ABJM quantum spectral curve at twist 1: algorithmic perturbative solution

We present an algorithmic perturbative solution of ABJM quantum spectral curve at twist 1 in sl(2) sector for arbitrary spin values, which can be applied to, in principle, arbitrary order of perturbation theory. We determined the class of functions -- products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions -- closed under elementary operations, such as shifts and partial fractions, as well as differentiation. It turns out, that this class of functions is also sufficient for finding solutions of inhomogeneous Baxter equations involved. For the latter purpose we present recursive construction of the dictionary for the solutions of Baxter equations for given inhomogeneous parts. As an application of the proposed method we present the computation of anomalous dimensions of twist 1 operators at six loop order. There is still a room for improvements of the proposed algorithm related to the simplifications of arising sums. The advanced techniques for their reduction to the basis of generalized harmonic sums will be the subject of subsequent paper. We expect this method to be generalizable to higher twists as well as to other theories, such as N=4 SYM.

Eventually, a detailed study of TBA equations for super spin chains corresponding to N = 4 SYM and ABJM models has led to their simplified alternative formulations in terms of Quantum Spectral Curve (QSC), a set of algebraic relations for Baxter type Qfunctions together with analyticity and Riemann-Hilbert monodromy conditions for the latter [77][78][79][80][81][82][83][84]. Within the quantum spectral curve formulation one can relatively easy obtain numerical solution for any coupling and state [85][86][87]. Also, QSC formulation allowed to construct iterative perturbative solutions for these theories at weak coupling up to, in principle, arbitrary loop order [81,88,89]. The mentioned solutions are however limited to the situation when the states quantum numbers are given explicitly by some integer numbers. Recently, in Ref. [90,91] we started developing techniques for the solution of QSC equations treating state quantum numbers as symbols. The first technique based on Mellin space transform [90] turned out to be quite complex to go for all-loop generalization. On the other hand, in Ref. [91] we suggested, that there should be relatively easy way to obtain a perturbative solution for the spectrum of twist 1 operators in sl (2) sector for ABJM model working directly in spectral parameter space. The goal of this paper is to present the algorithm for perturbative solution of ABJM quantum spectral curve at twist 1 in sl(2) sector to any loop order. The latter is based on the existence of a class of functions -products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions, which is closed under elementary operations, such as shifts and partial fractions, as well as differentiation. The introduced class of function is sufficient for finding solutions of involved inhomogeneous Baxter equations using recursive construction of the dictionary for the solutions of Baxter equations for given inhomogeneous parts. As an application of the proposed method we present computations of anomalous dimensions of twist 1 operators at six loop order. There is still a room for improvements of the proposed algorithm related to the simplifications of the arising sums and we plan to present advanced techniques for their reduction to the basis of generalized harmonic sums in one of our subsequent papers. The presented approach has the potential for generalizations to higher twists of operators, as well as to other theories such as N = 4 SYM and twisted N = 4 SYM and ABJM models. This paper is organized as follows. In the next section we give necessary details on ABJM quantum spectral curve equations putting emphasis on Pν-system. Section 3 contains all the details about our solution of Riemann-Hilbert problem for Pν-system, used for calculation of anomalous dimensions of twist 1 operators. Next, section 4 contains the results for anomalous dimensions up to six loop order and their discussion. Finally, in section 5 we come with our conclusion. Appendices and Mathematica notebooks contain some details of our calculation.

ABJM quantum spectral curve
As it was already mentioned in introduction, the ABJM model is the second most popular playground for testing AdS/CFT correspondence. It is a three-dimensional N = 6 Chern-Simons theory based on the product U(N) ×Û (N) of two gauge groups with opposite Chern-Simons levels ±k. In the planar limit, where N, k → ∞ so that the 't Hooft coupling λ = k N kept fixed, this theory has a dual description in terms of IIA superstring theory on AdS 4 × CP 3 . The field content of ABJM model consists of two gauge fields A µ andÂ µ , four complex scalars Y A and four Weyl spinors ψ A with matter fields transforming in the bi-fundamental representation of the gauge group. The global symmetries of ABJM theory with Chern-Simons level k > 2 are given by orthosymplectic supergroup OSp(6|4) [38,92] and the "baryonic" U(1) b [92]. The bosonic subgroups of OSp(6|4) supergroup are related to isometries of superstring background AdS 4 × CP 3 .
In the present paper we will be interested in the calculation of anomalous dimensions of twist 1 gauge-invariant operators from sl(2) sector for arbitrary spin values S. The latter are given by single-trace operators of the form [93]: where twist 1 corresponds to L = 1. The expressions for anomalous dimensions can be conveniently obtained by solving the corresponding spectral problem for OSp(6|4) spin chain. The most advanced framework for that at the moment is offered by quantum spectral curve (QSC) method. The latter is an alternative reformulation of Thermodynamic Bethe Ansatz (TBA) as a finite set of functional equations: Q-system. The most important advance is provided by the considerable simplification of the spectral problem calculations. In the case of ABJM model QSC formulation was introduced in Ref. [82,83], see also Ref. [89]. To perform actual calculations of anomalous dimensions we will use monodromy conditions for the part of ABJM Q-system known as Pν-system [82,83]. The latter consists of six functions P A , A = 1, . . . , 6 and eight (4 + 4) functions ν a , ν b , a, b = 1, . . . 4 satisfying nonlinear matrix Riemann-Hilbert problem [82,83]: where Here and in the followingf will denote a function f analytically continued around one of the branch points on the real axis. In addition, the P and ν -functions should satisfy extra constraints P 5 P 6 = 1 + P 2 P 3 − P 1 P 4 , (2.5) ν a ν a = 0, (2.6) Both P A and ν a , ν a are functions of spectral parameter u. The P A functions have a single cut on the defining Riemann sheet running from −2h to +2h (h is effective ABJM QSC coupling constant 1 ), while ν a , ν a functions have an infinite set of branch cuts located at intervals (−2h, +2h) + in, n ∈ Z and satisfy simple quasi-periodicity relations where P is a state dependent phase fixed from self-consistency of QSC equations [83]. To get QSC description of states in sl(2) sector (2.1) it is sufficient to consider Pν-system reduced to symmetric, parity invariant states. The reduced Pν-system is identified by constraints P 5 = P 6 = P 0 , ν a = χ ab ν b and is written as [82,83,89]: where and ν a satisfy now periodic/anti-periodic constraints (σ = ±1) where f [n] (u) = f (u + in/2) and the constraint for P functions takes the form In addition to the above analytical properties of P and ν functions it is required [83,89] that they are free of poles and stay bounded at branch points. The quantum numbers of spin chain states under consideration, that is twist L, spin S and conformal dimension ∆ are encoded in the behavior of P, ν functions at large values of spectral parameter u [82,83,89]: In contrast to N = 4 SYM, ABJM QSC coupling constant h is a nontrivial function of 't Hooft coupling constant λ, Ref. [25,94]. There is a conjecture for the exact form of h(λ), Ref. [95,96], made by a comparison with the localization results. and which serve as boundary conditions for the Riemann-Hilbert problem under study. The anomalous dimension γ, which is our main interest here, is given by γ = △ − L − S.

Solution of Riemann-Hilbert problem for Pν-system
To solve the Riemann-Hilbert problem for fundamental Pν-system it is convenient to add to original equations (2.8) -(2.9) their algebraic consequences [89].
We will also need the equations following from the sum of equations (2.8) and (3.15): where p A = (xh) L P A and is the Zhukovsky variable used to parameterize the single cut of P -functions on the defining Riemann sheet. In summary, the equations we are going to solve are given by and σν [2] , (3.24) In addition, there are simple consequences of a given cut structure for ν-functions, which will be used during solution. Namely, the following combinations of functions don't have cuts on the real axis. To find a perturbative solution of the above system of equations we will use expansion of ν a (u) functions in terms of QSC coupling constant h together with the following parametrization of P -functions [88,89] where we have also accounted for large u asymptotic of P functions (2.13). We would like to note, that, due to residual gauge symmetry of QSC equations, 2 the coefficients m i,k are some functions of spin S only and the mentioned gauge freedom can be also used to set A 1 = 1 and A 2 = h 2 . The analytical continuation of P-functions through the cut on real axis is simple and is given by [88]: In what follows we will consider perturbative solution in a special case of twist 1 operators, so from now on we put L = 1.

Sums of Baxter polynomials
Recently, in Ref. [91] we have suggested that the full all-loop solution of the Pµ-system (2.8) -(2.9) can be obtained in terms of linear combinations of products of rational (in spectral parameter u), Hurwitz -functions and Baxter polynomials and showed an explicit example at four-loop order. The purpose of this paper is to present explicit all-loop solution. To do that, let us first introduce the necessary notation. The expressions for Baxter polynomials are obtained as leading order solutions for ν where Q S (u) is given by (3.37) and α is some spin-dependent constant to be determined later.
Let us now introduce the following class of sums involving Baxter polynomials where w k are some weights. In the case, when the argument of Q S is u we will often drop it and simply write Q|w 1 (•) , w 2 (•) , . . . , w n (•) . We also introduce a shortcut Note that the w 1 , W -sums satisfy usual stuffle relations. Here and below we write weights w k (•) in several equivalent ways, w k (•) ≡ w k (j) ≡ w k and use W to denote arbitrary (maybe empty) sequence of weights. In addition, we will use the notation It turns out that weights at twist 1 can always be reduced to a set of canonical weights for which we introduce special notations: In Ref. [91] we have considered elementary operations on Baxter polynomials, such as shifts and partial fractions. The latter can be also extended to the sums of Baxter polynomials. In particular, the shift in spectral parameter u can be performed using (a = ±1): and Next, we have the rules for partial fractions (a = ±): and Finally we can shift the spin S of the Baxter polynomials using (3.50) Remarkably, the introduced class of functions, Q| . . . , is closed under differentiation. In order to prove this, let us first consider the sums Let us prove that these sums reduce to the linear combination of our standard sums (3.41). We proceed by induction over the depth of the sum. Let us first write where the lower sign is chosen if w 1 (j 1 ) contains the factor (−) j 1 . Then the denominator j 0 − j 1 cancels in the first term and, therefore, this term gives rise only to standard sums (3.41). The second term gives rise to the sums In order to transform these sums, we observe that This identity is proved by the substitution j 1 → j 0 + j 2 − j 1 . Then 55) and the inner sum is again of the form (3.51), but the depth is reduced. This proves the induction step. Now, using the differentiation formula 4 Since the inner sums in the last expression are both of the form (3.51), they can be expressed as a linear combination of the standard sums (3.41). Therefore, i∂ u Q| . . . can indeed be expressed as a linear combination of Q| . . . sums. In particular, the expansion of Q|W -sums at u = i/2 can be expressed in terms of W -sums. Let us summarize the results of the present subsection. We introduced the class of functions -products of rational functions in spectral parameter u with Q|W -sums (3.38) closed under elementary operations, such as argument shifts and partial fractions, as well as under differentiation. As we will see in next subsection this class of functions extended to products with Hurwitz functions is also sufficient for finding a perturbative solution of inhomogeneous Baxter equations.

Solutions of Baxter equations
The most complicated part in the perturbative solution of Riemann-Hilbert problem for 1 . The solution of these equations contains in general two pieces: the solution of homogeneous equation with arbitrary periodic coefficients and some particular solution of nonhomogeneous equation.

Homogeneous solution
The first homogeneous solutions of Baxter equations (3.58) and (3.59) are easy to find, they are given by 5 Here Φ per Q (u) and Φ anti Q (u) are arbitrary i-(anti)periodic functions of spectral parameter u. To find second solutions let us consider the following identity It is easy to see that the second homogeneous solution of the second Baxter equation (3.59) is given by Then the general solutions of first and second homogeneous Baxter equations are given by where Φ per a and Φ anti a are arbitrary i-periodic and i-anti-periodic functions in spectral parameter u. They have to be determined from the consistency conditions implied by the equations (3.20)-(3.28). We will parametrize their u dependence similar to Ref. [88,89] using the basis of i-periodic and i-anti-periodic combinations of Hurwitz functions defined as Note that P k (u) can be expressed via elementary functions: Then the functions Φ per a and Φ anti a are written as where the upper limits of summation depend on the order of perturbation theory as follows Here k = 1 for NLO, k = 2 for NNLO, and so on.

Dictionary for inhomogeneous solutions
To find particular solutions of Baxter equations (3.58) and (3.59) let us introduce the operators F ±1 which are right inverse of the Baxter operators B ±1 , Eqs. (3.58), (3.59), i.e., satisfy Note that they are nothing but the operators F S 1,2 introduced in our previous paper [91]. Our basic idea now is to compile a dictionary sufficient to treat all the functions appearing in the right-hand sides of Baxter equations.
Action of F ±1 on Q|W . We first act with the operators B ±1 on the functions Q (u) |w, W : Replacing w → 1 + · w in the first formula and w →1 + · w in the second we obtain the following entries in our dictionary: Action of F ±1 on Q|W ξ a 1 ,a 2 ,... . The basic idea of calculating F ±1 [ Q|W ξ A ] is to use the analogue of summation-by-part formulae from Ref. [88,89]. For any two functions f and g we have 7 Substituting g → F ±σ [g] and f = ξ a,A , we obtain In the present section σ = ±1 is an arbitrary sign, not to be confused with σ = (−) S entering (2.11) and other equations for Pν-system.
The operators F ±1 in the right-hand side of the above equations act on the 'simpler' objects because the number of indices of ξ-functions is reduced by one.
Let us present separately the formula (3.80) In order to find F σ Q| ξ σ|a|,A , we consider the identity We transform the right-hand side using the formulas (A.133) and obtain Then we obtain From now on we will present only the final entries of our dictionary as the derivations goes more or less along the same lines as before. The action of F ±1 on ξ A (u±i/2) a is the following Action of F ±1 on u n Q|W ξ A Finally, the right-hand side of the Baxter equations may also contain terms of the form u n Q|W ξ A with n = 1, 2. First, we use the same summation-by-part technique as before. Namely, we use Eq. (3.77) with g = u n Q|W to reduce the problem to the calculation of F σ [u n Q|W ].
In order to calculate F σ [u n Q|W ], it is convenient to introduce notation In terms of these functions we can easily express the required action: and we assume that w(j) is one of the canonical weights (3.42). Then we use the partialfractioning identities similar to where we choose the lower sign if w(j) contains (−) j factor and the upper sign otherwise. Then the first term in Eq. (3.94) is obviously a combination of canonical weights. The second term contains a shifted weight 1 ± (j − 1). Note that this shift is correlated with superscript {n} of Q functions. Therefore, we need the transformation rules for the sums of the form where w (•) is one of the canonical weights (3.42), and where1 is one of the four weights 1 + , 1 − ,1 + ,1 − . First, we note simple consequences of Eq. (3.49): for n > 0 we have The sums of the form (3.96) after the substitution of the definition (3.88) and shifting j 1 → j 1 + n are almost of the required form except for the upper limit of summation over j 2 which is j 1 + n − 1, i.e., is shifted by n. Then we can treat the missing/redundant terms in a recursive manner. For example, we have Then the second term is again of the same form as those in the right-hand side of Eq. (3.89). Note that special attention should be paid to the sums with depth less or equal to |n|. The full set of the reductions rules can be found in the code of the attached Mathematica file.

Constraints solution
Now, with the knowledge of how to find the solutions of two Baxter equations (3.58) and (3.59) in each order of perturbation theory we may proceed with the determination of constants in the anzats for P -functions together with additional φ (3.100) Also from the requirement of absence of poles in combinations (3.28) for ν

NLO
Following the above procedure step by step at NLO we get 9 1. First, from equation (3.24) : (3.103) 9 The expressions for q (1|2) 1,2 -functions can be found in accompanying Mathematica files.

Then, from equation (3.25)
: (1) 1 [1] function is given by and from the absence of poles in equations (3.28) we have

NNLO
At NNLO we were not reducing all w 1 (•) , w 2 (•) , . . . , w n (•) sums at intermediate steps to H and B -sums. Such reduction was performed only for the NNLO anomalous dimensions at the end. Moreover, this final reduction was not algorithmic -we just solved a system of equations for 768 spin values, which is the dimension of ourH basis 11 at weight 5 corresponding to NNLO. Still, our preliminary considerations show that the required algorithmic reduction at all steps is possible and, what is more important, it will make our algorithm much more efficient. The details of this reduction will be the subject of one of our subsequent papers. Following the steps of general procedure for constraints solution at NNLO we get 12

Anomalous dimensions
The expressions for the anomalous dimensions can be easily obtained from the corresponding expressions for A where 13 See appendix B for the definition ofH -sums.
and   The obtained results are in complete agreement with previous results at fixed spin values [89,97,98]. Note, that ourH -sums here can be further rewritten using cyclotomic or S-sums of Ref. [99,100] provided one extends the definition of the latter for the complex values of x i parameters. It is also possible to express them in terms of twisted η-functions introduced in Ref. [41]. Here, we see that the maximal transcendentality principle 14 [104,105] also holds for anomalous dimensions of ABJM theory with the account for finite size corrections up to six loop order and it is now natural to assume it is validity for ABJM model to all orders. That is the results for anomalous dimensions in each order of perturbation theory are expressed in terms ofH-sums of uniform weight w, where w = 3 at NLO and w = 5 at NNLO. In general, the size of the basis ofH-sums at weight w is equal to 3·4 w−1 and at NNNLO (w = 7) we should have 12288 such sums. Moreover, while discussing solution of NNLO constraints in subsection 3.3 we noted that at present we are missing automatic reduction of w 1 (•) , w 2 (•) , . . . , w n (•) sums, arising at different steps of our calculation, toH -sums, which makes intermediate expressions even larger. We are planing to address this latter issue in one of our subsequent publications. In addition it is desirable to construct Gribov-Lipatov reciprocity respecting basis [106][107][108] of generalized harmonic sums also for ABJM model. The latter in the case of N = 4 SYM is known to be much more compact compared to the original basis of harmonic sums and was used in Ref. [109][110][111][112][113] to simplify the reconstruction of the full spin S dependence of anomalous dimensions from the knowledge of anomalous dimensions at a set of fixed spin values.

Conclusion
In this paper we have presented an algorithmic perturbative solution of ABJM quantum spectral curve for the case of twist 1 operators in sl(2) sector of the theory. The solution treats operator spin S as a symbol and applies to all orders of perturbation theory. The presented solution is performed directly in spectral parameter u-space and effectively reduces the solution of multiloop Baxter equations given by inhomogeneous second order difference equations with complex hypergeometric functions to purely algebraic problem. The solution is based on the introduction of a new class of functions -products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions, which is closed under elementary operations, such as shifts and partial fractions, as well as differentiation. This class of functions is also sufficient for finding solutions of inhomogeneous Baxter equations involved. For the latter purpose we present recursive construction of F ±1 images for different products of Hurwitz functions with arbitrary indexes or fractions 1 (u±i/2) a with leading order Baxter polynomials or their sums. The latter are entering inhomogeneous pieces of multiloop Baxter equations at different orders of perturbative expansion in coupling constant. Similar to Ref. [88,89], where all the operations performed were closing on trilinear combinations of rational, η and P k -functions, all our operations are closing on fourlinear combinations of rational, η, P k and Q|W -functions. As a particular application of our method we have considered anomalous dimensions of twist 1 operators in ABJM theory up to six loop order. The obtained result was expressed in terms of generalized harmonic sums decorated by the fourth root of unity factors and introduced by us earlier. The results for anomalous dimensions respect the principle of maximum transcendentality. It should be noted, that there is still a room for improvements of the proposed algorithm related to the simplifications of arising sums at different steps of presented solution. The advanced techniques for their reduction toH-sums will be the subject of one of our subsequent papers.
We expect the presented method to be generalizable to higher twists as well as to other theories, such as N = 4 SYM. The developed techniques should be also applicable for solution of twisted N = 4 and ABJM quantum spectral curves with P functions having twisted non-polynomial asymptotic at large spectral parameter values, see [79,84,114] and references therein. The latter models received recently a lot of attention in connection with the advances in so called fishnet theories [115][116][117][118][119][120][121][122][123][124][125][126][127][128][129][130]. Moreover, similar ideas should be also applicable to the study of BFKL regime within quantum spectral curve approach [54][55][56] for N = 4 SYM. In the latter case we also have a perturbative expansion when both coupling constant g and parameter w ≡ S + 1, describing the proximity of operator spin S to −1 are considered to be small, while their ratio g 2 /w remains fixed.

A Hurwitz functions
We define Hurwitz functions entering the presented solution as Here A denotes the arbitrary sequence of indexes and ξ function without indexes is identical to unity. These are the shifted versions of Hurwitz functions introduced in [88,89] ξ A = η