independent of the dimension d . Unfortunately, it is notclear that the expressions obtained in this way are more correct than those of any other“gauge choice”.It is important to stress that the problem could have been avoided had we been workingin a constant dilaton background. In such a case we would have chosen to work withperiodic boundary conditions that do not make sense in the case at hand. Then, momentumconservation would not have been broken by the lattice. Moreover, the presence of the lineardilaton leads to yet another problem due to the anomalous form of the conservation law.The issue is not so much the fact that the sum of momenta is non-zero, as with the factthat it is imaginary. This is not a problem before the introduction of the lattice, since itonly results in the exponential term in coordinate space. However, with the introductionof the lattice actual imaginary terms pop-up. These terms produce further problems: aswe mentioned above, the action being cubic is not bounded from below, a problem that weresolve using a change of the contour of integration followed by an analytical continuation.– 30 –his procedure turns the real cubic terms to purely imaginary terms, which results inconvergence of the expressions. However, the imaginary terms become real now, whichbrings us back to the starting point, in which no numerical analysis is possible. One couldhope that the ambiguity in the form of the interaction term can be used in order to set tozero the imaginary part of the interaction. We examine the consequences of adding trivialparts to the action in section 4.8. We now want to use lattice simulations to calculate observables. The degrees of freedomare the fields found above using level truncation, up to some maximum total level l max ,not necessarily an integer. Explicitly, (1.5) can be written as l = l + α ′ p . (3.1)For our sine-expansion, p = πnL , and since we have only evaluated the level truncation upto l = 1 we must choose l <
2. The number of modes for the l = 0 fields is then n = ⌊ Lπ r l max α ′ ⌋ (3.2)and if l max > l = 1 fields is n = ⌊ Lπ r l max − α ′ ⌋ . (3.3)Thus given a lattice size L and a choice of ‘scheme’, our degrees of freedom will bea finite number of modes of one or more fields. We can read off the action from theappropriate expressions above, e.g. for scheme 4 and l < e S ratherthan e − S due to the way we Wick-rotated. In addition to the various ‘schemes’ describedabove, we have also carried out additional runs where we have removed the level-1 fieldsfrom the action. This is to try to assess whether the higher level fields are helping to tamethe instabilities.Looking at the action we see an immediate problem: The action has a cubic instability.To proceed, we consider the integral over each mode as a complex integral, and deformthe integration contour to be a straight line at an angle γ to the real axis. If we choose γ = π/
6, the cubic part of the action becomes pure imaginary and so the action is no longerunstable. In principle we could have chosen different phases, i.e. γ = ± π/ l and the momentum.However, taking the modes to be complex introduces another problem; the action alsobecomes complex and so cannot be interpreted as a weight for a Markov chain. Insteadwe simulate in the phase-quenched ensemble and reweight. That is, we split e S into anamplitude and a phase: e S = | e S | e iθ , (3.4)– 31 –nd calculate the expectation value of an observable O using the identity hOi = R O| e S | e iθ R | e S | e iθ (3.5)= hO e iθ i PQ h e iθ i PQ , (3.6)where the label PQ means the expectation value is evaluated in the phase-quenched en-semble, i.e. with the weight | e S | . This is a real, positive weight, so it can be used in aMonte Carlo simulation.We generate configurations in the phase-quenched ensemble using a Metropolis algo-rithm, chosen since it is simple to implement and to alter for the variety of different fieldcontents and actions we are concerned with. In the cases where we have Grassmann-oddfields we include their contribution by calculating the fermion determinant directly. Thiswould be expensive for a large number of modes (the cost scales as n ) but is reasonablefor the small number of modes in our simulations (we have at most n = 9). In any casesince the action is non-local the cost of evaluating it scales as n , even for the bosonicpart.Due to the phase-quenching, our errors increase as the imaginary part of the actionincreases, i.e. as we move to larger x . To some extent it is possible to compensate for thisby increasing the number of configurations in our simulations, but the number of configu-rations required increases exponentially with x so eventually this becomes impossible. Thepractical effect of this is that it gives an upper limit on the values of x at which we cansimulate; it will be very difficult to go much beyond this in future work.There is no general method known to avoid the exponentially large cost associated withcomplex actions. In some specific cases the complex Langevin method (see [88] for a recentreview), which does not have an exponential cost, can be used to bypass this ‘sign problem’.The complex Langevin method is not a panacea, however; in some cases it converges to thewrong limit [89]. We attempted to bypass the sign problem by implementing the complexLangevin method for our system. We found results in agreement with the conventionalMonte Carlo simulations at weak coupling, but disagreement at strong coupling, indicatingthat the complex Langevin method was converging to the wrong limit. Thus we did notpursue this method further.As discussed in section 2.1.1, the action can be reformulated so that the quadraticterms are non-local while the cubic terms are local. The two formulations are equivalentand therefore should give identical results. Confirming that this is the case is a usefuladditional check of the correctness of our code. We have carried out this check for severalsets of parameters and indeed found good agreement. The run time and statistical errorsare similar for both formulations, so there is no particular benefit from using either case.We have chosen to use the formulation with the non-locality in the cubic term, and all ourresults below are for that case. – 32 – .1 Observables The observables we measure are the action h S i , and the expectation values of the Grass-mann even fields and their squares. The specific field content is dictated by the choice ofscheme and level. For example, for scheme 4, we measure h T n i and h T n i for all l max , andalso h A n i and h A n i if l max >
1. Here the subscript n refers to the mode number, and wemeasure all modes present. We find that the h A n i are always consistent with zero, in somecases with very small errors, of order 10 − . This is because A only appears quadraticallyin the action — there are no terms linear or cubic in A . Hence we will not discuss h A n i further.One issue to be considered is whether or not to include the logarithm of the fermiondeterminant in the action when Grassmann-odd fields are present. (Here we refer to theaction considered as an observable, not to the action used for the update algorithm, whereof course the fermion determinant must be included.) The statistical weight used whenGrassmann-odd fields are present is det M e S B , (3.7)where det M is the fermion determinant and S B is the bosonic part of the action. This canbe rewritten as e ln det M + S B . (3.8)The question is whether to take S B or (ln det M + S B ) as the action. Neither of theseis obviously more physical than the other, but in the weak-coupling limit, S B will simplybe − per bosonic degree of freedom, whereas (ln det M + S B ) will contain additional L -dependent terms coming from det M . Because of this we have chosen to use S B as theaction observable. As described above, we define the theory by an analytical continuation of the integrationcontour, which is implemented by a rigid rotation in the complex plane. So far we con-sidered this rotation to be by an angle of π , which is exactly what is needed in order tomake the cubic part of the action purely imaginary. We define γ as the angle of rotation,i.e. γ = 0 is the original theory and γ = π is the angle that is needed for our analyticalcontinuation.Taking γ = π has a large numerical cost, since then the action has a large imaginarypart. Because of this, we use γ = 0 when this is possible, i.e. when the cubic term is small.In some cases we found that it is possible to use intermediate values of γ when the cubicterm is not too large; this is worth it because even a small decrease in γ from π gives alarge saving in computational cost.When the instability is small it is possible to compare results for different values of γ in the range 0 ≤ γ ≤ π in order to establish γ -independence. We have done this for severalsets of parameters and obtained good agreement. For example, at α ′ = 1, V = − q α ′ ,– 33 – = 20, x min = −
20, and l max = 1 .
6, we obtained h T i = − . − . i γ = 7 π , (3.9a) h T i = − . − . i γ = 15 π , (3.9b) h T i = − . − . i γ = π . (3.9c)These results are clearly in good agreement. In this case, we found out that the metasta-bility at γ = π is manifested only around 1 . × updates (the result above was obtainedfrom 10 updates), while we did not observe the metastability at γ = π . Our resultssuggest that as long as the metastability does not manifest itself the results are almost γ -independent. α ′ -independence α ′ , or equivalently l s (1.2), or m s (1.10), sets the scale for our simulations — for examplethe physical box size is l s L . A useful check on the code is that it gives the same resultsfor different α ′ when all physical quantities (box size, l max , g o , . . . ) are the same . Wehave carried out this check explicitly for the case α ′ = 1, L = 20, x min = − l max = 0 . α ′ = 1; thus the lattice units are equivalent tostring units. It is important to have reliable estimates of the statistical errors on our results, which weestimate using the jackknife method. This should provide accurate estimates, providedthat there are no large auto-correlation times in the data, i.e. provided that configurationswith large separation of lattice times are uncorrelated.In a typical simulation we evaluate about 10 updates, which we split into about100 bins for analysis. The jackknife analysis is supposed to work well provided thesebins are uncorrelated. Thus, the question is whether configurations 10 updates apartare correlated. Since we have only about 20 degrees of freedom or less, it would be verysurprising if this would have been the case.As an additional check we have analysed how accurate the error estimates are. First,we generated high statistics data (2 × configurations) for a particular set of parameters( α ′ = 1, V = − q α ′ , L = 20, x min = − l max = 0 . h T i with very small errors: h T i = − . − . i . We then carried outten independent low statistics runs (2 × configurations each) with the same parameters.Each of these gives an independent estimate of h T i with errors, and if the error estimateis correct these should all be consistent with the high-statistics result. For example, thefirst low-statistics run gave h T i = − . − . i , which is indeed consistent.Including both real and imaginary parts, this procedure gives 20 estimates with errors.11 of these are within 1 σ , 19 are within 2 σ , and all are within 3 σ of the high-statistics It is important to remember that, as described in 1.1, g o has dimension . – 34 –esult. This is completely consistent with the errors being estimated correctly, and indeedshows that any bias in the errors must be quite small. We focus our simulations on the issue of whether the theory becomes stable (or at leastless unstable) in the limit where l max and/or L go to infinity. Recall, that we interpretstability as the vanishing of the imaginary parts of the various expectation values. Wehave no reason to expect that those will vanish already at the low level we work with here,but we would like to observe a tendency of decreasing the imaginary parts as compared tothe real parts at least of some of the observables. In principle we would also like to take x max to infinity, but as explained above this will not be possible and we will have to becontent with taking it as far into the strong coupling region as we can. We expect thata sign of stability will be that the imaginary parts of observables go to zero, or at leastdecrease. Of course, it may be that this will work better for some observables than others.In particular, we might expect that it will work best for the lowest modes, which shouldbe less affected by the missing higher level modes and fields.The results below are mainly for scheme 4, where we have the most detailed results,organised roughly into sections dealing with the effect of varying a single parameter (e.g. L , V , . . . ) at a time. This is followed by briefer overviews of our results for the otherschemes. A simple test of our code is to look at the case of a single mode of the field T , where wecan compare to the analytical results of section 2.8.We can choose several sets of parameters that will give a single mode, for example L = 20, l max = 0 .
05, or L = 6, l max = 0 .
9, both with α ′ = 1 and V = − q α ′ . Theseshould give identical results, apart from an overall scale and shift in x due to the differentvalues of a and b . We have found that this is indeed the case.Another check is whether we obtain the correct limiting values. The case h S i = − for x min → −∞ is easy to check: for example, for L = 6 and l max = 0 . S = − . − . i ; x min = − S = − . . i ; x min = − . , (4.2)which is already very close.The case x min → ∞ is harder since the simulations become expensive in this limit.However, it is still doable, and for the same parameters we get S = − . − . i ; x min = − S = − . . i ; x min = − . (4.4)This is clearly going to the correct limit of − .– 35 – I m < T ^ > Re
Figure 1 : h T i in the complex plane for L = 6, l max = 0 . x min in the range − T approaches the origin from the correct direction,i.e. at an angle of π to the real axis. This is shown in Fig. 1, again for the case L = 6 and l max = 0 .
9. We see that indeed the results appear to be going towards the right asymptoticline. All other observables approach zero as well, except for the action, which as describedabove goes to − . (The reason the action is different is the factor of b in eq. (2.89), whichdiverges in this limit.) l max l max is the highest level allowed for fields in the simulation. Since we only include fieldsup to l = 1, it must be less than 2. Increasing l max means both allowing more fields (e.g.the A field only appears for l max >
1) and increasing the number of momentum modes ofeach field.Since increasing l max is like increasing the cutoff, we might hope that for high enough l max the results will become independent of l max (at least for some quantities). With thisin mind we have done some scans in l max , keeping all other parameters fixed.Generally we have concentrated on the observables h S i and h T i to simplify the pre-sentation. However results for the other observables are similar. Throughout this sectionwe fix α ′ = 1 and V = − q α ′ . L = 20We begin by describing our results for L = 20, where we have the most extensive results.This value of L is quite large, so there is a reasonable number of modes — 9 for T and6 for A up to l = 2. We have done runs for every l max between 0 and 2 which gives a– 36 – I m < S > maxlevel fullno level 1 (a) x min = − -0.25-0.2-0.15-0.1-0.05 0 0.05 0 0.5 1 1.5 2 I m < S > maxlevel fullno level 1 (b) x min = − -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0 0.5 1 1.5 2 I m < S > maxlevel fullno level 1 (c) x min = − -0.5-0.4-0.3-0.2-0.1 0 0.1 0 0.5 1 1.5 2 I m < S > maxlevel fullno level 1 (d) x min = − Figure 2 : ℑh S i for L = 20 for the full theory (red), and without level-1 fields (green), for x min = − , − , − , − A , B and C are not included in the simulation, to give some idea of the effect the level-1 fields have onthe physics.The strong coupling region begins around x min = −
20, and we are able to get resultswith reasonably small errors up to x min = −
18. We show results for the imaginary partsof the action and the T mode in Figs. 2 and 3. Other observables show similar behaviour.The general trends we see are quite clear. First, increasing l max increases ℑh S i , and to alesser extent ℑh T i as well. In particular, there is no evidence that the imaginary parts aregoing to zero as l max is increased. There is not a great difference between the runs withand without the level-1 fields; in some cases they make the imaginary parts smaller, and inother cases larger. We also see a general trend for the imaginary parts to be larger at larger x min ; this is not surprising as the destabilising cubic terms are becoming larger. Anotherpoint is that the results do not appear to become independent of l max as it is increased, sowe are not (yet?) seeing cutoff-independence.From all of these, we also see that generally the results are smooth in l max . This means– 37 – I m < T _1 > maxlevel fullno level 1 (a) x min = − -0.35-0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0 0.5 1 1.5 2 I m < T _1 > maxlevel fullno level 1 (b) x min = − -1-0.8-0.6-0.4-0.2 0 0 0.5 1 1.5 2 I m < T _1 > maxlevel fullno level 1 (c) x min = − -1.4-1.2-1-0.8-0.6-0.4-0.2 0 0.2 0 0.5 1 1.5 2 I m < T _1 > maxlevel fullno level 1 (d) x min = − Figure 3 : ℑh T i for L = 20 for the full theory (red), and without level-1 fields (green),for x min = − , − , − , − l max to see the trends, which we often do from now on,especially in section 4.3. L = 10This is a much smaller interval, so there are fewer modes, specifically 4 for T and 3 for A up to l = 2. There may potentially be a problem with having too few modes — e.g. onemight imagine that one needs many modes to see “continuum” physics. On the other handhaving fewer modes makes the simulations faster, so we can achieve smaller errors or goto stronger couplings. Note that because the numbers of level-0 and level-1 modes scaledifferently, there is no simple mapping between L = 10 and L = 20.Again, we have carried out simulations for every l max between 0 and 2 which gives adifferent number of modes. We have also added runs where the level-1 fields A , B and C are not included in the simulation, to give some idea of the effect the level-1 fields have onthe physics. – 38 – I m < S > maxlevel fullno level 1 (a) x min = − -0.2-0.15-0.1-0.05 0 0.05 0 0.5 1 1.5 2 I m < S > maxlevel fullno level 1 (b) x min = − -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0 0.5 1 1.5 2 I m < S > maxlevel fullno level 1 (c) x min = − Figure 4 : ℑh S i for L = 10 for the full theory (red), and without level-1 fields (green), for x min = − , − , − x min = −
10, and we are able to get resultswith reasonably small errors up to x min = −
8. We show results for the imaginary parts ofthe action and the T mode in Figs. 4 and 5. Roughly speaking, we would expect results ata given x min for L = 10 to match those at x min −
10 for L = 20, since both will extend thesame distance into the strong coupling region, so we have chosen the appropriate values of x min to allow this comparison to be made with the results of section 4.2.1. In fact this turnsout to be not exactly true, and we see no instability at all at x min = −
11 (the imaginaryparts are at most of order 10 − ), so we do not show any plots for this case.We see that the results look rather similar to those for L = 20. One potentiallyinteresting area in these plots is just above l max = 1, since the density of level-1 modesis much higher here for L = 20 than for L = 10. However, comparing the plots, nothinginteresting seems to happen in this range. Overall, it appears there is not much differencebetween L = 10 and L = 20, which at least suggests that the number of modes we arelooking at is not too small. – 39 – I m < T _1 > maxlevel fullno level 1 (a) x min = − -1.2-1-0.8-0.6-0.4-0.2 0 0.2 0 0.5 1 1.5 2 I m < T _1 > maxlevel fullno level 1 (b) x min = − -1.4-1.2-1-0.8-0.6-0.4-0.2 0 0.2 0 0.5 1 1.5 2 I m < T _1 > maxlevel fullno level 1 (c) x min = − Figure 5 : ℑh T i for L = 10 for the full theory (red), and without level-1 fields (green),for x min = − , − , − L = 30This is a large interval, with the number of modes and their density also becoming quitelarge - specifically we have n = 13 and n = 9 for l max = 2. If there is any effect thatrequires many modes, it would be surprising if it did not yet appear here. However, thelarge number of modes also makes L = 30 very expensive . The largest x min we havebeen able to reach is x min = −
30, corresponding to x max = 0 and so only at the beginningof strong coupling.Again, we have results for every l max between 0 and 2 which gives a different numberof modes. We show results for the imaginary parts of the action and the T mode in Figs. 6and 7. The overall results look rather similar, though in detail they are different; forexample, ℑh T i is now positive for some values of l max , which was not the case for smaller L . Unlike for the case of L = 20, x min = −
21, and L = 10, x min = −
11 we see that at The computational cost goes very roughly as n . – 40 – I m < S > maxlevel fullno level 1 (a) x min = − -0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0 0.5 1 1.5 2 I m < S > maxlevel fullno level 1 (b) x min = − Figure 6 : ℑh S i for L = 30 for the full theory (red), and without level-1 fields (green), for x min = −
31 and − -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 I m < T _1 > maxlevel fullno level 1 (a) x min = − -0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 I m < T _1 > maxlevel fullno level 1 (b) x min = − Figure 7 : ℑh T i for L = 30 for the full theory (red), and without level-1 fields (green),for x min = −
31 and − x min = −
31 there are rather large imaginary parts for the larger values of l max . Thereforeit looks like increasing L at constant x max is destabilising. This seems a bit counter-intuitive when we think about it another way: going from L = 20, x min = −
21 to L = 30, x min = −
31 is a change in x min while keeping x max fixed, so we are only adding a regionof extremely weak coupling — how can this make things less stable? Probably the answeris connected to the fact that increasing L means we have more modes, and thus a shift inone boundary affects the physics throughout the interval. Moreover, the non-locality of theaction might also source some influence of one boundary on the rest of the interval. Theseeffects lead to a worsening of the numerical sign problems and therefore to larger errors.– 41 – .3 Changing x min x min is a very important parameter as it controls the strength of the cubic terms and hencethe instabilities. We expect that for small x min there will be no instabilities and so theimaginary parts of observables will be zero. They will then increase, and finally, if theybehave as in the case of a single mode (see section 4.1) go to zero (except for the actionwhich should go to a finite value).Since we have established above that the observables have a rather smooth dependenceon l max , we have mostly concentrated on a few values of l max . We do not present resultsfor L = 30 in this section since, as discussed above, we have been unable to reach verystrong couplings in this case. L = 20We begin again with our results for L = 20, where we have the most data. For most ofthis scan in x min , we only use the values l max = { . , . , . , . , . } . We also include l max = { . , . } with level-1 fields removed. We have tried to reach the highest valuesof x min possible, in order to try to reach the extreme strong-coupling region where all theobservables go to zero. In practice this has only been possible for the lower values of l max .First we show a plot of h S i , in Fig. 8, showing both the behaviour of the imaginary partand the trajectory in the complex plane. We see that there is a similar trend for all l max .This even includes the single-mode l max = 0 .
05, except for a shift in x min . Presumablythe fact that there is a single mode means that there is less instability. Actually, this isthe opposite of what we would expect from the hope that “the instabilities become of zeromeasure” as the number of degrees of freedom increases.In the complex plane, it appears that the action executes a qualitatively similar ’loop’for all l max . In all cases the imaginary part only seems to become zero again when thetrajectory reaches its final point, although for the higher l max the data does not go farenough to be certain of this. Note that the large offsets between the loops for the different l max are simply due to the − / l max are rather similar — things appear to get neither better norworse as more modes are added.In some cases there appear to be cusps in the trajectories in Fig. 8, although this isnot quite clear. If they are present they are probably related to the loops and cusps seenat small L (see section 4.4 below.)The imaginary part of h T i has a similar behaviour, plotted in Fig 9. In this case thereturn to zero is slower, but this is partially compensated for by the fact that the errors aresmaller so we are able to go to higher x min . Again there is not much difference between thedifferent values of l max , except for l max = 0 .
05, where again the instability begins at higher x min . There are clear oscillations in the data; these correspond to cusps in the complexplane like those seen for the action. (These can be seen more clearly at smaller L , e.g. inFig. 13.) – 42 – I m < S > x_min 0.050.50.91.51.9991.5_nolevel11.999_nolevel1 (a) ℑh S i -0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2-8 -7 -6 -5 -4 -3 -2 -1 0 I m < S > Re 0.050.50.91.51.9991.5, no level 11.999, no level 1 (b) Trajectory of h S i in the complex plane.In all cases increasing x min corresponds toanticlockwise movement along the trajectory. Figure 8 : h S i for L = 20 as a function of x min for various l max . -2.5-2-1.5-1-0.5 0 0.5 1 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 I m < T _1 > x_min 0.050.50.91.51.9991.5_nolevel11.999_nolevel1 Figure 9 : ℑh T i for L = 20 as a function of x min for various l max . L = 10This is quite a small interval, but has the advantage that it is possible to get to strongercoupling than in the L = 20 case. We use the values l max = { . , . , . } , and also l max = 1 . h S i , which are plotted in Fig. 10. The main features arethe same as for L = 20, but now we can follow them to stronger coupling, giving us moreconfidence in what happens there. It is clearer that h S i is returning to the real axis for all l max . The cusps in the complex plane are still there, and the single-mode case (in this case l max = 0 .
2) again becomes unstable at larger x min than the others, although the difference– 43 – I m < S > x_min 0.20.91.91.9_nolevel1 (a) ℑh S i -0.4-0.3-0.2-0.1 0 0.1-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 I m < S > Re 0.20.91.91.9, no level 1 (b) Trajectory of h S i in the complex plane.In all cases increasing x min corresponds toanticlockwise movement along the trajectory. Figure 10 : h S i for L = 10 as a function of x min for various l max . -1.4-1.2-1-0.8-0.6-0.4-0.2 0 0.2 -10 -8 -6 -4 -2 0 2 I m < T _1 > x_min 0.20.91.91.9_nolevel1 Figure 11 : ℑh T i for L = 10 as a function of x min for various l max .seems smaller this time.For ℑh T i , plotted in Fig. 11, we can follow the behaviour almost back to ℑh T i = 0for some l max . In general, the behaviour is similar to that for L = 20. The oscillationsare still present, for example for l max = 0 .
9, though they are harder to see as we have notsampled so densely in x min . Some of the results in the sections above are rather complicated, due to the presence ofmany modes. This is probably necessary to reach the large-volume and continuum limits.However, it may also be useful to look at a small number of modes to try to interpolate– 44 – I m < S > x_min 0.91.21.9 (a) ℑh S i -0.2-0.15-0.1-0.05 0 0.05 0.1 0.15-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 I m < S > Re 0.91.21.9 (b) Trajectory of h S i in the complex plane.In all cases increasing x min corresponds toanticlockwise movement along the trajectory. Figure 12 : h S i for L = 6 as a function of x min for various l max .between the well-understood single mode case (see section 4.1) and the more complicatedcases with many modes. Another advantage is that by keeping the number of modes smallwe can go to stronger couplings.In this section we keep the number of modes small by taking the interval length L small, specifically 6 or less. Note that the smallest value of L we can take while keepingat least one mode below level 2 is L = π/ √ ≈ .
22, and that for
L < π we have only asingle mode.Another reason to look at small L is that it may be possible that the continuum andlarge-volume limits are tied together, such that we need to take L and l max to infinity whilekeeping their ratio l max /L fixed. Such a requirement seems natural, for example, from theperspective of T-duality. Since we are restricted to l max < L smalltoo, to keep this ratio at least moderately large. L = 6For L = 6 the maximum number of modes for l max below 2 is two for the T field and onefor the A field. Since the number of modes is small, we can look at all of them withoutthe plots becoming excessively complicated. We first plot h S i and h T i in Figs. 12 and 13respectively.The case l max = 0 . l max = 1 . l max = 1 . L . There are very large oscillations in ℑh S i ,and an inflection point in ℑh T i . From the right-hand panels of these figures we see thatthe oscillation in ℑh S i is actually a loop in the complex plane, and the inflection point in ℑh T i is actually a cusp.The remaining observables are T , T , T , and A (the superscripts are powers). There is also A , but this is always consistent with zero. We could have studied also expectation valuesof higher powers of the fields, but didn’t do that for simplicity. – 45 – I m < T _1 > x_min 0.91.21.9 (a) ℑh T i -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 I m < T _1 > Re
9, and T is smooth in this region, and the cusp in T is around x min = − .
0, where T is smooth. This possibly seems to indicate that they donot have a single cause.Another issue is whether these features are present at larger L . It is in fact quite likelythat they are present, but that they are more difficult to see. This is for two reasons: firstlythe large errors will obscure small features in the complex plane, and secondly it is notpossible to go to as strong couplings, where the features seem to show up. In fact thereare some indications for features like this at larger L in some of the plots, e.g. in Fig 10,and the oscillations in Fig. 9.For large x min nearly all the observables approach the origin. The exception is theaction, where we already saw for the case of a single mode that the action went to a finitevalue despite the fact that T and T go to zero.However, it is still interesting to ask how the observables approach the origin. Inparticular, do they approach along the real axis? If so this would be a good sign, since itwould mean that as the origin is approached the ratio of imaginary part and the real parttends to zero. This is not necessary, but could be an indication that we really approach agood limit, since for expectation values that approach a non-zero real limit this ratio tendsto zero regardless of the angle of approach. However, this does not seem to happen for anyof the quantities we measure; they all approach the origin from a complex direction. Thereis also no trend of this getting better as l max increases.– 46 – I m < S > x_min 0.91.9 (a) ℑh S i -0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2-1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 I m < S > Re 0.91.9 (b) Trajectory of h S i in the complex plane.In all cases increasing x min corresponds toanticlockwise movement along the trajectory. Figure 14 : h S i for L = 4 as a function of x min for various l max . -0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1-4 -3 -2 -1 0 1 2 3 4 I m < T _1 > x_min 0.91.9 (a) ℑh T i -0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 I m < T _1 > Re
15. This is almost the smallest interval we can havewhile keeping more than one mode — the second mode would go above level 2 at L = π .The modes present are in fact the same as at L = 4; it is just their masses and the cubicterms that are different. For example, evaluating the masses explicitly, we find for L = 4 m T = 0 . . . . (4.5) m A = 1 . . . . , (4.6)and for L = 3 . m T = 1 . . . . (4.7) m A = 1 . . . . . (4.8)So as we decrease L the masses increase, while coming closer together.We find that in general the results are rather similar to those for L = 4. As an examplewe show the action in the complex plane in Fig. 16, where clearly the differences are rathersmall. We find similar results for the other observables. This suggests that the changes inthe number of modes, rather than the changes in the parameters in the action, are moreimportant. Also, this is the largest l max /L we are able to reach, and nothing very helpfulseems to happen. In this section we do not present any results not already mentioned above. Instead weshow some plots comparing results at different L , mostly showing that there are no strongtrends in L . In particular, there is not much sign of instabilities reducing as L increases.This would be seen by either the imaginary parts of observables getting smaller, or by theapproach to the origin at large x max happening at a smaller angle to the real axis (or both).When comparing different L , it is not always clear what are the equivalent things tocompare. For example, if L is doubled, should the mode T be now compared to T , whichhas the same wavelength? Or should it still be compared to T , which is the lowest mode?We will mention issues of this sort as they become relevant below.First we compare h S i at two different values of l max for L = 6, 10 and 20, in Fig. 17.We see that in both cases there is a very large shift of h S i for the different L . However,most of this is just due to the fact that the number of modes is increasing and the freeaction is − / ℑh S i . Thereis also perhaps a tendency for the trajectories to become more complicated for higher L .We now turn to comparisons of the field T . First we consider the lowest mode T , forwhich we show a plot in Fig. 18. The cases L = 6, l max = 0 . L = 10, l max = 0 .
2– 48 – I m < S > Re L=3.15L=4
Figure 16 : Trajectory of h S i in the complex plane for L = 4 and L = 3 .
15, both for l max = 1 . x min corresponds to anticlockwise movement along the trajectory. -0.8-0.6-0.4-0.2 0 0.2 0.4-8 -7 -6 -5 -4 -3 -2 -1 I m < S > Re L=6, l_max=1.9L=10, l_max=1.9L=20, l_max=1.999 (a) l max = 1 . -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 I m < S > Re L=6, l_max=0.9L=10, l_max=0.9L=20, l_max=0.9 (b) l max = 0 . Figure 17 : Trajectory of h S i in the complex plane for L = 6 (red), L = 10 (green),and L = 20 (blue). Increasing x min corresponds to anticlockwise movement along thetrajectory.are both for the single-mode case: the reason they differ is that the mass is smaller forthe latter case. The other trajectories show what happens as l max is then increased for L = 10. We see that both the real and imaginary parts of T decrease, and the trajectoryalso becomes more complicated.Next we compare two modes with roughly the same wavelength, namely T at L = 6and T at L = 10, both for l max = 1 .
9. Their trajectories in the complex plane are plottedin Fig. 19. We see in this case that there is a large change in scale. Apart from this we seethat whereas the L = 6 trajectory has one loop, the trajectory for L = 10 has at least the– 49 – I m < T _1 > Re
Figure 18 : Trajectory of h T i in the complex plane for L = 6, l max = 0 . L = 10 with l max = 0 . l max = 0 . l max = 1 . x min corresponds to anticlockwise movement along the trajectory. -0.4-0.3-0.2-0.1 0 0.1 0.2-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 I m < T _ i > Re
Figure 19 : Trajectory of h T i in the complex plane for L = 6 (red), and of h T i for L = 10(green), both for l max = 1 .
9. Increasing x min corresponds to anticlockwise movement alongthe trajectory.start of a second.Keeping instead the mode number fixed, we plot h T i in Fig. 20 for L = 6, 10, and 20.We see that now there is quite good matching in the magnitudes of h T i . (The fact that thetrajectory looks smoother for some cases is simply because we have made measurements– 50 – I m < T _1 > Re
9. It also appears that the behaviourfor the two values of l max is similar, though unfortunately it is not possible to go to highenough x min with l max = 1 .
999 to see well the approach to the origin. For the same reason,we cannot really see if the trajectories are becoming more complicated.We now turn to comparisons of the h T n i . As an example we plot the lowest mode h T i in Fig. 21. The cases L = 6, l max = 0 . L = 10, l max = 0 . h T i is due to the mass decrease. The other twotrajectories then show what happens when L is kept fixed and l max is increased. We seethat h T i decreases, which roughly means that the fluctuations of h T i decrease. Also thetrajectories become more complicated, although they do not seem to be as complicated asthe trajectories of the h T n i we saw above.We find similar behaviour for the other modes h T n i and also for h A n i : generally thereare some complicated trajectories in the complex plane, with in some cases cusps or loopsappearing. There is no trend for imaginary parts to get smaller or the approach to theorigin to occur at a smaller angle to the real axis as we increase L or l max . h T n i Apart from looking at how an individual mode, say h T i moves in the complex plane as x min increases, we can get additional information by examining how all the modes movetogether. We show an example for L = 20 for a range of x min from −
21 to −
18, i.e. overthe transition from weak to strong coupling, with l max = 1 . h T n i all moving off the real axis as x min is– 51 – I m < T _1 ^ > Re
Figure 21 : As Fig. 18, but for h T i rather than h T i . -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 I m < t au > Re
Figure 22 : The h T n i in the complex plane for L = 20, l max = 1 . x min . Ineach case h T i is the point furthest to the left/bottom.increased, and then forming a complicated star-shaped pattern. We see similar behaviourfor other values of the parameters.We do not understand the full behaviour here, but some features can be understood.Firstly there is a rougly oscillatory behaviour, with h T n +1 i generally being on the oppositeside of the origin to h T n i . This is presumably because the cubic terms change sign when n increases by one. This in turn is because the cubic terms are proportional to integrals– 52 – < T ( x ) > x Real partIm part Figure 23 : The real (red) and imaginary (green) parts of h T ( x ) i for L = 20, l max = 1 . x min = − x max , and the sine-wave modes willchange sign in this region when n increases by one. Secondly there is a general decrease inmagnitude as n increases. This is presumably because the masses of the modes increase,constraining them to be closer to the origin. T ( x ) in position space Another way to look at the behaviour of the T n is to recall that they are the Fourier modesof the field T . Thus we can translate back to position space to obtain the expectation valueof T ( x ): h T ( x ) i = r L n X j =1 h T j i sin (cid:18) jπ ( x − x min ) L (cid:19) (4.9)This will in general be complex of course.The advantage of this is that the complicated behaviour of the T n in the complexplane can be packaged into a pair of functions, the behaviour of which may be easier tounderstand. We show an example in Fig. 23, where we plot h T ( x ) i for L = 20, l max = 1 . x min = −
19, which is one of the cases plotted in Fig. 22.The behaviour indeed appears simpler when plotted in this way. We see that T ( x ) = 0as both ends of the interval, as it must with our boundary conditions. Away from theboundaries it is very assymetric, with a maximum near the large- x end. This is notsurprising, since the cubic terms that pull T ( x ) away from zero are largest there. Notethat in this case T has 9 modes, so the shortest lengthscale that can appear is roughly L/ ≈ .
2, which is comparable to the distance from the peak of T to the boundary; thepeak is as far to the right as it can be. Indeed, we have found that this overall shape istypical. – 53 – R e < T ( x ) > xl_max=0.02l_max=0.2l_max=0.5l_max=0.9l_max=1.2l_max=1.55l_max=1.9 (a) Real parts -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 -30 -25 -20 -15 -10 -5 0 I m < T ( x ) > xl_max=0.02l_max=0.2l_max=0.5l_max=0.9l_max=1.2l_max=1.55l_max=1.9 (b) Imaginary parts Figure 24 : h T ( x ) i for L = 30, x min = −
31, and various values of l max .This in fact raises a general issue with our simulations. The characteristic lengthscaleover which we expect the fields to vary is something like 1 /V ≈ .
5. But the finest scale wecan probe is, say, half a wavelength of the highest mode. We have n = L √ l max /π modeson an interval of length L , so the highest mode has half-wavelength π/ √ l max . We can onlygo up to l max = 2, giving a minimum lengthscale of π/ √ ≈ .
2. To reach a lengthscale of0.5, we would need to go up to l max = 39, probably an impossible task.One way to look at this issue is to fix everything except l max and see how thingschange as l max increases, which corresponds to being able to look at smaller and smallerlengthscales. We have done this for the case of L = 30, x min = −
31 in Fig 24.We see that as l max increases features on smaller length-scales appear. However, theredoes not appear to be a smooth limit: h T ( x ) i changes significantly every time l max isincreased. In fact this should not be surprising as we know from section 4.2 that theobservables depend strongly on l max . This suggests that we may indeed not be resolvingsmall enough lengthscales. It is also possible to look at correlations among the T n , or equivalently between T ( x ) atdifferent points. These are related by h T ( x ) T ( x ) i = 2 L n X n,k =1 h T n T k i sin (cid:18) nπ ( x − x min ) L (cid:19) sin (cid:18) kπ ( x − x min ) L (cid:19) . (4.10)In principle, correlators could be used to extract the masses of the states in the theory.However, since everything is space-dependent, this will be complicated; the correlators willnot simply decrease as X k e − m k x . We have not attempted to extract masses but have simplylooked at a few correlators as a first step in this direction.We show an example in Fig. 25. Here we have fixed x and let x vary, and subtracted h T ( x ) ih T ( x ) i to show just the fluctuations. The results look reasonable: the largest– 54 – < T (- ) T ( x ) > - < T (- ) >< T ( x ) > x Re partIm part Figure 25 : Correlator h T ( x ) T ( x ) i − h T ( x ) ih T ( x ) i for fixed x = −
3, with x min = − L = 10, and l max = − . x = − x –dependent. In the full theory, the dilaton gradient V is fixed to V = − r α ′ . (4.11)Let us define v ≡ α ′ V . (4.12)Thus, the correct value for the dilaton slope is obtained at v = 25 . (4.13)However, in our level-truncated model we can choose any value, and it would be interestingto see whether anything special happens at v = 25. In particular, we might hope that theinstability will be minimised at this value.Another important value is v = 24. Here the mass-squared of the T field, given by α ′ m = v − v is further decreased. Hence for v < I m < S > V^2*6alpha’ FullNo level 1 (a) x min = − -0.4-0.3-0.2-0.1 0 0.1 0.2 24 26 28 30 32 34 36 I m < S > V^2*6alpha’ FullNo level 1 (b) x min = − -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 25 30 35 40 45 50 I m < S > V^2*6alpha’ FullNo level 1 (c) x min = − Figure 26 : ℑh S i as a function of v for L = 20, x min = − , − , −
18, and l max = 1 . . This quadratic instability cannot be controlled by our analyticcontinuation with γ = π/
6, and we find the simulations indeed become unstable. We havecarried out several scans in v which we describe below. L = 20Here we have results for l max = 1 . l max = 1 .
999 without level-1 fields.The imaginary parts of h S i and h T i are shown in Figs. 26 and 27 respectively. For bothobservables it appears that the instability decreases monotonically as v increases. It is alsoclear that v = 25 is no better than neighbouring values of v . L = 10We now turn to L = 10 where the errors are smaller. Here we use l max = 1 . l max below 2. We begin with x min = −
10 which is quite weakcoupling. We plot ℑh S i in Fig. 28 and ℑh T i in Fig. 29. Actually for finite L the lightest T -mode has mass-squared α ′ m = v − α ′ ( πL ) so the instability – 56 – I m < T _1 > V^2*6alpha’ FullNo level 1 (a) x min = − -2.5-2-1.5-1-0.5 0 0.5 24 26 28 30 32 34 36 I m < T _1 > V^2*6alpha’ FullNo level 1 (b) x min = − -2.5-2-1.5-1-0.5 0 0.5 25 30 35 40 45 50 I m < T _1 > V^2*6alpha’ FullNo level 1 (c) x min = − Figure 27 : ℑh T i as a function of v for L = 20, x min = − , − , −
18, and l max = 1 . v = 25. Forthe case x min = −
6, which is extremely strong coupling, we have done a more extensivescan in v , going up to v = 400. (Note the x -axis is compressed in these plots compared tothe earlier plots in this section.) Here we see the behaviour is more complicated: there aretwo minima in ℑh S i , and ℑh T i crosses zero and approaches zero from above for large v ,although this does not happen if level-1 fields are excluded. Disappointingly most of thisinteresting behaviour is at large v , far from v = 25. Much of the rather complicated behaviour seen above can be understood, at least qualita-tively, by considering how the quadratic and cubic terms depend on v .The behaviour of the quadratic (mass) terms is the more straightforward. The massof the T field (4.14) simply increases with v . starts slightly below v = 24. – 57 – I m < S > V^2*6alpha’ FullNo level 1 (a) x min = − -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 25 30 35 40 45 50 I m < S > V^2*6alpha’ FullNo level 1 (b) x min = − -0.25-0.2-0.15-0.1-0.05 0 0.05 50 100 150 200 250 300 350 400 I m < S > V^2*6alpha’ FullNo level 1 (c) x min = − Figure 28 : ℑh S i as a function of v for L = 10, x min = − , − , −
6, and l max = 1 . f i i i and g i i i . We have plotted them in Fig. 30 as functions of v for two ofthe sets of parameters above: L = 10, x min = −
10 and L = 10, x min = −
6. We see thatfor x min = −
10, the coefficients mostly decline throughout the range of v , whereas for x min = −
6, they peak around v = 100 and only then begin to decline.We can understand this analytically for the particular case of f . Taking the exactexpression and keeping only those terms which survive when v is large, one obtains f ∝ K − v/ e x max √ v . (4.15)For sufficiently large v the first term will dominate, so the coefficient will decline. Forsomewhat smaller v the second, growing, term will matter as well, and there will be a peakat v = x max / ln K ≈ x max . However we will not see this peak if x max is small — itwould be at small or negative v and we are only interested in v ≥
24 where the quadraticterms are stable . Presumably something similar happens for the other f i i i and the Also in this case we would have to take into account terms ignored in (4.15). – 58 – I m < T _1 > V^2*6alpha’ FullNo level 1 (a) x min = − -1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 0.2 25 30 35 40 45 50 I m < T _1 > V^2*6alpha’ FullNo level 1 (b) x min = − -1.2-1-0.8-0.6-0.4-0.2 0 0.2 50 100 150 200 250 300 350 400 I m < T _1 > V^2*6alpha’ FullNo level 1 (c) x min = − Figure 29 : ℑh T i as a function of v for L = 10, x min = − , − , −
6, and l max = 1 . -0.01-0.008-0.006-0.004-0.002 0 0.002 0.004 0.006 0.008 0.01 0 50 100 150 200 250 300 350 400v f_i1i2i3g_i1i2i3 (a) x min = − -1.5-1-0.5 0 0.5 1 1.5 0 50 100 150 200 250 300 350 400v f_i1i2i3g_i1i2i3 (b) x min = − Figure 30 : The f i i i and g i i i as functions of v for L = 10, x min = − , − i i i , and this accounts for the behaviour seen in Fig. 30.Putting all this together we can understand the observed pattern of stability. Forsmall x min the quadratic terms increase and the cubic terms decrease monotonically as v increases. Both these trends increase stability, so we see the imaginary parts of observablesdecrease. However, for larger x min at first both terms increase. Near v = 24 the increase inthe quadratic term should matter more, since below this value it is negative. Thus at firstthe instability decreases. Then the increase in the cubic terms becomes more important,and the instability increases. Finally the cubic terms peak and start decreasing, and theinstability decreases again. This matches the behaviour seen for x min = −
10. It does notfully explain the behaviour seen for x min = −
6, but this argument is rather rough andcould be refined — for example we have not considered the fermion determinant at all.To summarise, both our numerical results and our analysis of the behaviour of thecoefficients show that nothing special happens at v = 25. It is of course possible that thiswill change if we worked at higher l max or L , especially in light of the observation madein 4.6.1, according to which we are far from being able to sample space in a high enoughresolution as compared to the scale set by the dilaton slope. As discussed in section 2.9, we can add terms proportional to ( p + p + p − iV / S trivial = Z N X n , , =1 ( p + p + p − iV / p p p T ( p ) T ( p ) T ( p ) . (4.16)To keep this term simple, we have only considered it when only level-0 fields are present,i.e. for l max <
1. Note that the extra cubic coefficients due to this term are similar inmagnitude to the existing ones when Z is of order unity.We have found that adding this term can significantly affect the results. We show anexample of its effect in Fig. 31. Not surprisingly, the difference between simulations withand without (4.16) increase with Z . However, we are more interested in how this differencedepends on the other parameters, and in particular, whether it decreases in the continuumand/or large-volume limits.To analyse this, we have compared results with Z = 0 and Z = 1 with several sets ofparameters. In general, the full effect of (4.16) is complicated, but there are some cleartrends: • The differences between Z = 0 and Z = 1 increase at larger x min (stronger coupling).This is presumably simply because the new term increases exponentially in x , justlike the original cubic terms. • The absolute differences are roughly similar for different mode numbers. Since the h T n i are usually smaller for higher modes this means the relative differences increase. • The differences remain roughly constant as L increases.– 60 – Figure 31 : The h T n i in the complex plane for L = 10 and l max = 0 .
9, for various x min ,both with and without the term in (4.16).This last point is presumably because the dependence on L is mainly in the terms p i = ( n i π ) L , and the numerator and denominator cancel, since the highest mode number n i available is proportional to L (for fixed l max ).We conclude that we cannot really establish Z -independence and can only continuewith a rough principle of retaining the vertex in a form that is “as simple as possible”. Thereason for this awkward situation is the sine-expansion that we were forced to use in lightof the fact that the linear dilaton prevents us from using periodic boundary conditions.Moreover, at l ≥ We now turn to the other ‘schemes’, starting with scheme 1. This is the same as scheme 4except that the Grassmann-odd fields B and C are not present, but the Grassmann-evenfield B is added instead. The action in this scheme is given in (2.83) and (2.84). Note thatfor 0 ≤ l max < x min with the other parameters fixed to α ′ = 1, V = − q α ′ , L = 20, and with l max = 1 . x min ; in fact it isso much greater that the simulations become prohibitively expensive around x min = − x min covered in scheme 1is −
23 to − .
2, which does not overlap with the range −
22 to −
18 covered in scheme 4.– 61 – I m < S > x_min 1.999, scheme 11.999, scheme 4 Figure 32 : ℑh S i as a function of x min for L = 20, l max = 1 .
999 for scheme 1 (red) andscheme 4 (green). -1-0.8-0.6-0.4-0.2 0 0.2 -23 -22 -21 -20 -19 -18 I m < T _1 > x_min 1.999, scheme 11.999, scheme 4 Figure 33 : ℑh T i as a function of x min for L = 20, l max = 1 .
999 for scheme 1 (red) andscheme 4 (green).We plot the results for ℑh S i in Fig. 32 and for ℑh T i in Fig. 33, in both cases togetherwith the corresponding scheme 4 results. In both cases it is clear that the instability ismuch greater, or equivalently appears at smaller x min , for scheme 1. The shift in x forequivalent imaginary parts is about 2. This corresponds to a change in size of the cubicterms of roughly e V ≈ A and B , and the case where eachfield has only one mode: S = − (cid:16) m + (cid:16) πL (cid:17)(cid:17) A − α ′ B − V √ α ′ A B . (4.17)Completing the square, and taking α ′ = 1, one finds this has an almost massless modein the direction B = − V √ A , with mass only πL . Indeed, in the large-volume limit thismode becomes massless — presumably this is due to the fact that scheme 1 includes gaugedegrees of freedom.By itself this small but still positive mass would not lead to an instability. However, itdoes when the cubic terms are included. The reason is that there will be large fluctuationsof this mode, giving large values of the fields A and B . These will then lead to the cubicterms of the form AAT and T BB being large, much larger than they would be if this lightmode was not present. Since some of these cubic terms are unstable, these instabilitieswill be much larger than they would be without the light mode. As a partial check onthis mechanism, we have observed that in our simulations the modes A and B indeedfluctuate together, in exactly the direction B = − V √ A . Hence it appears that trying toincluded gauge degrees of freedom like this will lead to problems, and we did not continuefurther with simulations of scheme 1. In this section we briefly discuss our results for scheme 2. This is the same as scheme 4except that the Grassmann-odd fields B and C are not present. For 0 ≤ l max < x min for α ′ = 1, L = 20, V = − q α ′ , and l max = 1 . x min from −
22 to −
18. Theseare plotted together with the corresponding scheme 4 results in Figs. 34 and 35 for ℑh S i and ℑh T i respectively.We see that the results are very similar, with scheme 2 being perhaps slightly moreunstable. This includes a region where the instability is strong so it is not just a weak-coupling phenomenon. Furthermore, the other observables are also very similar betweenthe two schemes.In addition to the above scan, we also found only small differences between schemes 2and 4 for L = 10 at several values of V . Thus it appears that there is not much differencebetween schemes 2 and 4, at least up to l max = 2. In this work we performed the first quantum non-perturbative study of a string field theory.Our aim was to estimate the feasibility of the lattice approach. In principle, a lattice stringfield theory could be used to examine the validity of a given string field theory, as wellas to enable a numerical study of various non-perturbative aspects of string theory. One– 63 – I m < S > x_min 1.999, scheme 21.999, scheme 4 Figure 34 : ℑh S i as a function of x min for L = 20, l max = 1 .
999 for scheme 2 (red) andscheme 4 (green). -1.2-1-0.8-0.6-0.4-0.2 0 0.2 -22 -21.5 -21 -20.5 -20 -19.5 -19 -18.5 -18 I m < T _1 > x_min 1.999, scheme 21.999, scheme 4 Figure 35 : ℑh T i as a function of x min for L = 20, l max = 1 .
999 for scheme 2 (red) andscheme 4 (green).could hope to identify known as well as unknown solitons, to measure the mass of non-BPS objects , and to examine various dualities as well as other conjectures. It mightalso be useful for identifying and calculating generalizations of Ellwood invariants [92] (see Mass shifts in string theory can be studied even without an explicit string field theoretical formulation.However, such a formulation would make the search more systematic [90, 91]. Moreover, by studying stringfield theory on a lattice we could also identify mass shifts due to non-perturbative effects. – 64 –lso [93–97]).At this stage, however, our examination was of a much more preliminary nature. Weidentified the technical and fundamental obstructions towards a lattice approach in stringfield theory, suggested possible resolutions and examined lattice simulations in order tocheck whether the advocated methodology could work for the simplest possible model.While some of our results seem to suggest that our approach does make sense, otherspoint towards further obstacles that following studies will have to face. In particular wefound out that the lattice approach in the case of a linear dilaton theory must sample deepinto the strong interaction regime. However, such sampling is very expensive computa-tionally and one must look for a resolution of this problem. Possible directions include theexamination of theories without linear dilatons, as well as the introduction of a Liouvillewall that can potentially free us from the need to sample the strong coupling region. An-other source of high computational cost is the fact that the action is complex. It couldbe interesting to find other ways of dealing with the original action that do not involveanalytical continuation, or to find some other way to trade the analytically continued ac-tion by another, real, action. An interesting approach could be to work not with straightlines in the complex plane, but with Lefschetz Thimbles, following Witten’s suggestions onthe proper definition of the action of some Chern-Simons theories [98]. String field theorylooks, at least superficially, very similar to Chern-Simons theory and the idea of usingLefschetz Thimbles was already implemented in lattice field theory, see e.g. [99–101]. Itcould also be useful to find a way to implement the Langevin method for a lattice stringfield theory.A related issue is that of boundary conditions. First, the linear dilaton preventedus from using periodic boundary conditions and then it turned out that the obtainedfunctions tend to concentrate at the rightmost part of the working segment. Again, workingwith different backgrounds might be useful in order to avoid this whole state of affairs.However, working with such a background might be very difficult for reasons mentioned inthe introduction. The study of the theory with a Liouville wall, on the other hand, seemsto be relatively simple and would probably enable us to resolve some of the difficulties weare facing. It would also be interesting to see whether the current framework is sensitiveto some sort of a modification of the boundary conditions.There could have been other reasons to object the feasibility of our approach. To beginwith, we try to approximate an infinite number of fields by a truncation that takes intoaccount only modes from a finite number of fields. Furthermore, we approximate a non-local action using an expansion which retains only a finite number of derivatives. Whilesuch an approach is a standard one for the description of low energy physics, it is wellknown that it might be inadequate for a complete description of such a theory [77, 78].However, it is also known that a classical level truncation approach is often very accurateand useful in string field theory. Moreover, string field theory is expected to behave betterthan other non-local theories. On the other hand, we are dealing with a situation that ismore subtle than standard level truncation computations due to the presence of the lineardilaton background, the fact that we work with more general component fields than inthe usual case and, most importantly, the quantum nature of the analysis. We prefer to– 65 –ollow the footsteps of the original level truncation papers and examine these questionsexperimentally [58, 59].With the introduction of higher levels new problems might be encountered. One com-plication is related to the fact that in string field theory some of the auxiliary fields havekinetic terms with wrong signs. This problem could be avoided by explicitly integratingout all the auxiliary fields prior to any numerical analysis. Another problem is the ap-pearance of imaginary interaction terms. Such terms will become real after the analyticalcontinuation and will, therefore, lead to instabilities. It seems to us that the origin of theseterms is the breakdown of momentum conservation by the lattice together with the factthat this momentum conservation has an imaginary part in a linear dilaton background. Itmight be possible that the freedom of adding trivial terms, discussed in section 4.8, couldbe used for setting all the imaginary parts to zero. Exploring alternative theories and otherboundary conditions, as discussed above, might be useful also in this context.Another important issue, which we did not address in this work, is renormalization,namely comparing results at different levels. A related question is that of renormalizability.In standard field theory one of the advantages of the lattice approach is that it respectsthe gauge symmetry and is therefore not expected to imperil the renormalizability of thetheory. In string field theory, on the other hand, gauge symmetry mixes different levelsand is therefore broken by the lattice. While future works will have to study higher levels,they will also have to address the issues of gauge symmetry breaking, renormalizabilityand renormalization schemes. A possible approach that might be helpful in the context ofbreaking the gauge symmetry, as well as for other reasons, could be to construct a latticein the continuous κ -basis of string field theory [102–114]. We leave this approach to futurework.Finally, one could have objected the idea of using open string field theory for such astudy on the ground that it is not expected to describe the closed string moduli [115–117].A related objection is that the theory obeys only the classical master equation, while thequantum equation is divergent [49]. This observation suggests that the theory is not reallyconsistent at the quantum level. The difficulties with the quantum master equation mostprobably originate from the somewhat singular nature of Witten’s star product. They areprobably common also to other formulations based on this product, see e.g. [118, 119].The singularity of the star product seems to be related to the fact that it is used inorder to describe solely the open string sector. Indeed, one can consider continuous familiesof open-closed string field theories and Witten’s theory, which is obtained as a singularlimit of such an interpolation, is the only theory of the family whose master equation hasquantum singularities [120]. Problems with the quantum master equation suggest that thegauge symmetry might be broken at the quantum level. However, in the case at handit is known that at least in the Siegel gauge open string field theory leads to the correctcovering of moduli spaces and to correct expressions for all amplitudes [23, 24]. World-sheet open string theory is known to be renormalizable. Hence, as long as one uses thegauge fixed scheme we mostly used, the theory before the introduction of the lattice shouldbe consistent quantum mechanically. Nonetheless, it would be very useful in principle tohave a lattice formulation of closed string field theory [8], in which closed string moduli– 66 –ould also be varied. However, numerical analysis of this theory is extremely complicatedalready at the classical level [121–124].To summarize, while lattice string field theory could be a useful framework, there aremany obstacles on the way, some of which we dealt with in this work and others which stilllie ahead. It seems that a gauge fixed approach is the most promising one. But it remainsto be seen which theories can be studied and what are the most adequate backgrounds andboundary conditions. We hope that future studies will clarify these points. Acknowledgements
We benefited from discussing the matters presented in this paper with Gert Aarts, TedErler, Udi Fuchs, Carlo Maccaferri, Yaron Oz, Leonardo Rastelli, Martin Schnabl andBarton Zwiebach. We would further like to thank Udi Fuchs and Carlo Maccaferri forvaluable comments on a draft of the manuscript. The calculations for this work were,in part, performed on the University of Cambridge HPCs as a component of the DiRACfacility jointly funded by STFC and the Large Facilities Capital Fund of BIS.– 67 – eferences [1] J. Polchinski,
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