Lifting asymptotic degeneracies with the Mirror TBA
IITP-UU-11/21SPIN-11/16
Prepared for submission to JHEP
Lifting asymptotic degeneracies with the Mirror TBA
Alessandro Sfondrini a and Stijn J. van Tongeren a a Institute for Theoretical Physics and Spinoza Institute,Utrecht University, 3508 TD Utrecht, The Netherlands
E-mail:
[email protected] , [email protected] Abstract:
We describe a qualitative feature of the AdS × S string spectrum which isnot captured by the asymptotic Bethe ansatz. This is reflected by an enhanced discretesymmetry in the asymptotic limit, whereby extra energy degeneracy enters the spectrum.We discuss how finite size corrections should lift this degeneracy, through both perturbative(L¨uscher) and non-perturbative approaches (the Mirror TBA), and illustrate this explicitlyon two such asymptotically degenerate states. a r X i v : . [ h e p - t h ] J a n ontents Y -functions 15B Expansions for Y ◦ Q functions 16 In the decompactification limit, both the light-cone gauge fixed AdS × S superstringand its AdS/CFT dual [1] N = 4 super Yang-Mills theory have a description through anasymptotic Bethe ansatz . This description does not apply to either theory at finite size,where the only current non-perturbative description is through a set of equations knownas the mirror thermodynamic Bethe ansatz (TBA) equations for the superstring.The idea of applying methods from integrable relativistic models at finite size [4] tothe AdS/CFT correspondence was initiated in [5] and explored in detail in [6]. The mainstep in deriving the mirror TBA equations is the formulation of the string hypothesis [7],which was done for the present model in [8] by using the mirror version of the Bethe-Yang equations [9] for the AdS × S superstring. This was followed by a derivation ofthe canonical [10–12] and simplified [13] TBA equations that describe the ground state ofthe theory . These equations have been used to analyze the vanishing of the ground stateenergy of the theory at finite size [15]. Importantly, these ground state equations can beused to obtain equations for the excited states, through a contour deformation trick [16] For a review of integrability in the AdS/CFT correspondence see [2, 3]. The associated Y-system was conjectured in [14]. – 1 –nspired by the analytic continuation procedure of [17]. Using the contour deformationtrick, the mirror TBA equations have been used to reproduce perturbative results foundthrough L¨uscher’s approach [19–21], and to study certain states in the sl (2) sector inconsiderable detail [22–24], specifically at intermediate coupling in [25, 26]. The analyticproperties of the Y-functions [23, 27–29] are essential in determining these equations, andhave proved useful for further understanding of the mirror TBA equations. Following thesedevelopments, discontinuity relations can now be used to find the excited state equationsdirectly in the sl (2) subsector [30], giving results in complete equivalence with the contourdeformation trick where applicable. Moreover, the simplified TBA equations have recentlybeen brought to a quasi-local form [31], another step in the direction of obtaining a so-called non-linear integral equation (NLIE) description of the spectral problem at finite size.Additional steps in this direction had already been taken in [32, 33].It is well known that the asymptotic Bethe ansatz (ABA) captures the leading 1 /J corrections to the asymptotic energy spectrum, while it misses the exponential correctionsdue to the finite system size. Therefore the ABA misses quantitative information on thespectrum, as is for example clearly illustrated by wrapping corrections to scaling dimensionsin the gauge theory. Following an observation of [16], in this paper we will show that forthe AdS × S superstring finite size effects are not only quantitative in nature. In fact wewill demonstrate that the ABA also misses qualitative information on the spectrum, owingto a discrete symmetry enhancement of the model in the asymptotic limit, so that certainstates become degenerate asymptotically.This paper is organized as follows. In the next section we start by discussing thesymmetries and degeneracies of the asymptotic Bethe ansatz and explain the asymptoticsymmetry enhancement alluded to just above. Also, we will indicate how finite size effectsshould lift this degeneracy. After painting the general picture we illustrate these ideas byconsidering two concrete states with degenerate energies in the asymptotic limit. We willshow that these states have manifestly different TBA equations and explicitly compute thedifferent finite size corrections they receive, in line with the general discussion. In general the energy spectrum of string states is expected to have degeneracies, owing tothe superconformal symmetry of the model. Because of this symmetry, string states arrangethemselves in superconformal multiplets which each share a common energy. At the level ofthe (asymptotic) Bethe ansatz these degeneracies are reflected by the fact that solutions tothe Bethe-Yang equations only give the highest weight states of the underlying symmetryalgebra, familiar from e.g. the Heisenberg [34] and Hubbard model [35]. Completeness ofthe Bethe ansatz then follows by adding the states which lie in the same multiplet, whichthen by construction have the same energy. In the case of the asymptotic Bethe ansatzfor the AdS × S superstring however, there is additional degeneracy, degeneracy whicharises in the decompactification limit and should not be present in the complete model.This degeneracy occurs in the asymptotic Bethe ansatz due to enhanced symmetry in the The use of L¨uscher’s approach [18] in the AdS/CFT correspondence was first advocated in [5]. – 2 –symptotic limit, indicating qualitative features of the model that are not captured by theasymptotic solution.In the light-cone gauge the superstring has manifest su L (2 | ⊕ su R (2 |
2) symmetry,where the subscript L and R distinguish the two su (2 |
2) factors, conventionally called leftand right. By construction, the model possesses a Z symmetry, which we call left-rightsymmetry, interchanging the sets of left and right su (2 |
2) charges. This means that forevery state with a given set of su L (2 | ⊕ su R (2 |
2) charges, there exists a state with equalenergy, with left-right interchanged su (2 |
2) charges. It is this left-right symmetry which isenhanced in the asymptotic limit to a larger discrete group. At the level of the Bethe-Yangequations this enhancement is manifested by the fact that they allow more than just aninterchange of complete sets of left and right charges, actually allowing free redistributionof roots between the left and right sectors in certain cases. In the finite size model on thecontrary, there is no reason to assume states related by such a redistribution should havethe same energy, so we expect that finite size effects lift this asymptotic degeneracy. Wewill now consider these ideas in more detail.
The Bethe-Yang equation for the AdS × S superstring in the light-cone gauge is given by[9] 1 = e ip k J K I (cid:89) l =1 l (cid:54) = k S sl (2) ( p k , p l ) (cid:89) α = L,R K
II( α ) (cid:89) l =1 x − k − y ( α ) l x + k − y ( α ) l (cid:115) x + k x − k , k = 1 , . . . K I . (2.1)In addition to the rapidities of fundamental particles, this equation contains y ( α ) roots.Together with the w ( α ) roots ( α = L, R ) which enter in the auxiliary Bethe equations justbelow, these correspond to the su L (2 | ⊕ su R (2 |
2) symmetry of the model. The auxiliaryBethe equations consist of two independent sets of two coupled equations for the y and w roots, given by1 = K I (cid:89) l =1 y ( α ) k − x − l y ( α ) k − x + l (cid:115) x + l x − l K III( α ) (cid:89) l =1 ν ( α ) k − w ( α ) l + ig ν ( α ) k − w ( α ) l − ig , k = 1 , . . . , K II( α ) , α = L, R , (2.2)1 = K II( α ) (cid:89) l =1 w ( α ) k − ν ( α ) l + ig w ( α ) k − ν ( α ) l − ig K III( α ) (cid:89) l =1 l (cid:54) = k w ( α ) k − w ( α ) l − ig w ( α ) k − w ( α ) l + ig , k = 1 , . . . , K III( α ) , α = L, R , (2.3)where ν ( α ) k = y ( α ) k + 1 /y ( α ) k . The fact that we have two sets of identical left-right decoupledequations corresponds directly to the left-right symmetry mentioned earlier. At the level ofthe transfer matrix this is reflected by the fact that we have a transfer matrix for each sector, T ( L ) and T ( R ) ; both are su (2 | su (2 |
2) multiplets.As mentioned, solutions of the auxiliary equations (2.2-2.3) identify highest weight states– 3 –f the su (2) subalgebras of this symmetry algebra, labeled by the Dynkin labels ( s ( α ) , q ( α ) ).The weights are encoded in the excitations numbers as s ( α ) = K I − K II( α ) , q ( α ) = K II( α ) − K III( α ) , (2.4)where the excitation numbers satisfy K I ≥ K II( α ) ≥ K III( α ) , α = L, R . (2.5)
As indicated, the discrete left-right symmetry of the light-cone gauge fixed model can beenhanced in the asymptotic limit, giving a higher amount of degeneracy in the spectrum.This is the case when K III( α ) = 0, for states with a given total number of y roots; (cid:80) α K II( α ) = K II Tot > For any such state , the auxiliary Bethe equations reduce to K I (cid:89) l =1 y ( α ) k − x − l y ( α ) k − x + l (cid:115) x + l x − l = 1 , k = 1 , . . . , K II( α ) , α = L, R , (2.6)showing that we have one and the same equation for each of the y ( α ) k [16]. The numberof solutions we can pick for each y depends on the main excitation number K I , and ingeneral we must take care to only allow for regular configurations of roots. Nonetheless,the consequence of this degeneration is immediately clear: provided there is more than oneallowed solution for y ( α ) k we can freely redistribute any number of different y roots betweenthe left and right sectors without changing the main Bethe-Yang equation , because it itselfcontains a product over the left and right roots. This is the enhancement of the left-right symmetry of the finite size model in the asymptotic limit. Let us note that thecorresponding symmetry group acts on regular highest weight states only, and that thisaction cannot be extended to the other states.Two states differing by such a redistribution will have the same asymptotic momentum,hence energy, while there is no reason to assume their energies should be identical outsidethe asymptotic regime. Rather, it would actually be a surprising coincidence if theirenergies agreed. Stated more strongly, looking at the description of the finite size modelthrough the mirror TBA it should be conceptually clear that this symmetry is only presentasymptotically; the presence of w roots generically spoils this symmetry, and while anindividual state might have no w roots, in the mirror TBA such a state is described ininteraction with a thermal background containing all possible excitations. Note also thattwo such asymptotically equivalent states have manifestly different Dynkin labels, whilethey are not in the same superconformal multiplet. Indeed, such states correspond topotentially wildly different operators in N = 4 SYM. As just stated, we expect finite size corrections to lift this degeneracy of the asymp-totic spectrum, whether it be through the thermodynamic Bethe ansatz, or perturbatively– 4 –hrough L¨uscher corrections. How this happens is perhaps most immediately seen throughthe complete formula for the energy of a string state in the mirror TBA approach [10] E = K I (cid:88) k =1 E k − π ∞ (cid:88) Q =1 (cid:90) dv d ˜ p Q dv log(1 + Y Q ) , (2.7)where E k = i ˜ p ( u ∗ k ) gives the asymptotic contribution to the energy, while the second termarises from the finite size of the system. In this formula, the Y Q -functions are determinedthrough the mirror TBA equations, which intricately couple the auxiliary left and right Y -functions. Now in general there is no reason to expect that the TBA equations for twoof these asymptotically degenerate states should be the same, meaning they should receivedifferent finite size corrections, lifting the degeneracy of the asymptotic spectrum.Alternately, expanding the energy formula (2.7) to leading order around the asymptoticsolution (small Y Q -functions) gives a formula in direct agreement with L¨uscher’s approach E LO = − π ∞ (cid:88) Q =1 (cid:90) dv d ˜ pdv Y ◦ Q ( v ) . (2.8)Here Y ◦ Q , in the above expanded to leading order in the coupling constant, is given by thegeneralized L¨uscher’s formula [36] Y ◦ Q ( v ) = e − J ˜ E Q ( v ) T ( L ) ( v | (cid:126)u ) T ( R ) ( v | (cid:126)u ) (cid:89) k S Q ∗ sl (2) ( v, u k ) . (2.9)In this formula ˜ E Q ( v ) is the energy of a mirror Q -particle, S Q ∗ sl (2) ( v, u k ) denotes the sl (2) S -matrix with arguments in the mirror ( v ) and string regions ( u k ) and finally T ( L,R ) are theleft and right transfer matrices, given in appendix A. Clearly these corrections couple theleft and right sectors through the product of transfer matrices. Such transfer matrices, andmore importantly their products T ( L ) T ( R ) , will generically be different for two asymptoti-cally degenerate states, giving different perturbative finite size corrections, showing againthat the denegeracy of the asymptotic spectrum is lifted in the finite size theory.In what follows we will illustrate these ideas concretely for two different four-particlestates, both parametrized by two y roots. We will show how the full sets of mirror TBAequations describing these states are manifestly different (though naturally in an elegantsymmetric way), and explicitly compute different leading-order corrections to the energy,which hence lift the asymptotic degeneracy. Let us first introduce our states. We consider two states that have the same value for K I and (cid:80) α K II( α ) , but different valuesfor the individual K II( α ) . When K I = 2 there are no non-trivial level-matched solutions ofthe auxiliary equations, therefore we consider the states Θ and Ψ, as presented below intable 3. – 5 –tate K I K II( L ) K II( R ) K III( α ) WeightsΘ 4 2 0 0 [2 , J − , (2 , Ψ 4 1 1 0 [1 , J − , (3 , Table 1 . The two asymptotically degenerate states we consider. Note the manifestly differentexcitation numbers. For the readers’ convenience we have also presented the Dynkin labels of thestates, denoted by [ q L , p, q R ] ( s L ,s R ) . For either state we have four rapidities u i , level-matched, and two auxiliary roots y ( α ) i ,either both left or one left and one right. In both cases we take the four rapidities to comein pairs: u = − u > u = − u > y ( α ) i is the same.From (2.2), and imposing the rapidities to come in pairs, we find1 = (cid:89) i =1 y − x − i y − x + i , with x ± = − x ∓ , x ± = − x ∓ . (3.1)This admits two regular roots (in addition to y = 0 , ∞ ) that are opposite to each other, y = ± y o , where y o = (cid:115) x − x +1 ( x − − x +3 ) + x +3 x − ( x − − x +1 ) x − − x +1 + x − − x +3 . (3.2)Therefore we take y ( L )1 , = ± y o for state Θ, and y ( L )1 = + y o and y ( R )1 = − y o for state Ψ. Wecan now solve the two Bethe-Yang equations (2.1) for u and u at a given value of J , byplugging in the auxiliary roots, recalling that the sl (2) S -matrix is given by S sl (2) ( u , u ) = σ − x +1 − x − x − − x +2 − x − x +2 − x +1 x − , (3.3)where σ is the dressing factor.Solving the resulting equation analytically is not feasible. Therefore, we first considerthe limit g →
0, rescaling the rapidities such that they remain finite, u i → g u oi . Then theequations for u o and u o decouple, and both take the simple form1 = (cid:18) u ok + iu ok − i (cid:19) J +2 = ⇒ u ok = cot n k πJ + 2 , n k = 1 , . . . J + 1 , (3.4)where the sum of positive n k giving the string level of the state [37]. In order to have ageneric root configuration we focus on the case J = 4 at string level three, where we cansolve (2.1) numerically for arbitrary values of g , requiring that at small coupling u o = − u o = √ , u o = − u o = 1 √ . (3.5)In figure 1 the numerical solutions u ( g ) and u ( g ) are shown. Note again that thesesolutions are the same for both states. In solving (2.1) numerically, the representation of– 6 – (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231) u (cid:230) u g Figure 1 . The rapidities u and u obtained from Bethe-Yang equation at different values of g .Note that they asymptote to two as the coupling is increased. the dressing phase as presented in [26] is most convenient. It is worth remarking that atfinite values of g , no simple relation between u and u holds, despite what we see at weakcoupling in (3.5).Through the AdS/CFT duality, the Θ state corresponds to an operator schematicallyof the form Tr( D ¯ ψ ¯ ψZ ). The correspondent operator of the Ψ state is actually a linearcombination of two types of operators, namely Tr( D ¯ ψψZ ) and Tr( D W Z ). In theseexpressions, all excitations have the highest allowed charges. In order to obtain TBA equations for an excited state we use the contour deformationtrick, following [23]. A clear overview of this whole approach can be found in [16]. Inshort we assume that the ground state and excited state TBA equations differ only bythe choice of the integration contours. Upon deforming the integration contours of theexcited state TBA equations to the ground state ones, we pick up additional contributionswhenever there is a singularity in the physical strip of the rapidity plane. This leads to theappearance of new driving terms in the excited state TBA equations.In the present case, we do this for the left and right sectors, for both states. We denotethe Y -functions Y ( L ) M | w , Y ( L ) ± , Y ( L ) M | vw and Y ( R ) M | w , Y ( R ) ± , Y ( R ) M | vw for a given state in the left andright sectors respectively; the sectors are coupled by the Y Q functions. Below we discussthe analytic properties and related integration contours for both states in detail, followedby the resulting TBA equations. As discussed in [16], we will use the (left and right) asymptotic Y -functions to studythe analytic properties of the TBA equations. Their asymptotic construction is given inappendix A. Let us stress that all the Y -functions but Y Q are defined in a given sector, i.e.– 7 – M | w ≡ Y ( α ) M | w etc. Only Y Q couples the left and right sectors, and indeed asymptotically Y Q contains the product of left and right transfer matrices, as in (2.9).As shown in detail for the Konishi state in [23], the precise analytic structure ofasymptotic Y -function will depend on the coupling g . In general, when we increase g we can expect to encounter an asymptotic critical value [23] where some roots enter thephysical strip, so that we must include appropriate driving terms. This would make thediscussion more technically involved, but is of little relevance to understanding the liftingof degeneracies. We will therefore restrict our analysis to the small coupling region, belowthe first critical value of g .For both states, it turns out that roots of 1 + Y M | w , 1 + Y M | vw and 1 − Y − , and poles of Y + play an important role. In order to discuss this, let us fix some notation. Roots relatedto state Θ are described by script letters: (cid:37) M for Y M | w , (cid:37) for Y − and r M for Y M | vw . Theyare fixed by the conditions Y M | w ( (cid:37) M − i/g ) = − , Y − ( (cid:37) − i/g ) = 1 , Y M | w ( r M − i/g ) = − . (3.6)Similarly, we will use sans-serif letters to denote roots for state Ψ: ρ M for Y M | w , ρ for Y − and r M for Y M | vw , fixed by Y M | w ( ρ M − i/g ) = − , Y − ( ρ − i/g ) = 1 , Y M | w ( r M − i/g ) = − . (3.7)In addition, for both states in both sectors Y + asymptotically has poles at the rapiditiesshifted by i/g , Y + ( u i − i/g ) = ∞ , as in the Konishi case [23]. The relevant roots aresummarized in table 2.1 + Y ( L ) M | w Y ( R ) M | w − Y ( L ) − − Y ( R ) − Y ( L ) M | vw Y ( R ) M | vw Roots Θ ± (cid:37) M − i/g – ± (cid:37) − i/g – – ± r M − i/g Roots Ψ ρ M − i/g − ρ M − i/g − ρ − i/g ρ − i/g − r M − i/g r M − i/g Table 2 . Roots for Y -functions in the left and right sectors for states Θ and Ψ at small coupling.By definition we consider (cid:37) M , ρ M , r M , r M >
0. Asymptotically, one observes that (cid:37) M (cid:54) = ρ M and r M (cid:54) = r M . We expect that the roots, and hence the driving terms, distribute differently between theleft and right sectors for Θ and Ψ. This is indeed the case, as can be seen in table 2.On the real mirror line, the asymptotic Y -functions for state Θ are even, while forstate Ψ the Y -functions do not have a definite parity but satisfy Y ( L ) ( v ) = Y ( R ) ( − v ). We now apply the contour deformation trick to the simplified TBA equations of [10, 13].In order for the asymptotic solution to be a solution, we take the ground state TBA equa-tions and define the integration contour such that it goes slightly below the line − i/g , i.e. such that it encloses the poles of Y + at u i − i/g as well as the roots of table 2 between– 8 –tself and the real line. By taking the integration contour back to the real line, we find theappropriate driving terms, denoted D , and obtain the TBA equations. Below we list thedriving terms that appear for each state. The integration kernels and S -matrices whichenter in the equations below have been defined and are completely listed in [23]. As usual,for any kernel or S-matrix we define S ± ( v ) := S ( v ± i/g ). • M | w -strings; M ≥ Y | w = 0. The equation has the general formlog Y ( α ) M | w = log(1 + Y ( α ) M − | w )(1 + Y ( α ) M +1 | w ) (cid:63) s + δ M log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) s + D ( α ) M | w , (3.8)where D ( α ) M | w are the driving terms that differ in the left and right sector for each givenstate and α = L, R . For state Θ we have D ( L ) M | w ( v ) = − log S − ( ± (cid:37) M − − v ) − log S − ( ± (cid:37) M +1 − v ) , (3.9) D ( R ) M | w ( v ) = 0 , where the terms containing ± (cid:37) indicate the sum of two driving terms for opposite roots.For Ψ we have, instead, D ( L ) M | w ( v ) = − log S − ( ρ M − − v ) − log S − ( ρ M +1 − v ) , (3.10) D ( R ) M | w ( v ) = − log S − ( − ρ M − − v ) − log S − ( − ρ M +1 − v ) . • M | vw -strings; M ≥ Y | vw = 0log Y ( α ) M | vw ( v ) = − log(1 + Y M +1 ) (cid:63) s + log(1 + Y ( α ) M − | vw )(1 + Y ( α ) M +1 | vw ) (cid:63) s (3.11)+ δ M log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) s + D (0) M | vw + D ( α ) M | vw . When M = 1, we find a driving term that is independent of state and sector, arising fromthe poles of Y + , that is D (0) M | vw = − δ M (cid:88) i =1 log S − ( u i − v ) . (3.12)In addition to that, for state Θ we have D ( L ) M | vw ( v ) = 0 , (3.13) D ( R ) M | vw ( v ) = − log S − ( ± r M − − v ) − log S − ( ± r M +1 − v ) , – 9 –hereas for Ψ we have D ( L ) M | vw ( v ) = − log S − ( − r M − − v ) − log S − ( − r M +1 − v ) , (3.14) D ( R ) M | vw ( v ) = − log S − ( r M − − v ) − log S − ( r M +1 − v ) . • y -particles log Y ( α )+ Y ( α ) − = log(1 + Y Q ) (cid:63) K Qy + D (0) ratio , (3.15)log Y ( α ) − Y ( α )+ = 2 log 1 + Y ( α )1 | vw Y ( α )1 | w (cid:63) s − log (1 + Y Q ) (cid:63) K Q (3.16)+ 2 log(1 + Y Q ) (cid:63) K Q xv (cid:63) s + D (0) prod + D ( α ) prod . We expect both of these equations to pick up contributions from the exact Bethe equation Y ( u ∗ i ) = −
1, which will yield driving terms that do not depend on the state or sector.These are D (0) ratio ( v ) = − (cid:88) i =1 log S ∗ y ( u i , v ) , (3.17) D (0) prod ( v ) = − (cid:88) i =1 log (cid:0) S ∗ xv (cid:1) S (cid:63) s ( u i , v ) . where log (cid:0) S ∗ xv (cid:1) S (cid:63) s ( u, v ) ≡ (cid:90) ∞−∞ dt log S ∗ xv ( u, t ) S ( u − t ) s ( t − v ) . The contribution follows from the identitylog S ( u i − v ) − S ∗ xv (cid:63) s ( u i , v ) = − log ( S ∗ xv ) S (cid:63) s ( u i , v ) , (3.18)valid for real u i .In addition to the above driving terms, we have state-specific contributions in equation(3.15); for Θ we have D ( L ) prod ( v ) = 2 log S − ( ± (cid:37) − v ) , (3.19) D ( R ) prod ( v ) = − S − ( ± r − v ) , Here and afterwards, ∗ indicates analytic continuation to the string region. Also, S ∗ y ( u j , v ) ≡ S y ( u ∗ j , v ) is shorthand notation for the S-matrix with the first and second arguments in the string andmirror regions, respectively. The same convention is used for other kernels and S-matrices. – 10 –nd for Ψ we have D ( L ) prod ( v ) = 2 log S − ( ρ − v ) S − ( − r − v ) , (3.20) D ( R ) prod ( v ) = − S − ( r − v ) S − ( − ρ − v ) . • Q -particleslog Y Q ( v ) = − L TBA (cid:101) E Q + log (cid:0) Y Q (cid:48) (cid:1) (cid:63) (cid:16) K Q (cid:48) Q sl (2) + 2 s (cid:63) K Q (cid:48) − ,Qvwx (cid:17) + D (0) Q (3.21)+ (cid:88) α ∈{ L,R } (cid:18) log (cid:16) Y ( α )1 | vw (cid:17) (cid:63) s ˆ (cid:63) K yQ + log (cid:16) Y ( α ) Q − | vw (cid:17) (cid:63) s − log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) s (cid:63) K Qvwx + 12 log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) K Q + 12 log (cid:0) − Y ( α ) − (cid:1)(cid:0) − Y ( α )+ (cid:1) ˆ (cid:63) K yQ + D ( α ) Q . These are the TBA equations for Q -particles in the hybrid form of [23]. Summation overrepeated indices is understood. As before, we split the driving terms in a part independentof the specific state, that is D (0) Q , and sector dependent parts D ( α ) Q which will differ betweenΘ to Ψ. We then have D (0) Q ( v ) = (cid:88) i =1 (cid:16) − log S ∗ Q sl (2) ( u i , v ) + 2 log S (cid:63) K Qvwx ( u i , v ) − log S Qvwx ( u i , v ) (cid:17) , (3.22)where for any kernel K we definelog S (cid:63) K ( u, v ) = lim (cid:15) → + (cid:90) dt log S ( u − i/g − i(cid:15) − t ) K ( t + i(cid:15), v ) , (3.23)which is the same type of contribution as for the Konishi state.The left and right driving terms for Θ are D ( L ) Q ( v ) = log S (cid:63) K Qvwx ( ± (cid:37) , v ) −
12 log S − Q ( ± (cid:37) − v ) −
12 log S yQ ( ± (cid:37) − i/g, v ) , D ( R ) Q ( v ) = − log S ˆ (cid:63) K yQ ( ± r , v ) − log S − ( ± r Q − − v ) , (3.24)while for Ψ we have D ( L ) Q ( v ) = log S (cid:63) K Qvwx ( ρ , v ) −
12 log S − Q ( ρ − v ) −
12 log S yQ ( ρ − i/g, v ) − log S ˆ (cid:63) K yQ ( − r , v ) − log S − ( − r Q − − v ) , (3.25) D ( R ) Q ( v ) = log S (cid:63) K Qvwx ( − ρ , v ) −
12 log S − Q ( ρ − v ) −
12 log S yQ ( − ρ − i/g, v ) − log S ˆ (cid:63) K yQ ( r , v ) − log S − ( r Q − − v ) . – 11 –n the above, K ,Qvwx = 0, Y | vw = 0, meaning that the driving log S + ( v − r ) is not present.Let us stress that in order to check (3.21) on the asymptotic solution, L TBA needs tobe specified. We find that L TBA = J + 2 , (3.26)for both Θ and Ψ, just as for Konishi [23]. As discussed in [16], L TBA is the maximal J charge occurring in the conformal supermultiplet described by the TBA equations, and fora generic state that has full supersymmetry one indeed expects L TBA = J + 2. Nonetheless,there are examples of deformations of the superstring that break supersymmetry wheredifferent relations hold [38, 39]. As discussed, the finite-size energies of states Θ and Ψ depend on the allowed momenta. Inthe mirror TBA approach, these are found by analytically continuing the Q -particle TBAequations to the string region and imposing the exact Bethe equation Y ( u ∗ i ) = −
1, whichis the finite size quantization condition.The (logarithm of the) exact Bethe equation for a string rapidity u k is given by(2 n + 1) πi = iL TBA p k + log (cid:0) Y Q (cid:48) (cid:1) (cid:63) (cid:16) K Q (cid:48) ∗ sl (2) + 2 s (cid:63) K Q (cid:48) − , ∗ vwx (cid:17) + D (0)1 ∗ (3.27)+ (cid:88) α ∈{ L,R } (cid:32) log (cid:16) Y ( α )1 | vw (cid:17) (cid:63) (cid:0) s ˆ (cid:63) K y ∗ + s − (cid:1) − log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) s (cid:63) K ∗ vwx + 12 log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) K + 12 log (cid:0) − Y ( α ) − (cid:1)(cid:0) − Y ( α )+ (cid:1) ˆ (cid:63) K y ∗ + D ( α )1 ∗ , where the kernels have been analytically continued appropriately . As for the drivingterms, we get the state independent contribution D (0)1 ∗ ( u k ) = (cid:88) i =1 (cid:18) − log S ∗ ∗ sl (2) ( u i , u k ) + 2 log Res( S ) (cid:63) K ∗ vwx ( u i , u k ) (3.28) − u i − u k − ig ) x − j − x − k x − j − x + k . Coming to the state-dependent terms, for Θ we have D ( L ) Q ( u k ) = log S (cid:63) K ∗ vwx ( ± (cid:37) , u k ) −
12 log S − ( ± (cid:37) − u k ) −
12 log S y ∗ ( ± (cid:37) − i/g, u k ) , D ( R ) Q ( u k ) = − log S ˆ (cid:63) K y ∗ ( ± r , u k ) − log S ( ± r − v ) , (3.29) See the appendix of [23] for details. – 12 –hile for Ψ we have D ( L ) Q ( u k ) = log S (cid:63) K ∗ vwx ( ρ , u k ) −
12 log S − ( ρ − u k ) −
12 log S y ∗ ( ρ − i/g, u k ) , − log S ˆ (cid:63) K y ∗ ( − r , u k ) − log S ( − r − v ) , (3.30) D ( R ) Q ( u k ) = log S (cid:63) K ∗ vwx ( − ρ , u k ) −
12 log S − ( − ρ − u k ) −
12 log S y ∗ ( − ρ − i/g, u k ) , − log S ˆ (cid:63) K y ∗ ( r , u k ) − log S ( r − v ) . We used the short-handlog Res( S ) (cid:63) K ∗ vwx ( u, v ) = (cid:90) + ∞−∞ d t log (cid:104) S ( u − i/g − t )( t − u ) (cid:105) K ∗ vwx ( t, v ) , (3.31)and indicated the momentum of the magnon as p = i (cid:101) E Q ( z ∗ ) = − i log x s ( u + ig ) x s ( u − ig ) .Expanding the exact Bethe equation about the asymptotic Y -functions, we find, mod-ulo 2 πi , R k ≡ i p k + (cid:88) i =1 S ) (cid:63) K ∗ vwx ( u i , u k ) − u i − u k − ig ) x − j − x − k x − j − x + k (3.32)+ (cid:88) α ∈{ L,R } (cid:32) − log N ( α ) ∗ + log (cid:16) Y ( α )1 | vw (cid:17) (cid:63) (cid:0) s ˆ (cid:63) K y ∗ + s − (cid:1) − log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) s (cid:63) K ∗ vwx + 12 log 1 − Y ( α ) − − Y ( α )+ ˆ (cid:63) K + 12 log (cid:0) − Y ( α ) − (cid:1)(cid:0) − Y ( α )+ (cid:1) ˆ (cid:63) K y ∗ + D ( α )1 ∗ = 0 , where the expression is evaluated at u k . The terms log N ( α ) ∗ arise from the analyticcontinuation of N ( α ) ( v ) = K II( α ) (cid:89) i =1 y ( α ) i − x − ( v ) y ( α ) i − x + ( v ) (cid:115) x + ( v ) x − ( v ) , (3.33)that comes from the Bethe-Yang equations (2.1), appearing whenever K II α >
0. Equa-tion (3.32) can be verified numerically to ensure that the analytic continuation has beenperformed correctly.Since (3.21) contains a sum over the left and right sectors, the form of the resultingexact Bethe equation is the same for Θ and Ψ. We might wonder whether this gives samemomenta for both states, but this is of course not the case because the set of auxiliary Y -functions for the two states will be completely different. Indeed, even in the asymptoticcase, the numerical value of the two set of roots is different: (cid:37) M (cid:54) = ρ M and r M (cid:54) = r M .Finally, recall that the energy of each state is given by (2.7). Since we have seen thatthe two set of TBA equations of Θ and Ψ differ, we expect the energies E Θ and E Ψ to bedifferent as well. We will now show this explicitly by evaluating the first order wrappingcorrections to the energy in both cases. As in the Konishi case [20], this equation still holds for small perturbations around the solution of theBethe-Yang equation, { u k } . – 13 – .2 Wrapping corrections As shown above, the TBA equations for the two states we consider are not equivalent.Therefore, the resulting Y Q functions and hence the energies should be different, thus liftingthe degeneracy of the asymptotic Bethe ansatz. We will directly compute the leading orderwrapping corrections to the energy to see this explicitly, naturally finding different resultsfor the two states.The leading order wrapping correction to the energy can be conceptually seen to arisefrom L¨uscher corrections [18], or equivalently by perturbatively expanding the free energyof the mirror model [10], depending on your point of view.Using the asymptotic expression for the Y Q -functions, (2.9), we can compute the lead-ing order wrapping correction. To do so we evaluate our Y Q -functions to lowest order in g ,which give leading order wrapping interactions at seven loops. As expected the resulting Y Q -functions are manifestly different. The expanded Y Q -functions for either state are givenin appendix B. Recall that the leading order wrapping correction to the energy is given by E LO = − π ∞ (cid:88) Q =1 (cid:90) dv d ˜ pdv Y ◦ Q ( v ) . Integrating and summing the Y-functions for J = 4 yields the following explicit wrappingcorrection for our states, E Θ LO = − (cid:0) ζ (11) + ζ (9) − ζ (7) − ζ (5) + ζ (3) + (cid:1) g (3.34) ≈ − . g ,E Ψ LO = − (cid:0) ζ (11) + ζ (9) − ζ (7) − ζ (5) + ζ (3) + (cid:1) g (3.35) ≈ − . g . This shows explicitly how the degeneracy present in the asymptotic Bethe ansatz is liftedby finite size (wrapping) corrections, with Θ being the lighter state.
In this paper we have described a symmetry enhancement taking place for the AdS × S superstring in the asymptotic limit. Due to this enhancement certain states degenerate inthe asymptotic limit, as described through the asymptotic Bethe ansatz. This symmetryis not present in the finite size model, indicating a qualitative feature of the model thatis not captured by the asymptotic solution. We illustrated these ideas on a set of twoasymptotically degenerate states, by showing that they have manifestly different TBAequations, as well as explicitly computing their leading order wrapping corrections, clearlyshowing lifting of the asymptotic degeneracy. It would be interesting to verify these resulton the gauge theory side, where two unrelated sets of operators should have identical scalingdimensions exactly and only up to wrapping order.– 14 – cknowledgments We are grateful to Gleb Arutyunov for useful discussions, and to Sergey Frolov, Mariusde Leeuw and Ryo Suzuki for useful comments on the manuscript. The work by A.S. ispart of the VICI grant 680-47-602 of the Netherlands Organization for Scientific Research(NWO). The work by S.T. is a part of the ERC Advanced Grant research programme No.246974, “Supersymmetry: a window to non-perturbative physics” . A Transfer matrices and asymptotic Y -functions The eigenvalues of the transfer matrix T ( α ) Q, in the sl (2)-grading are known from [40, 41].The index α = L, R labels the sector; for clarity we suppress it from T Q, as well as fromthe auxiliary roots y, w that parametrize the eigenvalues. We have T Q, ( v ) = K II (cid:89) i =1 y i − x − y i − x + (cid:113) x + x − K II (cid:89) i =1 v − ν i + ig Qv − ν i − ig Q K I (cid:89) i =1 (cid:104) ( x − − x − i )(1 − x − x + i )( x + − x − i )(1 − x + x + i ) x + x − (cid:105) (A.1)+ Q − (cid:88) k =1 K II (cid:89) i =1 v − ν i + ig Qv − ν i + ig ( Q − k ) (cid:104) K I (cid:89) i =1 x ( v +( Q − k ) ig ) − x − i x ( v +( Q − k ) ig ) − x + i + K I (cid:89) i =1 1 − x ( v +( Q − k ) ig ) x − i − x ( v +( Q − k ) ig ) x + i (cid:105) K I (cid:89) i =1 x + − x + i x + − x − i v − v i − (2 k +1 − Q ) ig v − v i +( Q − ig − Q − (cid:88) k =0 K II (cid:89) i =1 v − ν i + ig Qv − ν i + ig ( Q − k ) K I (cid:89) i =1 x + − x + i x + − x − i (cid:114) x − i x + i v − v i − (2 k +1 − Q ) ig v − v i +( Q − ig K III (cid:89) i =1 w i − v + i (2 k − − Q ) g w i − v + i (2 k +1 − Q ) g − Q − (cid:88) k =0 K II (cid:89) i =1 v − ν i + ig Qv − ν i + ig ( Q − k − K I (cid:89) i =1 x + − x + i x + − x − i (cid:114) x − i x + i v − v i − (2 k +1 − Q ) ig v − v i +( Q − ig K III (cid:89) i =1 w i − v + ig (2 k +3 − Q ) w i − v + ig (2 k +1 − Q ) . The variable v = x + + 1 x + − ig a = x − + 1 x − + ig a (A.2)takes values in the mirror theory rapidity plane, so that x ± = x ( v ± ig a ) where x ( v ) isthe mirror theory x -function. Similarly, x ± j = x s ( u j ± ig ), where x s is the string theory x -function. Recall that x ( u ) = 12 ( u − i (cid:112) − u ) , x s ( u ) = u (cid:112) − /u ) . (A.3)Notice that the transfer matrix comes with a prefactor of N = (cid:81) K II i =1 y i − x − y i − x + (cid:113) x + x − encoun-tered already in the Bethe-Yang equations (2.1) and in (3.33). As discussed in [16], this isconsistent with the requirement that Y ∗ ( u k ) = − Y -functions as follows Y ( α ) M | w = T ( α )1 ,M T ( α )1 ,M +2 T ( α )2 ,M +1 , Y ( α ) − = − T ( α )2 , T ( α )1 , , Y ( α )+ = − T ( α )2 , T ( α )2 , T ( α )1 , T ( α )3 , , Y ( α ) M | vw = T ( α ) M, T ( α ) M +2 , T ( α ) M +1 , , (A.4) The general construction of the Y-functions in terms of transfer matrices is based on the underlyingsymmetry group of the model [42, 43]. For the string sigma model asymptotic Y-functions were presentedin [14]. In fact, this solution can be directly derived from the Bajnok-Janik formula [36] and the AdS/CFTY-system, see [16]. – 15 –n each sector, α = L, R . Recall that the Y Q functions are given asymptotically by (2.9). B Expansions for Y ◦ Q functions Taking the transfer matrix (A.1) and expanding in the coupling constant yields the follow-ing expressions for the left and right transfer matrices of state Θ T ( L ) Q, ( (cid:126)u | v ) = A Q [6 Q + Q ( u +5 u +18 v +2 ) + Q ( u − v ( u + u +10 ) − ( u + u +3 ) + u +18 v ) (B.1) − v ( u +7 u +22 ) − ( u +1 )( u +1 )( u + u +2 ) + v ( u + u ( u +6 ) + u +6 u +2 ) +6 v ] T ( R ) Q, ( (cid:126)u | v ) = AQ [ ( u + u +2 )( Q + Q ( u +3 u − v +2 ) + v ( u +3 u +2 ) − ( u +1 )( u +1 ) − v ) ] (B.2)where A Q = Q ( u +1 )( u +1 ) ( Q + v ) ( u +( Q − iv − )( u +( Q − iv − ) . The expansion of the individual transfer matrices is rather convoluted for the Ψ state, sowe present only the result for the product, given by T ( L ) Q, T ( R ) Q, ( (cid:126)u | v ) = A Q ( Q + v ) ( u + u +2 ) (cid:2) Q ( ( u + u +2 ) − v ) (B.3) + Q (cid:16) v ( u + u +2 ) +6 (cid:16) ( u + u ) − (cid:17) − v (cid:17) + Q (cid:0) v ( u + u +2 ) +2 v ( u − u ( u +4 ) + u − u − ) + ( u + u +2 )( u − u ( u +5 ) + u − u − ) − v (cid:1) +2 Q (cid:0) v ( u + u +2 ) − ( u +1 )( u +1 ) (cid:16) ( u + u ) − (cid:17) − v ( u +6 u ( u +8 ) +7 u +48 u +52 ) + v ( u + u +2 )( u + u ( u +2 ) + u +2 u +10 ) − v (cid:1) +17 v ( u + u +2 ) + ( u +1 ) ( u +1 ) ( u + u +2 ) − v ( u +6 u ( u +4 ) +5 u +24 u +20 ) + v ( u + u +2 )( u +2 u ( u +7 ) + u +14 u +22 ) − ( u +1 )( u +1 ) v ( u + u ( u +4 ) + u +4 u ) − v (cid:3) The S-matrix in the string-mirror region S ∗ Q sl (2) is found in [44] (see also [19]) and has thefollowing leading behavior in gS ∗ Q sl (2) ( u, v ) = − (cid:2) ( v − u ) +( Q +1) (cid:3)(cid:2) Q − i ( v − u ) (cid:3) ( u − i ) (cid:2) Q − − i ( v − u ) (cid:3) + O ( g ) . 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