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"Short"spinning strings and structure of quantum AdS_5 x S^5 spectrum

Using information from the marginality conditions of vertex operators for the AdS_5 x S^5 superstring, we determine the structure of the dependence of the energy of quantum string states on their conserved charges and the string tension proportional to lambda^(1/2). We consider states on the leading Regge trajectory in the flat space limit which carry one or two (equal) spins in AdS_5 or S^5 and an orbital momentum in S^5, with Konishi multiplet states being particular cases. We argue that the coefficients in the energy may be found by using a semiclassical expansion. By analyzing the examples of folded spinning strings in AdS_5 and S^5 as well as three cases of circular two-spin strings we demonstrate the universality of transcendental (zeta-function) parts of few leading coefficients. We also show the consistency with target space supersymmetry with different states belonging to the same multiplet having the same non-trivial part of the energy. We suggest, in particular, that a rational coefficient (found by Basso for the folded string using Bethe Ansatz considerations and which, in general, is yet to be determined by a direct two-loop string calculation) should, in fact, be universal.


Introduction and summary
Recent progress in understanding the integrable system that should be computing the spectrum of the maximally supersymmetric example of AdS/CFT duality makes it important to further develop a detailed matching of the Bethe ansatz predictions with quantum AdS 5 × S 5 string energies extracted from the perturbative string theory. While direct near-flat-space expansion of the quantum string theory determining the large tension (T = √ λ 2π ) expansion of quantum string energies with fixed quantum charges is still to be developed, here we shall follow the "semiclassical" approach suggested in [1] (see also [2]) and recently applied in [3,4,6,5] to demonstrate the matching of the numerical results of the TBA for the Konishi operator dimension interpolated from weak to strong coupling [7,8,9] with the perturbative string theory prediction for the corresponding string energy.
Our motivation is to further understand the structure of the dependence of the string energy on the string tension and its quantum numbers (spins) guided by the expected form of the string vertex operator marginality conditions [1,4] and recent progress on the Bethe ansatz side [10]. We shall consider several string states which belong (in the flat-space limit) to the leading Regge trajectory and for the lowest values of the spins or the lowest value of the string level represent states in the Konishi multiplet and discover the universality of some leading-order coefficients in the expansion of their energies.

General structure of the inverse tension expansion of the energy
Let us start with describing the general form of the dependence of the energy E of a string state on its quantum charges Q i in the large string tension expansion ( √ λ ≫ 1). 1 As follows from the structure of α ′ expansion of 2d anomalous dimensions of the corresponding AdS 5 ×S 5 string vertex operators [11,12], the solution of the marginality condition should give E = E(Q, √ λ) in the following general form [1,4] where Q i are supposed to be fixed in the limit √ λ ≫ 1. The highest power of charges in 1 ( √ λ) n term here is n + 2. This follows, e.g., from dimensional analysis, from the fact that higher order terms in 2d anomalous dimension operator may contain higher derivative operators (e.g., E 2 comes from SO(2, 4) Casimir originating from Laplacian on AdS 5 , etc.; see [12]) and also from the fact that, in any theory, an (n + 1)-loop Feynman graph renormalizing a (vertex) operator contains at most (n + 2) Wick contractions with fields in the (vertex) operator and thus contributes to its dimension terms like Q m /( √ λ) n with m ≤ n + 2. More explicitly, if we consider a string state with an orbital momentum J 3 ≡ J in S 5 and one extra oscillator number N (corresponding, e.g., to an intrinsic spin component due to an extended nature of the string) which determines the value of an effective string level then (1.1) is a consequence of the following 2d marginality condition 2 0 = N + 1 2 √ λ (−E 2 + J 2 + n 02 N 2 + n 11 N) 2 (n 01 NJ 2 + n 03 N 3 + n 12 N 2 + n 21 N) Including also some higher-order terms, the resulting expression for E 2 may be written as 3 E 2 = 2 √ λN + J 2 + n 02 N 2 + n 11 N + 1 √ λ n 01 J 2 N + n 03 N 3 + n 12 N 2 + n 21 N + 1 ( √ λ) 2 ñ 11 J 2 N +ñ 02 J 2 N 2 + n 04 N 4 + n 13 N 3 + n 22 N 2 + n 31 N + 1 ( √ λ) 3 ñ 01 J 4 N +ñ 21 J 2 N +ñ 12 J 2 N 2 + n 05 N 5 + ...
This expression follows under the assumption that in (1.2) E 2 enters only in the 1-loop 1 √ λ term. On general grounds, as E may be thought of as a global charge analogous to J, one might wonder if (1.2) should also contain terms like 1 ( √ λ) k (E k+1 + ... + E m N n + ...). However, terms depending only on E (or on E and J) should be 2d scheme-dependent (like higher powers of Laplacian in 2d anomalous dimension operator) and would also contradict BMN limit E = J in the absence of other charges (N = 0) leading to spurious 1 √ λ dependent solutions of the marginality condition; they should thus be absent in a scheme preserving target space supersymmetry. Terms in (1.2) involving both E and N like 1 ( √ λ) k E n N m with m + n ≤ k + 1, may be present, but in solving the marginality condition (1.2) for E in perturbative expansion in 1 √ λ they cannot modify the leading-order solution E 2 = 2 √ λN + ... and their perturbative treatment leads just to redefinitions of coefficients already present in eq. (1.3). Note also that the presence of the mixed terms J k N m terms reflects the fact that in curved space the center-of-mass and internal degrees of freedom do not in general decouple.
Expanding (1.3) in large √ λ for fixed N, J we get Substituting particular values of N and J into (1.3),(1.4) one can find the expansion of the corresponding quantum string state energy, i.e. the strong-coupling expansion of the dimension of the dual gauge-theory operator. Note that the first two terms in the r.h.s. of (1.3) have direct flat-space interpretation, so that N plays the role of string level and the spinning string states with maximal value of N for a given value on spin belong to the leading Regge trajectory. For example, N = 0 corresponds to massless (supergravity) states and N = 2 to states on the first excited string level which contains the Konishi long multiplet as its "floor" and also its "KK descendants" with higher values of J obtained by tensoring with [0, J, 0] representation [14]. The states in the Konishi multiplet that we will consider here correspond to N = 2, J = 2, see [1,4]. The goal is thus to determine the coefficients n km in (1.3). To achieve this one may use the observation [1,2] that a similar expansion of the string energy is also found by starting with a solitonic string carrying the same types of charges as the vertex operator representing a particular quantum string state and (i) first performing the semiclassical expansion √ λ ≫ 1 for fixed charge densities Q i = 1 √ λ Q i , i.e. (N , J ) = 1 √ λ (N, J), and then (ii) expanding E in small values of Q i . Indeed, the limit Q i = Q i √ λ → 0 should correspond to taking √ λ ≫ 1 for fixed values of the quantum charges Q i . Assuming that there is no order of limits problem, the same coefficients n km should be found in these two different approaches.
It should be noted that the quantum string sigma model 3) the first order N term is classical, J 2 + n 02 N 2 + n 11 N are 1-loop terms, etc., i.e. the coefficients n km , in general, appear at different loop orders in the two expansions. 4 Note also that while each ℓ-loop term in (1.3) is a polynomial of finite degree, (ℓ + 1), in the charges, this does not in general apply to the semiclassical expansion (1.8) where each term may contain an infinite series of terms in the small J , N expansion. To relate the two expansions one would need to reorganize or even resum them. 5 For example, the classical string energy term in (1.8) receives contributions from all higher loop orders in (1.3), etc. 6 Comparison of (1.9) or (1.4)-(1.7) to (1.3) shows that eq.(1.3) for the square of the energy provides a much more "economical" description of the spectrum. Computing the semiclassical expansion (1.9) directly one finds indeed many relations between the coefficients there in agreement with the general structure of E 2 in (1.3).
The expression for E 2 in (1.3) or in (1.8) may be formally organized as an expansion in small N which will then look like an expansion in powers of N: 4 Note that n ℓ1 (ℓ = 1, 2, ...) are still ℓ-loop coefficients in both expansions. 5 In particular, considering J ≫ N expansion will lead to inverse powers of J in the semiclassical expansion and thus will require a resummation to relate it to (1.3). 6 Note also that "non-analytic" terms [1] like B 2 , B 3 , ... in the large √ λ expansion of the energy .., which a priori could be present in the energy found by using semiclassical expansion, should not actually appear if this approach is consistent: they would lead to where for fixed J and large λ the coefficient functions h k are given by The corresponding expansion of E in small N for fixed J is then i.e. h 1 (λ, J) may be called, following [10], a "slope" function. In ref. [10] it was found exactly in the case of the folded string with spin S in AdS 5 (in this case N = S). While the coefficients in the "slope" function h 1 are expected, by analogy with the case in [10], to be rational (h 1 is determined [10] by the asymptotic Bethe ansatz and is also not sensitive to the phase) the coefficients in the next "curvature" function h 2 are already transcendental (as we shall discuss below n 12 contains ζ 3 ,ñ 12 contains ζ 5 , etc) and h 2 is expected to be sensitive to "wrapping" corrections.

Summary of results for the coefficients
Below we shall consider the examples of "small" semiclassical spinning string states discussed in [1,4] that fall into the class of states described by (1.3),(1.8), (1.9). They correspond to quantum string states with angular momentum J and few oscillator modes excited that are responsible for non-zero components of intrinsic spin. More specifically, we shall consider and compare the following solutions: 7 two folded string cases: (S, J) and (J ′ , J) and three rigid two-spin circular string cases: (J 1 = J 2 ≡ J ′ , J), (S 1 = S 2 ≡ S, J) and (S = J 1 ≡ J ′ , J). For lowest values of the winding numbers these represent (in the flat space limit) states on the leading Regge trajectory with the string level being N = S or N = J in the folded one-spin cases and N = 2J ′ or N = 2S in the circular two-spin cases. For example, for N = 2 these represent states on the first excited string level. In this case all states with fixed J (i.e. on a fixed KK level [14]) should belong to a single long P SU(2, 2|4) multiplet. 8 Furthermore, the string states with N = 2, J = 2 are dual to particular states in the Konishi multiplet on the gauge theory side [1,4].
As all operators in a given supermultiplet should have the same 4d anomalous dimension, that means that the corresponding string states should have the same target space energy (up to constant integer or half-integer shifts reflecting their positions in the supermultiplet; such shifts are ignored in (1.3)), i.e. the expression for E N =2 as a function of J and λ should be universal, with E N=2 (J = 2, λ) being equal to the dimension of the Konishi multiplet.
As follows from (1.3), this expected universality of the N = 2 value of the energy for any J and √ λ imposes the following invariance constraints on the coefficients of states within a supermultiplet:  In general, the highest loop order ℓ coefficient n ℓ1 in a ℓ originates from the slope function h 1 in (1.11) and thus should be rational (as was found for the (S, J) folded string state in [10]). 9 The subleading loop order coefficient n ℓ−1,2 (for ℓ > 1) originating from h 2 in (1.11) should already be transcendental -containing zeta-function ζ(2ℓ − 1) ≡ ζ 2ℓ−1 . Also, n ℓ−2,3 (for ℓ > 2) should contain ζ 2ℓ−1 , etc. Then the highest transcendentality term in a ℓ in (1.17) should contain ζ 2ℓ−1 .
Indeed, as we shall see below the 1-loop coefficients n 1k obey this pattern: n 12 contains ζ 3 , n 13 contains ζ 5 , etc. What is unclear at the moment is if the 2-loop and higher coefficients in h 2 , h 3 , ... (like n 22 , n 32 , ...) may contain other transcendental constants as well. 10 It would be important to carry out an explicit 2-loop computation of n 22 to clarify this question.
It is interesting to note that the weak-coupling expansion of the anomalous dimension of the Konishi multiplet states also contains ζ k constants at 4 and 5 loops (see, e.g., [17] and refs. there) while the transcendentality origin of higher loop coefficients here again appears to be an open question (an answer should follow from an analytic solution of TBA equations at weak coupling [7,8]). 9 In particular, for the (S, J) folded string state [10]: n 11 = −1, n 21 = − 1 4 , n 31 = − 1 4 , n 41 = − 25 64 , n 51 = − 13 16 , n 61 = − 1073 512 , etc. 10 For example, the 2-loop and higher order terms in the ln S coefficient of the large S limit of the folded string energy expanded in 1 √ λ contain Dirichlet beta function constants K = β(2), etc. (as well as ζ k ) [15,16].
Let us now summarize what is known [1,3,4,5,10,6] and what will be found below about the coefficients n km ,ñ km in (1.3) using the semiclassical approach. We will try to identify the general universality patterns in the structure of these coefficients. First, in all cases The universality of n 01 is in agreement with (1.15). This follows from the universal form of the "near-BMN" expansion of the classical string energy: where we assumed that N ≪ J ≪ √ λ. In other words, the first term in the semiclassical expansion of the slope function h 1 in (1.10) is universal: . The classical n 02 , n 03 and the leading 1-loop n 11 coefficients are also rational [1,3]. We find that in all cases 2n 02 + n 11 = 2 , (1.23) verifying the first universality relation in (1.15). The value ofñ 11 is determined by the term linear in N in the 1-loop semiclassical energy computed for fixed J and small N and then expanded in small J (see (1.9)). The results for the folded string [10,6] and the circular string results described below imply that in all cases n 11 = −n 11 ,n 11 = n 11 . (1.24) More generally, these results imply the universality (for the states on the leading Regge trajectory) of the J -dependence of the first two leading terms in the "slope" function h 1 in (1.10) expanded in the semiclassical limit √ λ ≫ 1 with J = J √ λ held fixed: We find also that the leading term in the semiclassical expansion of h 2 in (1.12) has the following general form The 1-loop coefficient n 12 in (1.3),(1.26) contains a transcendental ζ 3 part. This was first observed in the small-spin expansion of the folded string [2,18] and pulsating string [20] energy, indicating also that higher-order 1-loop terms should contain ζ 5 , etc., constants. The computation of n 12 for the circular 2-spin string with J 1 = J 2 ≡ J ′ (N = 2J ′ ) in [1] and for the folded spinning string (N = S) in [6] led to the exactly same coefficient of ζ 3 in n 12 , suggesting its universality, i.e. that 11 where n ′ 12 is a rational number depending on a particular string state on the leading Regge trajectory. The universality of the ζ 3 coefficient in (1.29) will be confirmed below also for the two other examples of the "small" circular string solutions: with two equal spins S 1 = S 2 in AdS 5 ; with one one spin in AdS 5 and one spin J 1 ≡ J ′ in S 5 with S = J ′ , N = 2S (in ref. [1] only n 11 was computed in these cases).
As was found in [10] from the exact computation of the "slope" function h 1 in (1.10) for the "ground-state" (S, J) state in sl(2) sector (corresponding to the folded (S, J) string), the 2-loop coefficient n 21 is rational and given by 12 Remarkably, as we shall find below, this constraint implies the same value (1.30) for the 2-loop coefficient n 21 also for the folded (J ′ , J), circular (J 1 = J 2 , J) and circular (S 1 = S 2 , J) strings. We thus suggest that this value n 21 = − 1 4 , like the value of the ζ 3 coefficient in (1.29), should again be the same for all the states on the leading Regge trajectory. 13 This universality of n 21 may help understand how to generalize the exact result of [10] for the function h 1 in (1.10) to states outside the sl(2) sector. While the direct 2-loop computation of n 21 is yet to be done for the circular string cases, the value (1.30) can be indirectly obtained from the knowledge of the 1-loop coefficients by using the expected universality of the subleading a 2 coefficient in the dimension of the Konishi state (1.19).
Note that in view of (1.29) and (1.31) the coefficients in the Konishi multiplet energy (1.17) take the following explicit form (1.32) 11 The ζ 3 coefficient is no longer universal for an m-folded string [6] but has simple m 2 dependence (see also section 2.2 below for the corresponding circular string case). 12 The simplicity of this coefficient may a priori be surprising as it should be given by some 2-loop world-sheet theory integral (with discrete sum over spatial momenta). 13 The universality of this subleading coefficient in the slope function is supported by the fact that while n 11 is sensitive to the curvature of subspace where string moves (i.e. it changes sign between the AdS 5 and S 5 cases) the 2-loop correction (determining, in particular, n 21 ) depends on the square of the curvature.
The universality of (a 1 ) J =2 = 1, i.e. the validity of (1.23) not only for the (S, J) folded [3] but also for the small circular string cases was already verified in [1,4].
Assuming the universality of the value of n 21 in (1.30) we get from (1.31) We shall explicitly confirm this relation (and thus the n 21 = − 1 4 prediction) in section 2 for the circular J 1 = J 2 and S 1 = S 2 cases. In the case of the circular S = J ′ string one has n 03 = − 1 2 and then (1.33) implies n ′ 12 = 5 8 . The direct computation of n ′ 12 in this case will be discussed in section 2.4 and Appendix C. As it will be explained in section 2.1, the result depends on a choice of a summation prescription over the fluctuation frequencies. One particular summation procedure discussed in Appendix C leads to n ′ 12 = 11 8 . While so far we were unable to identify a prescription leading to the value n ′ 12 = 5 8 consistent with the universality of (1.30), we believe it should exist. Further support of the universality of n 21 comes from the folded (J ′ , J) string discussed in Appendix D where we show that in this case n 03 = 1 8 and n ′ 12 = − 5 8 , in agreement with (1.33).
The 1-loop result for the (S, J) folded string in [6] (in eq. (B.5) there) and our present results for the circular and (J ′ , J) folded string cases all lead also to the following universal expression for the coefficientñ 12 in (1.3),ñ whereñ ′ 12 is a rational number depending on a particular state. Remarkably, like in the case of n 11 = −ñ 11 in (1.24), the ζ 3 term here is the same as in n 12 in (1.29), up to the sign. The coefficientñ 12 contributes to a higher subleading term a 4 in the Konishi dimension (1.17).
The value ofñ 12 can be found from the coefficient of the 1 2 √ λ N 2 J term in (1.9), i.e.
where we used (1.21). For example, for the (S, J) folded string the result of [6] gives (1.34) withñ ′ 12 = − 27 16 . The coefficient n 13 can be found also by starting with solutions with J = 0, expanding in small N and comparing to (1.4),(1.7) (see section 2): n 13 is present in the N 2 term in A 3 in (1.7) which appears at one loop order in the semiclassical expansion (as N 2 where n ′ 13 and n ′′ 13 are rational numbers. The coefficient of ζ 5 is again universal. In the semiclassical expansion of the energy at fixed J the coefficient n 13 first appears in the 1 term in (1.9), i.e. in the combination 2(n 13 −ñ 12 ) − n 01 n 12 + 3n 11 + (3ñ 01 −ñ 02 + 3 4 n 2 01 )n 11 + (3n 01 − n 02 )ñ 11  (1.36). Note that ζ 5 terms cancel out in this combination. The absence of ζ 5 in the coefficient of N 3 /J term is seen in the expression for the 1-loop energy for the AdS 5 folded string in [6]; we will also find that the same is true for the folded string in S 5 and the three circular string examples. As for the ζ 3 term in (1.37) appearing in the coefficient of N 3 /J in (1.9), the result of [6] and our results described in section 2 and Appendix D imply that it depends on a particular solution. Thus n ′′ 13 is not universal (we shall list its values for different solutions below). The results of [6] in the folded (S, J) string case lead to n 11 = −1, n ′ 12 = 3 8 , n ′′ 13 = 15 4 ,ñ ′ 12 = − 27 16 and thus n ′ 13 = − 9 16 . We expect the 3-loop slope coefficient n 31 to be rational for all states while the 2-loop coefficient n 22 to contain only ζ 3 as its highest transcendentality part, i.e.
Then the universality of the combination 8n 04 + 4n 13 + 2n 22 + n 31 in (1.16) is consistent with the universality of the ζ 5 coefficient in (1.36). Thus the next-order coefficient a 3 in the first excited string level state energy (1.20) should contain a ζ 5 part. Explicitly, as follows from the above discussion (cf. (1.27),(1.36)) the coefficients in the energy (1.17) for the states on the first excited string level take the form: The universality of a 3 implies that the coefficient of ζ 3 and thus n ′′ 13 + 1 2 n ′′ 22 should have stateindependent value. For the folded (S, J) string a 1 , a 2 in (1.39),(1.40) appeared in [3,6]. In this case the 3-loop coefficient n 31 can be inferred from the exact expression (A.2) for the "slope" h 1 in [10], i.e. n 31 = − 1 4 . Using also that for folded string solution n 04 = 31 64 and the value for n 13 in (1.36) given by n 13 = − 9 16 + 15 4 ζ 3 + 15 4 ζ 5 (see [6] and (D.28)) we conclude that for this state we should get To fix a 3 we thus need to know the 2-loop coefficient n 22 in h 2 in (1.12). As the folded string is an elliptic solution, the required direct 2-loop string computation appears to be hard. It should be easier to find n 22 for the rational circular J 1 = J 2 solution. In that case n 31 should be again rational, while (see (2.22) . The universality of this coefficient could be checked by an independent computation of n 22 by another circular string, e.g., S 1 = S 2 one.
It would be interesting also to extend the numerical TBA analysis in [9] to test the universal J dependence of a 3 and extract the value of n 22 for the folded string state. The J = 2, 3, 4 data in [9] suggests that n 22 ∼ −10.
Let us now list the values of few leading coefficients n km ,ñ km for various folded and circular spinning strings adding question marks next to the values that were not yet derived directly but are conjectured to be true on the basis of the universality of (1.30) (see also table in Appendix E). For the folded strings with one spin N in AdS 5 or S 5 and an S 5 orbital momentum J one finds: • folded string in AdS 5 with (S, J), N = S [2, 3, 6, 10]: [19,5]: The value of n 12 in (1.29),(1.44) andñ 11 will be determined below in Appendix D following the algebraic curve approach of [3,5,6]. For the circular strings with two equal spins in AdS 5 or S 5 and an S 5 momentum J one finds: ]. An intuitive explanation for this may be that since we are considering a near-flat-space expansion certain leading coefficients should be given just by sums of independent contributions of oscillators in different dimensions. Then to leading order the AdS 5 and S 5 directions should contribute similarly in the near-flat expansion, modulo signs due to opposite sign of the curvature.
It is useful also to add the corresponding expressions for the pulsating strings with N being the oscillation number (see [20] and refs. there): 15 • pulsating string in AdS 5 : • pulsating string in S 5 : As discussed in [20], for N = 2 the pulsating strings should also represent states on the first excited string level, i.e. in particular (for J = 2) states from the Konishi multiplet. With the above values of n km one indeed reproduces the coefficients in (1.32).
The rest of this paper is organized as follows. In the section 2 we first comment on the general strategy of computing one-loop correction to the energy of classical solitons and then use it to evaluate the one-loop contributions to the energy of the three "small" circular spinning strings. The necessary characteristic polynomials are collected, in a factorized form, in Appendix B. While the solutions with two spins in AdS 5 or with two spins in S 5 yield coefficients n km in line with the expectations and patterns outlined above, the rational terms in the result for the circular string solution with one spin in AdS 5 and one spin in S 5 are found to be ambiguous, depending on a choice of prescription for the summation of the characteristic frequencies. In Appendix C we compute the one-loop correction to the energy of the same small circular string solution using the algebraic curve approach and find a result consistent with a particular worldsheet summation prescription. In Appendix A we discuss the structure of the leading terms in the slope function h 1 [10] in the semiclassical expansion. The one-loop correction to the energy of folded string with spin in S 5 is found in Appendix D. Appendix E contains table with values of the leading coefficients discussed in this paper.

General comments on computation of one-loop correction
We will be interested in computing 1-loop corrections to the energy of rigid circular spinning strings in AdS 5 × S 5 . While these solutions are among the simplest ones being stationary and leading to fluctuation Lagrangian with constant coefficients this problem (addressed in the past, e.g., in [21,22,23,24,35,1]) turns out to be subtle. Expanding the string action near the solution and using a static gauge on fluctuations one ends up with a quadratic fluctuation operator ∆ 2 = diag(K B , K F ) for 8+8 coupled bosonic+fermionic fluctuation modes. Equivalent result for ∆ 2 (restricted to "physical" subspace) is found in the conformal gauge where 2 massless bosonic modes decouple and their contribution is cancelled against the conformal gauge ghost one. Since for all solutions we will consider the target-space time is proportional to the world sheet one, t = κτ , the 1-loop correction to the target space energy can be found as . Even though ln det∆ 2 is UV finite 16 , the computation of its finite part on 2d cylinder is potentially ambiguous -it may depend on how individual fluctuation modes are defined and how their contributions are combined together. One complication is that the space of bosonic fluctuations is multidimensional. Also, the lack of manifest Bose-Fermi 2d symmetry (like world-sheet supersymmetry in the NSR case) implies an extra ambiguity in choice of a consistent regularization. On general grounds, the choice of a prescription for computation of this quantum correction should be governed by the requirement of preservation of underlying symmetries of the theory (i.e. conserved charges, including "hidden" ones) which are "spontaneously broken" by a choice of a particular background we are expanding around. A practical implementation of this starting directly with the GS AdS 5 × S 5 string action remains a non-trivial task. 17 To give an example of possible ambiguities, consider a model where Here p 1 is an integer momentum in S 1 direction and the sum rules h r=1 c r = 0, h i=1 c r m 2 r = 0 ensure that E 2d is UV finite. The shifts a i and (integer) k r reflect particular choice of definitions of fluctuation modes. If one splits the sum over fluctuations into h separate 2d integrals and formally ignores the UV cutoffs in them one may shift the integration/summation variables so that to completely eliminate the dependence on a r , k r . However, if one first combines all the contributions into a single integrand the finite result will depend on a r , k r .
To evaluate similar 1-loop expressions one may choose to diagonalize ∆ 2 first to get its determinant over "flavor" indices as a product over roots of the corresponding characteristic polynomials, P B,F (p 0 ) ="det" One particular prescription for evaluating the resulting integral over p 0 is to first Wick-rotate it (which is equivalent to iǫ prescription p 0 → p 0 − iǫ). 18 Then performing the integral one gets a sum of absolute values of the characteristic frequencies Alternatively, one may also treat the worldsheet theory expanded to quadratic order around the classical solution as a collection of infinitely many coupled harmonic oscillators (found by expanding the 2d fluctuation fields in Fourier series in σ) and evaluate the corresponding vacuum energy using the 1-d Hamiltonian (operator) quantization method. As was discussed in [25,24], upon a diagonalization of the mixing, the contribution of each normal mode to the energy will enter in the sum with a sign s i = ±1 determined by a minor of the mixing matrix, i.e. in this case we get While this expression is equivalent to (2.3) in some standard simple cases, this need not be true in general. 19 The computation in 1-d Hamiltonian quantization setting may be sensitive to low values of p 1 when sign of ω i may fluctuate with p 1 and different treatments may correspond to different choices of oscillator vacuum for low (zero) modes. At the same time, the signs of sufficiently high mode number terms (i.e. with |p 1 | > n = finite number) cannot be sensitive to them. Indeed, since the mixing of modes is subleading (at most linear) in p 1 compared to the free kinetic term, the mixing can be ignored for large p 1 ; in particular, Since the transcendental (ζ 3 , ζ 5 , etc.) terms that may appear in the expression for the 2d energy can originate solely from a summation over infinite range of p 1 (the sum over any finite set of modes can only produce a rational number) it follows that the transcendental parts of the 2d energy should be controlled by the |p 1 | ≫ 1 limit and thus should not depend on a sign prescription. Moreover, fluctuations with high mode numbers have large 2d energy and thus probe only short worldsheet distances. 20 Their contribution is thus less sensitive to details of the classical solution which is chosen as an expansion point for the the worldsheet action 18 It is not clear a priori why the standard iǫ prescription should be preferred given that 2d Lorentz invariance is broken by the background. 19 The expression in (2.4) may be thought of also as a result of a generalized iǫ prescription: p 0 → p 0 − is i ǫ, with s i ω i =s i |ω i |,s 2 i = 1. 20 Classical scale invariance is broken by the background so this notion makes sense; "short distance" is measured with respect to the characteristic scale of the background which is set by the parameters of the solution.
(they will, however, be sensitive to the "topological" features of the solution, such as winding number). We may then expect that at least some of the coefficients of the transcendental terms in E 1 should be universal within a given Regge trajectory (parametrised by values of spins with fixed values of windings). This explains, in particular, the universality of the ζ 3 term in (1.29) and of the ζ 5 terms in (1.34) and (1.36).
The choice of signs s i may itself be sensitive to the definition of the fluctuation modes (related to shifts in fluctuation frequencies or choice of oscillator vacua that may also be different in different gauge choices). In general, one expects that the whole summation prescription should be determined by the requirement that the target space symmetry algebra is correctly realized on quantum string states. There are more practical physical conditions that are easier to verify, e.g., the vanishing of the one-loop correction to the energy in the limit in which all charges go to zero. The one-loop correction should also vanish in the limit in which the classical solution becomes supersymmetric (in cases where such limit exists) 21 e.g., one may require consistency with the BMN limit.
Another requirement one may impose is an analyticity in the smallest charge. Indeed, in the presence of a large charge one may expect that turning on another charge should be smooth; that is, the derivative of the energy with respect to the smallest charge evaluated at zero should not be singular. This translates into the absence in E 2d of fractional powers of small charges, Q α with α < 1. Such a requirement of the absence of "non-analytic" terms (see [1]) turns out to be consistent with the structure of the energy (1.3),(1.4) expected to follow from the marginality condition for the corresponding vertex operator.
2.2 Circular string with spins J 1 = J 2 and orbital momentum J We shall start with the "small" circular string in S 5 described by the following classical solution [21,23,1,4] (t = κτ , X k X k = 1) In the limit a → 0 this becomes a short string with small spin J ′ . m is a winding number which is to be set to 1 to get a state on the leading Regge trajectory. For ν = 0 the classical energy has the same expression as in flat space, E 0 = 2 √ mJ ′ . Expanding the classical string More generally, if we expand in small J ′ for fixed ρ 2 = J 2 /(4mJ ′ ), we find It is useful to perform the one-loop calculation in terms of the two independent semiclassical parameters a and ν. We will first expand in small a at fixed ν and then expand in ν. An important feature of this expansion is that all 1-loop integrals are then regularized in the IR by a non-zero value of ν or J and therefore a 2 and thus the spin J ′ will appear in the 1-loop world-sheet energy only in integer powers, E 2d = k f k a 2k . A further expansion in small J can then be carried out in the resulting coefficients. 22 Then Note that as the expansion of κ or the classical energy (2.9) contains inverse powers of J , terms of higher-order in J −1 in f i contribute to terms of lower order in the corresponding expansion of e i . Note also that in view of (2.1) we have To compute the 1-loop energy E 2d we need the quadratic fluctuation operators K B,F or the corresponding bosonic and fermionic characteristic polynomials. They can be extracted from [22] and are listed in Appendix B.1. As discussed in the previous subsection, we need also to choose an appropriate definition of ln detK B detK F or a quantization scheme in the Hamiltonian approach. Since in the present case the characteristic polynomials depend on p 0 only through p 2 0 , for each mode number p 1 we have a positive and a negative root which are equal in absolute value. In the Hamiltonian approach it is then natural to define the vacuum energy as a graded sum of the positive roots (cf. (2.4)). Such a prescription gives the same result as the path integral approach with the "standard" iǫ prescription leading to (2.3). We then find that the 22 Note that for fixed J the small J ′ expansions of a and κ (over which we are to divide E 2d to get E 1 in (2.1)) are given by one-loop correction to the energy vanishes in the limit J ′ → 0. This is a required feature since for J ′ = 0 (a = 0) and J = 0 the solution (2.6) reduces to a BMN geodesic. 23 Let us summarize the results for the 1-loop coefficients (2.10) in the J ′ ≪ J ≪ 1 expansion. Expanding E 2d first in a at fixed ν and then expanding the result in small ν we find for the coefficients f k in (2.10) (for m = 1): Then e 0 (J , 1) = 0 and Comparing this with the general expression for the energy (1.9) (here N = 2J ′ ) we conclude that the resulting values of n 11 , n 12 , n ′ 12 ,ñ 11 are as given in (1.29),(1.45). The values of n 11 and n 12 were already found in [1].

(2.15)
Together with a similar expression found in the (S, J) folded string case [10,6] this provides an evidence of the universality of the general expression in (1.25). Note that when formally expanded in large J , the function h 1 in (2.15) takes the following form: h 1 = 2J + λ J (1 + 2 J + ...) + .... Here the 2 J term is different by a factor of 2 from the result for the leading 1-loop finite size correction found in [26]. This disagreement should not, however, be surprising as the two expansions are derived in different limits (see also Appendix A). In the present case, relevant for "short" strings, we assumed that J ′ ≪ 1 and J is fixed. In contrast, the finite size correction calculation of [26] assumed the standard "fast string" limit of J ′ ≫ 1, J ≫ 1 with J ′ J being fixed and then taken to be small. 24 23 Let us note that to carry out the calculation in a path integral approach in the case of J = 0 one should write the p 0 integral as dp 0 ln detKB detKF = − dp 0 p 0 d dp0 ln detKB detKF . This integration by parts step here is legal as ln detKB detKF vanishes fast enough at infinity. The resulting rational function may then be expanded in J ′ and integrated without a difficulty. 24 Let us recall the distinction between the "small" and "large" circular 2-spin solutions [21,22]. The distinction is sharp at J ≡ J 3 = 0: (i) the solution is "small" if J 1 = J 2 = J ′ is such that J ′ < 1 2 (here J = 0 since ν = 0; this solution is stable); (ii) the solution is "large" if J ′ > 1 2 -(here J = 0 since a 2 = 1 2 ; this solution is unstable). For nonzero J the "small" solution may be defined by requiring that J 2 ≪ J ′ ; then its classical energy still starts with √ 4J ′ and thus scales as λ 1/4 for fixed J ′ . The "large" solution is the one with J ∼ J ′ and J ≫ 1 so that E 0 = J + 2J ′ + 1 J ǫ( J ′ J ) + .... It is stable if J ′ < 3 2 J . While the "small' and "large" cases are smoothly connected for the folded spinning string, that does not apply to the circular 2-spin case as the two expansions have different origins (a → 0 and a → 1 Let us now present the results for the 1-loop coefficients f k (ν, m) in (2.10) in the case of higher winding numbers m ≥ 1 (i.e. for states on subleading Regge trajectories): 25 Simple inspection shows that the coefficient of ζ 3 in f 2 grows like m 4 . This dependence is changed, however, after we express the parameters of the solution in terms of the spins, using, in particular, the relation a 2 = m −1 J ′ + O(J 2 ). The coefficients e k (J , m) in (2.10) are then found to be: As in the folded string case [6], the coefficient of ζ 3 in e 2 grows like m 2 , supporting the above argument for the universality of the transcendental terms. 26 It is possible to find higher orders in the small N = 2J ′ expansion of the one-loop correction (2.10) to the energy: We notice that through O(N 2 ) order all the transcendental terms are the same as in the case of the folded string in AdS 5 [6]; we will find them also to be the same for other two circular string solutions and the folded string in Let us now present the result for the 1-loop correction to the energy in the limit of small J ′ and fixed ρ 2 = J 2 4m √ λJ ′ . At fixed ρ and J ′ ≪ 1 the relation between the parameters of the solution and the charges is: We may use these expressions and f k in (2.10) given in (2.16) to find the fixed-ρ expansion of E 1 . Indeed, since a 2 ∝ J ′2 contains only positive powers of J ′ while κ and ν do not contain inverse powers of J ′ , higher orders in the small a and small ν expansion cannot affect lower orders. For m = 1 we then find (2.20) Taking the limit ρ → 0 we may read off the value of the coefficient n 12 in (1.4),(1.6) (here n 02 = 0) which is in agreement with (1.29),(1.45). It is possible also to determine the transcendental part of the next terms in the small J ′ expansion of the one-loop energy directly at J = 0, extending the ρ = 0 limit of the expression in (2.20) and showing that this limit can be safely taken in that equation: Comparing to (1.7) (where the transcendental part of the N 2 term is contained in n 13 − 1 4 n 02 n 12 ) we find the value of n 13 to be in agreement with (1.36) again with n ′ 13 = − 3 16 and n ′′ 13 = − 3 4 (here n 02 = 0).

Circular string with spins S 1 = S 2 and orbital momentum J
Let us now consider the small string with 2 equal spins in AdS 5 orbiting big circle in S 5 [21,23,1,4] Y 0 + iY 5 = √ 1 + 2r 2 e iκτ , Y 1 + iY 2 = r e i(wτ +mσ) , Y 3 + iY 4 = r e i(wτ −mσ) , X 1 + iX 2 = e iντ , w 2 = κ 2 + m 2 , κ 2 = 4m 2 r 2 + ν 2 , (2.23) Short string limit corresponds to r → 0 when the solution approaches its flat-space limit (for ν = 0). The parameter κ determined from the conformal gauge condition may be written as Below we shall consider the case of m = 1. For small S and small J we get the following "short" string expansion of the classical energy (E 0 = √ λE 0 ): In the limit of small S with fixed J we get As in the previous J 1 = J 2 case it is convenient to carry out the 1-loop calculation in terms of ν and r and then evaluate the result in the two limits: (i) small S with fixed J or (ii) small S with fixed ρ. As in (2.10) the 1-loop correction to the energy may be written as 3) in which we keep unspecified the signs of the terms that vanish in the r 2 ∼ S → 0 limit we found that expanding first in r and then in ν the expansion of the world-sheet energy E 2d in (2.28) contains the following terms We split the result into the contribution of few "low" modes (p 1 = 0, ±1, ±2) and the rest of "higher" modes. The coefficient q of the singular in ν → 0 contributions depends on the signs s p 1 of low fermionic frequencies which vanish at r = 0 for p 1 = ±1, i.e. q = 2 + s 1 + s −1 . There is thus a choice of a sign prescription that ensures the absence of unwelcome singular terms in 27 They can be obtained from those in the J 1 = J 2 case as the two solutions are related by an analytic continuation effectively interchanging the AdS 5 and S 5 parts, a 2 → −r 2 , κ → ν, etc. ν. The natural value for this coefficient is q = 0 as the complete two-dimensional energy of the solution, whose 1-loop part is E 2d above, is the right-hand side of eq. (1.2) and is therefore expected to contain only even powers of J = ν. Setting thus q = 0, the resulting values of the coefficients f k in (2.28) are Using that ν = J and (2.32) it follows that e k in (2.28) are given by Comparing to (1.9) (here N = 2S) we find, in agreement with (1.29),(1.46), that in the present case n 01 = 1, n 02 = 2, n 11 = −2,ñ 11 = 2, and The value of n 11 was previously found in [1]. The value n ′ 12 = 13 8 is the expected one, i.e. is in agreement with (1.33), implying the universality of the value of the energy for the corresponding (Konishi-multiplet) state with J = S = 2 on the lowest massive string level.
As in (2.18) we may determine the transcendental part of the higher order terms in the small S expansion of the energy (N = 2S) 28 : Taking the limit ρ → 0 we may read off again the value of the coefficient n 12 in (1.4),(1.6), (2.34). 29 Summing up the small ν expansion of the function f 1 (ν, 1) in (2.31) we may find the exact form of e 1 (J , 1) in (2.33): . (2.37) These expressions are just negative of the corresponding functions in the J 1 = J 2 case in (2.14), in agreement with the general expression (1.25) and the opposite signs of the n 11 coefficients in (1.45) and (1.46). One may also perform the computation of E 1 by setting J = 0 directly from the start. 30 While similarly to the J 1 = J 2 string case the characteristic polynomials here depend only on p 2 0 and thus for each mode number there are two roots equal in absolute value and opposite in sign, a sign prescription similar to that of the J 1 = J 2 case in which the one-loop energy is given by the graded sum of the positive roots of the characteristic polynomial (2.3) leads to an unwanted feature: a non-zero value for E 2d in the S → 0 limit (see also eq. (3.35) in [1]). As discussed in Appendix A of [1], this constant term may be removed by a specific reorganization of modes together with a change of integration variables, leading to a cancellation of the problematic term at the level of the p 0 integrand (so that a specific iǫ prescription was not necessary). The same result may be obtained by adjusting the sign of just one root of each of the two fermionic characteristic polynomials F 1 and F 2 which for p 1 = ±1 scale as √ S in the limit S → 0: their signs should be such that their contribution adds up to zero. 31 Then the "low" modes with p 1 = 0, ±1, ±2 contribute to the sum over the roots of the characteristic polynomial as: (2.39) Using that E 2d high = 2 3 r 2 + ( 2969 216 − 12ζ 3 )r 4 + O(r 6 ) we find which is the same as the ρ = 0 limit of (2.36). It is possible also to find the analog of (2.22), i.e. to determine the transcendental part of the next terms in the expansion of the one-loop energy of the S 1 = S 2 string at J = 0, extending (2.40) to next order: Comparing to (1.7) we conclude that the highest transcendental coefficient ζ 5 at the next order is again universal, leading to the expression for n 13 in (1.36) again with n ′ 13 = − 85 16 and n ′′ 13 = 15 4 .

Circular string with spins S = J ′ and orbital momentum J
The "mixed" AdS 5 × S 5 circular solution is described by (we set the two windings equal to 1) Note that this solution is "self-dual" under the analytic continuation interchanging AdS 5 and S 5 parts: κ ↔ ν, r ↔ ia, w ↔ −w ′ . The parameters κ and ν may be expressed in terms of the spins by solving the equations The classical energy has the following expansions As in the previous cases we shall carry out the 1-loop computation in terms of the parameters ν and r and then evaluate the result in the small S limit with fixed J or fixed ρ 2 = J 2 4S , i.e. we will define f k and e k as in (2.28). We will need the following small S expansions of the parameters: The corresponding characteristic polynomials are given in Appendix B.3. The summation prescription in (2.4) may be fixed as follows. All frequencies which are nonzero in the BMN limit (r → 0) are summed with uniform signs such that at p 1 ≫ 1 they contribute positively to the energy (this guarantees the vanishing of 1-loop correction to the BMN vacuum state). The signs of some remaining frequencies are fixed by requiring the absence of r 2 ν terms in the frequency sum. Few other signs are fixed by requiring that all O(r 2 ) terms vanish (such terms are expected to cancel due to opposite curvatures of AdS 5 and S 5 ). Then as in (2.29), (2.30) and (2.38) we may split the contribution of modes with p 1 = −2, . . . , 2 from that of the higher ones Here the parameter u represents the still unfixed sum of 4 bosonic p 1 = ±2 frequency signs; it can take values u = −4, −2, 0, 2, 4. Then f k in the analog of (2.28) are Expanding E 1 first in small S at fixed J and then in small J we get This gives n ′ 12 = ( 11 8 , 13 16 , 1 4 ) for u = (−4, −2, 0). The choice of n ′ 12 = 11 8 appears to be preferred in the algebraic curve approach that we discuss in Appendix C. None of these choices leads to n ′ 12 = 5 8 consistent with the universality of (1.30) observed for four other (two folded and two circular) examples of the solutions. This suggests that a consistent summation prescription in this S = J ′ case is yet to be identified.
Expanding in S for fixed ρ when where n ′ 12 is the same as in (2.53). Similarly to the J 1 = J 2 and S 1 = S 2 cases in (2.18),(2.35), the transcendental parts of the higher terms in the small S expansion of E 1 here are found to be (N = 2S) where q k are rational numbers. The coefficient of J N 2 term again leads to the same universal value ofñ 12 in (1.34) withñ ′ 12 = 2q 1 . At O(N 3 ) we should find that q 2 = 3 4 n 11 = 0 and that The absence of ζ 5 in N 3 /J term confirms again the universality of ζ 5 in n 13 in (1.36), the absence of ζ 3 implies that n ′′ 13 = 3 2 and the rational term fixes n ′ 13 = 2(q 1 + q 4 ) + 1 2 n ′ 12 . It is also possible to determine unambiguously the transcendental part of E 1 in the small S expansion at J = 0 (cf. (2.22),(2.41)) where k 3 is a rational number. This again leads to n 13 in eq. (1.36) with n ′ 13 = k 3 + 1 4 n ′
contribution, the second is 1-loop term, the third is 2-loop one, etc. The third line is found by rewriting the semiclassical result back in terms of J.
Starting with E = J 2 + h 1 (λ, J)N + ... in (1.10) and expanding it in semiclassical regime with fixed J and small N we get The 1-loop term − N 2J 1 1+J 2 was found directly in the semiclassical limit in [6]. As was mentioned in section 1, this term is universal, i.e. found also for other semiclassical states (see (1.25)). This expression can be expanded in several different limits and interpolates between some previously known results. If we assume that J ≫ 1, i.e. J 2 ≫ λ, then we get from (A.2) implying that the expansion of E in the large J, small N J limit is implying the values n 11 = −1,ñ 11 = 1 and n 21 = − 1 4 in (1.11),(1.43). This value for the 1loop coefficient n 11 in the small S semiclassical expansion (matching the one directly computed 32 For folded string the 1 J term was found in Appendix D of [28]. 33 To see that there is no linear in S/J ≡ N/J term in the "anomalous" part of the 1-loop correction one needs to differentiate this over M 2 (the first derivative vanishes). 34 As we have checked explicitly from the results in Appendix of [30], the same subleading 1/J 2 finite-size term as in (A.6) appears also for the circular (S, J) string state in the sl(2) sector (here J is the momentum along the circle in S 5 which the string is wound on). This suggests the universality of the terms given explicitly in (A.6) in the sl(2) sector.
using the algebraic curve approach in [3]) has the same origin in (A.2) as the 1 J string term in (A.6): both come from two different limits of the the 1-loop semiclassical term − 1 1+J 2 there. This confirms that this term should not be sensitive to wrapping (Luscher) corrections, being at the same time the origin of a finite-size (and even non-anomalous) term at large J. This also suggests that, like the coefficient of the − 1 J term, n 11 may be coming only from the zero-mode contributions in the near folded-string expansion. This supports the claim [10] that h 1 (λ, J) has its origin just in the asymptotic Bethe ansatz and is not even sensitive to the string phase.
One may expect to find similar expressions for the corresponding (J ′ , J) folded string state in the su(2) sector. Indeed, the folded string in S 5 is related to its AdS 5 counterpart by an analytic continuation [31], implying (up to signs) (E, S; J) → (E; J ′ , J), E = −J, S = J ′ , J = −E. In this case N = J ′ so we may expect to get similar relations as above up to some sign changes, i.e. 35 Changing the of sign of the subleading term in (A.8) compared to (A.2) has two implications: the signs of n 11 , ofñ 11 and of the leading 1 J term also change. Now n 11 = 1 = −ñ 11 as in (1.44) in agreement with [1,5] (see also Appendix D). 36 For large J we get where the (1+ 1 J ) term is in agreement with the result for the finite size corrections from the spin chain and the string sides (cf. eq.7.33,7.34 in [28]). As in the sl(2) sector case in (A.2),(A.5), the subleading term 1 J 2 in (A.9) should originate from the next (string 2-loop) term in h 1 in (A.8). The coefficient of this 1 J 2 term should be universal in the su(2) sector, i.e. the same also as for the circular string. Indeed, for the circular string in the su(2) sector we get (A.9) with the same terms in the bracket (1 + 1 J + 1 J 2 + ...), as one can see from [32] (these terms come from non-anomalous finite size contribution only). Such a correction in the near-BMN expansion was found also in [28]. It came out the same from the Bethe ansatz and the Landau-Lifshitz approach, so it should be a protected one. 37  In the case of "small" circular strings with 2 internal spins we again find (A.10) 35 Note that this analytic continuation is not useful if J is fixed, while E ∼ λ 1/4 ≫ 1 so there is no way of interchanging E and J. It still works at large J and thus large E and explains why the sign of first finite size correction changes: and then using that E = J + ... we get the required result. 36 The change of sign of the leading 1-loop string correction can be attributed to the change in sign of the curvature between AdS 5 and S 5 [1]. 37 The fact that it comes out of the Landau-Lifshitz approach means that one does not need the full superstring computation to reproduce it, provided one regularizes properly (in addition, only zero modes are expected to contribute to this term).
The characteristic polynomials for this circular string have been derived in [21,22]. The AdS 5 fluctuations have the standard BMN type form with mass κ (expressed in terms of the other independent parameters a and ν, see (2.6)) while the characteristic polynomial for the S 5 part is more complicated. Explicitly, 38 This structure from expansion of eq.(2.23) in [29] to linear order in N : again only the analytic spin chain side part or 0-mode string side part is contributing to it. It appears that the analytic finite size correction to the linear in N term is universal: 1 + 1 L in compact (su(2), etc.) sector and 1 − 1 L in noncompact (sl(2), etc) sector. Here L = J + N is total length, its difference from J is irrelevant to leading order in N . The sign difference is due to the analytic continuation between the sectors.
As discussed in [21,22], the determinant of the fermionic quadratic operator is the square of an operator expressed solely in terms of six-dimensional Dirac matrices. We note here that, due to the chirality of six-dimensional spinors, this determinant (over spinor indices) further factorizes: where F 1,2 are the corresponding fermionic characteristic polynomials Using the relations between the parameters of the solution, one can check that the product F 1 F 2 reproduces the fermionic characteristic polynomial in [22].

B.3 S = J ′ string
Here the AdS 5 bosonic characteristic polynomial can be directly extracted from [24] (from the expression found before using the conformal gauge constraint). 39 Then its S 5 counterpart can be found by using the "self-duality" property of the solution (2.42) under Including also a nontrivial winding number m, we end up with As in the previous cases here the fermionic operator can be put into a block-diagonal form where each block may be written in terms of the six-dimensional Dirac matrices. While the two blocks are not identical, parity invariance requires that their determinants are the same. The fact that six-dimensional spinors are chiral implies that the determinant of each block further factorizes as in (B.4), where now Let us comment on derivation of these expressions (that reduce to the ones in [24] for a = 1 in (2.42)). In the κ-symmetry gauge θ 1 = θ 2 the quadratic part of the fermionic Lagrangian is (see, e.g., [22,24] and refs. there) where D = d + 1 4 ω AB Γ AB is the usual spinor covariant derivative. For the solution (2.42) the 2d projected combinations e A α Γ A and ω AB α Γ AB are: 39 One can check directly that the massless mode decouples in the characteristic polynomial for three coupled AdS 5 fluctuation modes that follows from the fluctuation Lagrangian in eq. (4.13) in [24].
where Γ A are the 10-d Dirac matrices; one should project the quadratic operator onto its chiral part thus rendering it a 16 × 16 matrix. To evaluate the determinant of the quadratic fermionic operator we first notice that the matrices Γ 2 and Γ 3 in Γ * = iΓ 01234 in (B.17) do not appear elsewhere in the quadratic operator. The product Γ 23 may therefore be diagonalized; its diagonal entries are ±i. In this representation the quadratic operator is block-diagonal and each block may be obtained from (B.17) and (B.18) by using for Γ i and Γ ij the d=6 Dirac matrices and Γ * = ±Γ 014 . Since the sign of Γ * affects only the sign of the Wess-Zumino term which can also be changed by parity transformations, the determinants of the two blocks are equal and thus the 10d determinant is a perfect square, as in the first equation in (B.4). Since the 6d spinors are chiral, there exists a representation of the 6d Dirac matrices in which each block of the quadratic operator is itself block-diagonal. Thus, the determinant of each block further factorizes; each block is only a 4 × 4 matrix and its determinant can be easily evaluated leading to the two factors F 1 and F 2 in eq. (B.4) given by (B.15),(B.16).
Appendix C: One-loop energy of S = J ′ circular string from the algebraic curve approach Here we shall revisit the computation of the 1-loop correction to the energy of the S = J ′ circular string solution (2.42) discussed in section 2.4 using the algebraic curve approach [35,37] to determine the fluctuation frequencies.
In order to focus on a near flat space expansion in the short string limit we will consider the limit S → 0 for fixed ̺ ̺ = ν 2 √ S . (C.1) In section 2.4 in (2.46) we used instead Note also that

C.1 Quasimomenta
The quasimomenta can be obtained by explicit diagonalization of the monodromy matrix [35]; for the S 5 part the basic single cut quasimomenta vanishing at infinity are determined bỹ where the two rootsx 1 ,x 2 are given by: The four S 5 quasimomenta can be identified looking at the asymptotic x → ∞ behaviour of p(x) andp(x −1 ), which is related to the conserved global charges: For the AdS 5 quasimomenta, the basic function is given by: where thex i are: Again, comparing with the asymptotic, the identification of the quasimomenta goes as follows: p1(x) = −p(x −1 ) , p2(x) =p(x) , p3(x) = −p(x) , p4(x) =p(x −1 ) . (C.10)

C.2 Off-shell frequencies
Due to the symmetry of the circular string solution, all the fluctuation energies can be conveniently written as combinations of only two independent functions Ω A (x) = Ω23(x) and Ω S (x) = Ω23(x) [37]: Following [37], the two functions Ω A (x) and Ω S (x) can be uniquely fixed imposing the correct analytical and asymptotic properties for the perturbed quasimomenta p + δp: where the two functionsf (x) andf (x) are defined as with a suitable choice of the cuts.

C.3 One-loop energy
Given the above set of off-shell frequencies Ω I = Ω i,j , I ∈ {A, S, B 1,2,3,4 , F 1,2,3,4 }, the corresponding physical on-shell fluctuations energies associated to the (i, j) excitations with mode number n, are given by ω (n) I = ω (n) i,j = Ω i,j (x i,j n ) , (C.14) where, for any pair (i, j), x i,j n is determined as the solution of the equation The one-loop correction to the energy can be obtained as a sum over n and polarizations 40 This sum is sensitive to integer shifts in the labeling of the frequencies n → n + δ; following [35] here we propose to use the following choice: Then the final result in the short string limit has the same form as in (2.55))   Appendix D: One-loop energy of the (J ′ , J) folded string from the algebraic curve approach Here we shall derive the 1-loop coefficients in (1.44) in the small spin expansion of the energy of a folded string with spin J 1 = J ′ and orbital momentum J 3 = J representing a state in the su(2) sector on the dual gauge theory side. This will a direct counterpart of the computation done for the (S, J) folded string in [6].

D.1 Quasimomenta
The classical solution [38] for the folded string with spin J ′ and orbital momentum J in S 5 is related to the folded string with spin S in AdS 5 and orbital momentum J in S 5 by an analytical continuation [31] implying a relation between the string profiles and the global conserved charges In the algebraic curve approach the quasimomenta for the (J ′ , J) string can then be obtained by an analytical continuation of the quasimomenta for the (S, J) string given in [3]. According to [5] the S 5 quasimomentum p 2 as a function of the branch points is expressed in terms of the elliptic functions: where Re(a), Im(a) > 0, b = −a and The inversion symmetry provides the other sphere quasimomenta through the relations p 2 (x) = −p 3 (x) = −p 1 (x −1 ) = p 4 (x −1 ).
(D. 6) Since the motion in the AdS 5 part is trivial, the corresponding quasimomenta are simply