aa r X i v : . [ h e p - t h ] O c t Preprint typeset in JHEP style - HYPER VERSION arXiv:yymm.nnnnLPTENS 07/47
Complete 1-loop test of AdS/CFT
Nikolay Gromov
Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure et l’Universit´eParis-VI, Paris, 75231, France; St.Petersburg INP, Gatchina, 188 300, St.Petersburg,Russia; E-mail: [email protected]
Pedro Vieira
Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure et l’Universit´eParis-VI, Paris, 75231, France; Departamento de F´ısica e Centro de F´ısica do PortoFaculdade de Ciˆencias da Universidade do Porto Rua do Campo Alegre, 687, 4169-007Porto, Portugal; E-mail: [email protected]
Abstract:
We analyze nested Bethe ansatz (NBA) and the corresponding finite size cor-rections. We find an integral equation which describes these corrections in a closed form.As an application we considered the conjectured Beisert-Staudacher (BS) equations withthe Hernandez-Lopez dressing factor where the finite size corrections should reproducegeneric one (worldsheet) loop computations around any classical superstring motion in the
AdS × S background with exponential precision in the large angular momentum of thestring states. Indeed, we show that our integral equation can be interpreted as a sumover all physical fluctuations and thus prove the complete 1-loop consistency of the BSequations. In other words we demonstrate that any local conserved charge (including theAdS Energy) computed from the BS equations is indeed given at 1-loop by the sum ofcharges of fluctuations up to exponentially suppressed contributions. Contrary to all pre-vious studies of finite size corrections, which were limited to simple configurations insiderank 1 subsectors, our treatment is completely general. Keywords:
Duality in Gauge Field Theories. ontents
1. Introduction 22. Nested Bethe Ansatz and Bosonic Duality 53. Anomalies – finite size correction to Nested Bethe Ansatz equations 9
4. 1-loop shift 14
5. Bosonic duality 18 zippers
6. The AdS/CFT Bethe equations and the semiclassical quantization of thesuperstring on
AdS × S
7. Conclusions 38Appendix A: Transfer matrix invariance and bosonic duality for SU ( K | M ) supergroups 39Appendix B: Fluctuations for su ( n ) spin chains 41 – 1 – . Introduction Bethe equations [1] describe the scattering of the fundamental degrees of freedom of inte-grable 1 + 1 dimensional theories defined on some large circle of length L . The existence ofa large amount of conserved charges results in the factorizability property of the scatteringmatrix. Namely the full n particle S -matrix is completely fixed by the 2 particle scat-tering. Moreover in two dimensions this 2 to 2 scattering process conserves not only thetotal momentum but also the set of individual momenta. Then, for a large enough circle,the momenta of the several particles are quantized through the wave function periodicitycondition 1 = e ip k L L Y j = k S ( p k , p j ) (1.1)meaning that the (trivial) phase acquired by a particle with momentum p k while goingaround the circle equals the free propagation plus the scattering phases shifts (or timedelay in coordinate space) due to the passage through each of the other particles. Ingeneral Bethe equations are only asymptotically exact as L → ∞ otherwise wrappingeffects [2, 3] must be taken into account.Equation (1.1) is, however, describing particles with no isotopic degrees of freedom,that is S ( p k , p j ) is just a phase. In general, when we have some nontrivial symmetry group,this is not the case and, rather, we must solve the diagonalization problem | ψ i = e ip k L L Y j = k ˆ S ( p k , p j ) | ψ i where ˆ S ( p k , p j ) is now a matrix and | ψ i is the multi-particle wave function (for integrabletheories the number of particles is conserved). If the scattered particles transform undersome symmetry group we will obtain not just one equation like (1.1) but rather a set of n equations entangling the scattering of particles with momenta p k and p j in space-timewith the scattering of spin waves in the isotopic space.In this paper we will mainly consider the particular limit of low energies when thewave length of the spin waves are large and particles exhibit collective behavior which, insome important cases, can be associated with the classical motion of collective fields. Bystudying carefully this limit one can get important information about the quantization ofsome classical field theories.In terms of the Bethe ansatz equation this corresponds to a limit, first considered inthe condensed matter literature by Sutherland [4] in the study of the ferromagnetic limitof the Heisenberg chain and rediscovered and generalized in the context of AdS/CFT [5],where the Bethe roots u j ∼ cot( p j /
2) scale with the number of such roots and with the totalnumber of particles, u j ∼ K a ∼ L . In this limit the Bethe roots condense into disjoint cuts.Since there are several types of Bethe roots, one for each Bethe equation, the condensationof the Bethe roots for systems with n Bethe equations will generate some Riemann surfacewith n + 1 sheets as in figure 4. This resulting curve is in one-to-one correspondence with– 2 –he curves classifying classical solutions through the finite gap method [6, 7, 8, 9, 10]. Inthis way one finds the semi-classical spectrum of the theory.The next natural step is to compute the first quantum corrections to the semi-classicalspectrum, which from the Bethe ansatz point of view will correspond to the finite size (i.e.1 /L ) corrections. For the simplest Bethe equations of the form (1.1) these corrections,called anomalies, were known [11, 12, 13, 14, 15, 16, 17] but for nested Bethe ansatzequations the analysis is much more delicate due to the formation of bound states, calledstacks [18], which are the basic constituents of the cuts made out of more than one type ofBethe roots like the ones in figure 3. In this paper we develop the necessary tools to dealwith these richer systems with isotopic degrees of freedom.Particularly important tools are the so called dualities . One of them, the fermionic duality, is well studied [19, 20, 21, 22, 23, 18, 24] and has a clear mathematical meaning. Ifthe symmetry group under which the fundamental particles transform is a super group thenthere are several possible choices of NBA equations corresponding to the several possiblechoices of super Dynkin diagram which, for super-groups, is not unique. These equationsare related by some dualities associated with the fermionic nodes of the corresponding superDynkin diagram. In the scaling limit they correspond to the exchange of Riemann sheets.In this paper we also use an analogue of this duality, baptized bosonic duality, which existseven in the case of a purely bosonic symmetry. It is associated with the bosonic nodes ofthe Dynkin diagram.Next we apply our method to the recently conjectured Beisert-Staudacher (BS) Betheequations [25]. These equations contain a free parameter λ and should describe two systemsat the same time: four dimensional N = 4 SYM and type IIB super-strings in AdS × S ,two theories which are conjectured to be dual [26, 27, 28]. At weak coupling, λ ≪
1, we arein the perturbative regime of N = 4 SYM and the Bethe equations describe the spectrumof the planar dilatation operator which can be considered as a spin chain Hamiltonian[29, 30] with P SU (2 , |
4) symmetry. At strong coupling √ λ ∼ L ≫ AdS × S [8, 18] and the 1 / √ λ correctionsin the scaling limit correspond to the semi-classical quantization of such highly non-trivialquantum field theory.As it was stressed in our previous papers [31, 32] there are two completely differentways to compute the 1-loop correction to the quasi-classically quantized energy spectrum.One, straightforward but technically more involved, is to take the NBA equations, computeits spectrum and then expand it in powers of 1 / √ λ i.e. find its finite size corrections.Another way, more indirect one, is to pick some classical solution satisfying the semi-classical quantization condition, and quantize around it, i.e. find the spectrum of allpossible excitations of this solution. The one loop shift is then given by the zero energyoscillations and is equal to half of the graded sum of all excitation energies, like for a simpleset of independent one dimensional harmonic oscillators.Both calculations can be performed using the BS equations and it is a very nontrivialtest of the proposed equations that these two calculations give the same result. In fact forthe second calculation we do not even need the Bethe ansatz, since it is based completely onthe semi-classical quantization which, as shown in [31], can be performed relying uniquely– 3 –n the classical integrable structure of the theory – the algebraic curve [8, 9]. Moreover weexpect the second approach to give the exact result whereas the first one is only valid as longas one can trust the asymptotic BAE, which suffers from the wrapping effects mentionedabove. Indeed we found that the two results coincide not precisely, but only for large L/ √ λ with exponential precision. This is obviously a manifestation of the wrapping effectsconsidered in the AdS/CF T context in [33, 34, 35, 36, 37]. This exponential mismatch wasfirst observed in [38]. Finally we should stress that we follow
Figure 1:
For su ( K | M ) super algebras theDynkin diagram is not unique. The several possi-ble choices can be represented as the paths goingfrom the up right corner ( M, K ) to the origin al-ways approaching this point with each step. Theturns are the fermionic nodes whereas the straightlines correspond to the usual bosonic nodes. Dif-ferent paths will correspond to different sets ofBethe equations which are related by fermionicdualities which flip a left–down fermionic turninto down–left turn or vice-versa [24]. a constructive approach. That is we startfrom the classical integrable structure, thefinite gap curves. The curves can be de-scribed by some integral equations. We findhow to correct this equations in such a way that they will now describe not only theclassical limit √ λ → ∞ but also the 1 / √ λ corrections. Then we show that the inte-gral equations modified in this way coincideprecisely with the scaling limit expansion ofthe BS equations [25] with the Hernandez-Lopez phase [39] (up to some exponentiallysuppressed wrapping effects, irrelevant forlarge angular momentum string states)! Ourcomparison, being done at the functionallevel, is completely general.This paper is organized as follows. Insection 2 we introduce some notations, thenotion of stack and the bosonic duality. Insection 3 we derive, in two independent ways,an integral equation describing the finitesize corrections to the leading limit - usingthe dualities and using the transfer matri-ces. In section 6 we follow the constructive approach mentioned above to re-derive the sameintegral equation from the equations in the scaling limit. Section 5 contains some detailsabout the bosonic duality such as some theorems and examples – the reader interested onlyin the main results of the paper can skip this section. In section 6 we apply the methodsdeveloped in the previous sections to the study of the BS equations, compute the finitesize corrections and relate them with the quantum fluctuations of the theory. Appendix Ais devoted to the study of the invariance of the transfer matrices of su ( K | M ) supergroupsunder the bosonic duality and in Appendix B we derive an integral equation describing thesemi-classically corrected equations for su ( n ) spin chains.– 4 – . Nested Bethe Ansatz and Bosonic Duality In the first sections we stick mainly to the simple example of su (1 ,
2) spin chain, althoughour main motivation comes from its application to
AdS × S / N = 4 SYM correspondencewhere the symmetry group is P SU (2 , | AdS × S in section 6.For integrable rank r spin chains each quantum state is parameterized by a set { u a,j } of Bethe roots where a = 1 , . . . , r refers to the Dynkin node and j = 1 , . . . , K a where K a is the excitation number of magnons of type a . The Bethe equations from which we findthe position of these roots are then given by e iτ a u a,j + i V a u a,j − i V a ! L = − r Y b =1 Q b (cid:0) u a,j + i M ab (cid:1) Q b (cid:0) u a,j − i M ab (cid:1) (2.1)where Q a ( u ) = K a Y j =1 ( u − u a,j )are the Baxter polynomials, V a are the Dynkin labels of the representation considered and M ab the Cartan matrix. In fact, contrary to what happens for the usual Lie algebras, forsuper algebras the Dynkin diagram (and thus the Cartan matrix) is not a unique. Takefor example the su ( K | M ) super algebra. The different possible Dynkin diagrams can beidentified [24] as the different paths starting from ( M, K ) and finishing at (0 ,
0) (alwaysapproaching this point with each step) in a rectangular lattice of size M × K as in figure1. The turns in this path represent the fermionic nodes whereas the bosonic nodes arethose which are crossed by a straight line – see figure 1 (the index a goes along the pathas indicated). The Cartan matrix M ab is then given by M ab = ( p a + p a +1 ) δ ab − p a +1 δ a +1 ,b − p a δ a,b +1 where p a is associated with the link between the node a and a + 1 and is equal to +1 ( − g = diag (cid:16) e iφ , . . . , e iφ K , e iϕ , . . . , e iϕ M (cid:17) ∈ SU ( K | M ) (2.2)and the twists τ a , appearing in (2.1) and associated to a Dynkin node located at ( m, k ) inthe M × K network depicted in figure 1, are then given by [40] τ a = φ k − φ k +1 for a bosonic along a vertical segment of the path τ a = ϕ m +1 − ϕ m for a bosonic along a horizontal segment of the path τ a = ϕ m +1 − φ k + π for a fermionic node in a Γ like turn that is with p a − = − p a = 1 τ a = φ k +1 − ϕ m + π for a fermionic node with p a − = − p a = −
1– 5 – igure 2:
The middle node
Bethe roots u can condense into a line as depicted in figure 2a (Thespins in this spin chain transform in a non-compact representation and thus the cuts are topicallyreal. For the su (2) Heisenberg magnet the solutions are distributed in the complex plane as some umbrella shaped curves [5].). Roots of different types can form bound states, called stacks [18], asshown in figure 2b. The stacks behave as fundamental excitations and can also form cuts of stacksas represented in figure 2c. Notice that since g ∈ SU ( K | M ) we have P k φ k − P m ϕ m = 0 mod 2 π . We shall studythese Bethe equations with generic twists and we will see that the usual case ( τ a = 0) is infact quite degenerate.As mentioned above, we find already all the ingredients we will need for the studyof the BS equations in the simple example of a su (1 ,
2) spin chain in the fundamentalrepresentation described by the following system of NBA equations e iφ − iφ = − Q ( u ,j + i ) Q ( u ,j − i ) Q ( u ,j − i/ Q ( u ,j + i/ , j = 1 . . . K (2.3) e iφ − iφ u ,j − i u ,j + i ! L = − Q ( u ,j + i ) Q ( u ,j − i ) Q ( u ,j − i/ Q ( u ,j + i/ , j = 1 . . . K (2.4)The eigenvalues of the local conserved charges are functions of the roots u ,j only and aregiven by Q r = K a X j =1 ir − (cid:18) u ,j + i/ r − − u ,j − i/ r − (cid:19) . (2.5)We will often denote these roots carrying charges by middle node roots.First, consider only middle node excitations, K = 0 = K in the Sutherland scalinglimit [4] where u ∼ K ∼ L ≫
1. We shall always use x a,j = u a,j /L to denote the rescaledBethe roots in the scaling limit. Then, the Bethe equations in log form, to the leadingorder, can be cast as 2 πn j + φ − φ = 1 x ,j + 2 L X k = j x ,j − x ,k (2.6)where the integers n j come from the choice of the branch of the logs. These equations are exactly the same as for the su (3) spin chain except for the sign of the Dynkinlabels which makes the system simpler because the Bethe roots are in general real. – 6 – igure 3: In the scaling limit, to the leading order, the bosonic duality reads Q ≃ Q ˜ Q with Q a = Q K a k =1 ( u − u a ). Thus, if we start with the configuration in figure 3a where the K roots u form a cut of stacks together with K out of the K middle node roots u and apply the bosonicduality to this configuration, the K − K new roots ˜ u must be close to the roots u which werepreviously single while the cut of stacks in the left of figure 3a will become, after the duality, a cutof simple roots – see figure 3b. We see that we can think of the Bethe roots as positions of 2d Coulomb charges on aplane with an external potential equal for every particle plus an external force 2 πn j specificof each Bethe root. Thus, if we group the K mode numbers { n j } into N large groups ofidentical integers and consider the limit where both L and K are very large with K /L fixed, the Bethe roots will be distributed along N (real) cuts C A , each parameterized by aspecific mode number { n A } where A = 1 , . . . , N . Then the equations (2.6) can be writtenthrough the density of middle node roots x as2 πn A + φ − φ = 1 x + 2 /G ( x ) , x ∈ C A (2.7)where we introduce the resolvents G a ( x ) = Z ρ a ( y ) x − y , ρ a ( y ) = 1 L K a X j =1 δ ( x − x a,j ) (2.8)and where the slash of some function means the average of the function above and belowthe cut, /G ( x ) = ( G ( x + iǫ ) + G ( x − iǫ )). Let us also introduce some notation useful forwhat will follow. Defining the quasi-momenta as p = − x + G − φ ,p = − x − G + G − φ , (2.9) p = − x − G − φ , we can add the indices 23 to the mode number n A and to the cut C A in (2.7) and recastthis equation as 2 πn A = p/ − p/ , x ∈ C A . (2.10)– 7 –ext let us consider a state with only two roots u , ≡ u and u , ≡ v with differentflavors, that is K = K = 1. Bethe equations then yield u = 12 cot φ − φ + 2 πn L , v = u + 12 cot φ − φ n ∼ v ∼ u ∼ L and v = u + O (1)– the two Bethe roots form a bound state, called stack [18], and can be thought of as afundamental excitation – see figure 2 b . On the other hand we notice that, strictly speaking,for the usual untwisted Bethe ansatz with φ a = 0 the stack no longer exists.Since the stack in figure 2 b seems to behave as a fundamental excitation one mightwonder whether there exists a cut with K = K roots of type u and u , like in figure 2 c , dual to the configuration plotted in figure 2 a . To answer affirmatively to this question letus introduce a novel kind of duality in Bethe ansatz which we shall call bosonic duality .Indeed, as we explain in detail in section 5, given a configuration of K roots of type u and K roots of type u , we can write2 i sin ( τ / Q ( u ) = e iτ/ Q ( u − i/
2) ˜ Q ( u + i/ − e − τ/ Q ( u + i/
2) ˜ Q ( u − i/ , (2.12)where ˜ Q ( u ) = ˜ K Y j =1 ( u − ˜ u ,j ) , ˜ K = K − K , and τ = φ − φ . Moreover this decomposition is unique and thus defines unambiguouslythe position of the new set of roots ˜ u . Then, as we explain in section 5, the new set ofroots { ˜ u , u } is a solution of the same set of Bethe equations (2.1) with φ ↔ φ . Let us then apply this duality to a configuration like the one in figure 2 a where the roots u ∼ L are in the scaling limit and where there are no roots of type u , K = 0. To theleading order, we see that the ˜ u in (2.12) will scale like L so that the ± i/ Q ≃ ˜ Q , that is˜ u ,j = u ,j + O (1)and therefore we will indeed obtain a configuration like the one depicted in figure 2 c .Moreover the local charges (2.5) of this dual cut are exactly the same as those of theoriginal cut 2 a since they are carried by the middle node roots u which are untouchedduring the duality transformation.Finally, if we apply the duality transformation to some configuration like that in figure3 a in the scaling limit we find, by the same reasons as above, that Q ( u ) ≃ Q ( u ) ˜ Q ( u ).This means that the dual roots ˜ u will be close to the roots u which are not yet part of astack – the ones making the cut in the right in figure 3 a . Thus, after the duality, we willobtain a configuration like the one in figure 3 b .– 8 – igure 4: In the scaling limit the configurations in figure 3 condense into some disjoint segments,cuts, and we obtain a Riemann surface whose sheets are the quasi-momenta. In this continuouslimit the duality corresponds to the exchange of the Riemann sheets.
We conclude that, in the scaling limit with a large number of roots, the distributions ofBethe roots condense into cuts in such a way that the quasi-momenta p i introduced abovebecome the three sheets of a Riemann surface, see figure 4 a , obeying2 πn A ij = p/ i − p/ j , x ∈ C A ij . (2.13)when x belongs to a cut joining sheets i and j with mode number n A ij . The duality trans-formation amount to a reshuffling of sheets 1 and 2 of this Riemann surface so that asurface like the one plotted in figure 4 a transforms into the one indicated in figure 4 b .
3. Anomalies – finite size correction to Nested Bethe Ansatz equations
In this section we will study the leading 1 /L corrections to the scaling equations (2.13).Moreover since the charges of the solutions are expressed through middle node roots u andsince these roots are duality invariant it is useful to write the Bethe equations in terms ofthese roots only. Let us then consider a given configuration of roots condensed into somesimple cuts C and some cuts of stacks C . Then, to leading order, at cuts C we have1 x + 2 Z C − ρ ( y ) dyx − y + Z C ρ ( y ) dyx − y = 2 πn A + φ − φ , x ∈ C (3.1)because in a cut C we have ρ ≃ ρ + O (1 /L ). To study finite size corrections to thisequation two contributions must be considered. On the one hand when expanding the selfinteraction we get [11, 12, 13, 14, 15, 16] X j = k i log u ,k − u ,j − iu ,k − u ,j + i = 2 Z C − ρ ( y ) dyx − y + 2 Z C ρ ( y ) dyx − y + 1 L πρ ′ cot πρ As we shall see in the next section this interpretation can be made exact, and not only valid in thescaling limit. – 9 –here the 1 /L correction comes from the contribution to the sum from the roots separatedby O (1). On the other hand the auxiliary roots appear as X j i log u ,k − u ,j + i/ u ,k − u ,j − i/ − Z C ρ ( y ) x − y dy = − Z C ρ ( y ) x − y dy − Z C ρ ( y ) − ρ ( y ) x − y dy where the last term accounts for the mismatch in densities in cuts C and is clearly also a O (1 /L ) effect. Bellow we will compute this mismatch and find ρ ( x ) − ρ ( x ) = ∆ cot πiL = cot +21 − cot +23 πiL , x ∈ C (3.2)where ∆ f ≡ f ( x + i − f ( x − i
0) andcot ij ≡ p ′ i − p ′ j p i − p j . (3.3)Thus we find, for x ∈ C ,1 x + 2 Z C − ρ ( y ) dyx − y + Z C ρ ( y ) dyx − y = 2 πn A + φ − φ − L cot − Z C ∆ cot x − y dy πi (3.4)As explained before, if we apply the duality transformation, cuts C become cuts C andvice-versa and, to leading order, p ↔ p . Thus for cuts C we find precisely the sameequation (3.4) with 1 ↔
2, so that for x ∈ C x + 2 Z C − ρ ( y ) dyx − y + Z C ρ ( y ) dyx − y = 2 πn A + φ − φ − L cot − Z C ∆ cot x − y dy πi (3.5)These two equations describing the finite size corrections for the two types of cuts of the su (1 ,
2) spin chain are the main results of this section.In what follows we will derive this result from two different angles. Namely, we willfind this finite size corrections using a Baxter like formalism based on transfer matrices forthis spin chain in several representations and by exploiting the duality we mentioned inthe previous section. It will become clear that the generalization to other NBA equationsbased on higher rank symmetry groups is straightforward.
The central object in the study of integrable systems is the transfer matrix ˆ T ( u ). Thealgebraic Bethe ansatz formalism has the diagonalization of such objects as main goal andthe Bethe equations appear in the process of diagonalization (see [41] and references thereinfor an introduction to the algebraic Bethe ansatz). As functions of a spectral parameter u and of the Bethe roots u a,j these transfer matrices seem to have some poles at the positions recall that the Bethe roots u ,k belongs to a C cut and therefore is always well separated from u ,j roots which always belong to C cuts. – 10 –f the Bethe roots. On the other hand they are defined as a product of R operators whichdo not have these singularities. This means that the residues of these apparent poles mustvanish. These analyticity conditions (on the Bethe roots) turn out to be precisely the Betheequations, and thus, if we manage to obtain the eigenvalues of the transfer matrices, we canuse this condition of pole cancellation to obtain the Bethe equations without going throughthe algebraic Bethe ansatz procedure, see for example [42, 43, 44, 24]. For the su (1 ,
2) spinchain we have the following transfer matrices in the anti-symmetric representations: T ( u ) = e − iφ Q ( u − i ) Q ( u + i ) Q ( u + i ) Q ( u − i ) u − i u − i ! L (3.6)+ e − iφ Q ( u + i ) Q ( u + i ) u − i u − i ! L + e − iφ Q ( u − i ) Q ( u − i ) u − i u + i ! L ,T ( u ) = ¯ T (¯ u ) u − i u + i ! L , T ( u ) = u − i u + i ! L . One can easily see that the Bethe equations do follow from requiring analyticity of thesetransfer matrices.In [16] it was shown and emphasized that the
T Q relations are the most powerfulmethod to extract finite size corrections to the scaling limit of Bethe equations.In this section we will use the transfer matrices presented above along with the factthat, due to the Bethe equations, they are good analytical functions of u to find what arethe finite size corrections to this Nested Bethe ansatz. Since for generic (super) nestedBethe ansatz the transfer matrices in the several representations are known, this procedurecan be easily generalized for other NBA’s.The key idea to find the finite size corrections to NBA is to use the transfer matrices inthe various representations to define a new set of quasi-momenta q i as the solutions of analgebraic equation whose coefficients are these transfer matrices. For example, to leadingorder, T ( u ) ≃ e ip + e ip + e ip ,T ( u ) ≃ e i ( p + p ) + e i ( p + p ) + e i ( p + p ) ,T ( u ) ≃ e i ( p + p + p ) , so that if we define a set of exact quasimomenta q i by T ( u ) − e iq T ( u ) (cid:18) − L u (cid:19) + e iq T ( u ) (cid:18) − L u (cid:19) − e iq = 0 , (3.7) Exploiting the similarity between this definition equation and 4.1 in [24] we can easily generalize thisalgebraic equation to a more general su ( K | M ) super group. More precisely we identify e ∂ u ↔ e iq whichis obviously natural if we look at 4.2 in this same paper (see also Appendix A where we use this twoexpressions slightly modified to match our normalizations). We thanks V.Kazakov for pointing this out tous. – 11 –hen, to leading order, q i ≃ p i . Notice however that the coefficients in this equation haveno singularities except some fixed poles close to u = 0. Thus, defined in this way, thequasi-momenta q i constitute a 4 sheet algebraic surface (modulo 2 π ambiguities) such that q/ i − q/ j = 2 πn Aij , x ∈ C ij , (3.8)and, needless to say, this is an exact result in L , it is not a classical (scaling limit) leadingresult like (2.13)! On the other hand, the expansion at large L of the above algebraicequation yields q = p + 12 L (+ cot + cot ) q = p + 12 L ( − cot + cot ) q = p + 12 L ( − cot − cot ) , which follows from the expansion T ( u ) (cid:18) − L u (cid:19) = e ip + e ip + e ip − L (cid:2) e ip (2 p ′ − p ′ − p ′ ) + e ip ( p ′ − p ′ ) + e ip ( p ′ + p ′ − p ′ ) (cid:3) + O (cid:18) L (cid:19) T ( u ) (cid:18) − L u (cid:19) = e i ( p + p ) + e i ( p + p ) + e i ( p + p ) − L h e i ( p + p ) ( p ′ + p ′ − p ′ ) + e i ( p + p ) ( p ′ − p ′ ) + e i ( p + p ) (2 p ′ − p ′ − p ′ ) i + O (cid:18) L (cid:19) ,T ( u ) = e i ( p + p + p ) + O (cid:18) L (cid:19) . of the several transfer matrices. Then, to the first order in 1 /L the exact equation (3.8)gives, for the quasi-momenta p i introduced in (2.9), p/ − p/ = 2 πn A − L cot , x ∈ C (3.9) p/ − p/ = 2 πn A − L (cot +2 cot + cot ) , x ∈ C (3.10)where in (3.9) we use the fact that function cot − cot vanishes under the slash on thecut C since cot + ij = cot − kj , x ∈ C ik . (3.11)Equations (3.9),(3.10) are the finite size corrections we aimed at!Finally q must have no discontinuity at a cut C and therefore∆ p = 2 πi ( ρ − ρ ) = 1 L (cot +21 − cot +23 ) , x ∈ C . (3.12)Thus, replacing the quasi-momenta p i by its expressions in terms of resolvents (2.9) andrelating the density of auxiliary roots ρ to that of the middle node roots ρ through (3.12),we recover precisely (3.4) and (3.5) as announced.– 12 –e would like to stress the efficiency of the T Q relations. We were able to find the usual cot contributions (coming from the expansion of the log’s of the Bethe equations when theBethe roots are close to each other) plus the mismatch in densities of the different types ofroots making the cuts of stacks using only the fact that due to Bethe equations the transfermatrices in several representations were analytical functions of u . The computation donein this way is by far more economical than a brute force expansion of the Bethe equations.Finally let us make an important remark. To derive (3.5) from (3.10) one should usecot = − πi Z C ∪C ∆ cot x − y dy (3.13)which is clearly a valid relation if cot has only branch cuts as singularities. For generictwists and for small enough cuts C and C this is the case. Indeed, in the absence ofBethe roots we have no cuts at all and thus p − p = φ − φ . Suppose φ − φ = 2 πn .Then, by continuity, when we slowly open some cuts C and C then p − p will starttaking positive values around φ − φ without ever being zero. Thus, if the cuts are smallenough we will never get poles in cot . In the next section we will see that the stacksas described in [9] only exist when this assumption of absence of poles is right and aredestroyed when p − p reaches 2 πn . In this section let us re-derive the mismatch formula (3.2) using the bosonic duality (5.1).Besides the obvious advantage for what concerns our comprehension of having a secondderivation there are systems for which the Bethe equations are known but the algebraicformalism behind these equations is still not well developed (this is the case for example forthe
AdS/CF T
Bethe equations proposed by Beisert and Staudacher which we will studyin section 6).Denoting u ,i = u ,i − ǫ i , ˜ u ,i = u ,i − ˜ ǫ i , ǫ ∼ L → ∞ ) we getsin( τ /
2) = sin (cid:18) (cid:16) ˜ G − G + τ (cid:17)(cid:19) exp K X i =1 ǫ i u − u i + ˜ K X i =1 ˜ ǫ i u − u i , where τ = φ − φ . Taking the logarithm of this equation and differentiating with respectto u we get X ǫ i ( u − u i ) + X ˜ ǫ i ( u − u i ) = ˜ G ′ − G ′ L cot ˜ G − G + τ G − G − ˜ G !Thus we find G − G − ˜ G = ˜ G ′ − G ′ L cot ˜ G − G + τ ≃ G ′ − G ′ L cot G − G + τ L cot . – 13 –inally, by computing the discontinuity of this expression at the cuts C we will get the mismatch of the densities of the roots in a cut of stacks ρ − ρ = ∆ cot πiL = cot +21 − cot +23 πiL , which was the gap in the chain of arguments presented in the beginning of the section 3and leading to (3.4).Finally let us show that the bosonic duality amounts to a simple exchange of Riemannsheets in the scaling limit. Consider for example˜ p = − x + ˜ G − ˜ φ = − x + G − G − ˜ φ = p since, as we will see more carefully in the next section, ˜ φ , = φ , .
4. 1-loop shift
In [31] we explained how to obtain the spectrum of the fluctuation energies around anyclassical string solution using the algebraic curve by adding a pole to this curve. In particu-lar we reproduced in this way some previous results [45, 46, 47, 48] where the semi-classicalquantization around some simple circular string motions were considered by directly ex-panding the Metsaev-Tseytlin action [49] around some classical solutions and quantizingthe resulting quadratic action. Using the fact that one extra pole in the algebraic curvemeans one quantum fluctuation, we can compute the leading quantum corrections to theclassical energy of the state from the field theory considerations using the algebraic curvealone, as we mentioned in the introduction. This implies a nontrivial relation between fluc-tuations on algebraic curve and finite size corrections in Bethe ansatz as we will explain ingreater detail below. In this section we study this relation on the example of the su (1 , u ∼ K ∼ L ≫ E = K X j =1 ǫ j = K X j =1 u ,j + 1 / ∼ /L , around the ferromagnetic vacuum of the theory. In this limit the theory is well described bya Landau-Lifshistz model which is a field theory with coupling 1 /L [50, 51, 52]. Therefore avery nontrivial property relating fluctuations and finite size corrections in this NBA shouldhold: • Suppose we compute the energy shift δ E ijn due to the addition of a stack with modenumber n uniting sheets p i and p j to a given configuration with some finite cuts C . • Suppose on the other hand that we compute 1 /L energy expansion E = E (0) + L E (1) + . . . of the configuration with the finite cuts C . ∆ f = f + − f − , so that ρ = − ∆ G πi – 14 –rom the field theory point of view the first quantity corresponds to one of the fluctuationenergies around a classical solution parameterized by the configuration with the cuts C whereas the second quantity, E (1) , is the 1 -loop shift [53] around this classical solution withenergy E (0) . This 1-loop shift, or ground state energy, is given by the sum of halves of thefluctuation energies [53] E (1) = 12 X n X ij δ E ijn (4.1)In fact for usual (non super-symmetric) field theories this sum is divergent and needs to beregularized. We will see that (4.1) can be generalized and holds for arbitrary local charges Q (1) r = 12 X n X ij δ Q ijr,n . (4.2)Let us stress once more that from the Bethe ansatz point of view these quantities arecomputed independently and there is a priori no obvious reason why such relation betweenfluctuations and finite size corrections should hold. In this section we will show thatNested Bethe Ansatz’s describing (super) spin chains with arbitrary rank do indeed obeysuch property with some particular regularization procedure (for the Heisenberg su (2) spinchain a similar treatment was carried in [15]). Moreover we will see that the regularizationmentioned above also appears naturally from the Bethe ansatz point of view as someintegrals around the origin. In this section we will understand the interplay between fluctuations and finite size cor-rections in NBA’s in the scaling limit. For simplicity we are considering the su (1 ,
2) spinchain described in the previous sections. General su ( N ) is considered in Appendix B.Let us pick the leading order integral equation for the densities of the Bethe roots inthe scaling limit (3.1) and perturb it by a single stack, connecting p i with p j . Accordingto (2.8) this means simply implies ρ → ρ + L δ ( x − x ij ), where x ij is position of the newstack. Finally, the positions where one can put an extra stack, as it follows from the BAE(2.3,2.4), can be parametrized by one integer mod number np i ( x ij n ) − p j ( x ij n ) = 2 πn . (4.3)Therefore, for i = 2 , j = 3 the perturbed equation (3.1) reads1 x + 2 Z C − ρ ( y ) x − y + Z C ρ ( y ) x − y + 1 L x − x n = 2 πk + φ − φ , x ∈ C . (4.4)and this perturbation will lead to some perturbation of the density δρ ( y ), which will leadto the perturbation in the local charges (2.5) as δ Q r,n = Z δρ ( y ) y r dy + 1 L ( x n ) r , (4.5)the local charges of the fluctuation with polarization 23 and mode number n .– 15 –hus, by linearity, if we want to obtain the 1-loop shift (4.2) (or rather a large Nregularized version of this quantity where the sum over n goes from − N to N ) we have tosolve the following integral equation for densities1 x + 2 Z C − ρ ( y ) x − y + Z C ρ ( y ) x − y + N X n = − N L (cid:20) x − x n + 1 x − x n (cid:21) = 2 πk , x ∈ C . (4.6)and then the 1-loop shifted charges are given Q r = Z C ∪C ρ ( y ) y r dy + N X n = − N L (cid:20) x n ) r + 1( x n ) r (cid:21) (4.7)= Z C ∪C ρ ( y ) y r dy + N X n = − N L " I x n cot y r dy πi + I x n cot y r dy πi . (4.8)To pass from the first line to the second in the above expression we use that cot ij has polesat x ij n with unit residue. We will now understand how to redefine the density in such away that the second term is absorbed into the first one. We start by opening the contoursin (4.8) around the excitation points x ijn . These contours will then end up around thecuts C kl of the classical solution and around the origin. We will not consider the contouraround x = 0 – this contribution would lead to a regularization of the divergent sum inr.h.s. of (4.2). We will analyze it carefully in the super-string case, where it leads to theHernandez-Lopez phase factor. Then we get Q r = Z C ∪C ρ ( y ) y r dy + 12 L (cid:20) I C cot y r dy πi + I C cot y r dy πi (cid:21) (4.9)Noting that cot + ij = cot − kj , x ∈ C ik , (4.10)where the superscript + ( − ) indicates that x is slightly above (below) the cut, we can write Q r = Z C ∪C ρ ( y ) y r dy − L Z C ∪C ∆ cot y r dy πi (4.11)so that we see that it is natural to introduce a new density, “dressed” by the virtualparticles, ̺ = ρ − L ∆ cot πi (4.12)so that the expression for the local charges takes the standard form Q r = Z C ∪C ̺ ( y ) y r dy . Let us now rewrite our original integral equation (4.6) in terms of this dressed density.We will see that the integral equation we are constructing for this density by requiring a– 16 – igure 5:
Illustration of an identity used in the main text. proper semi-classical quantization will be precisely the equation (3.4) which is the finitesize corrected integral equation arising from the NBA for the spin chain! This will thusprove the announced property relating finite size corrections and 1-loop shift. Consider forexample the first summand in (4.6) (recall that x ∈ C ), X n x − x n = X n I x n cot x − y dy πi = cot + I C cot x − y dy πi = cot − Z C ∆ cot x − y dy πi , (4.13)Note that cot has branch cut singularities at C which we have to encircle when we blowup the contour, which leads to the second term. The first term comes from the pole at x = y . Finally, to write the second term as it is we used (4.10). Analogously (see figure 5for a pictorial explanation of the second equality) X n x − x n = I C cot x − y dy πi = cot / + Z C − ∆ cot x − y dy πi = cot / − Z C − ∆ cot x − y dy πi . (4.14)Then we note that (see (3.13))cot / = cot / = − Z C ∪C − ∆ cot x − y dy πi so that (4.6) reads1 x +2 Z C − ρ ( y ) x − y + Z C ρ ( y ) x − y + 12 L − Z C − ∆ cot x − y dy πi − Z C ∆ cot x − y dy πi = 2 πk + φ − φ which in terms of the redefined density ̺ becomes1 x + 2 Z C − ̺ ( y ) x − y + Z C ̺ ( y ) x − y + 1 L cot − Z C ∆ cot x − y dy πi = 2 πk + φ − φ which coincides precisely with (3.4) as announced above! Thus the finite size corrections tothe charge of any given configuration will indeed be equal to the field theoretical prediction,that is to the 1-loop shift around the classical solution.– 17 – . Bosonic duality In this section we will explain some details behind the bosonic duality (2.12) mentionedin section 2. There are two main steps to be considered. On the one hand we have to provethat for a set of K generic complex numbers u and K roots u obeying the auxiliaryBethe equations (2.3) it is possible to write ( τ = φ − φ )2 i sin ( τ / Q ( u ) = e iτ/ Q ( u − i/
2) ˜ Q ( u + i/ − e − iτ/ Q ( u + i/
2) ˜ Q ( u − i/ , (5.1)and that, in doing so, we define the position of a new set of numbers ˜ u . A priori this is notat all a trivial statement because we have a polynomial of degree K on the left whereason the right hand side we have only K − K parameters to fix. However, as we will see, if K equations (2.3) are satisfied it is possible to write Q ( u ) in this form. This will be thesubject of the section 5.1.Assuming (5.1) to be proved we can use this relation to show that in the original Betheequations we can replace the roots u by the new roots ˜ u with the simultaneous exchange φ ↔ φ . Indeed if we evaluate the duality at u = u ,j we find Q ( u ,j − i/ Q ( u ,j + i/
2) = e i ( φ − φ ) ˜ Q ( u ,j − i/ Q ( u ,j + i/ , meaning that in the equation (2.4) for the u roots we can replace the roots u by the dualroots ˜ u provided we replace φ ↔ φ . Moreover if we take u = ˜ u ,j ± i/ e iφ − iφ = − ˜ Q (˜ u + i )˜ Q (˜ u − i ) Q (˜ u − i/ Q (˜ u + i/ , which we recognize as equation (2.3) with K − K roots ˜ u in place of the K originalroots u and with φ ↔ φ . Finally evaluating (5.1) at u = u ,j ± i/ φ a . In section 5.3 somecurious examples of dual states will be given. In this section we shall prove that one can always decompose Q ( u ) as in (5.1) and thatthis decomposition uniquely fixes the position of the new set of roots ˜ u . In other words,let us show that we can set the polynomial P ( u ) ≡ e + i τ Q ( u − i/
2) ˜ Q ( u + i/ − e − i τ Q ( u + i/
2) ˜ Q ( u − i/ − i sin τ Q ( u )to zero through a unique choice of the dual roots ˜ u . Bazhanov and Tsuboi also found some similar duality in the study of the deformed U q ( sl (1 | – 18 – Consider first the case K = 0. Then it is trivial to see that we can always findunique polynomial ˜ Q = u K + P K n =1 a n u n − such that e + i τ ˜ Q ( u + i/ − e − i τ ˜ Q ( u − i/
2) = 2 i sin τ Q ( u ) . because this amounts to solving K linear equations for K coefficients a n with non-degenerate triangular matrix. • Next let us consider K ≤ K /
2. First we choose ˜ Q to satisfy K equations˜ Q ( u p ) = 2 ie − i τ sin τ Q ( u p − i/ Q ( u p − i ) ≡ c p , p = 1 , . . . , K these conditions will define ˜ Q ( u ) up to a homogeneous solution proportional to Q ( u ), ˜ Q ( u ) = Q ( u )˜ q ( u ) + K X p =1 Q ( u ) Q ′ ( u p )( u − u p ) c p where ˜ q ( u ) is some polynomial of the degree K − K . Now from (2.3) we noticethat with this choice of ˜ Q we have P ( u p + i/ Q ( u p + i/
2) = P ( u p − i/ Q ( u p − i/
2) = 0 , p = 1 , . . . , K and thus P ( u ) = Q ( u + i/ Q ( u − i/ p ( u )where p ( u ) = e i τ ˜ q ( u + i/ − e − i τ ˜ q ( u − i/ − i sin τ q ( u )and q is a polynomial. Thus we are left to the same problem as above where K = 0.For completeness let us note that we can write q ( u ) explicitly in terms of the originalroots u and u , q ( u ) = Q ( u ) Q ( u + i/ Q ( u − i/ − poleswhere the last term is a simple collection of poles at u = u p ± i/ q ( u ) is indeed a polynomial. • We can see that the number of the solutions of (2.3) with K = K and K = K − K is the same (see [41] for examples of states counting). Thus for each solution with K ≥ K / K ≤ K / K ≥ K / • Finally let us stress the uniqueness of the ˜ Q . If K > ˜ K we have nothing to showsince we saw explicitly above how the bosonic duality constrains uniquely the dual– 19 –olynomial ˜ Q . Let us then consider K < ˜ K and assume we have two differentsolutions ˜ Q and ˜ Q . Then from the duality relation (5.1) for either solution we find e i τ Q ( u − i/ (cid:16) ˜ Q ( u + i/ − ˜ Q ( u + i/ (cid:17) = e − i τ Q ( u + i/ (cid:16) ˜ Q ( u − i/ − ˜ Q ( u − i/ (cid:17) . Evaluating this expression at u = u ,j + i/ Q ( u ,j ) − ˜ Q ( u ,j ) = 0 sothat ˜ Q ( u ) − ˜ Q ( u ) = Q ( u ) h ( u ) and therefore e i τ h ( u + i/
2) = e − i τ h ( u − i/ h ( u ) – for large u we can neglect the i/ e iτ = 1 thus leading to a contradiction. In this section we will examine the transformation properties of the transfer matrices underthe bosonic duality. In Appendix A we consider this problem for the general su ( N | M )group. For now let us just take T for su (1 ,
2) from (3.6). Using (5.1) we can expressratios of Q ’s through ˜ Q and Q so that T ( u ) = e − iφ + 2 i sin τ e − i τ Q ( u − i ) Q ( u + i ) ˜ Q ( u + i ) + e − iτ ˜ Q ( u − i )˜ Q ( u + i ) ! Q ( u + i ) Q ( u − i ) u − i u − i ! L + e − iφ − i sin τ e + i τ Q ( u + i ) Q ( u + i ) ˜ Q ( u + i ) + e + iτ ˜ Q ( u + i )˜ Q ( u + i ) ! u − i u − i ! L + e − iφ Q ( u − i ) Q ( u − i ) u − i u + i ! L . We see that for τ = φ − φ the terms with sin τ cancel and we get the old expression for T with u replaced by ˜ u and φ ↔ φ .This simple transformation property of the transfer matrices automatically impliesthat the Riemann surface defined by the algebraic equation (3.7) is untouched under theduality transformation (to all orders in L ), so that the duality can cause at most somereshuffling of the sheets. However, as we will see in the next section, not necessarily thesheets as a whole are exchanged – this operation will be in general done in a piecewisemanner. In this section we will study some curious Bethe roots distributions for the twisted su (1 , su (2)Heisenberg chain, u ,j + i u ,j − i ! L = − Q ( u ,j + i ) Q ( u ,j − i ) . (5.2)– 20 –sing the first example we shall understand the importance of twists to stabilize big cutsof stacks like the ones depicted in figures 2 a , 2 b and explain how the stacks gets destroyedas we decrease the twists.We can dualize su (2) solutions of the twisted Heisenberg ring using the same duality(2.12) as before with Q ( u ) → u L . We will consider the dual solutions to the vacuumand to a 1-cut solution for the Heisenberg spin chain (5.2) as a prototype of the curioussolutions one would get. zippers In the previous sections we saw that the introduction of twists in the NBA equations areneeded to have a configuration with auxiliary roots u close to some momentum carryingroots u . In figure 6 we have two numerical solutions of the Bethe equations which arerelated by the bosonic duality. In either of them we see a configuration of Bethe rootswith a simple cut with middle roots only (in blue) and a cut of stacks (containing blue andyellow roots). In this situation it is clearly reasonable to think of stacks as bound statesof different types of roots and we see that they indeed condense into multicolor cuts. Figure 6:
The upper and the lower configuration of Bethe roots are dual to one another. Big bluedots are middle node roots u , yellow dots are auxiliary roots u . The formation of cuts of stacksis manifest for this situation where the twists are large (like π/
2) and the filling fractions are small.
We will examine what happens when we decrease the twists (or increase filling fractions,which is the same qualitatively). For simplicity we consider the configuration, dual to thesimple one cut solution ( K = K and K = 0) with no twist for the middle node roots, φ − φ = 0, and some generic twist φ − φ = τ for the auxiliary roots. Bosonic dualitywill leave untouched middle node roots u and create K new axillary roots u .In the upper left corner of figure 7 we applied the duality for some big twist τ = 4 . τ = 0 .
2. In this latter case the auxiliary (yellow) roots clearly do not form stacks together with the middle node (blue) roots!, rather they form a bubble,containing the original cut of roots u .To understand what happens in the scaling limit consider the position of n = 1fluctuation, given by (4.3), which would be a small infinitesimal cut between p and p .Clearly this probe cut would have no influence on the leading order algebraic curve for p i .In figure 7 the position of this virtual fluctuation is marked by a red crossed dot. When thetwist is big enough (and filling fraction is small enough) the fluctuation is to the left fromthe cut. When we start decreasing the twist the fluctuation approaches the cut (upperright picture on fig 7) and at this point we have at the same time p ( x n ) − p ( x n ) = 2 π For zero twist the duality becomes degenerate and we will see below that it needs to be slightly modified. – 21 – igure 7:
Disintegration of the stack configuration. When the twist is large (the top left corner)the auxiliary roots form bound states together with the middle node ones and constitute a cut ofstacks. As we decrease the twist fluctuation n = 1 (the red crossed dot) enters the cut of stacks(the top right corner) and subsequently partly disintegrate the cut of stacks forming some zipperlike configuration (the bottom left corner). At some very small value of the twist the configurationof Bethe roots bears no resemblance with a cut of stacks. and p ( x n ) − p ( x n ) = 2 π , which implies p − p = 0 so that equation (3.13) becomes wrong at this point. When wecontinue decreasing the twist the fluctuation passes through the cut and becomes a n = 0fluctuation. If we think of the fluctuation as being a small cut along the real axis we seethat density becomes negative after crossing the cut:0 < ρ fluc = − ∆( p − p )4 πi = − ∆( − p − p )4 πi = − ρ fluc This means that two branch points of the infinitesimal cut should not be connected directly,but rather by some macroscopical curve with real positive density! This curves z ( t ) can becalculated from the equation ρ ( z ) dz ∈ R + or p ( z ) − p ( z )2 πi ∂ t z = ± su (2)configuration [54]. In the scaling limit the black curve corresponds to the cut connecting p and p like on the figure 8.At first sight these figures seem to be defying our previous results. Indeed we checked inthe previous section that the transfer matrices themselves are invariant under the bosonicduality. Thus the algebraic curves obtained from (3.7) should be the same after and beforeduality and thus what one naturally expects is a simple interchange of Riemann sheets p ↔ p under the duality transformation. What really happens is a bit more tricky. The– 22 –uasimomenta are indeed only exchanged but this exchange operation is done in a piecewisemanner. That is,if we denote the new quasi-momenta by p newi and the old ones by p oldi andif we denote the bubble in figure 8 by R then we have p new = ( p old , outside R p old , inside R , p new = ( p old , outside R p old , inside R , p new = p old where the border of the region R can be precisely determined in the scaling limit asexplained above. Figure 8:
In the scaling limit the algebraiccurves for e ip j are the same before the dual-ity (blue cut only) and after the duality (whenthe auxiliary roots are created). The dualitycauses interchange of the sheets outside the bub-ble, while keeping the order untouched inside.This follows from the need of a positive densityfor the “virtual” cut. In other words the dualityis indeed only interchanging the sheets of the Rie-mann surface although it is interchanging them ina piecewise way. roots In this section we will consider an exampleof application of the bosonic duality to theHeisenberg magnet . The duality (2.12)can be applied to the roots u obeying (5.2)provided we replace Q ( u ) → u L . In factif we want to consider strictly zero twistwe need a new duality because that oneis clearly degenerate in this limiting case.The proper modified expression is in thiscase i ( ˜ K − K ) u L = Q ( u − i/
2) ˜ Q ( u + i/ − Q ( u + i/
2) ˜ Q ( u − i/ . and the number of dual roots is now L − K + 1. Contrary to what happened with non-zero twists, here, the dual solution is notunique. Indeed if ˜ K > K we can as well use˜ Q α ≡ α Q + ˜ Q . (5.3)All these solutions, parameterized by the constant α , have the same charges becausethe transfer matrix is invariant under this transformation – see appendix A. Notice thatif initially we have a physical state with K < L/ K > L/ Q out of the various solutions to (5.3) so that˜ Q α = u ˜ K + ˜ K − X l =0 c αl u l (5.4)becomes well defined through (5.3). We chose ˜ Q = ˜ Q to be the dual solution with c = 0.Consider for example the vacuum state for which Q = 1. Let us first take α to bevery large so that we can write α + ˜ Q ≃ α + ( xL ) L . (5.5) This section beneficed a lot from the insightful discussions with T. Bargheer and N. Beisert whom weshould thank. – 23 – igure 9:
Three configurations of Bethe roots dual to the ferromagnetic vacuum of the untwistedHeisenberg spin chain. For each physical solution (below half filling) of the Bethe equations thereis a one parameter ( α ) family of dual unphysical solutions. To the left, α is large and the rootsdistribute themselves along a circle with radius R α given by ( R α L ) L = α . Decreasing α the circlewill touch the fluctuations n = ±
1. Similarly to the previous section the virtual infinitesimal cutsbecome macroscopical bubble cuts with cusps at the position of the fluctuations. Intersection pointsof the new cuts with the circle are connected by condensates, which are logarithmic cuts on thealgebraic curve [54].
We see for large α the dual roots will be on a circle of radius | α | /L L . The correspondingconfiguration is present on the first picture on the figure 9. In this figure we also plotted acircle with this radius and one can see that the Bethe roots belong perfectly to the circle.Let us now understand this configuration from the algebraic curve point of view. Thethe quasi-momenta p = − p ≡ p = x − G , in the absence of Bethe roots, are simply givenby p = x . Let us find the curves with positive densities and mode number n = 0. Thedensity is given by ρ ( x ) = πi x and we have to find the curves where ρ ( x ) dx is real. It iseasy to see that the only possibility is the circle centered at the origin with an arbitraryradius. From the above arguments one can expect that for any α the roots will belong tosome circle. However, we analysed only the curves with zero mode number and as we see onthe figure 9 for smaller α ’s the circle develops four tails and two vertical lines. Along thesevertical lines the roots are separated by i (for L → ∞ ) forming the so called condensates or Bethe strings . The tails meet at the points where the virtual fluctuation is and thecorresponding curves are given by p ( z ) ± ππi ∂ t z = ± n = ± π jump log condensate with the Bethe roots separated by i/ igure 10: Dual configuration to 1-cut solution. Similar to the previous example for the large α the dual roots are distributed along the big circle and cut (first picture). When the α decreasesand the circle crosses the cut we have to choose another curve with the positive density (secondand third pictures).
6. The AdS/CFT Bethe equations and the semiclassical quantization ofthe superstring on
AdS × S The Beisert-Staudacher (BS) equations [25] are a set of 7 asymptotic [56] Bethe equations(the rank of the symmetry group
P SU (2 , | N = 4 SYM single trace operators with a large number of fields as wellas the energy of the dual string states . The perturbative gauge theory and the classicalstring regimes are interpolated by these equations through the t’Hooft coupling λ . In [57],based on an hypothesis for a natural extension for the quantum symmetry of the theory,Beisert found (up to a scalar factor) an S-matrix from which the BS equations would bederived. The scalar factor was then conjectured in [58, 59] from the string side – usingthe Janik’s crossing relation [60] – and in [61, 62] from the gauge theory point of view –based on several heuristic considerations [63]. From the gauge theory side these equationswere tested quite recently up to four loops [64, 65, 66]. From the string theory point ofview the scalar factor recently passed several nontrivial checks [67, 37, 68, 69] where severalloops were probed at strong coupling. Also at strong coupling, the full structure of the BSequations was derived up to two loops in [70, 71] in a particular limit [72] where the sigmamodel is drastically simplified.In this section we will check that the BS equations reproduce the 1–loop shift around any classical string soliton solution with exponential precision in the large angular momen- These large traces can be though of as spin chains and then the dilatation operator behaves like a spinchain Hamiltonian which turns out to be integrable [29, 30]. In this way Bethe equations appear naturallyfrom the gauge theory side. The existence of a finite gap description of the classical string motion [8, 9] lead to the belief thatthese equations ought to be the continuous limit of some quantum string Bethe equations. In other words,the Riemann surfaces present therein should in fact be the condensation of a large number of Bethe roots.Inspired by these finite gap constructions these quantum equations were proposed shortly after [55, 25]. – 25 –um in the string state. To do so our computation is divided into two main steps. On theone hand we will compute the 1 / √ λ corrections to Bethe equations in the scaling limit. Wewill have to use the technology developed in the previous sections in order to understandprecisely the several sources of corrections, the most subtle of all being the fine structureof the cuts of stacks which are generically present . At the end we will find out someintegral equation corrected by a 1 / √ λ term.On the other hand we start from the algebraic curve description of the string classicalmotion [8, 9]. The integral equations present in this finite gap formalism coincide with thescaling limit of the Bethe equations. Then we find how to correct this equations in sucha way that they will now describe not only the classical motion but also the semi-classicalquantization of the theory around any classical motion . For example we will find out howto modify the equations in such a way that they exhibit a very nontrivial property: the firstfinite corrections to any classical configurations equals the sum of quantum fluctuationsaround this same classical configuration. Then we show that, modified in this way, theintegral equations coincide precisely with the scaling limit expansion of the BS equationswith the HL phase [39] (up to some exponencially supressed wrapping effects, irrelevantfor large angular momentum string states)! In this way we establish that, to this order in1 / √ λ , the BS equations do provide the correct quantization of the system.These Bethe equations are a deformation of the equations (2.1) through the introduc-tion of the map x + 1 x = 4 πu √ λ , x ± + 1 x ± = 4 π √ λ (cid:18) u ± i (cid:19) . As explained in section 2 for superalgebras the choice of Bethe equations is not unique. In[25] four choices are presented. We need only to consider two of them , corresponding tothe diagram in figure 1 or to the reflected path along the diagonal going from the lowerleft to the upper right corner.Moreover we consider a twisted version of these equations for the same reasons men-tioned in the previous sections. In [76, 77] a similar kind of twists were introduced in thestudy of a set of deformations of N = 4 SYM and of the dual sigma model. Our twists seemto be a simple change in boundary conditions via the introduction of a constant matrixlike (2.2). It would be interesting to see if they can also be given a deeper physical inter-pretation following the lines of these works. We should stress that the twists are used hereas a technical tool which will simplify our analysis because, in particular, it allows us todeal with well defined stacks in a regime where the dualities are nothing but an exchangeof Riemann sheets. We will explain in section 6.7 that we can then safely analyticallycontinue the results to zero twist. In [73, 74, 75] the scaling limit of the SU (3) sector was considered. It would be interesting to use ourtreatment, including stacks, to compute explicitly the finite size corrections in this subsector following thelines of these papers. In [25] we consider η = η = η . – 26 –he BS equations then read e iηφ − iηφ = K Y j =1 u ,k − u ,j + i u ,k − u ,j − i K Y j =1 − /x ,k x +4 ,j − /x ,k x − ,j ,e iηφ − iηφ = K Y j = k u ,k − u ,j − iu ,k − u ,j + i K Y j =1 u ,k − u ,j + i u ,k − u ,j − i K Y j =1 u ,k − u ,j + i u ,k − u ,j − i ,e iηφ − iηφ = K Y j =1 u ,k − u ,j + i u ,k − u ,j − i K Y j =1 x ,k − x +4 ,j x ,k − x − ,j ,e iηφ − iηφ = x − ,k x +4 ,k ! ηL K Y j = k u ,k − u ,j + iu ,k − u ,j − i K Y j − /x +4 ,k x − ,j − /x − ,k x +4 ,j ! η − (cid:0) σ ( x ,k , x ,j ) (cid:1) η (6.1) × K Y j =1 − /x − ,k x ,j − /x +4 ,k x ,j K Y j =1 x − ,k − x ,j x +4 ,k − x ,j K Y j =1 x − ,k − x ,j x +4 ,k − x ,j K Y j =1 − /x − ,k x ,j − /x +4 ,k x ,j ,e iηφ − iηφ = K Y j =1 u ,k − u ,j + i u ,k − u ,j − i K Y j =1 x ,k − x +4 ,j x ,k − x − ,j ,e iηφ − iηφ = K Y j = k u ,k − u ,j − iu ,k − u ,j + i K Y j =1 u ,k − u ,j + i u ,k − u ,j − i K Y j =1 u ,k − u ,j + i u ,k − u ,j − i ,e iηφ − iηφ = K Y j =1 u ,k − u ,j + i u ,k − u ,j − i K Y j =1 − /x ,k x +4 ,j − /x ,k x − ,j . In fact, in order for the fermionic duality [25] (which we will review below) to exist, thetwists must not be completely independent but rather φ − φ + η K X j =1 i log x +4 x − = φ − φ ,φ − φ + η K X j =1 i log x +4 x − = φ − φ . (6.2)The energy (the anomalous dimension) can then be read from δD = √ λ π K X j =1 ix +4 ,j − ix − ,j ! . (6.3)To describe classical solutions (and to semi-classically quantize them) we should considerthe scaling limit where √ λ ∼ u ∼ K a ∼ L ≫ . In this limit we have x ± = x ± i α ( x ) + O (cid:18) λ (cid:19) – 27 –here α ( x ) ≡ π √ λ x x − . It is then useful to introduce the resolvents F a ( x ) = X j u − u a,j ,G a ( x ) = X j α ( x a,j ) x − x a,j , ¯ G a ( x ) = X j α (1 /x a,j ) x − /x a,j H a ( x ) = X j α ( x ) x − x a,j , ¯ H a ( x ) = X j α (1 /x )1 /x − x a,j and build with them eight quasi-momenta ( J = L/ √ λ ) p = + 2 π J x − δ η, +1 Q + δ η, − Q xx − η (cid:0) − H − ¯ H + ¯ H (cid:1) + φ p = + 2 π J x − δ η, − Q + δ η, +1 Q xx − η (cid:0) − H + H + ¯ H − ¯ H (cid:1) + φ p = + 2 π J x − δ η, − Q + δ η, +1 Q xx − η (cid:0) − H + H + ¯ H − ¯ H (cid:1) + φ p = + 2 π J x − δ η, +1 Q + δ η, − Q xx − η (cid:0) + H − H + ¯ H (cid:1) + φ p = − π J x − δ η, +1 Q + δ η, − Q xx − η (cid:0) − H + H − ¯ H (cid:1) + φ p = − π J x − δ η, − Q + δ η, +1 Q xx − η (cid:0) − H + H + ¯ H − ¯ H (cid:1) + φ p = − π J x − δ η, − Q + δ η, +1 Q xx − η (cid:0) − H + H + ¯ H − ¯ H (cid:1) + φ p = − π J x − δ η, +1 Q + δ η, − Q xx − η (cid:0) + H + ¯ H − ¯ H (cid:1) + φ (6.4)where G ( x ) ≡ − P ∞ n =0 Q n +1 x n . We can also write2 π √ λ δ D = Q . Then, to leading order, these quasi-momenta define an eight-sheet Riemann surface andthe BS equations read simply p/ i − p/ j = 2 πn ij in each of the cuts C ij uniting p i and p j .Finally, in this section we will usecot ij ≡ α ( x ) p ′ i − p ′ j p i − p j note that F a ( x ) = G a ( x ) + ¯ G a ( x ) = H a ( x ) + ¯ H a ( x ) . – 28 – igure 11: The several physical fluctuations in the string Bethe ansatz. The 16 elementary physicalexcitations are the stacks (bound states) containing the middle node root. From the left to theright we have four S fluctuations, four AdS modes and eight fermionic excitations. The bosonic(fermionic) stacks contain an even (odd) number of fermionic roots represented by a cross in the psu (2 , |
4) Dynkin diagram in the left.
In this section we will expand BS equations in the scaling limit for the roots belonging toa cut containing middle node roots x only. We do not assume that all the others cuts areof the same type, rather they can be cuts of stacks of several sizes. In the section 5.3 wewill generalize the results obtained in this section to an arbitrary cut, assuming, as in theprevious section, that the cuts are small enough and twists are not zero so that stacks arestable. We will discus in section 6.7 what happens when one takes all twists to zero.To leading order, the middle node equation (6.1) can be simply written as p/ − p/ = 2 πn while at 1–loop the first product in the r.h.s. of (6.1) corrects this equation due to1 i log K Y j = k (cid:18) u ,k − u ,j + iu ,k − u ,j − i (cid:19) ≃ /F ( x ) + α ( x ) πρ ′ ( x ) cot( πρ ( x )) (6.5)where ρ ( x ) = dkdu k . Expansion of the remaining terms in (6.1) will not lead to the appearanceof such anomaly like terms since the roots of another types are separated by ∼ x ,k .Thus we have simply2 πn = p/ − p/ − η α ( x ) πρ ′ ( x ) cot( πρ ( x )) , x ∈ C (6.6)In the next sections we will use dualities of the BS equations to get some extra informationabout cuts of stacks and generalize the above equation to any possible type of cut. Toachieve this we shall recast this equation in terms of the middle node roots x only. Obviously, the behavior of the Bethe roots will be as described in section 2 for a simplerexample of a su (1 ,
2) spin chain, that is, we will have simple cuts made out of x roots onlyand also cuts of stacks with x , x and x roots for example. Consider such cut of stacks.– 29 –learly, to be able to write the middle node equation (6.1) or (6.6) we need to computethe density mismatches ρ − ρ and ρ − ρ which are 1-loop contributions we must takeinto account if we want to write an integral equation for the middle node equation in termsof the density ρ of momentum carrying roots only. In this section we shall analyze thedualities present in the BS Bethe equations. By analyzing them in the scaling limit we willthen be able to derive the desired density mismatches. In [25] it was shown that the BS equations obey a very important fermionic duality. Sincewe chose to work with a subset of the possible Bethe equations, that is the ones with η = η = η present in [25], we should apply the duality present below not only to thefermionic roots x and x (as described below) but also to the Bethe roots x and x .Obviously the duality for x and x is exactly the same as for x and x and so we willfocus simply on the latter while keeping implicit that we always dualize all the fermionicroots at the same time.We construct the polynomial ( τ = η ( φ − φ )) P ( x ) = e + i τ K Y j =1 ( x − x +4 ,j ) K Y j =1 ( x − x − ,j )( x − /x − ,j ) − e − i τ K Y j =1 ( x − x − ,j ) K Y j =1 ( x − x +2 ,j )( x − /x +2 ,j ) (6.7)of degree K + 2 K which clearly admits x = x ,j and x = 1 /x ,j as K + K zeros . Theremaining K + 2 K − K − K roots are denoted by ˜ x ,j or 1 / ˜ x ,j depending on whetherthey are outside or inside the unit circle respectively, P ( x ) = 2 i sin( τ / K Y j =1 ( x − /x ,j ) ˜ K Y j =1 ( x − / ˜ x ,j ) K Y j =1 ( x − x ,j ) ˜ K Y j =1 ( x − ˜ x ,j ) (6.8)Then we can replace the roots x ,j , x ,j by the roots ˜ x ,j , ˜ x ,j in the BS equations providedwe change the grading η → − η and interchange the twists φ ↔ φ and φ ↔ φ . In fact,since we should also dualize the remaining fermionic roots, we should also change φ ↔ φ and φ ↔ φ and replace the remaining fermionic roots x and x .Since to the leading order x ± ≃ x each root will belong to a stack which must alwayscontain a momentum carrying root x . We have therefore ˜ K = K − K and ˜ K = K + K − K . Thus we label the Bethe roots as x ,j = x ,j − ǫ ,j , j = 1 , . . . , K ˜ x ,j = x ,j + K − ˜ ǫ ,j , j = 1 , . . . , ˜ K x ,j = x ,j − ǫ ,j , j = 1 , . . . , K x ,j = x ,j − ǫ ,j , j = 1 , . . . , K ˜ x ,j = x ,j + K − ˜ ǫ ,j , j = 1 , . . . , ˜ K we also have 1 /x has zeros because, due to (6.2), the equation for x ,j is the same as the equation for x ,j if we replace x ,j by 1 /x ,j . This is why the restriction (6.2) of the twists is so important. – 30 –ith ǫ ∼ / √ λ . Dividing (6.7) and (6.8) by Q K j =1 ( x − x ,j ) Q K j =1 ( x − x ,j )( x − /x ,j ) wehave e + i τ K Y j =1 x − x +4 ,j x − x ,j K Y j =1 x − x − ,j x − x ,j x − /x − ,j x − /x ,j − e − i τ K Y j =1 x − x +4 ,j x − x ,j K Y j =1 x − x +2 ,j x − x ,j x − /x +2 ,j x − /x ,j = 2 i sin( τ / K Y j =1 x − /x ,j x − /x ,j ˜ K Y j =1 x − / ˜ x ,j x − /x ,K + j K Y j =1 x − x ,j x − x ,j ˜ K Y j =1 x − ˜ x ,j x − x ,K + j (6.9)In this form it is easy to expand the duality relation in powers of 1 / √ λ . By expanding allfactors in (6.9) such as K Y j =1 x − x ± ,j x − x ,j = exp K X j =1 log x − x ± ,j x − x ,j ≃ exp ∓ i G ( x ) + K X j ǫ ,j x − x ,j , we findsin (cid:18) η ( p − p )2 (cid:19) = sin (cid:16) τ (cid:17) exp (cid:18) + X ǫ x − x + X ˜ ǫ x − x − X ǫ x − x (cid:19) × exp (cid:18) − X ǫ /x x − /x − X ˜ ǫ / ˜ x x − / ˜ x + X ǫ /x x − /x (cid:19) . Then, similarly to what we had in section 3.2 for the bosonic duality, we notice that α ( x ) ∂ x (cid:18)X ǫ x − x + X ˜ ǫ x − ˜ x − X ǫ x − ˜ x (cid:19) = H + H ˜3 − H − H , with a similar expression for the argument of the second exponential. Thus finally we get( H + H − H − H ˜3 ) + (cid:0) ¯ H − ¯ H − ¯ H ˜1 (cid:1) = − cot , or alternatively, using the x → /x symmetry transformation properties of the quasi-momenta, (cid:0) ¯ H + ¯ H − ¯ H − ¯ H ˜3 (cid:1) + ( H − H − H ˜1 ) = − cot . From this expressions we can deduce several properties of the density mismatches we wantedto obtain. For example, if we compute the discontinuity of (6.3.1) at a cut containing roots x , that is in a large cut of stacks C ,i> , we immediately get ρ − ρ = − ∆ cot πi , x ∈ C ,i> . (6.10)Proceeding in a similar way we find ρ − ρ = − ∆ cot πi , x ∈ C ,i> , (6.11) ρ − ρ = ρ − ρ ˜3 , x ∈ C ,i> ∪ C ,i> . (6.12)Let us now show that in the scaling limit the fermionic duality corresponds just to theexchange of the sheets { p i } of the Riemann surface. For illustration let us pick p and see– 31 – igure 12: Action of the duality on a long stack. By successively applying the fermionic and thebosonic dualities duality we can reduce the size of any large cut. One should not forget to changethe sign of the grading η after applying the fermionic duality. how it transforms under the duality. By definition the fermionic duality corresponds to thereplacement η → − η, H → H ˜1 , H → H ˜3 and φ ↔ φ , φ ↔ φ , so that p → π J x − δ η, − Q + δ η, +1 Q xx − − η (cid:0) − H ˜1 − ¯ H ˜3 + ¯ H (cid:1) + φ = p + η cot In the same way we get p → p + η cot , p → p − η cot , p → p − η cot , and since cot ij ∼ / √ λ we see that to the leading order the duality indeed just exchangesthe sheets. The bosonic nodes of the BS equations are precisely as in the usual Bethe ansatz discussed inthe first sections so that we can just briefly mention the results. The duality ( τ = η ( φ − φ )) e + i τ ˜ Q ( u − i/ Q ( u + i/ − e − i τ ˜ Q ( u + i/ Q ( u − i/
2) = 2 i sin τ Q ( u ) Q ( u )leads to ( H + H − H − H ˜2 ) + ( ¯ H + ¯ H − ¯ H − ¯ H ˜2 ) = cot (6.13)which implies ρ − ρ = + ∆ cot πi , x ∈ C ,i> As we already discussed in section 2 the bosonic duality also amounts to an exchangeof Riemann sheets. Indeed, under the replacement H → H ˜2 and φ ↔ φ , we find p → p − η cot , p → p + η cot which again, to the leading order in √ λ , is just the exchange of the sheets of the curve.– 32 – ,i C ,i C ,i πi ( ρ − ρ ) − ∆ cot πi ( ρ − ρ ) − ∆ cot +∆ cot πi ( ρ − ρ ) +∆ cot − ∆ cot − ∆ cot Table 1:
Densities missmatches
Using bosonic and fermionic dualities separately we already got some information aboutthe several possible mismatches of the densities inside the stack. To compute the missingmismatches we have to use both dualities together. For example suppose we want tocompute ρ − ρ in a cut C ,i> . We start by one such large cut of stacks (see figure 12a)and we apply the fermionic duality to this configuration so that we obtain a smaller cut asdepicted in figure 12b. For this configuration we can use (6.3.2) to get ρ − ρ ˜3 = + ∆ cot πi . However, from (6.12), this is also equal to the mismatch we wanted to compute, that is ρ − ρ = + ∆ cot πi , x ∈ C ,i> . To compute the last mismatch we apply the bosonic duality to get a yet smaller cut as infigure 12c for which we use (6.11) to get ρ ˜3 − ρ = − ∆ cot πi . Again, from (6.12), we can revert this result into a mismatch for the configuration beforeduality, that is ρ − ρ = − ∆ cot πi , x ∈ C ,i> . Let us then summarize all densities mismatches in table 1.
In this section we shall recast equation (6.6) or η π J x − δ η, +1 Q − δ η, − Q xx − /H − H − H − ¯ H − ¯ H = 2 πn + ηφ − ηφ − cot (6.14)in terms of the density ρ ( x ) of the middle roots x . To do so we only need to replace theseveral densities by the middle node density ρ ( x ) using the several density mismatchespresented in table 1. Defining H ij ( x ) ≡ Z C ij α ( x ) α ( y ) ρ ( y ) x − y dy – 33 –e can then rewrite equation (6.14) in terms of the middle node roots only, η π J x − δ η, +1 Q − δ η, − Q xx − /H + H + H − H − ¯ H − ¯ H = 2 πn + ηφ − ηφ − cot + X ≤ i ≤ ≤ j ≤ ( I i ij + I jij ) + X ≤ i ≤ ≤ j ≤ (¯ I i j + ¯ I ji ) (6.15)where x ∈ C and I klij ( x ) = ( − F kl Z C ij α ( x ) α ( y ) ∆ cot kl x − y dy πi , I kkij ( x ) ≡ , ¯ I klij ( x ) = I klij (1 /x ) . The several dualities amount to an exchange of Riemann sheets so that the cuts C ij → C i ′ j ′ with the subscripts in H ij changing accordingly. The middle roots x are never touched inthe process. Moreover to leading order p i ↔ p i ′ and thus the r.h.s. of (6.15) is also triviallychanged under the dualities. Therefore, as in section 3 (see (3.4) and (3.5)), we can nowtrivially write the corrected equation when x belongs to any possible type of cut of stacksby applying the several dualities to equation (6.15). In this section we shall find the integral equation (6.15) from the field theoretical point ofview like we did in section 4.1 and in appendix B. That is, we will find what the correctionsto the classical (leading order) equations [9] η π J x − δ η, +1 Q − δ η, − Q xx − /H − H − H − ¯ H − ¯ H = 2 πn + ηφ − ηφ , (6.16) should be in order to describe properly the semi-classical quantization of the string (and notonly the classical limit). We will find that this construction leads precisely to the integralequation (6.15) thus showing that the BS nested Bethe ansatz equations do reproduce the1-loop shift around any (stable) classical solution with exponential precision (in some largecharge of the classical solution). This section is very similar to section 4 and to AppendixB and thus we will often omit lengthy but straightforward intermediate steps. We assume i = 1 , . . . , j = 5 , . . . , ( − F of a virtual excitation for each possible modenumber n and polarization ij to each quasi-momenta. Notice that for this super-symmetricmodel the fluctuations can also be fermionic and indeed the grading ( − F equals +1 ( − ρ = ρ + δρ where ρ is the leading density, solution of the leading (classical)equation (6.16), while ρ obeys the corrected (semi-classical) equation. For example, if weconsider x ∈ C , , the starting point should be (see [32] for a similar analysis) − xδ η, − δ Q x − Z C α ( x ) α ( y ) δρ ( y ) x − y + Z C α ( x ) α ( y ) δρ ( y ) x − y + Z C α ( x ) α ( y ) δρ ( y ) x − y – 34 – Z C α (1 /x ) α ( y ) δρ ( y )1 /x − y − Z C α (1 /x ) α ( y ) δρ ( y )1 /x − y − Z C α (1 /x ) α ( y ) δρ ( y )1 /x − y + N X n = − N X i< α ( x ) x − x i n + X j> α ( x ) x − x jn − X i< α (1 /x )1 /x − x i n − X j> α (1 /x )1 /x − x jn = 0 (6.17)Then, by construction, the charges Q r = Z C ρ ( y ) y r dy + X n X ij ( − F ij α ( x ijn )2( x ijn ) r = Z C ρ ( y ) y r dy + X ij ( − F ij I x ijn cot ij y r dy πi (6.18)will take the 1 / √ λ corrected values. It is clear that, as before, we do not include thenew virtual excitations in the density ρ ( x ). Similarly to (4.12) and (7.11), if we want thecharges to have the standard form Q r = Z ̺ ( y ) y r dy we must redefine the density as ̺ = ρ + 14 πi X ij ′ ≥ ( − F jj ′ ∆ cot jj ′ . Now we want to go back to the integral equation (6.17) and rewrite it using the density δ̺ = ̺ − ρ . For example, for x ∈ C ,2 Z C α ( x ) α ( y ) δρ ( y ) x − y + Z C α ( x ) α ( y ) δρ ( y ) x − y + Z C α ( x ) α ( y ) δρ ( y ) x − y + N X n = − N X i ( − F i α ( x ) x − x i n + X j ( − F j α ( x ) x − x jn =2 Z C α ( x ) α ( y ) δ̺ ( y ) x − y + Z C α ( x ) α ( y ) δ̺ ( y ) x − y + Z C α ( x ) α ( y ) δ̺ ( y ) x − y + cot − X ij (cid:16) I iij + I j ij (cid:17) − X ij (cid:16) ¯ I i j + ¯ I j i + ¯ I iij + ¯ I jij (cid:17) where the identity ( − F i cot ,i = − X j (cid:0) I i j + I iij (cid:1) − X j (cid:16) ¯ I i j + ¯ I i ¯ ij (cid:17) , where ¯¯ i = i, ¯1 = 4 , ¯2 = 3, is being used. Now, when x ∈ C , we will get2 Z C α ( x ) α ( y ) δρ ( y ) x − y + Z C α ( x ) α ( y ) δρ ( y ) x − y + Z C α ( x ) α ( y ) δρ ( y ) x − y – 35 – N X n = − N X i ( − F i α ( x ) x − x i n + X j ( − F j α ( x ) x − x jn =2 Z C α ( x ) α ( y ) δ̺ ( y ) x − y + Z C α ( x ) α ( y ) δ̺ ( y ) x − y + Z C α ( x ) α ( y ) δ̺ ( y ) x − y − X ij (cid:16) I iij + I j ij − I i j − I j i (cid:17) Finally we can use the x to 1 /x symmetry to translate last equality into one for x ∈ C .Subtracting it from the previous equation we see that the 1 / √ λ corrected equation willcorrespond to adding − cot + X ij ( I iij + I iij + ¯ I i j + ¯ I j i )to the r.h.s. of (6.16) thus obtaining, after the identification ̺ = ρ , precisely the finitesize corrected equation (6.15) obtained from the NBA point of view! In the last section we showed that the one loop shift as a sum of all fluctuation energies(or others local charges) perfectly matches the finite size corrections in the NBA equations.However we systematically dropped the contours around the unit circle.For example, when we blow the contour in the last term of (6.18), we also get somecontribution from the singularities inside the unit circle. That is we will have an extracontribution to the charges given by an integral over the unit circle. Also, take (6.17) forinstance. To pass to the r.h.s we transformed the collections of poles into integrals overthe excitation points and then we blew the contour which became a collection of contourson the several existing cuts. Again we dropped the contribution from the integrals overthe unit circle which would lead to an extra 1 / √ λ term in the r.h.s. of (6.15). In ourprevious paper [32] we showed that this extra contribution matches precisely the extracontribution coming from the Hernandez-Lopez phase in the NBA!However, as we explained in [32], in order to obtain precisely the HL phase a preciseprescription for the labeling of the mode numbers of the fluctuations must be given.Moreover, in [32], we assumed that everywhere we can replace cot (cid:16) p i ( x ) − p j ( x )2 (cid:17) by i sign(Im x ) with exponential precision in L √ λ . This is reasonable for generic points in theunit circle, where the imaginary part of p i ( x ) − p j ( x ) is large, but one has to carefullyanalyze the neighbourhood of the real axis, where this imaginary part vanishes.Let us consider these two subtle points in greater detail. As we emphasized in [31] if we number the fluctuation charges Q ijn differently we mightobtain different results for the 1-loop shift, that is for the graded sums of these fluctuation Recently the HL phase was also found [78] in the study of the open string scattering of giant magnons[79]. – 36 –harges. Thus a precise prescription for the labeling of the quantum fluctuations is crucial.In the appendix A of [32] we found out that the contribution of the integrals of the previoussection does reproduce the HL phase provided we number the quantum fluctuations locatedat x ijn according to p i ( x ijn ) − p j ( x ijn ) = 2 π ( n − m i + m j )with some specific choice of m i . Moreover we also showed that for the same choice of m i thecontribution to the charges coming from the above mentioned integrals over the unit circleis zero. Using the x to 1 /x symmetries following from the definition of the quasi-momenta(6.4) plus the restriction (6.2) on the twists, we can redo the computation in the AppendixA of [32] to find that the condition on the m i now reads( m + m − m − m ) ( m + m − m − m ) = 0so that, in particular, m i = 0 does the job nicely. We see that, with the introduction of thesetwists and subsequent redefinition of the quasimomenta, the prescription for the labeling ofthe excitations becomes absolutely natural and algebraic curve friendly [31]. This answersthe question raised in [32] concerning the naturalness of the presciption needed to obtainthe HL phase [39] – see appendix A in [32]. Let us now us focus on the vicinity of x = 1 where we have the following expansion of thequasi-momenta p i ( x ) − p j ( x )2 = β ij x − . . . where β ij is usually of order L/ √ λ (and should be so for the asymptotical BAE to bevalid). We will consider the circle with radius x ijN +1 / ≃ πNβ ij , where N is some largecutoff in the sum of fluctuations (6.17). We want to estimate Z α ( x ) f ( x ) (cid:20) cot (cid:18) p i − p j (cid:19) + i sign(Im x ) (cid:21) ( p ′ i − p ′ j ) dx . This integral is dominated for x ≃ ± x ≃ Z α ( x ) f ( x ) (cid:20) cot (cid:18) p i − p j (cid:19) + i sign(Im x ) (cid:21) ( p ′ i − p ′ j ) dx = iπ f (1)6 β ij √ λ + O (cid:18) N (cid:19) which is zero under the sum over all polarizations. For example( − F β = − ( − F β . Thus we can indeed drop the cot’s when integrating over the unit circle and thus we finallyconclude that the one loop shift to any local charge computed from the BS equations withthe Hernandez-Lopez phase is indeed given by the sum of fluctuations as predicted by fieldtheoretical arguments. – 37 – .7 Zero twist and large fillings via analytical continuation
Although we always assumed the twists to be sufficiently large and the fillings to be suf-ficiently small we can always analytically continue the results towards zero twists or largefilling fractions. Let us briefly explain why. In the scaling limit, for large twists, the bosonicduality we introduced amounts to a simple exchange of sheets in some Riemann surface, p a ( x ) ↔ p b ( x ). As we saw in section 5.3 what happens when the twists start to becomevery small is that the quasi-momenta are still simply exchanged but in a piecewise manner,that is, we can always split the complex planes in some finite number of regions where thebosonic duality simply means p a ( x ) ↔ p b ( x ). Thus, from the e ip algebraic curve point ofview nothing special occurs for what analyticity is concerned and therefore we can safelyanalytically continue our findings to any value of the twists. Exactly the same analysisholds for the filling fractions. Moreover, for the usual Bethe system, we defined a set ofquasi-momenta, which constitute an algebraic curve to any order in 1 /L , and therefore wedon’t expect analyticity to break down at any order in 1 /L .We also preformed a high precision numerical check concluding that there is no sin-gularity when the configuration of the Bethe roots is affected by this partial reshuffling ofthe sheets and that finite size corrections are still related to the same sum of fluctuations,which are analytical functions w.r.t. the twists.
7. Conclusions
In this paper we studied generic nested Bethe ansatz (NBA) equations, the correspondingscaling limit and its leading finite size corrections. Let us summarize briefly our mainresults • We found out that the introduction of some extra phases, called twists, are crucialfor the formation of bound states of roots of different types, called in the literatureby stacks [18]. Strictly speaking without these twists the stacks do not exist. Seesections 2 and 5.3. • We understood how to use the bosonic duality between various systems of Betheroots which is present even in the absence of any fermionic symmetry. In the scalinglimit we showed that this duality amounts to a reshuffling of Riemann sheets of thealgebraic curve formed by the condensation of Bethe roots. See sections 2 and 5. • We explained how to write down the integral equation describing the leading finitesize corrections around generic NBA’s for (super) spin chains by using the transfermatrices for (super) group along with some
T Q relations. See section 3.6 • We provided an alternative derivation of this integral equation using an independentpath, namely using the dualities present in the Bethe equations allowing one to getrid of the several stacks and reduce the size of any cut by successive application ofseveral dualities. See section 3.2. – 38 –
We obtained the integral equation describing the finite size corrections to the Beisert-Staudacher equations [25] with the Hernandez-Lopez phase [11, 39] in the scaling limit(to do so we were forced to use the duality approach because at present the psu (2 , | ). See section 6. • In the scaling limit Beisert-Staudacher equations [25] describe the classical motion ofthe superstring on
AdS × S through the finite gap curves of [9]. Thus the integralequation we found should reproduce the 1-loop shift for all the charges around anyclassical string motion and this is obviously a very nontrivial check of the validity ofthe BS equations. We show that this equation indeed mimics the presence of a seaof virtual particles thus proving this general statement. See section 6.5. Acknowledgments
We would like to thank T. Bargheer, N. Beisert, J. Penedones, A, Rej, K. Sakai, M. Stau-dacher, A. Zabrodin and especially V. Kazakov for many useful discussions. The work ofN.G. was partially supported by French Government PhD fellowship, by RSGSS-1124.2003.2and by RFFI project grant 06-02-16786. N.G. thanks CFP, where part of this work wasdone, for the hospitality during his visit. N.G and P.V thank AEI Potsdam, where part ofthis work was done, for the hospitality during the visit. P. V. is funded by the Funda¸c˜aopara a Ciˆencia e Tecnologia fellowship SFRH/BD/17959/2004/0WA9. P.V. thanks PNPI,where part of this work was done, for the hospitality during his visit.
Appendix A: Transfer matrix invariance and the bosonic duality for SU ( K | M ) supergroups In this section we review the formalism of [24] which allows one to derive the transfer ma-trices of usual (super) spin chains in any representation. We will use this general formalismto prove the invariance under the bosonic dualities of all possible transfer matrices one canbuild. The transfer matrices presented in section 3.6 can be obtained trivially using thisformalism .As mentioned in section 2, for the standard SU ( K | M ) super spin chains (based on thestandard R –matrix R ( u ) = u + i P with P the super permutation) we can find the (twisted)transfer matrix eigenvalues for the single column young tableau with a boxes through the non-commutative generating functions [24, 40] ∞ X a =0 ( − a e ia∂ u T a ( u ) Q K,M ( u + ( a − K + M + 1) i/ e ia∂ u = −→ Y ( x,n ) ∈ γ ˆ V − x,n ( u ) (7.1) See section 6 in [80] for some attempts to fill this gap. We should mention that the transfer matrices in section 3.6 are not exactly the same we have in thisAppendix but can be obtained from these via a trivial rescaling in u which obviously does not spoil theinvariance of these objects. – 39 –here γ is a path starting from ( M, K ) and finishing at (0 ,
0) (always approaching thispoint with each step) in a rectangular lattice of size M × K as in figure 1 , x = ( m, k ) ispoint in this path and n = (0 , −
1) or ( − ,
0) is the unit vector looking along the next stepof the path. Each path describes in this way a possible Dynkin diagram of the SU ( K | M )super group with corners denoting fermionic nodes and straight lines bosonic ones, seefigure 1. Finally,ˆ V − m,k ) , (0 , − ( u ) = e iφ k Q k,m ( u + i ( m − k − / Q k,m ( u + i ( m − k + 1) / Q k − ,m ( u + i ( m − k + 2) / Q k − ,m ( u + i ( m − k + 0) / − e i∂ u ˆ V − m,k ) , ( − , ( u ) = (cid:18) e iϕ m Q k,m − ( u + i ( m − k − / Q k,m − ( u + i ( m − k + 0) / Q k,m ( u + i ( m − k + 1) / Q k,m ( u + i ( m − k − / − e i∂ u (cid:19) − where Q k,m is the Baxter polynomial for the roots of the corresponding node and { φ k , ϕ m } are twists introduced in the transfer matrix [40]. Let us then consider a bosonic node likethe one in the middle of figure 1 (the vertical bosonic node is treated in the same fashion).If the position of this node on the M × K lattice is given by ( m, k ) then it is obviousthat the only combination containing Q m,k in the right hand side of (7.1) comes from theproduct of ˆ V − m,k ) , ( − , ( u ) ˆ V − m +1 ,k ) , ( − , ( u ) which reads (cid:20) e iϕ m + ϕ m +1 Q k,m +1 ( u + i ( m − k + 2) / Q k,m +1 ( u + i ( m − k + 0) / Q k,m − ( u + i ( m − k − / Q k,m − ( u + i ( m − k + 0) /
2) + e i∂ u −− (cid:18) e iϕ m +1 Q k,m ( u + i ( m − k − / Q k,m ( u + i ( m − k + 1) / Q k,m +1 ( u + i ( m − k + 2) / Q k,m +1 ( u + i ( m − k + 0) /
2) ++ e iϕ m Q k,m − ( u + i ( m − k + 0) / Q k,m − ( u + i ( m − k + 2) / Q k,m ( u + i ( m − k + 3) / Q k,m ( u + i ( m − k + 1) / (cid:19) e i∂ u (cid:21) − (7.2)So, if we want to study the bosonic duality on the node ( k, m ) and its relation withthe invariance of several transfer matrices we need to study the last two lines of thisexpression. For simplicity let us shift u , omit the subscript k in the Baxter polynomials Q k,m − , Q k,m , Q k,m +1 and define the reduced transfer matrix as t ( u, ϕ m , ϕ m +1 ) ≡ e iϕ m +1 Q m ( u − i ) Q m ( u ) Q m +1 ( u + i/ Q m +1 ( u − i/
2) + e iϕ m Q m − ( u − i/ Q m − ( u + i/ Q m ( u + i ) Q m ( u ) . (7.3)Notice that the absence of poles at the zeros of Q m yields precisely the Bethe equationsfor this auxiliary node. Bosonic duality ⇒ Transfer matrices invariance
Thus, to check the invariance of the transfer matrices in all representations it suffices toverify that the reduced transfer matrix t ( u, ϕ m , ϕ m +1 ) is invariant under ϕ m ↔ ϕ m +1 and Notice that the path goes in opposite direction compared to the labelling a of the Baxter polynomial Q a used before. In the notation of this section Q k,m corresponds to the node is at position (m,k) in thislattice. ˆ Q , is normalized to 1. If we are considering a spin in the representation where the first Dynkin nodehas a nonzero Dynkin label then Q M,K will play the role of the potential term. In general the situation ismore complicated, see [24]. In any case we are mainly interested in the dualization of roots which are notmomentum carrying thus we need not care about such matters. – 40 – m → ˜ Q m where2 i sin (cid:18) ϕ m +1 − ϕ m (cid:19) Q m − ( u ) Q m +1 ( u ) = (7.4) e i ϕm +1 − ϕm Q m ( u − i/
2) ˜ Q m ( u + i/ − e − i ϕm +1 − ϕm Q m ( u + i/
2) ˜ Q m ( u − i/ . which can be easily verified. If suffices to replace, in t ( u, ϕ m , ϕ m +1 ) in (7.3), Q m ( u − i ) Q m ( u ) → e − i ( ϕ m +1 − ϕ m ) ˜ Q m ( u − i )˜ Q m ( u )+2 ie − i ϕm +1 − ϕm sin (cid:18) ϕ m +1 − ϕ m (cid:19) Q m − ( u + i/ Q m +1 ( u + i/ Q m ( u ) ˜ Q m ( u ) ,Q m ( u + i ) Q m ( u ) → e + i ( ϕ m +1 − ϕ m ) ˜ Q m ( u + i )˜ Q m ( u ) − ie − i ϕm +1 − ϕm sin (cid:18) ϕ m +1 − ϕ m (cid:19) Q m − ( u − i/ Q m +1 ( u − i/ Q m ( u ) ˜ Q m ( u ) , which are obvious consequences of the bosonic duality. Transfer matrix invariance ⇒ Bosonic duality
On the other hand suppose we have two solutions of Bethe equations, one of them character-ized by the Baxter polynomials { . . . , Q m − , Q m , Q m +1 , . . . } with twists { . . . , ϕ m , ϕ m +1 , . . . and another with { . . . , Q m − , ˜ Q m , Q m +1 , . . . } with twists { . . . , ϕ m +1 , ϕ m , . . . } for which thetransfer matrices are the same, that is t ( u, ϕ m , ϕ m +1 ) = ˜ t ( u, ϕ m +1 , ϕ m ) . (7.5)Then we can show that these two solutions are related by the bosonic duality (7.4). Indeedif we build the Wronskian like object W ( u ) ≡ e i ϕm +1 − ϕm Q m ( u − i/
2) ˜ Q m ( u + i/ Q m − ( u ) Q m +1 ( u ) − e − i ϕm +1 − ϕm Q m ( u + i/
2) ˜ Q m ( u − i/ Q m − ( u ) Q m +1 ( u ) . we can easily check that W ( u + i/ − W ( u − i/
2) = − e − i ϕm +1+ ϕm Q m ( u ) ˜ Q m ( u ) Q m − ( u − i/ Q m +1 ( u + i/ (cid:0) t ( u, ϕ m , ϕ m +1 ) − ˜ t ( u, ϕ m +1 , ϕ m ) (cid:1) = 0Since by definition W ( u ) is a rational function this means it must be a constant. Thusif ϕ m = ϕ m +1 we must have K m + ˜ K m = K m + K m +1 and the value of W can be readfrom the large u behavior. In this way we obtain precisely the bosonic duality (7.4). If ϕ m = ϕ m +1 then we see that K m + ˜ K m = K m + K m +1 + 1 and we will obtain a differentvalue for the constant W which will correspond to the untwisted bosonic duality describedin section 5.3.2. We would like to thank A.Zabrodin and V.Kazakov for sugesting this nice interpertation for the bosonicduality – 41 – ppendix B: Fluctuations for su ( n ) spin chains In this Appendix we consider a su ( n ) NBA with the Dynkin labels V a being +1 for aparticular a only (the generalization is obvious). This example is obviously more generalthan that considered in section 4.1 and can be a useful warmup for section 6.5 where wefind the integral equation describing the AdS × S su ( n ) NBA, in the classical limit, we will have n quasi-momenta each one above orbelow each of the n − . We label these quasi-momenta by p i ( p j ) with i, i ′ ( j, j ′ ) taking positive (negative) values for quasi-momenta above (below) the node for which V a = 0. Then let us mention how the equations in the previous section are generalized.We consider a middle node cut C , − . The analogue of equation (4.6) is now − x + X j Z C ,j δρ ( y ) x − y + X i Z C i, − δρ ( y ) x − y + N X n = − N L X i x − x i, − n + X j x − x ,jn = 0 (7.6)and the charges (4.7), (4.8), (4.9) and (4.11) become Q r − Z C ρ ( y ) y r dy = + X n X ij L x ijn ) r = + 12 L X ij J I x ijn cot ij y r dy πi (7.7)= + 12 L X ii ′ j I C i ′ j cot ij y r dy πi + 12 L X ijj ′ I C ij ′ cot ij y r dy πi (7.8)= − L X ii ′ j I C i ′ j cot ii ′ y r dy πi − L X ijj ′ I C ij ′ cot jj ′ y r dy πi (7.9)= − L Z C P i
4) transfer matrices for this (exotic) Bethe ansatz equations.).
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