PPreprint typeset in JHEP style - HYPER VERSION ITP-UU-07-50SPIN-07-37TCDMATH 07-15
On String S-matrix, Bound States and TBA
Gleb Arutyunov a ∗† and Sergey Frolov b † a Institute for Theoretical Physics and Spinoza Institute,Utrecht University, 3508 TD Utrecht, The Netherlands b School of Mathematics, Trinity College, Dublin 2, Ireland
Abstract:
The study of finite J effects for the light-cone AdS × S superstringby means of the Thermodynamic Bethe Ansatz requires an understanding of a com-panion 2d theory which we call the mirror model. It is obtained from the originalstring model by the double Wick rotation. The S-matrices describing the scatteringof physical excitations in the string and mirror models are related to each other byan analytic continuation. We show that the unitarity requirement for the mirror S-matrix fixes the S-matrices of both theories essentially uniquely. The resulting stringS-matrix S ( z , z ) satisfies the generalized unitarity condition and, up to a scalarfactor, is a meromorphic function on the elliptic curve associated to each variable z .The double Wick rotation is then accomplished by shifting the variables z by quarterof the imaginary period of the torus. We discuss the apparent bound states of thestring and mirror models, and show that depending on a choice of the physical regionthere are one, two or 2 M − solutions of the M -particle bound state equations sharingthe same conserved charges. For very large but finite values of J , most of thesesolutions, however, exhibit various signs of pathological behavior. In particular, theymight receive a finite J correction to their energy which is complex, or the energycorrection might exceed corrections arising due to finite J modifications of the Betheequations thus making the asymptotic Bethe ansatz inapplicable. ∗ Email: [email protected], [email protected] † Correspondent fellow at Steklov Mathematical Institute, Moscow. a r X i v : . [ h e p - t h ] D ec ontents
1. Introduction and summary 22. Generalities 7
3. Mirror S-matrix and supersymmetry algebra 13
4. Double Wick rotation and the rapidity torus 21
5. S-matrix on elliptic curve 32
6. Bethe ansatz equations 40 su (2 | su (2 | ⊕ su (2 |
7. Bound states of the
AdS × S gauge-fixed model 45
8. Bound states of the mirror model 579. Appendices 62 g expansions of solutions to the bound state equation 73
1. Introduction and summary
The conjectured duality between the maximally supersymmetric Yang-Mills theoryin four dimensions and type IIB superstring in the AdS × S background [1] is thesubject of active research. Integrability emerging on both sides of the gauge/stringcorrespondence [2, 3] proved to be an indispensable tool in matching the spectra ofgauge and string theories. Namely, it was shown that the problem of determiningthe spectra in the large volume (charge) limit, can be reduced to the problem ofsolving a set of algebraic (Bethe) equations. The corresponding Bethe equations arebased on the knowledge of the S-matrix which describes the scattering of world-sheetexcitations of the gauge-fixed string sigma-model, or alternatively, the excitations ofa certain spin chain in the dual gauge theory [4]-[9].Remarkably, the S-matrix is severely restricted by the requirement of invarianceunder the global symmetry of the model, the centrally extended psu (2 | ⊕ psu (2 | × S background. This would eventually allow one todescribe the finite-size spectrum of the corresponding model.The TBA approach was initially developed for studying thermodynamic proper-ties of non-relativistic quantum mechanics in one dimension [44] and further appliedto the computation of the ground state energy in integrable relativistic field theoriesin finite volume [41]. The method also was later extended to account for energies ofexcited states [45] (see also [46]).Implementation of the TBA approach consists of several steps. The primarygoal is to obtain an expression for the ground state energy of a Lorentzian theorycompactified on a circle of circumference L and at zero temperature. The startingpoint is the Euclidean extension of the original theory, put on a torus generatedby two orthogonal circles of circumferences L and R . The partition function ofthis theory can be viewed as originating from two different Lorentzian theories: the original one, which lives on a circle of length L at temperature T and has theHamiltonian H , or the mirror theory which is defined on a circle of length R = 1 /T at temperature ˜ T = 1 /L and has the Hamiltonian ˜ H . For Lorentz-invariant theories,the original and the mirror Hamiltonians are the same. However, in general and inparticular for the case of interest here, the two theories need not be the same. Takingthe thermodynamic limit R → ∞ one ends up with the mirror theory on a line andat finite temperature, for which the exact (mirror) Bethe equations can be written.Thus, computation of finite-size effects in the original theory translates into theproblem of solving the infinite volume mirror theory at finite temperature.Although taking the thermodynamic limit simplifies the system, a serious com-plication arises due to the fact that the mirror theory could have bound states whichmanifest themselves as poles of the two-particle mirror S-matrix. Thus, the com-plete spectrum would consist of particles and their bound states; the latter should– 3 –e thought of as new asymptotic particles. Having identified the spectrum, one hasto determine the S-matrix which scatters all asymptotic particles. It is this S-matrixwhich should be used to formulate the system of TBA equations.As is clear from the discussion above, the mirror theory plays a crucial role inthe TBA approach. In this paper we will analyze the mirror theory in some detail.First, we will explain its relation to the original theory. Indeed, given that the modelin question is not Lorentz invariant, the mirror and the original Hamiltonians are notthe same. However, since the dispersion relation and the S-matrix can be deducedfrom the 2-point and 4-point correlation functions on the world-sheet, and since thecorrelation functions in the mirror theory are inherited from the original model byperforming a double Wick rotation, it follows that the mirror dispersion relationand the mirror S-matrix can be obtained from the original ones by the double Wickrotation. Here we will meet an important subtlety. To explain it, we first have torecall the basic properties of the string S-matrix.As was shown in [47], the psu (2 | ⊕ psu (2 | S ( p , p ), whichdepends on real momenta p and p of scattering particles, obeys • the Yang-Baxter equation S S S = S S S • the unitarity condition S ( p , p ) S ( p , p ) = I • the physical unitarity condition S ( p , p ) S † ( p , p ) = I • the requirement of crossing symmetry C − S t ( p , p ) C S ( − p , p ) = I , where C is the charge conjugation matrix.The first three properties naturally follow from the consistency conditions of theassociated Zamolodchikov-Faddeev (ZF) algebra [48, 49], while the last one reflectsthe fact that the particle-to-anti-particle transformation is an automorphism of theZF algebra [47]. The unitarity and physical unitarity conditions imply the followingproperty S ( p , p ) = S † ( p , p ) . One should bear in mind that the S-matrix is defined up to unitary equivalenceonly: unitary transformations (depending on the particle momentum) of a basis of– 4 –ne-particle states correspond to unitary transformations of the scattering matrixwithout spoiling any of the properties listed above.The mirror S-matrix ˜ S (˜ p , ˜ p ) is obtained from S ( p , p ) by the double Wickrotation. The above-mentioned subtlety lies in the fact that only for a very specialchoice of the one-particle basis the corresponding mirror S-matrix remains unitary.As we will show, this problem can be naturally attributed to the properties of thedouble Wick rotation for fermionic variables. Upon the basis is properly chosen toguarantee unitarity of the mirror S-matrix, the only freedom in the matrix structure of S ( p , p ) reduces to constant, i.e. momentum-independent, unitary transformations .There is another interesting explanation of the interrelation between the originaltheory and its mirror. As was shown in [12], the dispersion relation between theenergy and momentum of a single particle can be naturally uniformized in terms ofa complex variable z living on a torus with real and imaginary periods equal to 2 ω and 2 ω , respectively. Since z plays the role of the generalized rapidity variable, itis quite natural to think about the S-matrix as the function S ( z , z ), which for realvalues of the generalized rapidity variables coincides with S ( p , p ). In other words,the S-matrix admits an analytic continuation to the complex values of momenta. Itappears that the unitary momentum-dependent freedom in the matrix structure ofthe S-matrix gets fixed if we require the analytic continuation to be compatible withthe requirement of • generalized unitarity S ( z ∗ , z ∗ ) (cid:2) S ( z , z ) (cid:3) † = I , which can be thought of as the physical unitarity condition extended to the general-ized rapidity torus. The unitarity and the generalized unitarity further imply S ( z ∗ , z ∗ ) = (cid:2) S ( z , z ) (cid:3) † In fact, the last equation is equivalent to the standard requirement of hermitiananalyticity for an S-matrix in two-dimensional relativistic quantum field theories.Thus, the S-matrix which admits the analytic continuation to the generalizedrapidity torus compatible with the requirement of hermitian analyticity is essentiallyunique. Of course, it satisfies all the other properties listed above, including crossingsymmetry. As we will show, the mirror S-matrix is obtained from S ( z , z ) consideredfor real values of z , z by shifting z , z by quarter of the imaginary period˜ S ( z , z ) = S (cid:0) z + ω , z + ω (cid:1) . Of course, there is always a freedom of multiplying the S-matrix by an overall (momentum-dependent) phase. – 5 –here is a close analogy with what happens in relativistic models. In the lattercase the physical region is defined as the strip 0 ≤ Im θ ≤ π , where θ = θ − θ is the rapidity variable. A passage to the mirror theory corresponds to the shift θ k → θ k + i π , i.e. to the shift by the quarter of imaginary period . Of course, forrelativistic models, due to Lorentz invariance, the S-matrix depends on the differenceof rapidities and, therefore, it remains unchanged under the double Wick rotationtransformation. Also, in our present case the notion of the physical region is notobvious and its identification requires further analysis of the analytic properties ofthe string S-matrix.Having identified the mirror S-matrix, we can investigate the question about thebound states. We first discuss the Bethe equations for the gauge-fixed string theorywhere the existence of the BPS bound states is known [50]. No non-BPS bound statesexist, according to [50]-[52]. We find out, however, that the number of solutions of theBPS bound state equations depends on the choice of the physical region of the model,and for a given value of the bound state momentum there could be 1, 2 or 2 M − M -particle bound states sharing the same set of global conserved charges. It is unclearto us whether this indicates that the actual physical region is the one that containsonly a single M -particle bound state or it hints on a hidden symmetry of the modelresponsible for the degeneracy of the spectrum. These solutions behave, however,differently for very large but finite values of L ; most of them exhibit various signs ofpathological behavior. In particular, they might have complex finite L correction tothe energy, or the correction would exceed the correction due to finite L modificationsof the Bethe equations thus making the asymptotic Bethe ansatz inapplicable. Inthe weak coupling limit, i.e. in perturbative gauge theory, and for small enoughvalues of the bound state momentum only one solution reduces to the well-knownBethe string solution of the Heisenberg spin chain. It is also the only solution thatbehaves reasonably well for finite values of L . Therefore, it is tempting to identifythe physical region of the string model as the one that contains this solution only.By analyzing the Bethe equations for the mirror theory, we show that boundstates exist and that they can be regarded as “mirror reflections” of the BPS boundstates in the original theory. No other bound states exist, in agreement with theresults by [52]. Given the knowledge of bound states, the next step would be toconstruct the S-matrix which describes scattering of all asymptotic particles includingthe ones which correspond to bound states. In principle, such an S-matrix can beobtained by the fusion procedure [53, 54] applied to the “fundamental” S-matrix weadvocate here. This is the bootstrap program whose discussion we will postpone forthe future.The paper is organized as follows. The next section contains the discussion ofthe double Wick rotation, the mirror dispersion relation and the mirror magnon. In The shift of θ by the half-period corresponds to the crossing transformation. – 6 –ection 3 we discuss the supersymmetry algebra and the construction of the mirror S-matrix. In section 4 we analyze the double Wick rotation on the generalized rapiditytorus as well as various possible definitions for the physical region. In section 5the properties of the string S-matrix defined on the generalized rapidity torus arediscussed. We also prove here the unitarity of the scalar factor in the mirror theory.In section 6 we present various versions of the Bethe equations in the original andmirror theory pointing out that the Bethe equations based on the su (2 | ⊕ su (2 | × S gauge-fixedmodel and its mirror theory. Section 9 consists of several appendices.
2. Generalities
In this section we discuss how the vacuum energy of a two-dimensional field theoryon a circle can be found by considering the Thermodynamic Bethe Ansatz for amirror model obtained from the field theory by a double Wick rotation. We followthe approach developed in [41].
Consider any two-dimensional field-theoretic model defined on a circle of circumfer-ence L . Let H = (cid:90) L d σ H ( p, x, x (cid:48) ) (2.1)be the Hamiltonian of the model, where p and x are canonical momenta and coor-dinates. They may also include fermions but in this section we confine ourselves tobosonic fields only. We will refer to the action corresponding to the Hamiltonian H as to the Minkowski action, however it does not have to be relativistic invariant.We want to compute the partition function of the model defined as follows Z ( R, L ) ≡ (cid:88) n (cid:104) ψ n | e − HR | ψ n (cid:105) = (cid:88) n e − E n R , (2.2)where | ψ n (cid:105) is the complete set of eigenstates of H . By using the standard pathintegral representation [55], we get Z ( R, L ) = (cid:90) D p D x e R R dτ R L dσ ( ip ˙ x −H ) , (2.3)where the integration is taken over x and p periodic in both τ and σ . Formula(2.3) shows that − (cid:82) R dτ (cid:82) L dσ ( ip ˙ x − H ) can be understood as the Euclidean actionwritten in the first-order formalism. Indeed, integrating over p in the usual first-order– 7 –ction (cid:82) R dτ (cid:82) L dσ ( p ˙ x − H ), we get the Minkowski-type action, and the Euclideanaction is obtained from it by replacing ˙ x → i ˙ x which is equivalent to the Wickrotation τ → − iτ .Let us now take the Euclidean action, and replace x (cid:48) → − ix (cid:48) or, equivalently, dothe Wick rotation of the σ -coordinate σ → iσ . As a result we get the action where σ can be considered as the new time coordinate. Let (cid:101) H be the Hamiltonian withrespect to σ (cid:101) H = (cid:90) R d τ (cid:101) H ( (cid:101) p, x, ˙ x ) , (2.4)where (cid:101) p are canonical momenta of the coordinates x with respect to σ .We will refer to the model with the Hamiltonian (cid:101) H as to the mirror theory. Ifthe original model is not Lorentz-invariant then the mirror Hamiltonian is not equalto the original one, and the Hamiltonians H and (cid:101) H describe different Minkowskitheories.The partition function of the mirror model is given by (cid:101) Z ( R, L ) ≡ (cid:88) n (cid:104) (cid:101) ψ n | e − e HL | (cid:101) ψ n (cid:105) = (cid:88) n e − e E n L , (2.5)where | (cid:101) ψ n (cid:105) is the complete set of eigenstates of (cid:101) H . Again, by using the path integralrepresentation, we obtain (cid:101) Z ( R, L ) = (cid:90) D (cid:101) p D x e R R dτ R L dσ ( i e px (cid:48) − e H ) . (2.6)Finally, integrating over (cid:101) p , we get the same Euclidean action and, therefore, weconclude that (cid:101) Z ( R, L ) = Z ( R, L ) . (2.7)Now, if we take the limit R → ∞ , then log Z ( R, L ) ∼ − RE ( L ), where E ( L ) is theground state energy. On the other hand, log (cid:101) Z ( R, L ) ∼ − RLf ( L ), where f ( L ) is thebulk free energy of the system at temperature T = 1 /L with σ considered as thetime variable. This leads to the relation E ( L ) = Lf ( L ) . (2.8)To find the free energy we can use the thermodynamic Bethe ansatz because R >> (cid:101) H . Although the light-conegauge-fixed string theory on AdS × S is not Lorentz invariant, (cid:101) H (cid:54) = H , it is stillnatural to expect that there is a close relation between the two systems because theirEuclidean versions coincide. – 8 – potential problem with the proof that (cid:101) Z ( R, L ) = Z ( R, L ) is that the integra-tion over p and (cid:101) p produces additional measure factors which may be nontrivial. Thecontribution of such a factor is however local, and one usually does not have to takeit into account. We will assume throughout the paper that this would not cause anyproblem. The dispersion relation in any quantum field theory can be found by analyzing thepole structure of the corresponding two-point correlation function. Since the corre-lation function can be computed in Euclidean space, both dispersion relations in theoriginal theory with H and in the mirror one with (cid:101) H are obtained from the followingexpression H + 4 g sin p E , (2.9)which appears in the pole of the 2-point correlation function. Here and in whatfollows we consider the light-cone gauge-fixed string theory on AdS × S which hasthe Euclidean dispersion relation (2.9) in the decompactification limit L ≡ P + → ∞ [5, 6, 9, 10]. The parameter g is the string tension, and is related to the ’t Hooftcoupling λ of the dual gauge theory as g = √ λ π .Then the dispersion relation in the original theory follows from the analyticcontinuation (see also [39]) H E → − iH , p E → p ⇒ H = 1 + 4 g sin p , (2.10)and the mirror one from H E → (cid:101) p , p E → i (cid:101) H ⇒ (cid:101) H = 2 arcsinh (cid:16) g (cid:112) (cid:101) p (cid:17) . (2.11)Comparing these formulae, we see that p and (cid:101) p are related by the following analyticcontinuation p → i arcsinh (cid:16) g (cid:112) (cid:101) p (cid:17) , H = (cid:114) g sin p → i (cid:101) p . (2.12)We note that the plane-wave type limit corresponds to taking g → ∞ with (cid:101) p fixed,in which case we get the standard relativistic dispersion relation (cid:101) H pw = 1 g (cid:112) (cid:101) p . (2.13)The expression above suggests that in this limit it is natural to rescale (cid:101) H by 1 /g or, equivalently, to rescale (cid:101) τ = iσ by g . This also indicates that the semi-classical– 9 –imit in the mirror theory should correspond to g → ∞ with (cid:101) p/g fixed, so that thedispersion relation acquires the form (cid:101) H sc = 2 arcsinh (cid:18) | (cid:101) p | g (cid:19) . (2.14)We will show in the next subsection that the mirror theory admits a one-solitonsolution whose energy exactly reproduces eq.(2.14).In what follows we need to know how the parameters x ± introduced in [5] areexpressed through (cid:101) p . By using formulae (2.12), we find x ± ( p ) → g (cid:32)(cid:115) g (cid:101) p ∓ (cid:33) ( (cid:101) p − i ) (2.15)and, as a consequence, ix − − ix + → g (1 + i (cid:101) p ) . Note that these relations are well-defined for real p , but one should use them with cau-tion for complex values of p . In section 4 we introduce a more convenient parametriza-tion of the physical quantities in terms of a complex rapidity variable z living on atorus [12]. In this parametrization the analytic continuation would simply correspondto the shift of z by the quarter of the imaginary period of the torus. In this section we will derive the dispersion relation for the “giant magnon” in thesemi-classical mirror theory. This will provide further evidence for the validity of theproposed dispersion relation (2.14).Consider the classical string sigma-model on AdS × S and fix the generalizeduniform light-cone gauge as in [56, 57]. The gauge choice depends continuously on aparameter a with the range 0 ≤ a ≤
1. The gauge-fixed Lagrangian in the generalized a -gauge can be obtained either from the corresponding Hamiltonian [59, 57] by usingthe canonical formalism or by T-dualizing the action in the direction canonicallyconjugate to the light-cone momentum P + [26]. Its explicit form in terms of theworld-sheet fields is given in appendix 9.1. To keep the discussion simple, in whatfollows we will restrict our analysis to the a = 1 gauge .We are interested in finding a soliton solution in the mirror theory, which isobtained from the original theory via the double Wick rotation with further exchangeof the time and spacial directions˜ σ = − iτ , ˜ τ = iσ , (2.16) Recall that unlike to the case of finite P + the dispersion relation of the giant magnon in theinfinite volume limit P + = ∞ was shown to be gauge independent [38]. – 10 –here σ, τ are the variables parametrizing the world-sheet of the original theory.Recall that the giant magnon can be thought of as a solution of the light-conegauge-fixed string sigma-model described by a solitonic profile y ≡ y ( σ − vτ ), where y is one of the fields parametrizing the five-sphere and v is the velocity of the soliton .In the infinite P + limit this soliton exhibits the dispersion relation (2.10), where p coincides, in fact, with the total world-sheet momentum p ws carried by the soliton.Owing to the same form of the dispersion relation in the dual gauge theory, thisgives a reason to call this soliton a “giant magnon” [37]. For our further discussionit is important to realize that if, instead of taking the field y from the five-sphere, wewould make a solitonic ansatz z ≡ z ( σ − vτ ), where z is one of the fields parametrizingAdS , we would find no solutions exhibiting the dispersion (2.10). As we will nowshow, in the mirror theory the situation is reversed: this time the giant magnonpropagates in the AdS part, while there is no soliton solution associated to the five-sphere.Take the string Lagrangian (9.1) in the gauge a = 1 and put all the fields tozero except a single excitation z from AdS . Upon making the double Wick rotation(2.16), the corresponding mirror action can be written as follows S = g (cid:90) r − r d˜ σ d˜ τ (cid:32) − (cid:112) z − z (cid:48) + (1 + z ) ˙ z z (cid:33) ≡ (cid:90) r − r d˜ σ d˜ τ L . (2.17)Here r is an integration bound for ˜ σ and ˙ z ≡ ∂ ˜ τ z , z (cid:48) ≡ ∂ ˜ σ z . Although our goal is toidentify the mirror magnon configuration in the decompactification limit, i.e. when r → ∞ , for the moment we prefer to keep r finite.To construct a one-soliton solution of the equations of motions corresponding tothe action (2.17), we make the following ansatz z = z (˜ σ − v ˜ τ ) . Our further discussion follows closely [38]. Plugging the ansatz into (2.17), we obtainthe reduced Lagrangian, L red = L red ( z, z (cid:48) ), which describes a one-particle mechanicalsystem with ˜ σ treated as a time variable. Introducing the canonical momentum π conjugate to z , we construct the corresponding reduced Hamiltonian H red = πz (cid:48) − L red , which is a conserved quantity with respect to time ˜ σ . Fixing H red = 1 − ω , where ω is a constant, we get the following equation to determine the solitonic profile( z (cid:48) ) = 1 + z − ω − v − v z . (2.18) See [38] and appendix 9.1 for more details. – 11 –he minimal value of z corresponds to the point where the derivative of z vanishes,while the maximum value is achieved at the point where the derivative diverges z min = (cid:114) ω − , z max = (cid:114) v − , v < ω < . The range of ˜ σ is determined from the equation r = (cid:90) r d˜ σ = (cid:90) z max z min d z | z (cid:48) | = √ − v E (cid:16) arcsin( z (cid:112) ω / (1 − ω )) , η (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) z max z min , where we have introduced η = v ω − ω − v . Here E stands for the elliptic integral of thesecond kind. We see that the range of σ tends to infinity when ω →
1. Thus, ω → (cid:101) H = p z ˙ z − L , where p z = ∂ L ∂ ˙ z is the momentum conjugate to z with respect to time ˜ τ . For oursolution in the limiting case ω = 1 we find p z = − v | z | (cid:112) − v (1 + z ) . The energy of the soliton is then (cid:101) H = g (cid:90) ∞−∞ d σ (cid:101) H = 2 g (cid:90) z max z min d z | z (cid:48) | (cid:101) H = 2 g arcsinh √ − v | v | . To find the dispersion relation, we also need to compute the world-sheet momen-tum p ws , the latter coincides with the momentum ˜ p of the mirror magnon consideredas a point particle. It is given by˜ p = p ws = − (cid:90) ∞−∞ d σ p z z (cid:48) = 2 (cid:90) z max z min =0 d z | p z | = 2 √ − v | v | . Finally, eliminating v from the expressions for (cid:101) H and ˜ p we find the following disper-sion relation (cid:101) H = 2 g arcsinh | ˜ p | . To consider the semi-classical limit g → ∞ , one has to rescale the time as ˜ τ → ˜ τ /g so that the energy (cid:101) H → g (cid:101) H will be naturally measured in units of 1 /g . Under thisrescaling the momentum ˜ p scales as well, so that the dispersion relation takes theform (cid:101) H = 2 arcsinh | ˜ p | g , (2.19)which is precisely the previously announced expression (2.14) for the energy of themirror magnon. – 12 – . Mirror S-matrix and supersymmetry algebra The S-matrix in field theory can be obtained from four-point correlation functions byusing the LSZ reduction formula. Since the correlation functions can be computedby means of the Wick rotation, it is natural to expect that the mirror S-matrix isrelated to the original one by the same analytic continuation (cid:101) S ( (cid:101) p , (cid:101) p ) = S ( p , p ) , (3.1)where we replace p i in the original S-matrix by (cid:101) p i by using formulas (2.12). Just asthe original S-matrix, the resulting mirror S-matrix should satisfy the Yang-Baxterequation, unitarity, physical unitarity, and crossing relations for real (cid:101) p k .On the other hand, the original S-matrix is su (2 | ⊕ su (2 |
2) invariant and thestates of the light-cone gauge-fixed AdS × S string theory carry unitary represen-tations of the symmetry algebra su (2 | ⊕ su (2 | su (2 | ⊕ su (2 | It is obvious that the double Wick rotation preserves the bosonic symmetry SU(2) .To understand what happens with the supersymmetry generators it is instructiveto apply the double Wick rotation to fermions. We consider the quadratic part ofthe light-cone gauge-fixed Green-Schwarz action depending on the fermions η in theform given in [57] L = iη † a ˙ η a − (cid:16) η a η (cid:48) − a − η † a η (cid:48)† − a (cid:17) − η † a η a = iη † a ˙ η a − H . (3.2)Here, a = 1 , , ,
4, and we set κ = 1 and rescale σ in the action from [57] so that (cid:101) λ disappears.Computing again the partition function of the model and using the path integralrepresentation, we get Z ( R, L ) = (cid:90) D η † D ηe R R dτ R L dσ ( − η † a ˙ η a −H ) . (3.3)We note that fermionic variables here are anti-periodic in the time direction: η ( τ + R ) = − η ( τ ) . Would fermions be periodic in the time direction, the corresponding path integralwould coincide with Witten’s index Tr( − F e − HR , where F is the fermion number– 13 –58]. Since in the mirror model τ plays the role of the spatial direction, the mirrorfermions are always anti-periodic in the spacial direction of the mirror model. Onthe other hand, the periodicity condition in the time direction of the mirror modelcoincides with a fermion periodicity condition in the spacial direction of the originalmodel. In particular, if the fermions of the original model are periodic then thepartition function of the original model is equal to the Witten’s index of the mirrormodel.After the first Wick rotation the Lagrangian takes the form L = − η † a ˙ η a − (cid:16) η a η (cid:48) − a − η † a η (cid:48)† − a (cid:17) − η † a η a . (3.4)Note that the fermions in this Euclidean action are not anymore hermitian conjugateto each other.Let us now perform the following change of the fermionic variables η a = i √ (cid:16) ψ † − a − ψ a (cid:17) , η † a = i √ (cid:0) ψ † a + ψ − a (cid:1) . (3.5)Computing (3.4), we get L = − ψ † a ψ (cid:48) a − (cid:16) ψ a ˙ ψ − a − ψ † a ˙ ψ † − a (cid:17) − ψ † a ψ a . (3.6)It is the same action as (3.4) after the interchange τ ↔ σ and ψ → η , and this showsthat the double Wick rotation should be accompanied by the change of variables (3.5).Note, that in terms of ψ ’s the supersymmetry algebra has the standard form with theusual unitarity condition. Thus, we expect that the supersymmetry generators willbe linear combinations of the original ones. One may assume that in the interactingtheory (beyond the quadratic level) one would take the same linear combinations.To summarize, the consideration above seems to indicate that the symmetry alge-bra of the mirror theory should correspond to a different real slice of the complexified su (2 | ⊕ su (2 |
2) algebra. Moreover, one might expect that the unitary representa-tion of the AdS × S string model could be chosen in such a way that its analyticcontinuation by means of formulae (2.12) would produce a unitary representation ofthe mirror model. Let us recall that the centrally extended su (2 |
2) algebra consists of the bosonicrotation generators L ab , R αβ , the supersymmetry generators Q αa , Q † aα , and three– 14 –entral elements H , C and C † . The algebra relations are (cid:2) L ab , J c (cid:3) = δ bc J a − δ ba J c , (cid:2) R αβ , J γ (cid:3) = δ βγ J α − δ βα J γ , (cid:2) L ab , J c (cid:3) = − δ ca J b + 12 δ ba J c , (cid:2) R αβ , J γ (cid:3) = − δ γα J β + 12 δ βα J γ , { Q αa , Q † bβ } = δ ab R αβ + δ βα L ba + 12 δ ab δ βα H , { Q αa , Q βb } = (cid:15) αβ (cid:15) ab C , { Q † aα , Q † bβ } = (cid:15) ab (cid:15) αβ C † . (3.7)Here in the first two lines we indicate how the indices c and γ of any Lie algebragenerator transform under the action of L ab and R αβ . For the AdS × S stringmodel the supersymmetry generators Q αa and Q † aα , and the central elements C and C † are hermitian conjugate to each other: ( Q αa ) † = Q † aα . The central element H ishermitian and is identified with the world-sheet light-cone Hamiltonian. It was shownin [10] that the central elements C and C † are expressed through the world-sheetmomentum P as follows C = i g ( e i P − e iξ , C † = − i g ( e − i P − e − iξ , g = √ λ π . (3.8)The phase ξ is an arbitrary function of the central elements, and reflects the ob-vious U(1) automorphism of the algebra (3.7): Q → e iξ Q , C → e iξ C . In ourprevious paper [47] we fixed the phase ξ to be zero to match the gauge theory spinchain convention [9] and to simplify the comparison with the explicit string theorycomputation of the S-matrix performed in [26]. As we will see in a moment, if wewant to implement the double Wick rotation under which P → i (cid:101) H , H → i (cid:101) P onthe algebra level then we should choose ξ = − P /
4. This choice makes the centralelements C and C † to be hermitian and equal to each other C = C † = − g sin P . (3.9)As we discussed above, the symmetry algebra of the mirror theory should cor-respond to a different real slice of the complexified su (2 | ⊕ su (2 |
2) algebra. Thismeans that we should give up the hermiticity condition for the algebra generators andconsider a linear transformation of the generators which is an automorphism of thecomplexified su (2 | ⊕ su (2 |
2) algebra. The transformation (3.5) suggests to considerthe following change of the supersymmetry generators which manifestly preservesthe bosonic SU(2) symmetry (cid:101) Q αa = 1 √ (cid:0) Q αa − i (cid:15) ac Q † cγ (cid:15) γα (cid:1) , (cid:101) Q † aα = 1 √ (cid:0) Q † aα − i (cid:15) αβ Q βb (cid:15) ba (cid:1) . (3.10) A possibility of this choice was noticed in [10]. – 15 –hen, by using the commutation relations (3.7), we find { (cid:101) Q αa , (cid:101) Q † bβ } = δ ab R αβ + δ βα L ba + i δ ab δ βα ( C + C † ) , (3.11) { (cid:101) Q αa , (cid:101) Q βb } = (cid:15) αβ (cid:15) ab
12 ( C − C † + i H ) , { (cid:101) Q † aα , (cid:101) Q † bβ } = (cid:15) ab (cid:15) αβ
12 ( C † − C + i H ) . Now we see that if we want to interpret the change of the supersymmetry generatorsas a result of the double Wick rotation then we should choose the central elements C , C † to be of the form (3.9) because with this choice the algebra relations (3.11)take the form { (cid:101) Q αa , (cid:101) Q † bβ } = δ ab R αβ + δ βα L ba − δ ab δ βα ig sin P , { (cid:101) Q αa , (cid:101) Q βb } = (cid:15) αβ (cid:15) ab i H , { (cid:101) Q † aα , (cid:101) Q † bβ } = (cid:15) ab (cid:15) αβ i H , (3.12)and performing the analytic continuation P → i (cid:101) H , H → i (cid:101) P , we obtain the mirror algebra { (cid:101) Q αa , (cid:101) Q † bβ } = δ ab R αβ + δ βα L ba + g δ ab δ βα sinh (cid:101) H , { (cid:101) Q αa , (cid:101) Q βb } = − (cid:15) αβ (cid:15) ab (cid:101) P , { (cid:101) Q † aα , (cid:101) Q † bβ } = − (cid:15) ab (cid:15) αβ (cid:101) P . (3.13)Note that after the analytic continuation has been done we can impose on the newsupersymmetry generators (cid:101) Q and new central elements (cid:101) H , (cid:101) P the same hermiticitycondition as was assumed for the original generators. It is also clear that the algebra(3.13) implies the mirror dispersion relation (2.11). The symmetric choice of the central charges (3.9) differs from the one we made in [47].The S-matrix corresponding to the symmetric choice (3.9) coincides, however, withthe string S-matrix in [47]. Indeed, this choice simply corresponds to multiplicationof Q and Q † by e − i P / and e i P / , respectively, which apparently does not changethe invariance condition for the S-matrix. On the other hand, the string S-matrixalso depends on the parameters η ’s which reflect the freedom in the choice of abasis of two-particle states. This freedom was partially fixed in [47] by requiring thestring S-matrix to satisfy the standard Yang-Baxter equation. This still allows oneto change the basis of one-particle states, or, in other words to change the basis ofthe fundamental representation of su (2 | η ’s basically uniquely. – 16 –o this end, we compute the action of the generators (cid:101) Q and (cid:101) Q † on the funda-mental representation of su (2 | Q αa | e b (cid:105) = a δ ab | e α (cid:105) , Q αa | e β (cid:105) = b (cid:15) αβ (cid:15) ab | e b (cid:105) , Q † aα | e β (cid:105) = d δ αβ | e a (cid:105) , Q † aα | e b (cid:105) = c (cid:15) ab (cid:15) αβ | e β (cid:105) (3.14)we get (cid:101) Q αa | e b (cid:105) = (cid:101) a δ ab | e α (cid:105) , (cid:101) Q αa | e β (cid:105) = (cid:101) b (cid:15) αβ (cid:15) ab | e b (cid:105) , (cid:101) Q † aα | e β (cid:105) = (cid:101) d δ αβ | e a (cid:105) , (cid:101) Q † aα | e b (cid:105) = (cid:101) c (cid:15) ab (cid:15) αβ | e β (cid:105) , (3.15)where (cid:101) a = 1 √ a + ic ) , (cid:101) b = 1 √ b + id ) , (cid:101) c = 1 √ c + ia ) , (cid:101) d = 1 √ d + ib ) . (3.16)and (cid:101) Q αa , (cid:101) Q † aα are defined by eqs.(3.10). The unitarity of the representation afterthe analytic continuation requires( c + ia ) = b ∗ − id ∗ . (3.17)The parameters of the original unitary representation before the analytic continua-tion are given by a = (cid:114) igx − − igx + e i ( ξ + ϕ ) , b = − x − (cid:114) igx − − igx + e i ( ξ − ϕ ) ,d = (cid:114) igx − − igx + e − i ( ξ + ϕ ) , c = − x + (cid:114) igx − − igx + e − i ( ξ − ϕ ) , (3.18)where ξ ∼ p and ϕ ∼ p are real, and the parameters x ± satisfy the following complexconjugation rule ( x + ) ∗ = x − . (3.19)After the analytic continuation, ξ , ϕ and p become imaginary (so that (cid:101) p is real) and( x + ) ∗ = 1 x − . (3.20)Taking this into account and computing (3.17), we find that the analytically contin-ued representation is unitary for any choice of ξ if e iϕ = (cid:114) x + x − = e i p . (3.21)This means that the S-matrix which is unitary for real p and real (cid:101) p is obtained fromthe string S-matrix, see eq. (8.7) in [47], by choosing η = η ( p ) e i p , η = η ( p ) , (cid:101) η = η ( p ) , (cid:101) η = η ( p ) e i p , (3.22)– 17 –here we have introduced η ( p ) = e i p (cid:112) ix − ( p ) − ix + ( p ) . (3.23)Up to a scalar factor the S-matrix reads as [47] S ( p , p ) = x − − x +1 x +2 − x − η η ˜ η ˜ η (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + ( x − − x +1 )( x − − x +2 )( x − + x +1 )( x − − x +2 )( x − x − − x +1 x +2 ) η η ˜ η ˜ η (cid:16) E ⊗ E + E ⊗ E − E ⊗ E − E ⊗ E (cid:17) − (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + ( x − − x +1 )( x − − x +2 )( x − + x +2 )( x − − x +2 )( x − x − − x +1 x +2 ) (cid:16) E ⊗ E + E ⊗ E − E ⊗ E − E ⊗ E (cid:17) + x − − x − x +2 − x − η ˜ η (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + x +1 − x +2 x − − x +2 η ˜ η (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + i ( x − − x +1 )( x − − x +2 )( x +1 − x +2 )( x − − x +2 )(1 − x − x − )˜ η ˜ η (cid:16) E ⊗ E + E ⊗ E − E ⊗ E − E ⊗ E (cid:17) + i x − x − ( x +1 − x +2 ) η η x +1 x +2 ( x − − x +2 )(1 − x − x − ) (cid:16) E ⊗ E + E ⊗ E − E ⊗ E − E ⊗ E (cid:17) + x +1 − x − x − − x +2 η ˜ η (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + x +2 − x − x − − x +2 η ˜ η (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) , (3.24) where E ji with i, j = 1 , . . . , × p ’s. The analytically continued S-matrix ˜ S ( (cid:101) p , (cid:101) p ) is thenobtained from (3.24) by simply substituting p → i arcsinh 12 g (cid:112) (cid:101) p , (3.25)c.f. section 2.2. One can verify that this matrix is also unitary for real (cid:101) p ’s : (cid:101) S ( (cid:101) p , (cid:101) p ) (cid:101) S † ( (cid:101) p , (cid:101) p ) = I . (3.26)The only subtlety here is that the string S-matrix also depends on a scalar factor,which has been omitted so far. Thus, one should separately check that this factorremains unitary after the analytic continuation. This will be discussed in section 5.2.– 18 –n exact relation between the S-matrix, S AFZ , found in [47] and the S-matrix(3.24) is given by the following transformation S ( p , p ) = G ( p ) G ( p ) S AFZ ( p , p ) G ( p ) − G ( p ) − , where G ( p ) = diag(1 , , e i p , e i p ). It is amusing to note that a similar transformationhas been recently introduced in [60], but with a very different motivation. Namely,as was shown in [60], the graded version of S ( p , p ) coincides with the Shastry R-matrix [61] for the one-dimensional Hubbard model [62]-[64]. In section 4 we willgive another interesting interpretation to our choice (3.22) which is based on therequirement of generalized unitarity. We will also show there that this choice of η ’s makes the S-matrix (3.24) and, therefore, the Shastry R-matrix a meromorphicfunction on the z -torus.To summarize, in order to have a unified description of the symmetry algebraof the AdS × S light-cone gauge-fixed string theory and its mirror sigma-modelwe should make the symmetric choice of the central charges (3.9), and choose thefundamental representation of the centrally-extended su (2 |
2) with the parameters a, b, c, d given by a = d = (cid:114) igx − − igx + (cid:114) H + 12 ,b = c = − (cid:114) ig x + − ig x − = − (cid:114) H − . (3.27)Taking into account (3.16), (3.19) and (3.20), it is easy to check that both the originaland the mirror (analytically-continued) representations are unitary with respect totheir own reality conditions. Let us stress that the parameters a, b, c, d have thesame dependence on x ± in the original and mirror theories. We simply regard x ± asfunctions of p in the original model, and as functions of (cid:101) p in the mirror one. Formulas (3.27) define how the algebra generators of the original and mirror theoriesact on one-particle states of the theory. We also need to know their action on anarbitrary multi-particle state. The simplest way to have a unified description of theiraction is to use the Hopf algebra structure of the unitary graded associative algebra A generated by the even rotation generators L ab , R αβ , the odd supersymmetrygenerators Q αa , Q † aα and two central elements H and P subject to the algebrarelations (3.7) with the central elements C and C † expressed through the world-sheetmomentum P by the formula (3.9). We will be using the Hopf algebra introducedin [47] which is basically equivalent to the Hopf algebras discussed in [65], see also The finite-size correction to the dispersion relation found in [32] involves the coefficients a , a and a of S AFZ (see [47] for notation) which are unaffected by this transformation. – 19 –66] for further discussion of algebraic properties of the centrally extended su (2 | (cid:15) : A → C , is defined by (cid:15) (id) = 1 , (cid:15) ( J ) = 0 , (cid:15) ( Q ) = 0 , (cid:15) ( Q † ) = 0 , (3.28)and the co-product is given by the following formulas ∆( J ) = J ⊗ id + id ⊗ J , ∆( Q αa ) = Q αa ⊗ e i P / + e − i P / ⊗ Q αa , (3.29)∆( Q † aα ) = Q † aα ⊗ e − i P / + e i P / ⊗ Q † aα , where J is any even generator. Here we use the graded tensor product, that is forany algebra elements a, b, c, d ( a ⊗ b )( c ⊗ d ) = ( − (cid:15) ( b ) (cid:15) ( c ) ( ac ⊗ bd ) , where (cid:15) ( a ) = 0 if a is an even element, and (cid:15) ( a ) = − a is an odd element of thealgebra A .It is interesting to note that the antipode S is trivial for any algebra element,that is S ( J ) = − J , S ( Q ) = − Q , S ( Q † ) = − Q † . (3.30)This action of the antipode arises for the symmetric choice (3.9) of the central ele-ments C and C † only.The co-product is obviously compatible with the hermiticity conditions one im-poses on the algebra generators in the AdS × S string theory, and this ensuresthat the tensor product of two unitary representations is unitary. To check if theco-product is also compatible with the hermiticity conditions one imposes on thealgebra generators of the mirror model we compute the co-product action on thesupersymmetry generators (cid:101) Q , (cid:101) Q † ∆( (cid:101) Q αa ) = (cid:101) Q αa ⊗ cosh (cid:16) (cid:101) H (cid:17) + cosh (cid:16) (cid:101) H (cid:17) ⊗ (cid:101) Q αa + i(cid:15) ad (cid:101) Q † dδ (cid:15) δα ⊗ sinh (cid:16) (cid:101) H (cid:17) − i sinh (cid:16) (cid:101) H (cid:17) ⊗ (cid:15) ad (cid:101) Q † dδ (cid:15) δα , (3.31)∆( (cid:101) Q † aα ) = (cid:101) Q † aα ⊗ cosh (cid:16) (cid:101) H (cid:17) + cosh (cid:16) (cid:101) H (cid:17) ⊗ (cid:101) Q † aα − i(cid:15) αδ (cid:101) Q δd (cid:15) da ⊗ sinh (cid:16) (cid:101) H (cid:17) + i sinh (cid:16) (cid:101) H (cid:17) ⊗ (cid:15) αδ (cid:101) Q δd (cid:15) da . To derive these expressions from the ones given in [47] one should rescale the supersymmetrygenerators in [47] by e ± i P / . – 20 –ince in the mirror theory (cid:101) H is hermitian, the co-product is also compatible with thehermiticity conditions of the mirror theory. This guarantees that an su (2 | A ( (cid:101) p )and A † ( (cid:101) p ) which create asymptotic states of the mirror model. The relations canbe then used to determine the antiparticle representation, and to derive the crossingrelation following the steps in [47]. A simple computation gives (cid:101) Q αa A † ( (cid:101) p ) = A † ( (cid:101) p ) Q αa cosh (cid:16) (cid:101) H (cid:17) + cosh (cid:16) (cid:101) H (cid:17) A † ( (cid:101) p )Σ (cid:101) Q αa (3.32)+ iA † ( (cid:101) p ) (cid:0) (cid:15) ad Q dδ (cid:15) δα (cid:1) sinh (cid:16) (cid:101) H (cid:17) − iA † ( (cid:101) p ) sinh (cid:16) (cid:101) H (cid:17) Σ (cid:0) (cid:15) ad (cid:101) Q † dδ (cid:15) δα (cid:1) , (cid:101) Q † aα A † ( (cid:101) p ) = A † ( (cid:101) p ) Q aα cosh (cid:16) (cid:101) H (cid:17) + cosh (cid:16) (cid:101) H (cid:17) A † ( (cid:101) p )Σ (cid:101) Q † aα (3.33) − iA † ( (cid:101) p ) (cid:0) (cid:15) αδ Q δd (cid:15) da (cid:1) sinh (cid:16) (cid:101) H (cid:17) + i sinh (cid:16) (cid:101) H (cid:17) A † ( (cid:101) p )Σ (cid:0) (cid:15) αδ (cid:101) Q δd (cid:15) da (cid:1) , where Q αa and Q aα are the matrices of the symmetry algebra structure constantscorresponding to the fundamental representation (3.27) and Σ = diag(1 , , − , −
4. Double Wick rotation and the rapidity torus
The universal cover of the parameter space describing the fundamental representa-tion of the centrally extended su (2 |
2) algebra is an elliptic curve [12]. Indeed, thedispersion formula H − g sin p , (4.1)which originates from the relation between the central charges of the fundamentalrepresentation, can be naturally uniformized in terms of Jacobi elliptic functions p = 2 am z , sin p z, k ) , H = dn( z, k ) , (4.2)where we introduced the elliptic modulus k = − g = − λπ <
0. The correspondingelliptic curve (the torus) has two periods 2 ω and 2 ω , the first one is real and the Our convention for the elliptic modulus is the same as accepted in the
Mathematica program,e.g., sn( z, k ) = JacobiSN[ z, k ]. Since the modulus is kept the same throughout the paper we willoften indicate only the z -dependence of Jacobi elliptic functions. – 21 –econd one is imaginary2 ω = 4K( k ) , ω = 4 i K(1 − k ) − k ) , where K( k ) stands for the complete elliptic integral of the first kind. The dispersionrelation is obviously invariant under the shifts of z by 2 ω and 2 ω . The torusparametrized by the complex variable z is an analog of the rapidity plane in two-dimensional relativistic models.In this parametrization the real z -axis can be called the physical one for theoriginal string theory, because for real values of z the energy is positive and themomentum is real due to1 ≤ dn( z, k ) ≤ √ k (cid:48) , z ∈ R , where k (cid:48) ≡ − k is the complementary modulus.We further note that the representation parameters x ± , which are subject to thefollowing constraint x + + 1 x + − x − − x − = 2 ig , (4.3)are expressed in terms of Jacobi elliptic functions as x ± = 12 g (cid:16) cn z sn z ± i (cid:17) (1 + dn z ) . (4.4)This form of x ± follows from the requirement that for real values of z the absolutevalues of x ± are greater than unity: | x ± | > z ∈ R . Note also that for real valuesof z we have Im( x + ) > x − ) < x ± are periodic with theperiod ω , the range of the variable Re z can be restricted to the interval from − ω / ω / − π ≤ p ≤ π .Postponing an extensive discussion of the bound states till section 7, we note herethat the latter problem requires consideration of complex values of particle momenta.According to eq.(4.2), a rectangle − ω / ≤ Re( z ) ≤ ω / − ω / i ≤ Im( z ) ≤ ω / i is mapped one-to-one onto the complex p -plane. By this reason, it is tempting tocall this rectangle by the physical region in the complex z -plane , and, therefore, torestrict the allowed values of the z -coordinates of the particles forming a bound stateby this region. An advantage of adopting such a choice is that all the bound states In relativistic field theories treated in terms of the rapidity θ = θ − θ , the physical region isdefined as a strip 0 < Im θ < π and it incorporates the bound states. Correspondingly, the physicalregion of an individual particle is Im θ ∈ ( − π/ , π/
2) and it covers the complex p -plane (with acut) through the relation p = sinh θ . – 22 – + | | | x I x <> - < || ||xx +- >11| |||xx <<11 +- | || |xx >11 +- > -1 -0.5 0.5 1-224 Im + xIm -< 0 x Im xIm + -<0 x Im + -<0 xIm + - 0 x > > > Figure 1:
On the left figure the torus is divided by the curves | x + | = 1 and | x − | = 1 intofour non-intersecting regions. The middle figure represents the torus divided by the curvesIm( x + ) = 1 and Im( x − ) = 1, also in four regions. The right figure contains all the curvesof interest. would have positive energy. We will see, however, that this is not the only option,and there are other two regions in the complex z -plane which could equally deservethe name “ physical ”. As it will become clear later on, counting the degeneracy ofthe bound states drastically depends on the choice of a physical region.Each solution of eq.(4.3) corresponds to a point of the half-torus, i.e. of therectangle − ω / ≤ Re( z ) ≤ ω / − ω / i ≤ Im( z ) ≤ ω / i . In what follows wewill be loosely referring to this rectangle as the torus. The torus covers the complex p -plane twice. Since the space of solutions of eq.(4.3) is mapped one-to-one on thetorus, the latter could be also chosen as the physical region. Such a choice is howeverproblematic because half of all the states would have negative energy, i.e. the regionwould contain both particles and anti-particles, as well as bound states and anti-bound states. We point out, however, that there exist positive energy solutions ofthe bound state equations with some of the particles falling outside of the rectangle − ω / ≤ Re( z ) ≤ ω / − ω / i ≤ Im( z ) ≤ ω / i that covers the complex p -planeonce.Constraint (4.3) implies that if a pair ( x + , x − ) satisfies it then (1 /x + , x − ),( x + , /x − ) and (1 /x + , /x − ) also do. Each of these four pairs corresponds to adifferent point on the torus. Taking into account that for any complex number w if | w | > | /w | <
1, and if Im( w ) > /w ) <
0, one can divide thetorus into four non-intersecting regions in the following two natural ways, see Fig.1: We made slightly asymmetric choice for Im( z ) to achieve better visual clarity. – 23 – -1 -0.5 0.5 1-224 -1 -0.5 0.5 1-224-1 -0.5 0.5 1-224-1 -0.5 0.5 1-224 g= / g= g= -1 -0.5 0.5 1-224 Figure 2:
Divisions of the torus by the curves | x ± | = 1 (upper figures) and by the curvesIm x ± = 0 (lower figures) for g = 1 / g = 1 and g = 50. The red curves are | x − | = 1, andthe pink ones are | x + | = 1. The coordinates x and y are the rescaled real and imaginaryparts of z : x = Re( ω z ), y = Re( ω z ). In the limit g → ∞ the curves | x ± | = 1 andIm x ± = 0 are related by the shift z → z + ω . • {| x ± | > } , {| x ± | < } , {| x + | < , | x − | > } and {| x + | > , | x − | < } ; thedivision is done by the curves | x ± | = 1. • { Im( x ± ) > } , { Im( x ± ) < } , { Im( x + ) > , Im( x − ) < } and { Im( x + ) < , Im( x − ) > } ; the division is done by the curves Im( x ± ) = 0.The shape of the regions depends on the value of the coupling constant g , seeFig.2. Quite remarkably, in the strong coupling limit g → ∞ two divisions of the– 24 –orus produced by the red ( | x ± | = 1) and green (Im( x ± ) = 0) curves become relatedto each other through a global shift by ω / | x ± | = 1 intersectwith the curves Im( x ± ) = 0, see Fig.1. These points are z = ± ω + n +14 ω , n = − , − , ,
1. It is known [18] that these points are the branch points of the one-loopcorrection [16] to the dressing phase. It is unclear, however, if they remain the branchpoints of the exact dressing phase. One could try to use the integral representation[52] of the BES dressing phase [14] to understand this issue. In fact, all currentlyavailable representations for the dressing phase are defined for | x ± | ≥
1, and this isanother reason to figure out the location of the curves | x ± | = 1 on the z -torus.Both divisions play an important role in the analysis of the bound states ofstring and mirror theories. To understand the meaning of the equations | x ± | = 1and Im( x ± ) = 0, it is convenient to use another parameter u which is similar tothe rapidity parameter of the Heisenberg spin chain. In terms of x ± it is defined asfollows u = x + + 1 x + − ig = x − + 1 x − + ig . (4.5)By using eqs.(4.5) and (4.4), one can express the rapidity u as a meromorphic functionon the torus u = cn z dn zg sn z . (4.6)It is not difficult to check that the eight special points on the torus are mappedonto the four points on the u -plane with coordinates u = ± ± ig , while the points z = ± ω / u = 0, and the points z = ± ω / ω / ± i u = ±∞ ± i ∞ .A special role of the points u = ± ± ig can be also understood by expressing x ± in terms of u x + = 12 (cid:32) u + ig ± (cid:115)(cid:18) u − ig (cid:19) (cid:18) u + 2 + ig (cid:19)(cid:33) ,x − = 12 (cid:32) u − ig ± (cid:115)(cid:18) u − − ig (cid:19) (cid:18) u + 2 − ig (cid:19)(cid:33) . (4.7)Thus, on the u -plane there are four branch points with coordinates u = ± ± ig corresponding to x ± = ± x ± ) = 0. Therefore, we can naturally choose thecuts either connecting the points − ± ig and 2 ± ig , or going from ±∞ to ± ± ig along the horizontal lines. Let us determine what values of x ± correspond to the– 25 – -4 -2 2 4 z u A B A D C D B C Figure 3:
On the left figure the upper and lower curves correspond to | x + | = 1 + 0and | x − | = 1 + 0, respectively. The map z → u ( z ) folds each of these curves onto thecorresponding cut on the u -plane. lines u = u ± ig with u real. We see that u = x + + 1 x + , x + = 12 (cid:18) u ± (cid:113) u − (cid:19) , if u = u − ig ,u = x − + 1 x − , x − = 12 (cid:18) u ± (cid:113) u − (cid:19) , if u = u + ig . It is clear that points x ± and 1 /x ± of the complex x ± -plane correspond to the samepoint u of the u -plane. Then, the points of the circle | x + | = 1 map to points u in theinterval [ − − ig , − ig ] , while the points of | x − | = 1 correspond to u ∈ [ − ig , ig ].On the other hand, the points of the lines Im( x + ) = 0 and Im( x − ) = 0 correspond topoints u outside the intervals [ − − ig , − ig ] and [ − ig , ig ], respectively. Notealso that if one chooses a definite sign in eq.(4.7) then the interval [ − ∓ ig , ∓ ig ]maps onto a half of a unit circle in the x ± -plane. One has to use both signs to coverthe unit circles | x ± | = 1 and real lines Im( x ± ) = 0.To determine the location of the upper and lower edges of the u -plane cuts[ − ∓ ig , ∓ ig ] on the x ± -planes, we introduce a small real parameter (cid:15) and write x ± = e (cid:15) e iϕ , | x ± | = e (cid:15) , Im( x ± ) = e (cid:15) sin ϕ , u ≈ ϕ ∓ ig + 2 i(cid:15) sin ϕ . (4.8)We see that the upper edges [ − ∓ ig + i , ∓ ig + i
0] are mapped either outside theupper halves or inside the lower halves of the circles | x ± | = 1, and the lower edges– 26 – − ∓ ig − i , ∓ ig − i
0] are mapped either outside the lower halves or inside theupper halves of the circles | x ± | = 1, and vice verse:[ − ∓ ig + i , ∓ ig + i ⇐⇒ (cid:26) | x ± | = 1 + 0 , Im( x ± ) > | x ± | = 1 − , Im( x ± ) < , (4.9)[ − ∓ ig − i , ∓ ig − i ⇐⇒ (cid:26) | x ± | = 1 + 0 , Im( x ± ) < | x ± | = 1 − , Im( x ± ) > . (4.10)As we discussed above, the z -torus can be divided into four non-intersecting re-gions by the curves | x ± | = 1. Now it is easy to show that each of the regions is mappedone-to-one onto the u -plane with the two cuts. Let us consider for definiteness theregion with | x ± | >
1. Then, according to the discussion above, the boundaries of theregion with | x + | = 1 + 0 , Im( x + ) > | x + | = 1 + 0 , Im( x + ) < − − ig , − ig ] in the u -plane, respectively.In the same way the boundary of the region with | x − | = 1 is mapped onto the upperand lower edges of the cut [ − ig , ig ], see Fig.3.Another way to understand how different copies of the u -plane are glued togetheris to consider any of the curves | x ± ( z ) | = 1 and shift its variable z by a small positive (cid:15) in the imaginary direction. For the image of the corresponding shifted curve on the u -plane one obtains Im u ( z + i(cid:15) ) = ∓ g + (cid:15) Re (cid:18) ∂u∂z (cid:19) + . . . , (4.11)where Re (cid:0) ∂u∂z (cid:1) is computed along | x ± | = 1. Further analysis shows that along anyof the curves | x ± | = 1 the expression Re (cid:0) ∂u∂z (cid:1) is positive for − ω < Re z < ω andnegative otherwise. This determines how the edges of the cuts | x ± | = 1 are mappedonto the edges of the corresponding cuts on the u -plane (see Fig.3 for an example ).To summarize, any region confined between the curves | x ± | = 1 is mapped under z → u ( z ) onto a single copy of the u -plane with a point at infinity added, i.e. onto theRiemann sphere. Extended to the whole torus, this map defines a four-fold coveringof the Riemann sphere by the torus which has eight ramification points : a genericpoint on the u -plane has four images belonging to the four regions. There are twocuts on each copy of the u -plane1) [ − i/g, i/g ]2) [ − − i/g, − i/g ]which are images of the curves | x − | = 1 and | x + | = 1, respectively.In the same way we can determine the images of the upper and lower edges ofthe u -plane cuts ( −∞ , − ∓ ig ] , [2 ∓ ig , ∞ ) on the x ± -planes. We again introduce a In agreement with the Riemann-Hurwitz formula. – 27 – x + | >>>
1| x | - + | | < - | + | 1 |x | < | x - < || | x + > | x | - < CB B C
11 22
Figure 4:
Four copies of the u -plane (the Riemann sphere) glued together through thecuts to produce the torus of the kinematical variable z . We indicated four branch points B , and C , which are images of those on Fig.3. small real parameter (cid:15) and write x ± = re i(cid:15) , | x ± | = | r | , Im( x ± ) ≈ r(cid:15) , u ≈ r + 1 r ∓ ig + i(cid:15) ( r − r ) . (4.12)We see that the upper edges ( −∞ , − ∓ ig + i , [2 ∓ ig + i , ∞ ) are mapped eitheronto the upper edge of the intervals ( −∞ , − , [1 , ∞ ) or the lower edge of the interval[ − , −∞ , − ∓ ig − i , [2 ∓ ig − i , ∞ ) are mapped eitheronto the lower edge of the intervals ( −∞ , − , [1 , ∞ ) or the upper edge of the interval[ − ,
1] of the real lines Im( x ± ) = 0, and vice verse:( −∞ , − ∓ ig + i ∪ [2 ∓ ig + i , ∞ ) ⇐⇒ (cid:26) Im( x ± ) = +0 , | x ± | > x ± ) = − , | x ± | < , (4.13)( −∞ , − ∓ ig − i ∪ [2 ∓ ig − i , ∞ ) ⇐⇒ (cid:26) Im( x ± ) = +0 , | x ± | < x ± ) = − , | x ± | > , (4.14)Again, dividing the z -torus into four non-intersecting regions by the curves Im( x ± ) =0, we see that each of the regions also maps one-to-one onto the u -plane with thetwo cuts. This gives a different (but equivalent) four-fold covering of the Riemannsphere by the torus.When a point on the z -plane runs along the curve | x + | = 1 or | x − | = 1 its imagecovers the corresponding interval on the u -plane twice. To appreciate this fact, let– 28 –s note that if z is, e.g., on the curve | x + | = 1 then the points z ± = − z ± ω ω | x + ( z ) | = x + ( z ) x − ( z ∗ ), we have | x + ( z ± ) | = x + (cid:16) − z ± ω ω (cid:17) x − (cid:16) − z ∗ ± ω − ω (cid:17) = 1 | x + ( z ) | = 1 , where we have used the properties of Jacobi elliptic functions under the shifts byquarter-periods. In the same way one finds that if z lies on a curve | x − | = 1 thenthe points z ± belong to another copy of | x − | = 1 which is obtained from the originalone by the shift by ω . Finally, using the properties of the Jacobi elliptic functionsit is easy to show that u ( z ± ) = u ( z ), i.e. the points z and z ± have one and the sameimage on the u -plane.It is clear that the half of the torus and, therefore, the complex p -plane is mappedonto the u -plane twice. The coordinate u is real for real z , and in this case we caneasily express it in terms of p [5] u ( p ) = 1 g cot p (cid:114) g sin p . (4.16)In the limit g → u -plane, namely the one which maps to the region | x ± | > z -torus. There are certain advantages of such a choice which we willdiscuss later on. The main disadvantage is, however, that the region | x ± | > p -plane.It is interesting to see what happens with our three candidates for the physicalregion in the limits g → ∞ and g →
0. In the limit g → ∞ the periods of the torushave the following behavior ω → log gg , ω → iπ g if g → ∞ . (4.17)To keep the range of Im( z ) finite, we rescale z as z → z/ (2 g ), and the momentum as p → p/g . Then the dispersion relation (4.1) takes the relativistic form H − p = 1,the variable z plays the role of θ because p = sinh z , and we have • The torus degenerates to the strip with − π < Im( z ) < π and −∞ < Re( z ) < ∞• The half-torus corresponding to the complex p -plane degenerates to the stripwith − π/ < Im( z ) < π/ −∞ < Re( z ) < ∞ – 29 – The region | x ± | > u -plane degenerates to thestrip with − π/ < Im( z ) < π/ −∞ < Re( z ) < ∞ We see that both the half-torus and the region | x ± | > g → ω → π , ω → i log g if g → . (4.18)We see that all the three regions degenerate into the strip with − π/ < Re( z ) < π/ −∞ < Im( z ) < ∞ . The properties of the S-matrix arising in the limit g → The z -torus can be also used to describe the mirror model. Since we know therelation between p = 2 am z and the mirror momentum ˜ p , we can express ˜ p in termsof z . Indeed, the equality 2 am z = 2 i arcsinh 12 g (cid:112) (cid:101) p (4.19)implies (cid:101) p = − i dn z . (4.20)The energy in the mirror theory takes the form (cid:101) H = 2 arccoth √ k (cid:48) dn z . (4.21)The formulae above show that real values of z correspond to imaginary ˜ p . Nowwe would like to understand for which values of z the corresponding values of (cid:101) p arereal. One can see that if we shift the variable z by ω / z → z + ω /
2, that is if wewrite (cid:101) p = − i dn (cid:16) z + ω , k (cid:17) ≡ √ k (cid:48) sn z cn z , (4.22)then for real values of the shifted variable z the corresponding values of (cid:101) p are real aswell. We also recognize here a close analogy with the relativistic case – making thedouble Wick rotation corresponds to the shift by a quarter-period on the rapidityplane. The function cn( z, k ) has zeroes at z = − ω and z = ω (and dn( z, k ) haspoles at z = ( − ω + ω ) and z = ( ω + ω )) which explains the apparent absenceof the periodicity in (cid:101) p . Thus, when the shifted variable z runs from − ω to ω themomentum ˜ p monotonically increases from −∞ to + ∞ and it passes though zero for z = 0. – 30 –ne further finds that the parameters x ± are expressed in terms of the shiftedparameter z of the mirror model as follows x ± = − i √ k (cid:48) ∓ dn z √− k dn z (cid:16) i √ k (cid:48) sn z cn z (cid:17) . (4.23)We can now find how x ± are expressed in terms of the mirror momentum. Indeed,since (cid:16) cn z sn z (cid:17) = − k − dn z , we deduce from eq.(4.23) that x ± = 12 g (cid:32)(cid:115) g (cid:101) p ∓ (cid:33) ( (cid:101) p − i ) . This, of course, agrees with the formula (2.15).The variables x ± of the mirror theory obey a relation x + x − = (cid:101) p − i (cid:101) p + i which implies that | x + x − | = 1 for (cid:101) p real.It is also not difficult to show that the dispersion relation in the mirror theorytakes the form (2.11) (cid:101) H = 2 arccoth √ k (cid:48) dn z = 2 arccoth (cid:115) − k (cid:101) p = 2 arcsinh 1 √− k (cid:112) (cid:101) p . This completes the proof that the double-Wick rotation corresponds to a shift of the z variable by a quarter of the imaginary period of the torus, and the real axes of theshifted z corresponds to real values of the momentum of the mirror theory. Finally, it is useful to express the rapidity u in terms of the shifted parameter z of the mirror model and (cid:101) p . We have u = 2 cn (cid:0) z + ω , k (cid:1) dn (cid:0) z + ω , k (cid:1) √− k sn (cid:0) z + ω , k (cid:1) = − i √ k (cid:48) dn (cid:0) z + ω , k (cid:1) √− k dn (cid:0) z, k (cid:1) . Then one can check that the points z = ± ω / ± i u = ±∞ ± i ∞ .The coordinate u is real for real z , and in this case we can express it in terms of (cid:101) pu = 2 (cid:101) p √− k (cid:115) − k (cid:101) p = (cid:101) pg (cid:115) g (cid:101) p . After having performed the shift, one can do various physically equivalent transformations of theshifted z -variable preserving the axes of real z . Particular useful examples of these transformationsare z → z + ω , z → − z + ω , z → − z ± ω . – 31 –gain, there are three choices of the physical region. It is the half-torus corre-sponding to the complex (cid:101) p -plane, the whole torus, and the region Im( x ± ) < u -plane. The third choice is different from the one made forthe string theory, and is motivated by the analysis of the bound states of the mirrormodel.
5. S-matrix on elliptic curve
The dispersion relation (4.1) is naturally parametrized by the elliptic curve. Withoutimposing the unitarity condition for the S-matrix, the phase η in (3.23) can be chosenin an arbitrary way, for instance, η ( p ) = 1. In the latter case, the S-matrix (3.24) iswell defined on the elliptic curve but it is non-unitary. It is therefore tempting to as-sume that the unitary S-matrix also admits an analytic continuation into the complex z -plane. To find such a continuation one has to resolve the branch cut ambiguitiesarising due to the η -factor in the S-matrix (3.24): η ( p ) = e i p (cid:112) ix − ( p ) − ix + ( p ).This can be done in the following way. First, we recall the elliptic parametrization(4.4) which gives η ( p ) = e i p (cid:112) ix − ( p ) − ix + ( p ) = 1 √ g e i am z √ z == 1 √ g (cid:112) (1 + dn z )(cn z + i sn z ) . (5.1)Second, by using the following formulae (recall k = − g )1 + dn z = 2 dn z − k sn z , cn z + i sn z = (cid:0) cn z + i sn z dn z (cid:1) − k sn z relating elliptic functions to those of the half argument, we can resolve the branchcut ambiguities by means of the relation e i p (cid:112) ix − ( p ) − ix + ( p ) = √ √ g dn z (cid:0) cn z + i sn z dn z (cid:1) g sn z ≡ η ( z ) (5.2)valid in the region − ω < Re z < ω and − ω /i < Im z < ω /i . Further, we noticethat the non-local dependence of η ’s on the momentum of another particle enters as e i p = e i am z and, therefore, can be naturally treated as e i p = cn z + i sn z .Thus, we define an analytic continuation of the S-matrix onto the rapidity torusfor each of the complex variables z and z by means of eq.(3.24), where the variables η , and ˜ η , are given by η = η ( z )(cn z + i sn z ) , η = η ( z ) , ˜ η = η ( z )(cn z + i sn z ) , ˜ η = η ( z ) . – 32 –n this way we completely resolved the branch cut ambiguities of the S-matrix (3.24)and defined it as a meromorphic function on the elliptic curve (for each z -variable). Itis remarkable to observe that such a continuation becomes possible due to additionalphase factors, e i p , introduced in the previous section to guarantee unitarity of themirror theory.Let us now analyze the basic properties of the elliptic S-matrix. One can checkthat it satisfies the Yang-Baxter equation and the usual unitarity requirement S ( z , z ) S ( z , z ) = I . (5.3)Further, it obeys the generalized unitarity condition : S ( z ∗ , z ∗ ) (cid:2) S ( z , z ) (cid:3) † = I . (5.4)Here “ † ” means hermitian conjugation. For z and z real the last condition reducesto the requirement of physical unitarity. In fact, one can see that the elliptic S-matrixis compatible with the generalized unitarity condition only due to our specific choicefor the phase factors discussed above. Then, unitarity and generalized unitarityimply hermitian analyticity: S ( z ∗ , z ∗ ) = (cid:2) S ( z , z ) (cid:3) † .Let us now compute monodromies of the S-matrix (3.24) over the real and imag-inary periods. We find S ( z + 2 ω , z ) = Σ S ( z , z )Σ = Σ S ( z , z )Σ ,S ( z + 2 ω , z ) = Σ S ( z , z )Σ = Σ S ( z , z )Σ . (5.5)Hence, the S-matrix exhibits the same monodromies over real and imaginary cyclesand it is a periodic function on a double torus with periods 4 ω and 4 ω . HereΣ = Σ ⊗ I and Σ = I ⊗ Σ, where Σ is defined in section 3.4, and the S-matrixcommutes with the product Σ ⊗ Σ. Note that Σ is in the center of the groupSU(2) × SU(2).Second, we establish the monodromy properties w.r.t. shifts by half-periods.Under the shift by the real half-period we get S ( z + ω , z ) = (cid:0) V ⊗ Σ (cid:1) S ( z , z ) (cid:0) V − ⊗ I (cid:1) , (5.6)where V = diag (cid:0) e − iπ , e − iπ , e iπ , e iπ (cid:1) .The shift by the imaginary half-period corresponds to the crossing symmetrytransformation [12]. To discuss it, we multiply the S-matrix (3.24) with a scalarfactor S to produce the string S-matrix obeying crossing symmetry S ( z , z ) = S ( z , z ) S ( z , z ) . (5.7)We then find that with a proper choice for S ( z , z ) the string S-matrix exhibits thefollowing crossing symmetry relations C − S t ( z , z ) C S ( z + ω , z ) = I , C S t ( z , z ) C − S ( z − ω , z ) = I , (5.8)– 33 –nd also C − S t ( z , z ) C S ( z , z − ω ) = I , C S t ( z , z ) C − S ( z , z + ω ) = I . (5.9)Here t denotes transposition in the first matrix space and C is a constant chargeconjugation matrix C = (cid:18) σ i σ (cid:19) , (5.10)where σ is the Pauli matrix. The compatibility of eqs.(5.8) and (5.9) with (5.5) isguaranteed by the identity C Σ = C − which is equivalent to C = Σ.The crossing symmetry relations lead to the following equations for the scalarfactor S [12] S ( z , z ) S ( z + ω , z ) = f ( z , z ) , S ( z , z ) S ( z , z − ω ) = f ( z , z ) , (5.11)where the function f is expressed through x ± as follows f ( z , z ) = (cid:16) x − − x − (cid:17) ( x − − x +2 ) (cid:16) x +1 − x − (cid:17) ( x +1 − x +2 ) . (5.12)One can easily check that the function f ( z , z ) obeys the following properties f ( z , z ) f ( z + ω , z ) = 1 = f ( z , z ) f ( z , z + ω ) , f ( z + ω , z + ω ) = f ( z , z ) , which are, however, incompatible with the assumption that the scalar factor is ananalytical function of z , z .Another important property of the string S-matrix (5.7) is that it remains in-variant under the simultaneous shift of z and z by ω : S ( z + ω , z + ω ) = S ( z , z ) . (5.13)This follows from the fact that both the S-matrix (3.24) and the scalar factor S areinvariant under the shift. This property together with the crossing relations (5.8),(5.9) implies S t ,t ( z , z ) = C C S ( z , z ) C − C − = C − C − S ( z , z ) C C , where t and t mean the transposition in the first and in the second matrix spaces,respectively. This is in opposite to [47], where the charge conjugation matrix was found to depend on thesign of the particle momentum. This dependence is, in fact, spurious and it gets removed by aproper resolution of the branch cut ambiguities we propose here. – 34 –ssuming that the above-mentioned properties of the S-matrix (3.24) are sharedby S , we can now see that the string S-matrix allows one to define consistently anelliptic analog of the ZF algebra, i.e. A ( z ) A ( z ) = S ( z , z ) A ( z ) A ( z ) ,A † ( z ) A † ( z ) = A † ( z ) A † ( z ) S ( z , z ) , (5.14)where the creation and annihilation ZF operators are now functions of the complexvariable z . In addition, away from the line z = z we can impose the followingrelation between the creation and annihilation operators A ( z ) A † ( z ) = A † ( z ) S ( z , z ) A ( z ) . (5.15)As usual, the absence of cubic and higher relations for the ZF operators is guaranteedby the Yang-Baxter equation for S . Furthermore, the validity of relations (5.14),(5.15) for all values of z and z is due to unitarity condition (5.3).Transposing the second equation in (5.14) in the first matrix space we get( A † ( z )) t A † ( z ) = A † ( z ) S t ( z , z )( A † ( z )) t , On the other hand, shifting in eq.(5.15) the variable z by the imaginary half-periodwe obtain A ( z + ω ) A † ( z ) = A † ( z ) S ( z + ω , z ) − A ( z + ω ) . Since the string S-matrix satisfies the crossing relation we see that the algebra struc-ture is compatible with the following identification A ( z + ω ) = C − A † ( z ) t , A † ( z − ω ) = − A ( z ) t C . (5.16)Analogously, we establish A ( z − ω ) = C A † ( z ) t , A † ( z + ω ) = − A ( z ) t C − . (5.17)These relations together with the monodromy properties (5.5) of the S-matrix furtherimply A ( z + 2 ω ) = Σ A ( z ) , A † ( z + 2 ω ) = A † ( z )Σ ,A ( z + 2 ω ) = Σ A ( z ) , A † ( z + 2 ω ) = A † ( z )Σ . This means that the bosonic operators are unchanged under the shift around thetorus while fermionic ones acquire the minus sign. Thus, the monodromy propertiesof the S-matrix imply the spin structure ( − , − ) for the fermionic ZF operators.– 35 –inally, the generalized unitarity condition (5.4) allows one to impose the fol-lowing hermiticity conditions on the ZF operators:[ A i ( z )] † = A † i ( z ∗ ) for 0 < | Im z | < ω i ;[ A i ( z )] † = − A † i ( z ∗ ) for ω i < | Im z | < ω i . (5.18)The hermiticity condition for the ZF creation and annihilation operators in the anti-particle region ω / i < | Im z | < ω /i is compatible with the hermiticity conditionfor the ZF operators in the particle region 0 < | Im z | < ω / i and the identifications(5.16) and (5.17). It is clear from the discussion above that the S-matrix of the mirror theory is obtainedfrom the string S-matrix just by the shift of the z -variables by ω / (cid:101) S ( z , z ) = S ( z + ω , z + ω . (5.19)The momentum of the mirror theory is expressed in terms of the variable z byeq.(4.20) and is real for real values of z , and the generalized unitarity of the mirrorS-matrix in terms of the shifted coordinates z takes the usual form (cid:104) (cid:101) S ( z , z ) (cid:105) † (cid:101) S ( z ∗ , z ∗ ) = I . (5.20)This just follows from the generalized unitarity of the string S-matrix and relation(5.13) which is a consequence of the crossing equations (cid:104) (cid:101) S ( z , z ) (cid:105) † = S ( z ∗ − ω , z ∗ − ω S ( z ∗ + ω , z ∗ + ω (cid:101) S ( z ∗ , z ∗ ) . In fact, since both the S-matrix (3.24) and the scalar factor S satisfy the generalizedunitarity condition and relation (5.13), the same holds for the mirror theory.It is of interest to understand how the dressing factor of the mirror theory trans-forms under the complex conjugation. To this end we recall that in the a = 0light-cone gauge the scalar factor of the string S-matrix can be written in the form[18] S ( z , z ) = s ( z , z ) σ ( z , z ) , s ( z , z ) = x − − x +2 x +1 − x − − x +1 x − − x − x +2 . (5.21)Here the gauge-independent dressing factor σ ( z , z ) has the following structure [6]1 i ln σ ( z , z ) ≡ θ ( z , z ) = ∞ (cid:88) r =2 ∞ (cid:88) s = r +1 c r,s ( g ) (cid:104) q r ( z ) q s ( z ) − q r ( z ) q s ( z ) (cid:105) , (5.22) It is easy to check that the additional a -dependent factor does not break any of the propertiesof the S-matrix. – 36 –here q r ( z ) = ir − (cid:2) ( x + ) − r − ( x − ) − r (cid:3) are the local conserved charges. At any order ofthe perturbative expansion in powers of 1 /g the sums in r and s define the convergentseries for | x ± | > | x ± | > Thus, the S-matrix is by construction well-definedonly in the region where | x ± | > and it should be analytically continued for othervalues of x ± . The string theory dressing factor satisfies the generalized unitarity condition thatfollows from the fact that under the complex conjugation the variables x ± transformas [ x ± ( z )] † = x ∓ ( z ∗ ). In the mirror theory the variables x ± depend on the shiftedcoordinate z and, as a result, satisfy the following complex conjugation rule (cid:104) x ± ( z + ω (cid:105) † = 1 x ∓ ( z ∗ + ω ) . By using this rule one can easily check that the factor s ( z , z ) in (5.21) transformsunder the complex conjugation as follows[ s ( z ∗ , z ∗ )] † s ( z , z ) = (cid:18) x − x +2 x +1 x − (cid:19) , (5.23)where x ± i = x ± ( z i + ω ). Taking into account that the scalar factor S of the mirrortheory satisfies the generalized unitarity condition, we find the complex conjugationrule for the dressing factor of the mirror theory[ σ ( z ∗ , z ∗ )] † σ ( z , z ) = (cid:18) x +1 x − x − x +2 (cid:19) . (5.24)In particular, for real values of z ’s corresponding to real (cid:101) p ’s the dressing factor of themirror theory is not unitary.It is interesting to note that the scalar factor can be split into a product of twofactors satisfying the generalized unitarity condition in both string and mirror theory S ( z , z ) = x − − x +2 x +1 − x − x +1 x − − x − x +2 − × x − x +2 x +1 x − σ ( z , z ) . (5.25)Another interesting splitting is given by S ( z , z ) = u − u − ig u − u + ig × (cid:32) − x +1 x − − x − x +2 (cid:33) σ ( z , z ) . (5.26)This splitting is useful for analyzing the bound state spectrum of the mirror model.Knowing the series representation for the dressing phase in the original the-ory [13], it is interesting to understand what is precisely the source of its unitaritybreakdown in the mirror theory. – 37 –o clarify this issue, we recall that the dressing phase can be conveniently writtenin terms of a single function χ ( x , x ) [18] θ ( z , z ) = χ ( x +1 , x +2 ) − χ ( x +1 , x − ) − χ ( x − , x +2 ) + χ ( x − , x − ) −− χ ( x +2 , x +1 ) + χ ( x − , x +1 ) + χ ( x +2 , x − ) − χ ( x − , x − ) , which admits the following strong coupling expansion χ ( x , x ) = g ∞ (cid:88) n =0 χ ( n ) ( x , x ) (cid:16) g (cid:17) − n . Here χ (0) ( x , x ) = − x − x x − x log x x − x x is the leading AFS factor [6]. The next-to-leading contribution is [16]: χ (1) ( x , x ) = − π Li √ x − / √ x √ x − √ x − π Li √ x + 1 / √ x √ x + √ x + 12 π Li √ x + 1 / √ x √ x − √ x + 12 π Li √ x − / √ x √ x + √ x . (5.27)All higher terms are rational functions of x , x [13]. As we will now show, theunitarity breakdown of the dressing phase is due to the leading AFS contributiononly, the Hern´andez-L´opez term (5.27), as well as all higher order terms do notinfluence the unitarity condition.To simplify the notations in what follows we only consider the case of real z ’s inthe mirror theory. It is easy to see that the complex conjugate of the function χ (0) is given by (cid:2) χ (0) ( x ± , x ± ) (cid:3) ∗ = − χ (0) ( x ∓ , x ∓ ) − iπ + 1 x ∓ + ( iπ − x ∓ − (cid:16) x ∓ − x ∓ (cid:17) log x ∓ x ∓ . Using this formula for computing the leading value θ AFS , we find that the contributionof non-logarithmic terms cancels out and we get θ ∗ AFS = θ AFS + g x − − x − x − x − (1 − x − x − ) log x − x − + g x +1 − x +2 x +1 x +2 (1 − x +1 x +2 ) log x +1 x +2 − g x − − x +2 x − x +2 (1 − x − x +2 ) log x − x +2 + g x − − x +1 x +1 x − (1 − x +1 x − ) log x +1 x − . Using identity (4.3), it is easy to show that all logarithmic terms are neatly combinedto produce the following answer θ ∗ AFS = θ AFS + i log (cid:16) x +1 x − x − x +2 (cid:17) , (5.28)– 38 –hich coincides with the logarithmic form of eq.(5.24).Since we have shown that the shift of the phase under the complex conjugationoccurs due to the leading contribution, all the higher order terms in the expansion of θ must be real functions. To convince oneself that this is indeed the case, we considerthe next-to-leading term in the strong coupling expansion of θ . As was shown in [13],this term admits the following representation θ HL = ψ ( q +1 − q +2 ) − ψ ( q +1 − q − ) − ψ ( q − − q +2 ) + ψ ( q − − q − ) . (5.29)Here the function ψ ( q ) is ψ ( q ) = 12 π Li (1 − e iq ) − π Li (1 − e iq + iπ ) − i − e iq + iπ ) + π , (5.30)where the variables q ± are related to x ± through e iq ± = x ± + 1 x ± − . (5.31)Taking into account the conjugation rule in the mirror theory, eq.(3.20), we obtain( q ± ) ∗ = − q ∓ − π . (5.32)Since θ HL depends on the difference of two q (cid:48) s , the shift by π arising upon the complexconjugation will cancel out. Thus, taking the complex conjugate we obtain θ ∗ HL = ¯ ψ ( q − − q − ) − ¯ ψ ( q − − q +2 ) − ¯ ψ ( q +1 − q − ) + ¯ ψ ( q +1 − q +2 ) , (5.33)where the function ¯ ψ ( q ) is defined as¯ ψ ( q ) = 12 π Li (1 − e iq ) − π Li (1 − e iq − iπ ) − i − e iq − iπ ) + π . (5.34)Taking into account the following transformation property of the dilogarithm func-tion Li (1 − e iq − iπ ) = Li (1 − e iq + iπ − πi ) = Li (1 − e iq + iπ ) + 2 πi log(1 − e iq ) , we find that ¯ ψ ( q ) = ψ ( q ) + π . Since the shift by π in the previous formula does not contribute to θ ∗ HL , we concludethat θ ∗ HL = θ HL . Finally, by working out several higher order terms χ ( k ) , one caneasily check that they always lead to the real functions θ , in accord with eqs.(5.24)and (5.28).Thus, we have shown that under the double Wick rotation the scalar factorremains unitary, while the dressing factor does not; the non-unitarity of the dressingfactor is only due to the leading contribution θ AFS , which is another distinguishedproperty of θ AFS . Concluding this section, we note that it would be interesting tounderstand whether the BES factor [14] exhibits the same kind of non-unitaritybehavior in the mirror theory. – 39 – . Bethe ansatz equations
In this section we discuss the nested Bethe equations for the light-cone string theoryon AdS × S and its mirror model. These equations based on the su (2 | ⊕ su (2 | P satisfying e iP/ = 1, the set of equations foundin these papers coincides with the one previously obtained in [8, 9] by using thespin chain description of the gauge theory. It appears, however, that in the sectorwith odd winding number, where e iP/ = −
1, the Bethe equations by [60, 67] differfrom the ones derived from the gauge theory. The origin of this disagreement canbe traced back to the fact that in the light-cone gauge the fermions of the stringsigma model are anti-periodic in the odd winding number sector [68, 56], and thischanges the periodicity conditions for wave functions which one imposes to get theBethe equations. Indeed, in the light-cone gauge one of the fields, an angle φ whichparametrizes the five-sphere, appears to be unphysical and it is solved in terms of(transversal) physical fields. In particular, the equation of motion for φ implies φ (2 π ) − φ (0) = P .
Since φ enters into parametrization of the five-sphere via e iφ , the closed string peri-odicity condition for physical fields gives rise to the winding sectors φ (2 π ) − φ (0) = 2 πm , where m is an integer. Now, we recall that fermions of the original string sigma-models are charged w.r.t. the U(1) isometry acting on φ as φ → φ + const. Also,the Wess-Zumino term in the sigma-model action contains e iφ , i.e. it is non-local interms of physical fields. To uncharge the fermions under the U(1) isometry, as wellas to make the Wess-Zumino term local, one has to redefine the fermions as ψ → e i φ ψ . Thus, the redefined fermions acquire the periodicity properties which do depend onthe winding sector ψ (0) = e i P ψ (2 π ) = e iπm ψ (2 π ) , i.e. they are periodic in the even winding sector and they are ant-periodic in the oddwinding sector [68, 56].As a result, the Bethe equations obtained in [60, 67] correctly describe the light-cone string theory in the sector with periodic fermions only. Changing the boundaryconditions for fermions to anti-periodic, one derives a new set of Bethe equationswhich does agree with the gauge theory one for physical states satisfying e iP/ = − .1 BAE for a model with the su (2 | -invariant S-matrix The asymptotic states of both the original and the mirror theory are constructedby applying the ZF operators A † M ˙ M to the vacuum state. The indices M and ˙ M are associated with two factors of the centrally-extended su (2 | ⊕ su (2 |
2) algebra;the latter being the symmetry algebra of the light-cone string theory [9, 10]. For ourpresent purpose it is convenient to think about the ZF operator as being a product oftwo (anti)commuting operators each transforming in a fundamental representation of su (2 | A † M ˙ M ∼ A † M A † ˙ M . Since the string S-matrix is a tensor product of two su (2 | su (2 | | Ψ (cid:105) = (cid:88) Ψ M ··· M K I A † M ( p ) · · · A † M K I ( p K I ) | (cid:105) , (6.1)where K I is a number of particles in the asymptotic state and p i are their momenta.Denote by N ( M ) the number of particles of type M (that is number of indices oftype M ) occurring in the wave function (6.1). Obviously, K I = N (1) + N (2) + N (3) + N (4) . Since the scattering is elastic, the number of particles K I is a conserved quantity.The form of the Bethe equations derived through the nesting procedure of thecoordinate Bethe ansatz depends on the choice of the initial reference state. Due tothe su (2) bosonic symmetry there are two inequivalent choices for a model with the su (2 | su (2 | K I bosonic operators A † on the vacuum: A † ( p ) . . . A † ( p K I ) | (cid:105) . Then, we define K II+ = 2 N (2) + N (3) + N (4) , K III = N (2) + N (4) . It appears that in the scattering process not only K I but also these numbers areconserved [9]. By this reason, the values of K II+ and K III are the same for any termin the sum (6.1). In particular, K II+ plays the role of the fermionic number, becausein the background of the A † -particles A † counts for two fermions. The number K III has a similar meaning. – 41 –hen the asymptotic Bethe equations based on the su (2 | R and with (anti)-periodic fermions can bewritten in the form [9, 60, 67] e ip k R = K I (cid:89) l =1 l (cid:54) = k S ( p k , p l ) x + k − x − l x − k − x + l (cid:115) x + l x − k x − l x + k K II+ (cid:89) l =1 x − k − y l x + k − y l (cid:115) x + k x − k ( − (cid:15) = K I (cid:89) l =1 y k − x + l y k − x − l (cid:115) x − l x + l K III (cid:89) l =1 v k − w l + ig v k − w l − ig (6.2)1 = K II+ (cid:89) l =1 w k − v l − ig w k − v l + ig K III (cid:89) l =1 l (cid:54) = k w k − w l + ig w k − w l − ig . Here (cid:15) = 0 for a sector with periodic fermions and (cid:15) = 1 for a sector with anti-periodicfermions, x ± k depend on the momentum p k of the model, y l and w l are auxiliary rootsof the second and third levels, respectively, and v = y + y .On the other hand, if one chooses a “fermionic” reference state created by K I fermionic operators A † : A † ( p ) . . . A † ( p K I ) | (cid:105) , then, one should define K II − = 2 N (4) + N (1) + N (2) , K III = N (2) + N (4) , because these numbers are also conserved in the scattering process. Then, K II − playsthe role of the bosonic number, because in the background of the A † -particles A † counts for two bosons.Then the corresponding Bethe equations take the following form e ip k R = ( − (cid:15) K I (cid:89) l =1 l (cid:54) = k S ( p k , p l ) K II − (cid:89) l =1 x + k − y l x − k − y l (cid:115) x − k x + k ( − (cid:15) = K I (cid:89) l =1 y k − x + l y k − x − l (cid:115) x − l x + l K III (cid:89) l =1 v k − w l + ig v k − w l − ig (6.3)1 = K II − (cid:89) l =1 w k − v l − ig w k − v l + ig K III (cid:89) l =1 l (cid:54) = k w k − w l + ig w k − w l − ig . Equations (6.3) can be derived either by using the nesting procedure of the coordi-nate Bethe ansatz (see appendix 9.3.2 for an example) or by applying the duality– 42 –ransformation discussed in [8] to eqs.(6.2). Comparing the two sets of Bethe equa-tions (6.2) and (6.3), we see that only the first lines in two sets are different. Letus stress, however, that in general K II − (cid:54) = K II+ . We further note that the bosonicreference state corresponds to, say, η = 1 and the fermionic one to η = −
1, where η and η are the gradings introduced in [8]. su (2 | ⊕ su (2 | -invariant string S-matrix The Bethe equations based on the su (2 | ⊕ su (2 | su (2 | su (2 | ⊕ su (2 | A † ( z ) . . . A † ( z K I ) | (cid:105) , η = η = 1 ; A † ( z ) . . . A † ( z K I ) | (cid:105) , η = η = − , and fermionic reference states are A † ( z ) . . . A † ( z K I ) | (cid:105) , η = − η = 1 ; A † ( z ) . . . A † ( z K I ) | (cid:105) , η = − η = − , where for the original theory the z -variables lie on the real line, while for the mirrortheory they have Im z = ω / i , and we also indicated the corresponding gradings.To discuss the bound states of the light-cone string sigma model, it is convenientto choose as the reference state the one created by the bosonic operators A † . Thesereference states are dual to gauge theory operators from the su (2) sector. Thenthe corresponding Bethe equations based on the su (2 | ⊕ su (2 | e ip k J = K I (cid:89) l =1 l (cid:54) = k (cid:34) S ( p k , p l ) x + k − x − l x − k − x + l (cid:115) x + l x − k x − l x + k (cid:35) (cid:89) α =1 K II( α ) (cid:89) l =1 x − k − y ( α ) l x + k − y ( α ) l (cid:115) x + k x − k ( − (cid:15) = K I (cid:89) l =1 y ( α ) k − x + l y ( α ) k − x − l (cid:115) x − l x + l K III( α ) (cid:89) l =1 v ( α ) k − w ( α ) l + ig v ( α ) k − w ( α ) l − ig (6.4)1 = K II( α ) (cid:89) l =1 w ( α ) k − v ( α ) l − ig w ( α ) k − v ( α ) l + ig K III( α ) (cid:89) l =1 l (cid:54) = k w ( α ) k − w ( α ) l + ig w ( α ) k − w ( α ) l − ig . Here we take into account that the string sigma model in the a = 0 light-cone gaugeis defined on a circle of length J , α = 1 , su (2 |
2) and y ( α ) l and w ( α ) l are auxiliary roots of the second and third levels, respectively, and v = y + y .– 43 –or the reader’s convenience we point out that the excitation numbers in the setof Bethe equations are related to the ones used in [8] as follows( K III(1) , K
II(1) , K I , K II(2) , K
III(2) ) = ( K , K + K , K , K + K , K ) , and the Dynkin labels [ q , p, q ] of su (4) and [ s , s ] of su (2) ⊕ su (2) ⊂ su (2 ,
2) areexpressed in terms of the excitation numbers by the following formulas q = K I − K II(1) , s = K II(1) − K III(1) ,p = J − K I + 12 ( K II(1) + K II(2) ) , s = K II(2) − K III(2) ,q = K I − K II(2) . (6.5)To analyze the bound states of the mirror theory, it is more convenient, however,to choose as an initial reference state the one created by the operators A † . Thereason is that the operators A † create states from the sl (2) sector, and, as we haveseen, it is this sector which gives rise to mirror magnons. Analogously, there are M -particle bound states made only out of the A † -type particles.If we choose in the mirror theory the above-described reference state then thecorresponding Bethe equations take the form e i e p k R = K I (cid:89) l =1 l (cid:54) = k [ S ( (cid:101) p k , (cid:101) p l )] (cid:89) α =1 K II( α ) (cid:89) l =1 x + k − y ( α ) l x − k − y ( α ) l (cid:115) x − k x + k − K I (cid:89) l =1 y ( α ) k − x + l y ( α ) k − x − l (cid:115) x − l x + l K III( α ) (cid:89) l =1 v ( α ) k − w ( α ) l + ig v ( α ) k − w ( α ) l − ig (6.6)1 = K II( α ) (cid:89) l =1 w ( α ) k − v ( α ) l − ig w ( α ) k − v ( α ) l + ig K III( α ) (cid:89) l =1 l (cid:54) = k w ( α ) k − w ( α ) l + ig w ( α ) k − w ( α ) l − ig . Note that in the mirror model we do not have ( − (cid:15) in the middle equation becausethe fermions are always anti-periodic with respect to ˜ σ . In terms of excitation Let us note in passing that in recent papers [70, 71] the anomalous dimension of the operatorTr F L was computed by using the asymptotic Bethe ansatz with an understanding that in the large L limit one may trust the corresponding result to an arbitrary loop order. One can notice, however,that the excitation pattern of Bethe roots for the operator is ( K III(1) , K
II(1) , K I , K II(2) , K
III(2) ) =(0 , L − , L − , L − , L −
2) with J = , and, therefore, one would expect the breakdownof the asymptotic ansatz due to the finite size effects already at two loops. It may happen thatthe asymptotic ansatz could still be used to determine the leading L behavior of the anomalousdimension of Tr F L if the finite-size corrections are subleading at large L , but this is currentlyunknown. We are grateful to R. Janik and M. Martins for drawing our attention to this point. – 44 –umbers, the Dynkin labels read now as follows q = K II(1) − K III(1) , s = K I − K II(1) ,p = J −
12 ( K II(1) + K II(2) ) + K III(1) + K III(2) , s = K I − K II(2) ,q = K II(2) − K III(2) . (6.7)
7. Bound states of the
AdS × S gauge-fixed model Bound states arise as poles of the multi-particle S-matrix corresponding to complexvalues of the particle momenta, see e.g. [69]. In the thermodynamic limit theyare described by string-like solutions known as “Bethe strings”. In this section wediscuss in detail the bound states of the string sigma-model. They have been alreadyanalyzed in [50, 51]. The main outcome of this analysis is that the M -particle boundstates comprise into short (BPS) multiplets of the centrally extended su (2 | ⊕ su (2 | − π ≤ p ≤ π , but have a rather intricatestructure. Moreover, depending on the choice of the physical region for a given valueof the bound state momentum there could be 1, 2 or 2 M − M -particle bound statessharing the same set of global conserved charges: Q r = (cid:80) Mi =1 q r ( z i ). It is unclearto us whether this indicates that the actual physical region is the one that containsonly a single M -particle bound state (it is the one with | x ± | >
1) or it is a sign of ahidden symmetry of the model responsible for the degeneracy of the spectrum.
Let us consider a bound state made of two excitations from the su (2) sector of thestring sigma-model. In terms of the ZF creation operators we can think about thisstate as A † ( p ) A † ( p ) | (cid:105) , where the particle momenta p and p are complex. We find it convenient toparametrize the momenta as follows p = p − iq , p = p iq , Re q > , (7.1)where p is the real total momentum of the bound state. When q is real then p and p are complex conjugate to each other and the energy of the corresponding boundstate being the sum of the (complex) energies of individual particles is obviously real.Interestingly, as we will show below, there necessarily exists a branch of BPS bound– 45 –tates which corresponds to complex values of q with Re q >
0. Such solutions canbe reinterpreted as solutions parametrized by a new real variable q : q → Re q andfor which the real parts of p and p are not anymore equal to each other. Of course,one has to check that the energy of these solutions is real.The first equation in the set of the Bethe equations [6] takes the form e i ( p/ q ) L e Re q L = e iP K I (cid:89) l =2 x +1 − x − l x − − x + l − x +1 x − l − x − x + l σ l , (7.2)where P = p + p + · · · + p K I and L = J + K I with J being one of the global chargescorresponding to the isometries of the five-sphere.We see that for large L the l.h.s. is exponentially divergent. Then, there shouldexist a root p such that for Re q > (cid:0) x − − x +2 (cid:1) (cid:16) − x − x +2 (cid:17) ∼ e − Re q L . (7.3)In the infinite L limit eq.(7.3) becomes (cid:0) x − − x +2 (cid:1) (cid:16) − x − x +2 (cid:17) = 0 , (7.4)which is equivalent to x − − x +2 = 0 or 1 − x − x +2 = 0 . (7.5)The first equation x − − x +2 = 0 (7.6)implies that the central charges corresponding to the two-particle bound state satu-rate the BPS condition [50] H = 2 + 4 g sin p . (7.7)On the contrary, solutions of the second equation in (7.5) do not saturate the BPSbound, and as was argued in [52], this pole of the S-matrix does not correspond to abound state.It is easy to see that equation (7.6) is equivalent to vanishing the following fourthorder polynomial in the variable t = cos p e q ( t − e q )(1 − e q t ) + g ( t − − e q t + e q ) = 0 . (7.8) We assume here and in what follows that the dressing factor σ is non-singular on solutions ofthe bound state equation. For any p there are two solutions for x − and, therefore, for x + = e ip x − . The fourth orderpolynomial is universal and it does not depend on which solution for x − we take. – 46 –he equation has four solutions which can be cast to the following simple form e q = (cid:16)(cid:113) g sin p ± (cid:17) (cid:16) cos p (cid:113) g sin p ± (cid:113) cos p − g sin p (cid:17) g sin p , (7.9)where any choice of the ± sign is possible.Analysis of eq.(7.9) immediately shows that solutions corresponding to real val-ues of q exist if and only if the total momentum p does not exceed a critical value p cr determined by sin p cr g (cid:16)(cid:112) g − (cid:17) . (7.10)For any given p obeying | p | < p cr equation (7.8) has four real roots for q , two of themare positive and the other two are negative. According to our assumption Re q > They are given by the formula e q ± = (cid:16)(cid:113) g sin p + 1 (cid:17) (cid:16) cos p (cid:113) g sin p ± (cid:113) cos p − g sin p (cid:17) g sin p . (7.11)Various expansions of eq.(7.11) for small and large values of g can be found in Ap-pendix 9.4.It turns out that from the two positive roots only the smaller one, q − , falls insidethe region confined by the curves | x ± | = 1. We therefore arrive at the conclusionthat inside the region | x ± | > p and q , and itexists if and only if | p | < p cr , ≤ q < log 2 g + (cid:113) (cid:112) g − (cid:112) g − . (7.12)The second solution with q = q + lies outside the region with | x ± | > − ω / i < Im( z ) < ω / i ; the latter maps onto the complex p -plane,see section 4. Both solutions have the same values of all global conserved charges Q r = q r ( z ) + q r ( z ) = ir − (cid:2) ( x +1 ) − r − ( x − ) − r (cid:3) because x +1 and x − are the same onboth solutions.We see that if we choose the physical region to be the one with | x ± | > | p | < p cr . This region, however, does not coverthe whole complex p -plane. One the other hand, if the physical region is the half ofthe torus corresponding to the p -plane, then there are two solutions with the same The solutions with negative q correspond to bound states of anti-particles with negative energy. The energy of the corresponding bound state is
E < (cid:113) (cid:112) g + 2 . – 47 –nergy and other conserved charges. Finally, if one considers solutions on the z -torusthen there are four solutions but only two of them have positive energy.Continuing above the critical value, | p | > p cr , two solutions (7.11) acquire imag-inary parts and become complex-conjugate to each other, or, equivalently, the realparts of p and p become different. Thus, we see that the BPS bound states naturallysplit into two families depending on whether the total momentum is below (the firstfamily) or above (the second family) the critical value p cr . The two complex conjugate roots give two different solutions beyond criticality: p ± = p ± Im q − i Re q , p ± = p ∓ Im q + i Re q , Re q > . (7.13)We can choose either ( p +1 , p +2 ) or ( p − , p − ) as a possible solution of the BPS condition(7.6). Note that the second solution is the complex conjugate of the first one. A re-markable fact to be proven below is that both solutions lie precisely on the boundaryof the region defined by the curves | x ± | = 1.Now if we adopt the physical region (sheet) to be | x ± | > | x ± | = 1, then it should contain only one solution from the second BPS family.Indeed, we do not expect the doubling of the number of BPS bound states movingbeyond the critical point. The second solution can be then naturally interpreted aslying on the boundary of another (unphysical) sheet joint to the physical one throughthe cut. It is unclear however what is the precise origin for such an asymmetry. Apossible explanation would be the absence of parity invariance of the string sigma-model but a concrete implication of this fact is unknown to us.To visualize the singularities of the string S-matrix and also to verify that energyis real for the second BPS family, it is instructive to analyze eqs.(7.5) in terms ofthe generalized rapidity variables z and z associated to the first and the secondparticles, respectively. It is not hard to see that the first family of the BPS statescorresponds to imposing the reality condition z ∗ = z . In this case, eqs.(7.5) areequivalent to Im( x − ) = 0 or | x − | = 1 , (7.14)where the first equation defines the first BPS family. Solving the bound state equa-tion for z , one gets a curve in the torus. The part of the curve that lies insidethe region | x ± | > B OC , and thecorresponding momentum p has Im( p ) = − q − . The variable z = z ∗ of the sec-ond particle runs along another (conjugate) green curve B OC , which can be alsoviewed as describing solutions of the equation Im( x +2 ) = 0 for z . The dashed curveson Fig.5a, which are outside the region | x ± | >
1, represent solutions of the equa-tions Im( x − ) = Im( x +2 ) = 0 for z , z corresponding to the momentum p withIm( p ) = − q + . – 48 – DCBA D CB A
A O OA B C DA B C D
12 1 1 12 2 21 1 112 22 2
B C DA B C D b) c)a)
Figure 5:
Two-particle bound states of string theory. Figure a) describes the first BPSfamily corresponding to p < p cr . The green curves are Im( x − ) = 0 for Im( z ) < x + ) = 0 for Im( z ) >
0. For any p < p cr there are two solutions: the first one isrepresented by the continuous curves B OC (1st particle) and B OC (2nd particle), thesecond one corresponds to the dashed curves A B ∪ C D (1st particle) and A B ∪ C D (2nd particle). Figure b) describes the second BPS family corresponding to p > p cr .Again, for any p > p cr there are two solutions B C ∪ A B ∪ C D and B C ∪ A B ∪ C D . Figure c) represents one of the four possibilities to smoothly connect solutions fromthe first and the second BPS families. Here the variable z of the 1st particle is on thecurve A B OC D . When z runs along the curve from A to D the real part of themomentum Re( p ) increases monotonically from − π to π . At the same time, the variable z corresponding to the 2nd particle encloses the curve A B OC D . To describe the second family of the BPS states corresponding to the complexvalues of q one has to take z = − z ∗ + ω ω . (7.15)In this case x + ( z ) = x + (cid:16) − z ∗ + ω ω (cid:17) = 1 x + ( z ∗ ) = 1[ x − ( z )] ∗ , (7.16)where we have used the properties of Jacobi elliptic functions under the shifts byquarter-periods. Hence, due to the BPS equation x − = x +2 , the points z and z lieon the curves | x − | = 1 and | x + | = 1, respectively.As was discussed above, there are two different ways to choose the second BPSfamily which is equivalent to deciding what is the physical sheet. Consider the point– 49 – corresponding to the first particle, Fig.5c. When it moves along the curve B OC corresponding to the first BPS family and reaches, e.g., the point C then there aretwo possibilities to continue its path along the curve | x − | = 1: either one movesalong C D or along C B . In the case when z moves along the curve C D , thesecond point z follows the path C D . In the opposite situation, when z movesalong C B , the point z follows C B . Similar discussion applies to continuing thefirst BPS family beyond B . Obviously, for the second family z and z are notcomplex conjugate anymore, rather they obey the relation (7.15). The bound stateenergy H = ig ( x − − x +1 ) − z -plane z cr = ± ω ± ω x − ) = 0 and | x − | = 1 or Im( x +2 ) = 0 and | x +2 | = 1 aresimultaneously satisfied. These are the critical points where two BPS families meet.The most transparent description of the bound states is achieved in terms of therapidity variable u introduced in section 4, rather than in terms of momentum p orthe variable z . Indeed, in terms of u eq.(7.3) becomes (cid:0) x − − x +2 (cid:1) (cid:16) − x − x +2 (cid:17) = u − u − ig = 0 , (7.18)i.e. the rapidity variables u and u of the first and the second particle lie on astraight line running parallel to the imaginary axis. Moreover, for the first BPSfamily the variables u , are subject to the following conjugation rule u ∗ = u which,together with eq.(7.18) allows one to conclude that u , = u ± ig , u ∈ R . (7.19)This is a typical pattern of “Bethe string”. One can further see that for the firstBPS family corresponding to p ≤ p cr the variable u is restricted to satisfy | u | ≥ , u , cr = ± ig , (7.20)where u , cr is a critical value of rapidity u for which the first BPS family ceased toexist. Under the map to the u -plane the four critical points z cr are mapped to thefour branch points on the u -plane (see Fig.3 in section 4) u cr = ± ± ig . (7.21)– 50 –et us now turn to the second BPS family. First, by using eq.(4.5) and theproperties of the elliptic functions, we derive u ∗ = x − (cid:16) − z + ω ω (cid:17) + 1 x − (cid:16) − z + ω + ω (cid:17) + ig = u . (7.22)We see that for both families of BPS states the conjugation rule for u ’s is the oneand the same. By this reason, a solution to the BPS condition is always representedby the Bethe string (7.19). However, one finds that for the second family a solutionexists for | u | ≤ u -plane both families of BPS states admit auniform description in terms of the Bethe string with u running over the whole realline. The consideration of the two-particle bound states can be easily generalized to the M -particle case. The corresponding set of bound state equations reads [50] x − j − x + j +1 = 0 , j = 1 , . . . , M . (7.23)The total momentum of a state satisfying these equations is given by e ip = x +1 x − x +2 x − · · · x + M x − M = x +1 x − M , and the energy of the state is H M = M (cid:88) i =1 (cid:0) − − igx + i + igx − i (cid:1) = − M − igx +1 + igx − M . (7.24)Both the energy and momentum depend on the values of x +1 and x − M only. Sincethe energy is real, x − M must be the complex conjugate of x +1 : ( x − M ) ∗ = x +1 . In fact,a simple but important observation is that any global conserved charge of a stateobeying (7.23) depends only on x +1 and x − M : Q r = M (cid:88) i =1 q r ( z i ) = M (cid:88) i =1 ir − (cid:2) ( x + i ) − r − ( x − i ) − r (cid:3) = ir − (cid:2) ( x +1 ) − r − ( x − M ) − r (cid:3) . Another important consequence of eqs.(7.23) is that the coordinates x +1 and x − M satisfy the following quadratic constraint x +1 + 1 x +1 − x − M − x − M = 2 Mg i . (7.25)– 51 –his is the same constraint as the one satisfied by x ± (4.3) with g → g/M , and weget immediately the dependence of x +1 and x − M on the total real momentum p x + = e i p g sin p (cid:16) M + (cid:114) M + 4 g sin p (cid:17) ,x − = e − i p g sin p (cid:16) M + (cid:114) M + 4 g sin p (cid:17) , (7.26)and, using (7.24), the BPS energy formula H M = M + 4 g sin p . Moreover, we see that the set of global conserved charges Q r is the same for anysolution of (7.23) with a given total momentum p .It is also easy to see that the number of different solutions with a real momentum p and positive energy is equal to 2 M − because for a given x + there are two different x − solving the constraint (4.3), see the diagram below for M = 4 x +1 −→ x − = x +2 −→ x − = x +3 −→ (cid:26) x − = x +4 x − = x +4 x − = x +3 −→ (cid:26) x − = x +4 x − = x +4 x − = x +2 −→ x − = x +3 −→ (cid:26) x − = x +4 x − = x +4 x − = x +3 −→ (cid:26) x − = x +4 x − = x +4 −→ x − To have all these solutions one would have to allow the parameters z i of the particlesto be anywhere on the z -torus, in particular, some of them would be in the anti-particle region with | x ± | < | x ± | > x + only one solution for x − satisfies the condition | x − | ≥
1. For M even it is alsonecessary to specify what parts of the boundaries | x ± | = 1 belong to the regionbecause if the momentum of a bound state exceeds a critical, g − and M -dependent,value then there are several solutions of the bound state equations with | x − M/ | = | x + M/ | = 1.Finally, if the parameters z i of the particles belong to the half of the toruscorresponding to the complex p -plane, then one can show that for any M there aretwo solutions of the bound state equations. In general for a given momentum p there are two solutions of the constraint (7.25), and therecould be any sign in front of the square root in (7.26). The positive sign guarantees the positivityof the energy. – 52 –ust as for the case of two-particle bound states, the simplest description of M -particle bound states is provided by the u -plane where a solution is given by theBethe string u j = u + ( M − j + 1) ig , j = 1 , . . . , M . (7.27)We can choose one and the same map of the u -plane with the cuts described insection 4 onto the region of the z -torus with | x ± | > p there is just a single M -particle boundstate that falls inside the physical region. Its structural description however becomesrather involved. It is of interest to analyze finite-size corrections to the energy of the BPS boundstates, and to see what restrictions on the dressing factor could be derived from thecondition that the energy corrections are real. To this end, we consider two-particlestates in the su (2) sector described by the following two equations, see (7.2) (cid:18) x +1 x − (cid:19) J = Σ x +1 − x − x − − x +2 − x +1 x − − x − x +2 , (cid:18) x +1 x +2 x − x − (cid:19) J = 1 , (7.28)where Σ = x − x +2 x +1 x − σ is the unitary factor that appeared in the splitting (5.25) of thescalar factor, and J is one of the global charges corresponding to the isometries ofthe five-sphere. The variables x ± i also satisfy the constraint (4.3). These equationsare supposed to be valid asymptotically for large values of J , and have to be modifiedfor finite J .We will analyze these equations for large values of J in the vicinity of a boundstate satisfying the bound state equation x − = x +2 and having a fixed total mo-mentum p = πmJ where m is an integer. The quantization condition for the totalmomentum follows from the second equation in (7.28).Let x ± i denote the values of x ± i satisfying the bound state equation and thesecond equation in (7.28). Then, (cid:16) x − x +1 (cid:17) J ∼ e − qJ exponentially decreases at large J ,and we can look for a solution of the form x ± i = x ± i (cid:32) (cid:18) x − x +1 (cid:19) J y ± i (cid:33) . Expanding the equations (7.28) and the constraint (4.3) in powers of y ± i , we find a– 53 –ystem of linear equations for y ± i . The solution of the system is given below y − = 4Σ ( ig ( x − − x +2 ) + x − x +2 ) (cid:0) i x +1 x +2 + g (cid:0) x +1 (cid:0) ( x +2 ) + 1 (cid:1) − x +2 (cid:1)(cid:1) g ( g x − − ( g + 2 i x − ) x +1 ) (cid:0) ( x +2 ) − (cid:1) y +1 = 4Σ x +1 ( ig ( x − − x +2 ) + x − x +2 ) g ( g x − − ( g + 2 i x − ) x +1 ) (cid:0) ( x +2 ) − (cid:1) y − = − i Σ x − ( g ( x +1 − x +2 ) + i x +1 x +2 ) g (( g + 2 i x − ) x +1 − g x − ) (cid:0) ( x +2 ) − (cid:1) y +2 = 4Σ ( g ( x +1 − x +2 ) + i x +1 x +2 ) (cid:0) g x − ( x +2 ) − g + i x − ) x +2 + g x − (cid:1) g ( − ig ( x − − x +1 ) − x − x +1 ) (cid:0) ( x +2 ) − (cid:1) , where Σ is evaluated on the solution to the bound state equation. The leadingcorrection to the energy of the state is easily found by expanding E = E + E , E i = 1 + igx + i − igx − i = − − igx + i + igx − i . (7.29)By using E i = 1 + igx + i − igx − i , we obtain δE = (cid:18) x − x +1 (cid:19) J Σ i ( x − (2 x +1 − x +2 ) − x +1 x +2 )( g ( x − − x +1 ) − i x − x +1 ) (cid:0) ( x +2 ) − (cid:1) . (7.30)On the other hand by using E i = − − igx + i + igx − i , we get δE = (cid:18) x − x +1 (cid:19) J Σ i x − x +1 x +2 ( x − + x +1 − x +2 )( g ( x − − x +1 ) − i x − x +1 ) (cid:0) ( x +2 ) − (cid:1) . (7.31)Even though the expressions look different they coincide on solutions to the boundstate equation. In what follows we will be using the simpler eq.(7.31). Note also thatthe perturbation theory breaks down at p = p cr . Due to the quantization conditionfor the momentum p it may happen only at special values of the coupling constant g depending on m/J .It is clear that the energy correction cannot be real for any choice of the dressingfactor Σ . The imaginary part of the correction depends on the branch of the boundstate under consideration.In the first case with Im( x − ) = Im( x +2 ) = 0 and the total momentum smallerthan the critical value (7.10), the parameters x ± i satisfy the complex conjugation rule( x ± ) ∗ = x ∓ , and we get δE − δE ∗ = (cid:32)(cid:18) x − x +1 (cid:19) J Σ − (cid:18) x +2 x − (cid:19) J Σ ∗ (cid:33) i x − x +1 x +2 ( x − + x +1 − x +2 )( g ( x − − x +1 ) − i x − x +1 ) (cid:0) ( x +2 ) − (cid:1) . – 54 –aking into account that (cid:16) x − x +1 (cid:17) J = (cid:16) x +2 x − (cid:17) J , we conclude that in this case the correc-tion is real only if the dressing factor is real Σ = Σ ∗ . This property of the dressingfactor can be easily shown by using the representation (5.22) for the dressing phase.In the second case with | x − | = | x +2 | = 1 and the total momentum exceedingthe critical value (7.10), the parameters x ± i satisfy the complex conjugation rule( x +1 ) ∗ = x − , ( x − ) ∗ = 1 / x +2 , and we obtain δE − δE ∗ == 4 x − x +1 (cid:0) Σ ∗ ( x +2 x − ) − J (2 − ( x − + x +1 ) x +2 ) − Σ ( x +1 ) − J ( x +2 ) J +1 ( x − + x +1 − x +2 ) (cid:1) ( ig ( x − − x +1 ) + 2 x − x +1 ) (cid:0) ( x +2 ) − (cid:1) . We see that the imaginary part of the correction would vanish only ifΣ ∗ = Σ ( x +2 ) J +1 ( x − + x +1 − x +2 )(2 − ( x − + x +1 ) x +2 ) . Since the last equation depends on J and on a particular bound state solution, itcannot be satisfied for any choice of the dressing factor. The complex energy of thestate would mean that the Hamiltonian of the model is not hermitian for finite J .One might conclude from this result that the S-matrix poles with | x − | = | x +2 | = 1are spurious and do not correspond to bound states, and, therefore, should be omit-ted. That would mean, however, that for any value of the total momentum the boundstates satisfying the equations Im( x − ) = Im( x +2 ) = 0 would disappear as soon as thecoupling constant g reaches a critical (momentum-dependent) value. This seems tocontradict to the statement that the bound states are BPS. We believe that sucha conclusion might be erroneous and the result above indicates, in fact, that theasymptotic Bethe ansatz cannot be used to analyze the finite-size corrections to theenergy of bound states with the total momentum exceeding the critical value (7.10).To show that this is indeed the case, let us recall that, as was shown in [38],at large values of the string tension g and the charge J the dispersion relation re-ceives finite-size corrections of the order e − J/ ( g sin p/ . On the other hand, the energycorrection we computed above is of the order e − qJ where q is the imaginary partof the momentum p . It depends on the total momentum p and the string tension g . By using eq.(9.35), it is not difficult to determine the large g dependence of themomenta p and p of a bound state p = cos p g sin p − ig sin p + O ( 1 g ) , p = p − cos p g sin p + ig sin p + O ( 1 g ) . (7.32)The second solution of eq.(7.8) (with q >
0) is related to (7.32) as p → p ∗ , p → p ∗ that is one exchanges the real parts of momenta p i . A surprising result of thecomputation is that q is equal to g sin p , and, therefore, e − qJ is exactly equal to the– 55 –agnitude of the finite-size correction to the dispersion relation. That means thatcomputing the finite J correction to the energy of such a bound state one has totake into account the necessary modifications of the asymptotic Bethe ansatz. As aresult of these modifications, one should be able to get a real finite-size correctionto the energy of a bound state carrying momentum exceeding the critical value. Infact, this would be a non-trivial check of finite J “Bethe” equations.The analysis performed above raises the question if one can use the asymptoticBethe ansatz to compute the corrections to the energy of the bound states withmomenta smaller than the critical value. At large g we can again compare the valueof q with g sin p . If q is less than g sin p then the energy correction (7.31) is bigger thanthe correction due to finite J modifications of the asymptotic Bethe ansatz, and wecan trust (7.31). Since p cr = 2 / √ g at large values of g one should consider a boundstate with momentum p of the order 1 / √ g . The large g dependence of the momenta p and p of a bound state is easily found by using eq.(9.36) p = p − i ± (cid:113) − p g gp + · · · , p = p i ± (cid:113) − p g gp + · · · , (7.33)leading for p < p cr to the following two real solutions for qq ± = 2 1 ± (cid:113) − p g gp . (7.34)Comparing these values with g sin p ≈ gp , we see that q − < gp and q + > gp . Thus, theasymptotic Bethe ansatz can be used to analyze finite J corrections to the energy ofa bound state with momentum smaller than p cr for the bound state with q − only.Actually, the fact that the energy correction (7.31) to the bound state with q + is smaller than the corrections due to finite J modifications of the asymptotic Betheansatz raises a question if these solutions correspond to the actual bound states. Itmay happen that finite J Bethe equations would not have any solution that wouldreduce to the solution with q + in the limit J → ∞ .A similar analysis can also be performed for small values of g . Then we expectthat the finite J effects (in gauge theory they are due to the wrapping interactions)become important at order g J , and therefore we could trust the asymptotic Betheansatz and the energy correction (7.31) only if q < − g .The leading small g dependence of q of the bound state solutions with the mo-mentum smaller than p cr is given by eqs.(9.37), (9.38) q + = − g + · · · , q − = − log cos p · · · . (7.35)We see immediately that again only the solution with the smaller imaginary part ofthe momentum q − satisfies the necessary condition. The energy correction to the– 56 –tate with q + is of order g J that is exactly the order of wrapping interactions, andthe asymptotic Bethe ansatz again breaks down for the state.Finally, the leading small g dependence of q of the bound state solutions withthe momentum exceeding p cr is given by (9.39) q ± = − log g ± iα + · · · , (7.36)where α is related to the momentum p as follows p = π − g cos α . We see that the real part of q ± is smaller than − g , and therefore one couldconclude that one might use the asymptotic Bethe ansatz for the states in this regime.This, however, leads to the problem of the complex energy of these states discussedabove. As before the only resolution of the problem we see is the breakdown ofthe asymptotic Bethe ansatz. This would imply, however, that for these states thewrapping interactions become important already at the order g J . The fact that ingauge theory these states are not dual to gauge-invariant operators does not seemto be important for this conclusion. One could for example scatter a bound statecarrying momentum p = π which always exceeds the critical momentum p cr with anelementary one carrying momentum − π so that the total momentum would be zero,and such a state would be dual to a gauge-invariant operator. We would still expectthe finite J corrections to this state to be of the order g J . Another puzzling propertyof the states with p > p cr is that in the limit g → p cr = π , and cannot be seen in the perturbative gauge theory.
8. Bound states of the mirror model
Let us now consider in a similar fashion bound states of the mirror model. In thiscase one should consider mirror particles of type A † .We begin our consideration with two-particle bound states, and let the complexmomenta of the two particles be (cid:101) p = p − iq , (cid:101) p = p iq , Re q > , where p is the total momentum of the mirror bound state.The first equation in (6.6) takes the form e ipR/ e qR = σ x − − x +2 x +1 − x − − x +1 x − − x − x +2 , (8.1)where we set all auxiliary roots to 0. Assuming that the dressing factor does notvanish, we conclude that for Re q > R → ∞ the following boundstate equation should hold x +1 − x − = 0 . (8.2)– 57 –he second factor in the denominator of the Bethe equation (8.1) may also vanishbut the energy of the corresponding state does not satisfy the BPS condition. Weexpect that, just as a similar factor in the string theory, the pole due to this factordoes not correspond to a bound state.By using eqs. (2.15) which express x ± as functions of (cid:101) p , we find that eq.(8.2) isequivalent to − g q + t − q t (2 − t ) + q t = 0 , (8.3)where t ≡ p . This equation gives the following two solutions with a positive realpart of q : q = (cid:114) g t ± (cid:114) − t + g t = (cid:115) g p ± (cid:115) − p g p . (8.4)Solutions for q are real provided the expression under the square root is nonnegative,and this implies the following restriction on the total momentum of the bound state | p | ≤ p cr ≡ √ (cid:113) − (cid:112) g . (8.5)For an exact inequality we have two positive solutions q ± , and when the bound onthe momentum is saturated the solution is obviously unique q − < q cr < q + , q cr = 1 √ (cid:113) (cid:112) g . (8.6)It is interesting to notice that the dependence of q ± on the momentum of the boundstate is smoother at p = 0 than the one for string theory bound states. We see fromeq.(8.4) that q − reaches its minimum, and q + reaches its maximum at p = 0 q min − = (cid:112) g − g , q max+ = (cid:112) g + g , p = 0 . (8.7)In string theory the corresponding values are 0 and ∞ .To find what curves in the z -torus correspond to the two solutions with real q we take into account that in this case (cid:101) p ∗ = (cid:101) p , and the reality condition for x ± inthe mirror theory is (cid:0) x ± (cid:1) ∗ = 1 /x ∓ . Thus the bound state equation (8.2) reduces tothe following equivalent conditions | x +1 | = 1 ⇐⇒ | x − | = 1 , being represented by the two curves in the z -torus that bound the yellow regionwith | x + | < , | x − | > The energy of the bound state is (cid:101) E cr = 2 arcsinh √ g (cid:113) (cid:112) g . – 58 –he horizontal line passing through the point z = ω . Let us recall that hermitianconjugation in the mirror theory is defined with respect to this line, see section 4.It is not difficult to check that the parts of the curves | x +1 | = 1 , | x − | = 1 that areinside the region Im( x ± ) < q − of eq. (8.2). Theother parts of the curves correspond to the second solution with q = q + , see Figure1. Just as it was for string theory bound states, both solutions have the same valuesof all global conserved charges Q r = q r ( z ) + q r ( z ) = ir − (cid:2) − ( x − ) − r + ( x +2 ) − r (cid:3) .We see that if we want to have only one bound state with | p | < p cr in a physicalregion, then we should choose the physical region to be the one with Im( x ± ) < | x ± | = 1 as it is for string theory. We will seein a moment that the region Im( x ± ) ≤ | p | > p cr described by the solutions with complex q .Above the critical value, | p | > p cr , the two solutions (8.4) acquire imaginaryparts and become complex conjugate to each other. It is convenient to denote thecorresponding solutions as follows q ± = (cid:115) g p ± i p (cid:115) − g p (4 + p ) . (8.8)We see that the real part of q ± is a decreasing function of p , and its minimum valueis 1. On the contrary the imaginary part of q ± is an increasing function of p and itbehaves as ± p/ p . As a result, the two complex momenta (cid:101) p ± = p ± Im q − i Re q , (cid:101) p ± = p ∓ Im q + i Re q , Re q > , have the following large p behavior (cid:101) p = p − i , (cid:101) p = i ; (cid:101) p − = − i , (cid:101) p − = p + i . A remarkable fact is that both solutions lie precisely on the boundary of theregion Im( x ± ) ≤
0. To see this we notice that, just as it was for string theory boundstates, the coordinates z and z of the solutions with the complex values of q arerelated by eq.(7.15) z = − z ∗ + ω ω . (8.9)Then, one can easily show that x − ( z ) = x − (cid:16) − z ∗ + ω ω (cid:17) = x − ( z ∗ ) = [ x + ( z )] ∗ , and, therefore, the bound state equation x +1 = x − is equivalent to Im( x + ( z )) =Im( x − ( z )) = 0. We plot the corresponding curves in Figure 6.– 59 – B C DA B C D A B C DO B CA BB C DC a) b) c)
A D
Figure 6:
Bound states of the mirror theory. Figure a) represents the first BPS family: forany p with | p | < p cr there are two solutions corresponding to q − (the curves B C for the 1stparticle and B C for the 2nd one, respectively) and to q + (the curves A B ∪ C D for the1st particle and A B ∪ C D for the 2nd one, respectively). Figure b) represents the secondBPS family which is also doubly degenerate: it is given by either A B ∪ C D ∪ B D orby B C ∪ A B ∪ D C . Figure c) corresponds to one of the four possibilities to connectthe first and the second BPS family: when the variable z of the 1st particle runs alongthe curve A B C D the real part of its momentum increases from −∞ to + ∞ . At thesame time, the variable z of the 2nd particle encloses the curve A B C D . Thus, we have shown that these solutions lie on the boundary of the regionIm( x ± ) ≤
0, and, therefore, the region contains bound states with any value of thetotal momentum and could be considered as the physical one for the mirror model.It is also necessary to specify what part of the boundary of the region Im( x ± ) ≤ u -plane where the bound state equation reduces to x +1 − x − = 0 = ⇒ u − u = 2 ig . As was discussed in section 4, eqs. | x +1 | = | x − | = 1 describing a bound state withthe momentum not exceeding the critical value p cr and with a real q give a Bethestring solution with the real part of u lying in the interval [ − , u , = u ∓ ig , − ≤ u ≤ . On the other hand, values of u lying outside the interval [ − ,
2] correspond tosolutions of eqs. Im( x +1 ) = Im( x − ) = 0. The momentum (cid:101) p = (cid:101) p ( u ) is a multi-valued– 60 –unction of u , and one should choose a proper branch of the function to get the rightvalues of the momenta (cid:101) p , (cid:101) p of the bound state. This fixes the cuts in the u -planewhich run from ±∞ to ± ∓ ig , and also the boundaries of the region Im( x ± ) ≤ z -plane which is mapped onto the u -plane with these cuts.The discussion of bound states of M particles of type A † basically repeats theone in section 7. One finds a system of equations x + j − x − j +1 = 0 , j = 1 , . . . , M − . (8.10)In terms of the variable u the Bethe string solution reads as u j = u − ( M − j + 1) ig , j = 1 , . . . , M , (8.11)and has the energy E = log x − x + M = 2 arcsinh 12 g (cid:112) M + (cid:101) p , (8.12)where (cid:101) p = (cid:101) p + . . . + (cid:101) p M is a total (real) momentum of the bound state.Depending on a choice of the physical region, the system (8.10) could have one,two or 2 M − solutions. All solutions have the same global conserved charged. Theybehave, however, differently for very large but finite values of R , and the solutionswhich are not in the region Im( x ± ) < R correction to the energy, or thecorrection would exceed the correction due to finite R modifications of the Betheequations thus making the asymptotic Bethe ansatz inapplicable. Acknowledgements
We thank Marija Zamaklar for collaboration at the early stage of the project. Weare grateful to N. Beisert, N. Dorey, P. Dorey, D. Hofman, R. Janik, M. de Leeuw, J.Maldacena, R. Roiban and M. Staudacher for valuable discussions. The work of G. A.was supported in part by the RFBI grant N05-01-00758, by the grant NSh-672.2006.1,by NWO grant 047017015 and by the INTAS contract 03-51-6346. The work ofS.F. was supported in part by the Science Foundation Ireland under Grant No.07/RFP/PHYF104 and by a one-month Max-Planck-Institut f¨ur GravitationsphysikAlbert-Einstein-Institut grant. The work of G. A. and S. F. was supported in partby the EU-RTN network
Constituents, Fundamental Forces and Symmetries of theUniverse (MRTN-CT-2004-512194). – 61 – . Appendices
The Lagrangian density of the gauge-fixed sigma-model in the generalized a -gauge[56, 57] can be written in the following form [26] L = − (cid:112) G ϕϕ G tt (1 − a ) G ϕϕ − a G tt √W + a G tt + (1 − a ) G ϕϕ (1 − a ) G ϕϕ − a G tt , (9.1)where W ≡ − (1 − a ) G ϕϕ − a G tt (cid:104)(cid:16) G ϕϕ G tt (cid:17) ∂ α X · ∂ α X − (cid:16) − G ϕϕ G tt (cid:17) (cid:16) ˙ X · ˙ X + X (cid:48) · X (cid:48) (cid:17) (cid:105) + ((1 − a ) G ϕϕ − a G tt ) G ϕϕ G tt (cid:0) ( ∂ α X · ∂ α X ) − ( ∂ α X · ∂ β X ) (cid:1) . Here X = ( y i , z i ), where y i , i = 1 , . . . , z i are fields parametrizing four directions in AdS . The fields X in the La-grangian above are contracted with the help of the metric ds = − G tt dt + G zz dz + G ϕϕ dϕ + G yy dy . Here G tt = (cid:18) z − z (cid:19) , G zz = 1 (cid:0) − z (cid:1) , G ϕϕ = (cid:18) − y y (cid:19) , G yy = 1 (cid:0) y (cid:1) , where we had used the notation z ≡ z i z i and y ≡ y i y i . Here we describe the properties of the “one-loop” S-matrix which is obtained fromthe S-matrix (3.24) upon taking the limit g →
0. We will work in the ellipticparametrization discussed in section 4.1. According to eq.(4.18), in this limit Jacobielliptic functions degenerate into the corresponding trigonometric ones and we findthe following trigonometric S-matrix: – 62 – ( z , z ) = e − i ( z − z ) cot z − cot z + 2 i cot z − cot z − i (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) − e − i ( z − z ) i cot z − cot z − i (cid:16) E ⊗ E + E ⊗ E − E ⊗ E − E ⊗ E (cid:17) − (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) − i cot z − cot z − i (cid:16) E ⊗ E + E ⊗ E − E ⊗ E − E ⊗ E (cid:17) + e iz cot z − cot z cot z − cot z − i (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + e − iz cot z − cot z cot z − cot z − i (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + e − i ( z − z ) i cot z − cot z − i (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) + e − i ( z − z ) i cot z − cot z − i (cid:16) E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E (cid:17) . (9.2) The relations between the z -variable, momentum and the rescaled rapidity u → gu transform in the limit g → p = 2 z , u = cot z = cot p . (9.3)Surprisingly enough, this S-matrix cannot be written in the difference form, i.e. asa function of one variable being the difference of a properly introduced spectralparameter. By construction, this S-matrix satisfies the usual Yang-Baxter equation S ( z , z ) S ( z , z ) S ( z , z ) = S ( z , z ) S ( z , z ) S ( z , z ) , (9.4)as one can also verify by direct calculation. On the other hand, at one-loop thereis another “canonical” S-matrix which is a linear combination of the graded identityand the usual permutation: S can12 = u − u u − u − i I g + 2 iu − u − i P . (9.5)This S-matrix satisfies the same Yang-Baxter equation (9.4).The results of [47] imply that the two one-loop S-matrices, (9.2) and (9.5) arerelated through the following transformation S can ( z , z ) = U ( z ) (cid:104) V ( z ) V ( z ) S ( z , z ) V − ( z ) V − ( z ) (cid:105) U − ( z ) , (9.6)where we have introduced the diagonal matrices U ( z ) = diag( e iz , e iz , , , (9.7) V ( z ) = diag( e i z , e i z , e − i z , e − i z ) . (9.8)– 63 –he transformation by V is just a gauge transformation which always preserves theYang-Baxter equation. On the other hand, transformation by U is a twist, thatgenerically transforms the usual Yang-Baxter equation into the twisted one and viceversa [47]. Indeed, S can12 is nothing else but the one-loop limit of the spin chain S-matrix [9]; the latter obeys the twisted Yang-Baxter equation [47]. Note also thatthe twist U does not belong to the symmetry group SU(2) × SU(2) of the stringS-matrix.To understand why at one loop the Yang-Baxter equation is preserved underthe twisting, we first write the Yang-Baxter equation for S can by using the relation (9.5) U ( z ) S U − ( z ) U ( z ) S U − ( z ) U ( z ) S U − ( z ) == U ( z ) S U − ( z ) U ( z ) S U − ( z ) U ( z ) S U − ( z ) , (9.9)which can be reshuffled as follows U ( z ) S U ( z ) U ( z ) S U − ( z ) U − ( z ) S U ( z ) == U ( z ) U ( z ) S U − ( z ) S U ( z ) S U − ( z ) U − ( z ) . (9.10)It is clear now that we will get the usual Yang-Baxter equation for S provided itobeys the following relation [ S, U ⊗ U ] = 0 , (9.11)where U is an arbitrary diagonal matrix . One can easily verify that both S-matrices,(9.2) and (9.5), do indeed satisfy this relation. At higher orders in g the relation(9.11) does not hold anymore. The corresponding “all-loop” S-matrix (3.24) satisfiesonly a weaker condition[ S, G ⊗ G ] = 0 , G ∈ SU(2) × SU(2) , (9.12)which is nothing else but the invariance condition for the string S-matrix. As aconsequence, the Yang-Baxter equation is preserved by the twist transformationonly at the one-loop order.As a final remark, we note that it would be interesting to understand how thederivation of the Hirota difference equations for the canonical S-matrix [72] could beextended to the “twisted” S-matrix (9.2). This might shed some light on constructionthe fusion procedure for the all-loop S-matrix (3.24). The gauge transformation by the matrix V decouples from the Yang-Baxter equation. – 64 – .3 BAE with nonperiodic fermions9.3.1 Bethe wave function and the periodicity conditions In any asymptotic domain Q with x Q (cid:28) x Q (cid:28) · · · (cid:28) x Q N where N ≡ K I and Q , . . . , Q N is a permutation of 1 , , . . . , N , the wave function of N particles withflavors i , i , . . . , i N can be written as a superposition of plane waves with momenta p > p > · · · > p N Ψ Q i ··· i N ( x , . . . , x N ) = (cid:88) P A P|Q i ··· i N e i p P · x Q , (9.13)where the sum runs over all permutations of the momenta p i . The scalar product p P · x Q is defined as p P · x Q ≡ (cid:80) Nk =1 p P k x Q k , and for any two permutations P and Q it satisfies p P · x Q = p PQ − · x I = (cid:80) Nk =1 p ( PQ− k x k where I is the trivial permutation.The amplitude A P|Q i ··· i N is related to the probability of finding the particle withthe flavor i k (the i k -th particle in what follows) carrying the momentum p ( PQ− k atthe position x k . That means that the index i k is attached to the coordinate x k . Asa result the wave function (9.13) should satisfy the following symmetry condition forany two indices k, m Ψ Q i ··· i k ··· i m ··· i N ( x , ..., x k , ..., x m , ..., x N ) == ( − ) (cid:15) ik (cid:15) im Ψ P km Q i ··· i m ··· i k ··· i N ( x , ..., x m , ..., x k , ..., x N ) , (9.14)where P km is the permutation of k and m , and (cid:15) i = 0 if the i -th particle is bosonand (cid:15) i = 1 if the i -th particle is fermion, that is one takes the minus sign if both the i k -th and i m -th particles are fermions, and the plus sign otherwise.In any two domains Q and Q the amplitudes A P|Q i ··· i N and A P|Q i ··· i N of the sameplane wave (that is p P · x Q = p P · x Q ) are related through the S-matrix. The relationcan be easily found by representing the amplitudes as the following products of theZF operators A P|Q i ··· i N ∼ ± A † i Q ( p P ) · · · A † i Q N ( p P N ) , (9.15)and then by using the ZF algebra to relate the amplitudes in the domains Q and Q . The + / − sign in this formula is related to the even/odd number of permutationsof fermions by the permutation Q . To understand the origin of this formula let usrecall that the indices i k are attached to the coordinates x k which explains the orderof A † i Q k . The dependence of A † i Q k of the momentum follows from the coupling p P k x Q k in the exponential of the wave function (9.13).To proceed it is convenient to use matrix notations. We introduce the simplepermutation P = E ij ⊗ E ji which permutes the spaces V and V but does nottouch the momenta p i so that S = P S ( p , p ) P , the graded permutation P g =– 65 – − (cid:15) i (cid:15) j E ij ⊗ E ji , and the graded two-particle S-matrix S g which can be written in theform S g = I g S where I g = ( − (cid:15) i (cid:15) j E ii ⊗ E jj is the graded identity. We also define S g = P S ( p , p ) P I g = P S ( p , p ) P g so that the unitarity condition S g S g = I is fulfilled.Then we multiply the wave function (9.13) and (9.15) by the tensor product of N rows E i ⊗ E i ⊗ · · · ⊗ E i N ≡ (cid:0) E E · · · E N (cid:1) i i ··· i N , and (9.13) takes the formΨ Q ( x , . . . , x N ) = (cid:88) P A P|Q e i p P · x Q , (9.16)where A P|Q ∼ A †Q ( p P ) · · · A †Q N ( p P N ) I g Q , and the index Q k refers to the location of the row E Q k , and I g Q is the product ofgraded identities which can be found by representing the permutation Q as a productof Y simple permutations P km : Q = P k m · · · P k Y m Y , and then I g Q = I gk m · · · I gk Y m Y .Now the ZF algebra can be used to express the amplitudes A P|Q with Q ≡ P − Q fixed in terms of the amplitude A I|Q . In particular the amplitudes A P|P are ex-pressed in terms of the incoming amplitude A I|I ∼ A † ( p ) · · · A † N ( p N ). The corre-sponding terms in the wave function can be used to derive the periodicity conditions.To find the relations, it is convenient to represent A P|Q ∼ A †Q ( p P ) · · · A †Q N ( p P N ) I g Q = A †P ( p P ) · · · A †P N ( p P N )( QP − ) ··· N I g Q , A I|Q ∼ A † ( p ) · · · A † N ( p N ) · ( Q ) ··· N I g Q = A † ( p ) · · · A † N ( p N )( P − Q ) ··· N I g Q , where ( QP − ) ··· N is the permutation matrix that acting on the tensor product E P ⊗· · · ⊗ E P N produces E Q ⊗ · · · ⊗ E Q N . Now we use the ZF algebra to find the relation A P|Q = A † · · · A † N · S P ···P N ( p P , . . . , p P N )( QP − ) ··· N I g Q = A I|Q I g Q ( Q − P ) ··· N S P ···P N ( p P , . . . , p P N )( QP − ) ··· N I g Q , where S P ···P N ( p P , . . . , p P N ) is the multi-particle S-matrix.In particular, we find that A P|P = A I|I S P ···P N ( p P , . . . , p P N ) I g P ≡ A I|I S g P ···P N ( p P , . . . , p P N ) , where S g P ···P N ( p P , . . . , p P N ) is the graded multi-particle S-matrix. Note that it isnot a product of two-particle graded S-matrices.This formula can be used to find the set of periodicity conditions. We write thepart of the wave function with the plane wave e ip k x k Ψ( x , . . . , x N ) = (cid:88) P A P|P e i p P · x P θ ( x P < . . . < x P N )= A I|I (cid:88) P S g P ···P N ( p P , . . . , p P N ) e i p P · x P θ ( x P < . . . < x P N ) . (9.17)– 66 –he periodicity conditions readΨ( x , . . . , x k = 0 , . . . , x N ) = Ψ( x , . . . , x k = L, . . . , x N ) W k , where the diagonal matrix W is equal to the identity matrix if the fermions areperiodic, and it is W = ( − (cid:15) i E ii if the fermions are anti-periodic. For the su (2 | W = Σ = diag(1 , , − , −
1) for anti-periodic fermions.By using eq.(9.17), we getΨ( x , . . . , x k = 0 , . . . , x N ) = A I|I (cid:88) P : P = p k S gk P ···P N ( p k , p P , . . . , p P N ) e i p P · x P θ ( x P < . . . < x P N ) , Ψ( x , . . . , x k = L, . . . , x N ) = e ip k L A I|I (cid:88) P : P N = p k S g P ···P N − k ( p P , . . . , p P N − , p k ) W k e i p P · x P θ ( x P < . . . < x P N − ) , Comparing the terms, we obtain A I|I (cid:16) S gk P ···P N ( p k , p P , . . . , p P N ) − e ip k L S g P ···P N k ( p P , . . . , p P N , p k ) W k (cid:17) = 0 . (9.18)To compute the S-matrices, we use their definitions A † k ( p k ) A †P ( p P ) · · · A †P N ( p P N ) = A † · · · A † N · S k P ···P N ( p k , p P , . . . , p P N ) ,A †P ( p P ) · · · A †P N ( p P N ) A † k ( p k ) = A † · · · A † N · S P ···P N k ( p P , . . . , p P N , p k ) . Then we use the ZF algebra to order the product A †P ( p P ) · · · A †P N ( p P N ) A †P ( p P ) · · · A †P N ( p P N ) = A † · · · A † k − A † k +1 · · · A † N · S P ···P N ( p P , . . . , p P N ) , and finally we get the multi-particle S-matrices A † k A †P ( p P ) · · · A †P N ( p P N ) = A † · · · A † N · S k,k − S k,k − · · · S k · S P ···P N A †P ( p P ) · · · A †P N ( p P N ) A † k = A † · · · A † N · S k +1 ,k S k +2 ,k · · · S Nk · S P ···P N . Thus, for S P ···P N = 1 eq.(9.18) takes the form A I|I (cid:0) S k,k − · · · S k I gk,k − · · · I gk − e ip k L S k +1 ,k · · · S Nk I gk +1 ,k · · · I gNk W k (cid:1) = 0 (9.19)or, equivalently, A I|I (cid:0) e ip k L − S k,k − · · · S k I gk,k − · · · I gk W k I gkN · · · I gk,k +1 S kN · · · S k,k +1 (cid:1) = 0 . (9.20)It is possible to show that the same equations follow if S P ···P N (cid:54) = 1 which uses theidentity S km I gkn I gmn = I gkn I gmn S km , and also that the terms in the wave function withthe plane wave e i p P · x Q lead to the same equations.– 67 –he consistency condition for the system of equations (9.20) requires that thematrices T k ≡ S k,k − · · · S k I gk,k − · · · I gk W k I gkN · · · I gk,k +1 S kN · · · S k,k +1 mutually commute. Naturally, we expect that the matrices T k should be related tothe monodromy matrix T ( p A ) = − Str A W A S fAN ( p A , p N ) S fA,N − ( p A , p N − ) · · · S fA ( p A , p ) , (9.21)where S fjk is the fermionic R -operator defined, e.g., in eq.(102) of [73]. The authorsof [73] use index notations to define the operator. It is more convenient, however, touse the matrix notations and the usual convention for S jk to work with the operator.One can check that it can be written in the following form S fjk ( p j , p k ) = (cid:40) I gj ··· N I gk ··· N I gjk S jk ( p j , p k ) I gj ··· N I gk ··· N if j < k ; I gj ··· N I gk ··· N S jk ( p j , p k ) I gjk I gj ··· N I gk ··· N if j > k . (9.22)Here I gjk is the graded identity and I gj ··· N ≡ I gj,j +1 I gj,j +2 · · · I gjN . To prove the formula, one should use the following representation for the gradedprojection operators (cid:101) E βjα eq.(28) of [73] (cid:101) E βjα = I gj ··· N E βjα I gj ··· N . There are two natural choice for the index A in (9.21), that is A = 0 or A = N + 1.The choice leading to T k appears to be A = N + 1 > k . Then we get S fAk ( p A , p k ) = I gk ··· N S Ak ( p A , p k ) I gAk I gk ··· N . Now we compute the following product S fAk ( p A , p k ) S fA,k − ( p A , p k − ) = I gk ··· N S Ak I gAk I gk ··· N I gk − ··· N S A,k − I gA,k − I gk − ··· N = I gk ··· N I gk − ··· N I gk − ,k S Ak I gAk I gk − ,k S A,k − I gA,k − I gk ··· N I gk − ··· N = I gk ··· N I gk − ··· N I gk − ,k S Ak S A,k − I gAk I gA,k − I gk − ,k I gk ··· N I gk − ··· N , (9.23)where we used the identity S A,k − ( p A , p k ) I gk − ,k I gAk = I gk − ,k I gAk S A,k − ( p A , p k ) . – 68 –he following generalization of the formula (9.23) can be proven by using the math-ematical induction S fAk ( p A , p k ) S fA,k − ( p A , p k − ) · · · S fA,k − n ( p A , p k − n ) == I gk ··· N · · · I gk − n ··· N I gk − ,k I gk − ··· k · · · I gk − n ··· k × (9.24) × S Ak · · · S A,k − n I gAk · · · I gA,k − n I gk ··· N · · · I gk − n ··· N I gk − ,k I gk − ··· k · · · I gk − n ··· k . To get the monodromy matrix, we set k = N and n = N − I gN − ··· N · · · I g ··· N I gN − ,N I gN − ··· N · · · I g ··· N = I , we find the following drastic simplification T ( p A ) = − Str A W A S AN · · · S A I gAN · · · I gA . Now we choose p A = p k and use the fact that S Ak ( p k , p k ) = − P Ak . Recalling thatour goal is to show that T ( p k ) = T k , we have T ( p k ) = Str A W A S AN · · · S A,k +1 P Ak S A,k − · · · S A I gAN · · · I gA = Str A P Ak W k S kN · · · S k,k +1 · S A,k − · · · S A I gAN · · · I gA = Str A P Ak S A,k − · · · S A I gA,k − · · · I gA · W k S kN · · · S k,k +1 · I gAN · · · I gAk . Now we use that S kN · · · S k,k +1 · I gAN · · · I gAk = I gAN · · · I gAk · S kN · · · S k,k +1 to get T ( p k ) = Str A S k,k − · · · S k I gk,k − · · · I gk I gkN · · · I gk,k +1 P Ak I gAk W k S kN · · · S k,k +1 . The supertrace can be easily takenStr A P Ak I gAk = Tr (( − (cid:15) c I ⊗ E cc ) (cid:0) E ab ⊗ E ba (cid:1) (cid:16) ( − (cid:15) f (cid:15) g E ff ⊗ E gg (cid:17) = ( − (cid:15) a + (cid:15) a E aa = I , and, therefore, we show that T ( p k ) = T k . Since T ( u ) T ( v ) = T ( v ) T ( u ) for any u and v , we have shown that the periodicity equations (9.20) are consistent. In framework of the algebraic Bethe Ansatz twisted boundary conditions for Hubbard-likemodels have been studied in [63]. – 69 – .3.2 Two-particle Bethe equations
To see how the formulas of the previous subsection work let us consider a two-particlewave function given byΨ ij ( x , x ) = (cid:40) A | ij e i p x + i p x + A | ij e i p x + i p x if x < x A | ij e i p x + i p x + A | ij e i p x + i p x if x < x . (9.25)According to (9.15), we can identify A | ij ∼ A † i ( p ) A † j ( p ) , A | ij ∼ ( − ) (cid:15) i (cid:15) j A † j ( p ) A † i ( p ) . It is clear that the amplitudes A | ij and A | ij correspond to the in- and out-states,respectively. By using the ZF algebra we find A † j ( p ) A † i ( p ) = S lkji ( p , p ) A † k ( p ) A † l ( p ) ⇒ A | ij = ( − ) (cid:15) i (cid:15) j S lkji ( p , p ) A | kl . In a similar way we get A | ij ∼ A † i ( p ) A † j ( p ) , A | ij ∼ ( − ) (cid:15) i (cid:15) j A † j ( p ) A † i ( p ) , and A † j ( p ) A † i ( p ) = S lkji ( p , p ) A † k ( p ) A † l ( p ) ⇒ A | ij = ( − ) (cid:15) i (cid:15) j S lkji ( p , p ) A | kl . The amplitudes A | ij and A | ij are not independent. By the symmetry condition(9.14) they are related to each other as follows A | ij = ( − ) (cid:15) i (cid:15) j A | ji = S lkij ( p , p ) A | kl ⇒ A | ij = S lkij ( p , p ) A | kl . The wave function (9.25) can be written in the matrix form by multiplying it by therow E i ⊗ E j and summing over i, j . Then we getΨ( x , x ) = (cid:26) A ( e i p x + i p x + S P e i p x + i p x ) if x < x A ( P g e i p x + i p x + S g e i p x + i p x ) if x < x , (9.26)where we recall that P g = ( − (cid:15) i (cid:15) j E ij ⊗ E ji is the graded permutation and S g is thegraded S-matrix S g = P S ( p , p ) P I g = P S ( p , p ) P g .The (quasi)-periodicity condition can be easily imposedΨ( x , x ) = Ψ( x + L, x ) W , Ψ( x , x ) = Ψ( x , x − L ) W , x < x , where the matrix W is equal to I for periodic boundary conditions and to Σ foranti-periodic boundary conditions for fermions. By using the wave e ip k x k this leadsto the following equations A (cid:0) − e ip L S g W (cid:1) = 0 , A (cid:0) − e − ip L S g W (cid:1) = 0 , – 70 –r by using the wave e ip x + ip x to A (cid:0) S P − e ip L P g W (cid:1) = 0 , A (cid:0) S P − e − ip L P g W (cid:1) = 0 . These two sets of the periodicity conditions are obviously equivalent because P g W = W P g . Let us also mention that the equations are compatible if the matrices W S g and S g W commute, and this follows from unitarity S g S g = I and W W S g = S g W W . Let us now see how the nesting procedure works for the case of one A † bosonand one A † fermion. Consider the system of equations A | = S ( p , p ) A | + S ( p , p ) A | , A | = S ( p , p ) A | + S ( p , p ) A | . (9.27)Assuming that S klij are matrix elements of the string S-matrix S , we get A | = S ( p , p ) (cid:104) x − − x − x +1 − x − e i p A | − x +1 − x − x +1 − x − η ( p ) η ( p ) e i p e i p A | (cid:105) , A | = S ( p , p ) (cid:104) x +2 − x − x − − x +1 η ( p ) η ( p ) A | + x +2 − x +1 x − − x +1 e − i p A | (cid:105) , (9.28)where S ( p , p ) is the scalar prefactor.For the amplitudes of interest the general Bethe equations e − ip L A | ij = ( − (cid:15)(cid:15) i + (cid:15) i (cid:15) j S lkji ( p , p ) A | kl (9.29)read as follows e − ip L A | = S ( p , p ) A | + S ( p , p ) A | = A | ,e − ip L A | = ( − (cid:15) (cid:104) S ( p , p ) A | + S ( p , p ) A | (cid:105) = ( − (cid:15) A | , (9.30)where we have used eqs.(9.27). Note that in eq.(9.29) the multiplier ( − (cid:15)(cid:15) i takesinto account the boundary conditions for fermions: (cid:15) = 0 for periodic fermions and (cid:15) = 1 for anti-periodic ones, respectively.The system (9.28) can be solved in two different ways depending on the choiceof the first level vacuum [9]. Below we present both solutions. • Regarding A . . . A as the first level vacuum, we first choose the followingansatz A | = f ( p ) S ( p ) , A | = f ( p ) , A | = S ( p , p ) f ( p ) S ( p ) , A | = S ( p , p ) f ( p ) , (9.31)– 71 –here S ( p , p ) is the corresponding element of the string S-matrix. One caneasily show that this ansatz indeed solves the system (9.28) provided we take f ( p ) = e i p η ( p ) x + − x − y − x − , S ( p ) = e i p y − x − y − x + . According to eqs.(9.31), the last formulae give e − ip L f ( p ) S ( p ) = S ( p , p ) f ( p ) ,e − ip L f ( p ) = ( − (cid:15) S ( p , p ) f ( p ) S ( p )and we derive the corresponding Bethe equations e ip L = S ( p , p ) S ( p ) , ( − (cid:15) = S ( p ) S ( p ) . • If we choose A . . . A as the first level vacuum, we modify the ansatz for thecorresponding amplitudes as follows A | = f ( p ) , A | = f ( p ) S ( p ) . A | = S ( p , p ) f ( p ) , A | = S ( p , p ) f ( p ) S ( p ) . (9.32)Note that S ( p , p ) = − S ( p , p ). This time satisfaction of eqs.(9.28) requiresone to choose f ( p ) = η ( p ) e − i p yy − x − , S ( p ) = − e − i p y − x + y − x − . The Bethe equations (9.30) read e − ip L f ( p ) = S ( p , p ) f ( p ) S ( p ) ,e − ip L f ( p ) S ( p ) = ( − (cid:15) S ( p , p ) f ( p ) , and, therefore, we find e ip L = ( − (cid:15) S ( p , p ) S ( p ) ≡ ( − (cid:15) S ( p , p ) x +1 − yx − − y e − i p , ( − (cid:15) = S ( p ) S ( p ) . This completes consideration of our simple example illustrating the dependence ofthe Bethe equations on the periodicity conditions for fermions.– 72 – .4 Large/small g expansions of solutions to the bound state equation The four general solutions of the bound state equation (7.8) are e q = (cid:16)(cid:113) g sin p + 1 + 1 (cid:17) (cid:16) cos p (cid:113) g sin p + 1 ± (cid:113) cos p − g sin p (cid:17) g sin p , (9.33) e q = (cid:16)(cid:113) g sin p + 1 − (cid:17) (cid:16) cos p (cid:113) g sin p + 1 ± (cid:113) cos p − g sin p (cid:17) g sin p , (9.34)where only the first two solutions (9.33) correspond to states with positive energy.The large g dependence of q of the bound state solutions with momentum ex-ceeding p cr is obtained by expanding (9.33) in powers of 1 /g with the bound statemomentum p kept fixed q ± = 1 g sin p − g sin p ± i (cid:18) p − cos p g sin p (cid:19) + O ( 1 g ) . (9.35)To find the large g dependence of q of the bound state solutions with momentumsmaller than p cr one should take into account that p cr → / √ g as g → ∞ , andtherefore one should consider a bound state with momentum p of the order 1 / √ g and keep the product p √ g fixed in the large g expansion q ± = 2 1 ± (cid:113) − p g gp − g p − p g ± (cid:113) − p g . (9.36)The small g dependence of q of the bound state solutions with momentum smallerthan p cr is obtained by expanding (9.33) at small g with the bound state momentum p kept fixed q + = − g + log 4 cos p sin p + g p ) tan p O ( g ) , (9.37) q − = − log cos p g p p O ( g ) . (9.38)To find the small g dependence of q of the bound state solutions with the mo-mentum exceeding p cr , one should take into account that p cr → π − g as g → p as follows p = π − g cos α , and keep α fixed in the expansion. Then we get q ± = − log g g α ) ± i (cid:18) α + g α − g α (cid:19) + O ( g ) . (9.39) We assume here that p ∈ (0 , π ). – 73 – eferences [1] J. M. Maldacena, “The large N limit of superconformal field theories andsupergravity,” Adv. Theor. Math. Phys. (1998) 231 [Int. J. Theor. Phys. (1999)1113], hep-th/9711200.[2] J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-Mills,”JHEP (2003) 013, hep-th/0212208.[3] I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the AdS × S superstring,” Phys. Rev. D (2004) 046002, hep-th/0305116.[4] V. A. Kazakov, A. Marshakov, J. A. Minahan and K. Zarembo, “Classical /quantum integrability in AdS/CFT,” JHEP (2004) 024, hep-th/0402207.[5] N. Beisert, V. Dippel and M. Staudacher, “A novel long range spin chain and planarN = 4 super Yang-Mills,” JHEP (2004) 075, hep-th/0405001.[6] G. Arutyunov, S. Frolov and M. Staudacher, “Bethe ansatz for quantum strings,”JHEP , 016 (2004), hep-th/0406256;[7] M. Staudacher, “The factorized S-matrix of CFT/AdS,” JHEP (2005) 054,hep-th/0412188;[8] N. Beisert and M. Staudacher, “Long-range PSU(2,2 |
4) Bethe ansaetze for gaugetheory and strings,” hep-th/0504190.[9] N. Beisert, “The su (2 |
2) dynamic S-matrix,” hep-th/0511082.[10] G. Arutyunov, S. Frolov, J. Plefka and M. Zamaklar, “The off-shell symmetryalgebra of the light-cone AdS × S superstring,” hep-th/0609157.[11] N. Beisert, “The Analytic Bethe Ansatz for a Chain with Centrally Extended su (2 | (2007) P017, nlin.si/0610017.[12] R. A. Janik, “The AdS × S superstring worldsheet S-matrix and crossingsymmetry,” Phys. Rev. D (2006) 086006, hep-th/0603038.[13] N. Beisert, R. Hernandez and E. Lopez, “A crossing-symmetric phase for AdS × S strings,” hep-th/0609044.[14] N. Beisert, B. Eden and M. Staudacher, “Transcendentality and crossing,”hep-th/0610251.[15] N. Beisert and A. A. Tseytlin, “On quantum corrections to spinning strings andBethe equations,” Phys. Lett. B (2005) 102, hep-th/0509084.[16] R. Hernandez and E. Lopez, “Quantum corrections to the string Bethe ansatz,”JHEP (2006) 004, hep-th/0603204. – 74 –
17] L. Freyhult and C. Kristjansen, “A universality test of the quantum string Betheansatz,” Phys. Lett. B (2006) 258, hep-th/0604069.[18] G. Arutyunov and S. Frolov, “On AdS × S string S-matrix,” Phys. Lett. B (2006) 378, hep-th/0604043.[19] Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, “The four-loopplanar amplitude and cusp anomalous dimension in maximally supersymmetricYang-Mills theory,” hep-th/0610248.[20] M. K. Benna, S. Benvenuti, I. R. Klebanov and A. Scardicchio, “A test of theAdS/CFT correspondence using high-spin operators,” hep-th/0611135.[21] A. V. Kotikov and L. N. Lipatov, “On the highest transcendentality in N = 4SUSY,” hep-th/0611204.[22] L. F. Alday, G. Arutyunov, M. K. Benna, B. Eden and I. R. Klebanov, “On thestrong coupling scaling dimension of high spin operators,” JHEP (2007) 082,hep-th/0702028.[23] I. Kostov, D. Serban and D. Volin, “Strong coupling limit of Bethe ansatzequations,” hep-th/0703031.[24] P. Y. Casteill and C. Kristjansen, “The Strong Coupling Limit of the ScalingFunction from the Quantum String Bethe Ansatz,” hep-th/0705.0890.[25] J. Maldacena and I. Swanson, “Connecting giant magnons to the pp-wave: Aninterpolating limit of AdS × S ,” hep-th/0612079.[26] T. Klose, T. McLoughlin, R. Roiban and K. Zarembo, “Worldsheet scattering inAdS × S ,” hep-th/0611169.[27] T. Klose and K. Zarembo, “Reduced sigma-model on AdS × S : one-loop scatteringamplitudes,” JHEP (2007) 071, hep-th/0701240.[28] N. Gromov and P. Vieira, “Constructing the AdS/CFT dressing factor,”hep-th/0703266.[29] R. Roiban, A. Tirziu and A. A. Tseytlin, “Two-loop world-sheet corrections inAdS × S superstring,” JHEP (2007) 056, arXiv:0704.3638 [hep-th].[30] T. Klose, T. McLoughlin, J. A. Minahan and K. Zarembo, “World-sheet scatteringin AdS × S at two loops,” hep-th/0704.3891.[31] V. Giangreco Marotta Puletti, T. Klose and O. Ohlsson Sax, “Factorized world-sheetscattering in near-flat AdS × S ,” arXiv:0707.2082 [hep-th].[32] R. A. Janik and T. Lukowski, “Wrapping interactions at strong coupling – the giantmagnon,” arXiv:0708.2208 [hep-th]. – 75 –
33] R. Roiban and A. A. Tseytlin, “Strong-coupling expansion of cusp anomaly fromquantum superstring,” arXiv:0709.0681 [hep-th].[34] B. Basso, G. P. Korchemsky and J. Kotanski, “Cusp anomalous dimension inmaximally supersymmetric Yang-Mills theory at strong coupling,” arXiv:0708.3933[hep-th].[35] S. Schafer-Nameki, M. Zamaklar and K. Zarembo, “Quantum corrections to spinningstrings in AdS × S and Bethe ansatz: A comparative study,” JHEP (2005)051; S. Schafer-Nameki and M. Zamaklar, “Stringy sums and corrections to thequantum string Bethe ansatz,” JHEP (2005) 044; S. Schafer-Nameki, Exactexpressions for quantum corrections to spinning strings, Phys. Lett. B , 571(2006), hep-th/0602214; S. Schafer-Nameki, M. Zamaklar and K. Zarembo, “Howaccurate is the quantum string Bethe ansatz?,” hep-th/0610250.[36] A. V. Kotikov, L. N. Lipatov, A. Rej, M. Staudacher and V. N. Velizhanin,“Dressing and Wrapping,” hep-th/0704.3586.[37] D. M. Hofman and J. M. Maldacena, “Giant magnons,” hep-th/0604135.[38] G. Arutyunov, S. Frolov and M. Zamaklar, “Finite-size effects from giant magnons,”hep-th/0606126.[39] J. Ambjorn, R. A. Janik and C. Kristjansen, “Wrapping interactions and a newsource of corrections to the spin-chain / string duality,” Nucl. Phys. B (2006)288, hep-th/0510171.[40] J. Teschner, “On the spectrum of the Sinh-Gordon model in finite volume,”hep-th/0702214.[41] A. B. Zamolodchikov, “Thermodynamic Bethe ansatz in relativistic models. Scalingthree state Potts and Lee-Yang models,” Nucl. Phys. B (1990) 695.[42] C. Destri and H. J. de Vega, Phys. Rev. Lett. (1992) 2313.[43] V. V. Bazhanov, S. L. Lukyanov and A. B. Zamolodchikov, Nucl. Phys. B (1997) 487, hep-th/9607099.[44] C. N. Yang and C. P. Yang, “Thermodynamics of a one-dimensional system ofbosons with repulsive delta-function interaction,” J. Math. Phys. (1969) 1115.[45] P. Dorey and R. Tateo, “Excited states by analytic continuation of TBA equations,”Nucl. Phys. B (1996) 639, hep-th/9607167.[46] M. J. Martins, “Complex excitations in the thermodynamic Bethe ansatz approach,”Phys. Rev. Lett. (1991) 419.[47] G. Arutyunov, S. Frolov and M. Zamaklar, “The Zamolodchikov-Faddeev algebra forAdS × S superstring,” JHEP (2007) 002, hep-th/0612229. – 76 –
48] A. B. Zamolodchikov and A. B. Zamolodchikov, “Factorized S-matrices in twodimensions as the exact solutions of certain relativistic quantum field models,”Annals Phys. (1979) 253.[49] L. D. Faddeev, Sov.Sci.Rev.Math.Phys. 1C(1980) 107.[50] N. Dorey, “Magnon bound states and the AdS/CFT correspondence,” J. Phys. A (2006) 13119, hep-th/0604175.[51] H. Y. Chen, N. Dorey and K. Okamura, “The asymptotic spectrum of the N = 4super Yang-Mills spin chain,” JHEP (2007) 005, hep-th/0610295.[52] N. Dorey, D. M. Hofman and J. Maldacena, “On the singularities of the magnonS-matrix,” hep-th/0703104.[53] H. Y. Chen, N. Dorey and K. Okamura, “On the scattering of magnon boundstates,”JHEP (2006) 035, hep-th/0608047.[54] R. Roiban, “Magnon bound-state scattering in gauge and string theory,” JHEP (2007) 048, hep-th/0608049.[55] L.D. Faddeev and A.A. Slavnov, “Gauge fields: an introduction to quantum theory,”1991, Addison-Wesley PC, Redwood, CA, US, 217 pp[56] G. Arutyunov and S. Frolov, “Uniform light-cone gauge for strings in AdS × S :Solving su (1 |
1) sector,” JHEP (2006) 055, hep-th/0510208.[57] S. Frolov, J. Plefka and M. Zamaklar, “The AdS × S superstring in light-conegauge and its Bethe equations,” J. Phys. A (2006) 13037, hep-th/0603008.[58] E. Witten, “Constraints On Supersymmetry Breaking,”Nucl. Phys. B (1982) 253.[59] G. Arutyunov and S. Frolov, “Integrable Hamiltonian for classical strings onAdS × S ,” JHEP (2005) 059, hep-th/0411089.[60] M. J. Martins and C. S. Melo, “The Bethe ansatz approach for factorizable centrallyextended S-matrices,” Nucl. Phys. B (2007) 246, hep-th/0703086.[61] B. S. Shastry, “Exact integrability of the one-dimensional Hubburd-model”,Phys.Rev.Lett 56 (1986) 2453.[62] P. B. Ramos and M. J. Martins, “Algebraic Bethe Ansatz Approach For TheOne-Dimensional Hubbard Model,” J. Phys. A (1997) L195, hep-th/9605141.[63] P. B. Ramos and M. J. Martins,“The quantum inverse scattering method forHubbard-like models”, Nucl. Phys. B [FS] (1998) 413-470.[64] F. H. L. Essler, H. Frahm, F. G¨ohmann, A. Kl¨umper and V. Korepin, “Theone-dimensional Hubbard model”, Cambridge University Press , 2005. – 77 –
65] C. Gomez and R. Hernandez, “The magnon kinematics of the AdS/CFTcorrespondence,” hep-th/0608029; J. Plefka, F. Spill and A. Torrielli, “On the Hopfalgebra structure of the AdS/CFT S-matrix,” hep-th/0608038.[66] A. Torrielli, “Classical r-matrix of the su (2 |
2) SYM spin-chain,” Phys. Rev. D (2007) 105020, hep-th/0701281; N. Beisert, “The S-Matrix of AdS/CFT andYangian Symmetry,” PoS SOLVAY (2006) 002 [arXiv:0704.0400 [nlin.SI]];T. Matsumoto, S. Moriyama and A. Torrielli, “A Secret Symmetry of the AdS/CFTS-matrix,” JHEP (2007) 099, [arXiv:0708.1285 [hep-th]]; N. Beisert andF. Spill, “The Classical r-matrix of AdS/CFT and its Lie Bialgebra Structure,”arXiv:0708.1762 [hep-th].[67] M. de Leeuw, “Coordinate Bethe Ansatz for the String S-Matrix,” hep-th/0705.2369.[68] L. F. Alday, G. Arutyunov and S. Frolov, “New integrable system of 2dim fermionsfrom strings on AdS × S ,” JHEP (2006) 078, hep-th/0508140;“Green-Schwarz strings in TsT-transformed backgrounds,” JHEP (2006) 018,hep-th/0512253.[69] L. D. Faddeev, “How Algebraic Bethe Ansatz works for integrable model,”hep-th/9605187.[70] A. Rej, M. Staudacher and S. Zieme, “Nesting and dressing,” J. Stat. Mech. (2007) P08006, hep-th/0702151.[71] M. Beccaria and V. Forini, “Anomalous dimensions of finite size field strengthoperators in N=4 SYM,” arXiv:0710.0217 [hep-th].[72] V. Kazakov, A. Sorin and A. Zabrodin, “Supersymmetric Bethe ansatz and Baxterequations from discrete Hirota dynamics,” hep-th/0703147.[73] F. G¨ohmann and V. E. Korepin, “Solution of the quantum inverse problem,” J.Phys. A (2000) 1199, hep-th/9910253.(2000) 1199, hep-th/9910253.