Quantum Spectral Curve of \gamma-twisted {\cal N}=4 SYM theory and fishnet CFT{}^{*}

Abstract

We review the quantum spectral curve (QSC) formalism for the spectrum of anomalous dimensions of SYM, including its -deformation. Leaving aside its derivation, we concentrate on the formulation of the “final product” in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the -system – the full system of Baxter -functions of the underlying integrable model. The algebraic structure of the -system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for -functions organized into the Hasse diagram. When supplemented with analyticity conditions on -functions, it fixes completely the set of physical solutions for the spectrum of an integrable model. First we demonstrate the spectral equations on the example of and Heisenberg (super)spin chains. Supersymmetry occurs as a simple “rotation” of the Hasse diagram for a system. Then we apply this method to the spectral problem of AdS/CFT-duality, describing the QSC formalism. The main difference with the spin chains consists in more complicated analyticity constraints on -functions which involve an infinitely branching Riemann surface and a set of Riemann-Hilbert conditions. As an example of application of QSC, we consider a special double scaling limit of -twisted SYM, combining weak coupling and strong imaginary twist. This leads to a new type of non-unitary CFT dominated by particular integrable, and often computable, 4D fishnet Feynman graphs. For the simplest of such models – the bi-scalar theory – the QSC degenerates into the -system for integrable non-compact Heisenberg spin chain with conformal, symmetry. We describe the QSC derivation of Baxter equation and the quantisation condition for particular fishnet graphs – wheel graphs, and review numerical and analytic results for them.

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Quantum Spectral Curve of -twisted SYM theory and fishnet CFT

1

V. Kazakov]Vladimir Kazakov  

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1 Introduction

In the past 40 years, a multitude of super-symmetric conformal quantum field theories (CFT) in four dimensions has been discovered and studied[1]. Typically, they are various deformations of super-Yang-Mills theories, with supersymmetries (see [2] for modern classification). On the other hand, well identified non-supersymmetric and/or non-gauge CFTs in four dimensions are rare species. Apart from a rather exotic Banks-Zaks theory [3] or critical Potts model[4] there are hardly known examples which are explicitly constructed and well understood.2

Even more rare are the integrable four-dimensional CFT’s. The SYM theory is the emblematic example of such a theory: it is conformal for gauge group for any but integrable only in the large , t’ Hooft limit (see the review[5] and references therein).3 It was long believed that its large global super-conformal symmetry is responsible for the integrability. However, under specific deformations breaking the supersymmetry partially or entirely, the theory seems to retain its integrability.

An important, and rather general class of such deformations is the -twist [1, 6, 7]. It breaks the global symmetry to , i.e. only three Cartan subgroups are left from -symmetry and preserves, at least on the tree level, the 4D conformal symmetry. The last one could be endangered by various conformal anomalies[8] but, remarkably, it survives when adding to the action a well defined set of double-trace counter-terms [9, 10, 11, 12, 13] and tuning the double-trace couplings to certain critical values. The critical double-trace couplings are, in general, complex functions of the ’t Hooft coupling which parameterizes the whole family of these non-unitary CFTs.

All the quantum integrability properties known from the undeformed SYM seem to survive as well this -deformation[13]. In particular, the quantum spectral curve (QSC)[14, 15, 16] – the most advanced formalism of AdS/CFT integrability, giving a comprehensive solution of the problem of spectrum of anomalous dimensions of local (and some non-local) operators – remains valid the -deformation, with minor modifications, and describes this non-unitary CFT precisely at the (complex) critical line [13]. The QSC method already found numerous applications in the study of planar spectrum of SYM theory [17, 18, 19, 20, 21, 22, 23, 24, 16, 25, 26, 27, 28] (see also recent review [29] and references therein).

Recently, Ö.Gürdogan and the author proposed in [30] a special double scaling (DS) limit of the -deformed SYM, combining the weak coupling limit and large imaginary values of the -parameters. It gives rise to a new 4D non-unitary CFT where the gauge interactions decouple and only chiral 4-scalar and Yukawa interactions are left. They are also expected to inherit the integrability properties of their “mother” theory – the -deformed SYM. In the simplest case, when only one double-scaling coupling is kept non-zero, it becomes a simple theory of two interacting complex scalars (referred to in what follows as the “bi-scalar theory”). Nevertheless, it is still a non-trivial interacting CFT but in addition it is integrable in planar limit! Its integrability, unlike the integrability of its “mother” theory, has a clear origin: its perturbation theory for various correlation functions is dominated by the “fishnet” Feynman graphs. This means that sufficiently large planar graphs have in the bulk the shape of regular square lattice. It was noticed long ago[31] that such a graph defines an integrable 2D statistical-mechanical spin system with symmetry, which is a four-dimensional conformal group. Thus the integrability of the bi-scalar theory is tightly related with the integrability of the conformal, non-compact Heisenberg spin chain. The theory of integrable non-compact spin chains has a long history [32, 33, 34, 35, 36, 37, 38] following the fundamental works of L.D. Faddeev and the Leningrad school (see [39] and references therein). It had also a few important applications, such as BFKL approximation in high-energy, Regge limit in QCD[35, 36]. Many of these and other old results on non-compact integrable spin chains appear to be very helpful in the study of non-perturbative dynamics of the bi-scalar theory and the other CFT’s from the family of chiral CFT obtained in the DS limit from -deformed SYM [40].

In this work, we will review the formalism of quantum spectral curve (QSC) for the spectrum of anomalous dimensions of the planar SYM theory, including its -deformed version (named below as QSC). We will concentrate ourselves on the minimal set of basic propositions when formulating the QSC equations, leaving aside its derivation. The algebraic part of QSC, entirely dictated by the global symmetry (broken to by -deformation) and quantum integrability, is most conveniently formulated in terms of the -system - a set of Baxter’s -functions of spectral parameter . -functions are organized into the Hasse diagram, reflecting the fact that they are Grassmannian coordinates and they obey certain Plücker relations. The analytic part of QSC construction consists of description of the structure of Riemann surfaces of -functions, where the main element is an infinite “ladder” of equally spaced “Zhukovsky” quadratic cuts at , where is the ’t Hooft coupling. The large asymptotics of -functions and their specific monodromy properties around the Zhukovsky cuts conclude the formulation of spectral problem. Roughly speaking, QSC represents a system of non-linear Riemann-Hilbert equations. We will start with the section 2 where we demonstrate the similar formalism and the emergence of the supersymmetric -system on the example of Heisenberg spin chain where the analyticity constraints look much simpler. Then, in section 3, we will describe the QSC for -deformed SYM (AdS/CFT duality). In section 4, we will describe the CFTs emerging in the DS limit of this theory, and in particular the so-called bi-scalar theory dominated by integrable “fishnet” graphs. We will review the results obtained for these models from QSC, such as the exact computation of certain multi-loop “wheel” graphs and discuss the equivalence of the bi-scalar theory to the conformal integrable Heisenberg spin chain. The section 6 is devoted to conclusions and unsolved problems.

2 Spectrum of Heisenberg spin chain from Baxter -functions

In this section, we will give an alternative formulation of the well known solution for the spectrum of compact, Heisenberg spin chain which will be useful for the generalization to AdS/CFT integrability. We will avoid the direct use of standard Bethe equations since in the sigma-model on background, which describes the string side of the duality, the notion of Bethe roots is a tricky and not very invariant issue. We will rather rely on the full system of Baxter -functions, forming a Grassmannian, spectral parameter dependent structure in the dimensional space. The -functions are most naturally classified by the vertices of Hasse diagram – the -dimensional hypercube. Specifying the analyticity properties of these -functions w.r.t. the spectral parameter one can classify and efficiently study the spectrum. The supersymmetric generalization of this picture in terms of Hasse diagram, from bosonic group to to supersymmetric group will be essentially a “rotation” of the hypercube when imposing specific determinant relations (“determinat” flow) and analyticity conditions. Many of the details, missing in this short overview of spin chains from the point of view of -functions, can be found in [41, 42, 16, 43] and in the recent review [29].

2.1 Spectrum of spin chain via -functions on Hasse diagram

Let us start from the Heisenberg spin chain with spins belonging to the bosonic group and twisted boundary conditions. This spin-chain is defined through the hamiltonian

(1)

where the spin at each site takes the values , the permutation acts on a pair of spins as and the twist is a fixed element of Cartan subgroup.4 Explicitly, in components, various terms in (1) mean

(2)
(3)

To formulate the solution for the spectrum of this hamiltonian we introduce a set of Baxter -functions of spectral parameter with a single index

(4)

each of them being a polynomial of spectral parameter times a twist-dependent exponential factor. The positive integers are in fact the Cartan charges of the residual symmetry left after breaking the original symmetry by twisting.

Let us also define a natural object – the multi-index -functions – by the following Wronskian formula:

(5)

where we denoted by capital letter a subset of the full set of indices. By definition, all indices in this subset are different and ordered from left to right. Any permutation of indices in (5) can only change the overall sign by factor . It was also natural to introduce the “empty set” -function in denominator of (5), but the reasons which will be clear below, when we will discuss the Plücker relations (12). In total, we have different -functions, but they are obviously interrelated since they are given in terms of only single index functions.

To fix all the roots of these -functions, and thus to find all the eigenvalues of the above hamiltonian, it is enough to find all solutions of the following equation[44, 45]

(6)

where, according to (5),

(7)

is the Wronskian of the full set of single index -functions and is the Vandermonde determinant of twist eigenvalues. Here and below we use the notations for standard shifts of arguments of the functions: .

Once one finds a solution of (6), the corresponding energy – the eigenvalue of the hamiltonian (1) – is given by the familiar formula

(8)

where we used the index -functions given by -determinant according to the formula (5)5. The answer does not depend on the choice of (this symmetry is related to the so called particle-hole duality).

It is natural to attach all these -functions to the vertices of the -dimensional hypercube which is called in this occasion the Hasse diagram. For example, in the simplest case of spin chain () we have the set of 4 -functions: which we place at 4 vertices of the square, as shown on Fig.1(left).

Figure 1: Examples of Hasse diagrams for Q-system of Baxter functions of integrable models with symmetry (on the left) and GL(3) symmetry (on the right). The arrow shows the direction of “determinant flow”: the -functions with indices are determinants (5) of single index -functions, i.e. increasing in size with the increase of the level .

The upper vertex is occupied by , which is connected by two edges with and , which, in turn, are connected by two edges with  6. The spectral equation (6) takes the form

(9)

with the twist . Imposing the specific analyticity condition – the “polynomiality” of -functions (4) – we obtain the usual Bethe equation for the roots of :

(10)

We used for that two relations (9) at the roots of , shifted from the original one by , and divided one over another. A similar equation for the roots of leads to the same spectrum given by (8). In the formula for energy (8) we can use either or .

For , the Hasse diagram is 3D cube Fig.1(right). It is convenient to orient the cube in such a way that two of the vertices connected by the main diagonal appear to be the upper and the lower ones. We place again at the top vertex, the single-indexed – at the vertices adjacent to it, then, say, – on the next level, at the vertex adjacent to both and , etc. One could pictorially think of the Hasse diagram as of the “globe”, where the level of -functions with given number of indices is like a “latitude”, the -vertex and the -vertex – like the “north and south poles”, respectively.

The generalization to any is straightforward. All levels from top to bottom are ordered w.r.t. the number of indices in the corresponding functions. This induces a natural direction in parameterization (5) which we will call “determinant flow”. At a given , the collection of functions on a particular -level with forms an -dimensional linear subspace representing the Plücker coordinates of a point on the Grassmannian defined on the linear space . As was pointed out in [16], the quantum integrability, constraining the spectra of various integrable models, from spin chains to quantum field theories, is based on the following abstract relation between these Plücker coordinates:

(11)

following of course from (5). The Grassmannian structure of quantum integrability was first pointed out in [46] on the example of transfer-matrices and Hirota bi-linear finite difference equations.

We also present on Fig.2 two other important examples. On Fig.2(left) we depicted the Hasse diagram for the system, relevant for the conformal or for the R-symmetry subgroups of . On Fig.2(right) the Hasse diagram for the system is presented. As we will see, the last one is closely related to the full symmetry group of SYM theory.

Figure 2: Hasse diagrams for -system of Baxter functions of integrable models with symmetry (on the left) and symmetry (on the right) are hypercubes with the dimension of the rank of the symmetry of integrable quantum system. The arrow shows the direction of determinant flow according to (5). The Hasse diagram will be the same as for the super-group of AdS/CFT duality (up to a certain “rotation” of direction of the determinant flow and the details of analyticity structure).

The determinant flow (5) leads to the following Plücker relation (which is also called the QQ-relation in the AdS/CFT integrability literature) between the four -functions adjacent to the same two-dimensional face of the Hasse diagram (shown on Fig.3):

Figure 3: QQ-relations (Plücker identities for the Grassmannian) (12). They emerge at any 2-dimensional face of the hypercube of Hasse diagram as a consequence of the determinant relations (5).
(12)

where is a particular vertex on Hasse diagram. Notice, that the introduction of arbitrary function in denominator of (6) was necessary for satisfying the -relations (12) on the whole Hasse diagram, including the -vertex.

To obtain the standard nested Bethe ansatz equations, one has to choose a set of -functions along a “meridian” of the Hasse diagram, say, . Then one can use the above Plücker relations along the faces adjacent to this “meridian” (on the same side of it) and exclude all other -functions at the roots of the “meridional” -functions by the trick similar to the one which led us to the Bethe equation (10[47, 48, 42].

2.2 Spectrum of supersymmetric spin chain via -functions

The Hamiltonian of the super-spin chain has the following form

(13)

where the super-spin at each site takes two kinds of values

(14)

They correspond to two different gradings: . The super-permutation acts on a pair of spins as and the twist is a fixed group element 7.

The supersymmetric generalization of the above picture in terms of -functions can be nicely and easily presented as a specific “rotation” of Hasse diagram, when imposing the analyticity (“polynomiality”) conditions. Namely, for case we can preserve the same determinant flow (5) along the Hasse diagram as for bosonic case. But to fix the analyticity conditions we choose, instead of , a pair of -functions at the extremes of a different main diagonal of the hypercube, one on the level , another on the level For example, we can pick and .8 This supersymmetrization procedure is shown for the example on Fig 4.

Figure 4: Supersymmetrization of Hasse diagram of a QQ system of rank-3 on the example of rational Heisenberg spin chains: we pass from spin chain to spin chain by rotating the direction of the determinant flow, i.e. imposing different analyticity (polynomiality) conditions ((15)-(16) instead of (4)) and fixing a different pair of diametrally opposed -functions different in each case. The determinant flow is the same in both cases.

To find the spectrum of the Heisenberg super-spin chain we impose the following analyticity (polynomial times exponential for twist) conditions on the -functions:

(15)
(16)

where the hat over in the l.h.s. of the first equation means that the corresponding index is missing from the sequence. For the example of Fig. 4 such three -functions in (15),(16) are and , respectively. The positive integers are the Cartan super-charges of the residual symmetry left after breaking the original symmetry by twisting. To fix all the roots of these -functions, and thus to find all the eigenvalues of the above hamiltonian of super-spin chain of length , it is enough to find all solutions of the -system with the following conditions imposed

(17)

Once one finds a solution of (17), the corresponding energy – an eigenvalue of the hamiltonian (13) – is given by the familiar formula, through the -functions neighboring the “momentum-carrying” -function on Hasse diagram

(18)
(19)

where the answer does not depend on the choice of or .

We see that the scheme of solution for supersymmetric case is almost identical to the previous, bosonic case. But the “rotation” of the Hasse diagram in such a way leads to the dramatic change of analyticity properties. For example, the known function of eq.(17) cannot be expressed through the basic functions (15),(16) as a simple determinant, as in the bosonic case, but rather has to be found by solving a chain of Plücker relations (12), which leads to more complicated formulas. One can also derive the corresponding supersymmetric Bethe ansatz equations [49, 50] directly from the QQ-relations (12) as it was done in section 5 of [47] in less invariant notations. 9

The -functional approach has a long history [51, 52, 46, 53, 47, 54, 55, 56, 42, 57, 58, 59, 60] and it has been developed in the form described above in the series of papers [15, 16, 61, 42, 62, 63, 48] where the reader can find many more details. This approach is not only aesthetically attractive, it also appeared to be more efficient in certain explicit computations compared to more conventional Bethe equations for rational spin chains [26, 27]. The construction in terms of -system, based on Hasse diagram, presented above, can be applied for more complicated quantum integrable systems, such as non-compact (super)-spin chains and 2D sigma models in a finite volume. In this case, the analyticity conditions should be modified, since the -functions, or at least a part of them, cannot be parameterized by polynomials anymore. However, the solutions for the spectrum in such problems can be still formulated in terms of certain analyticity conditions on the set of -functions for which the algebraic structure of -system is entirely defined by the symmetry group. This approach was successfully applied for example for the study of spectrum of the principal chiral field model on a finite space-circle [64, 65].   In the next section, we will use the Q-system approach to formulate, in the most concise and general way, the Quantum Spectral Curve (QSC) equations  [14, 15] - a system of non-linear functional equations for computation of anomalous dimensions of arbitrary local operators, at any coupling, in the planar limit of Super-Yang-Mills (SYM) theory.

3 Quantum spectral curve for twisted N=4 SYM

In this section we will give a concise formulation of the quantum spectral curve (QSC) for the spectrum of anomalous dimensions of local operators in planar Super-Yang-Mills (SYM) theory, first introduced in  [14, 15], including its twisted version [16, 20]. It will be based on the -system10 approach described above. We will first make precise the algebraic structure of this /CFT -system, based on the super-conformal symmetry of the model. Then we will describe the analyticity properties of the underlying -functions and the Riemann-Hilbert “sewing” conditions allowing to completely fix the system of equations for the physical solutions.

Let us stress that we don’t give here any derivation of the /CFT QSC. We only formulate the final mathematical formalism, ready for further applications. Until the last chapter devoted to a particular application of QSC to the chiral double limit of -twisted SYM, we avoid, on purpose, the discussion of any consequences of QSC equations and of secondary details, concentrating only on the basic foundations of QSC construction. For the derivation, details and numerous consequences, the reader can turn to the original papers [14, 15, 16], to the recent review [29] as well as to the already rich literature of its generalisations and applications [17, 18, 19, 20, 21, 22, 23, 24, 16, 25, 26, 27, 28].

3.1 Algebraic structure of the AdS/Cft -system

The Hasse diagram for the AdS/CFT -system is similar to the one for the super-spin chain described in the previous section. It represents an 8-dimensional hypercube with the -functions attached to its vertices, as shown on Fig.2(right). The -functions have the same determinant flow as described by eq.(5). Let us note that the -system obeys a certain residual symmetry corresponding to two bosonic subgroups of the symmetry. This algebraic symmetry refers to the linear transformations of, separately, 4 functions with 3 indices and 4 functions with 5 indices . Another, “gauge” symmetry of the -system, due to the homogeneity of -relations, consists of the rescalings of -functions (there are two such rescaling parameters, see [15]).

But we should demand for QSC even more: we impose two to unit value

(20)

at any spectral parameter . The first of these conditions can be achieved by rescalings. But the second one, the -function diametrally opposed on Hasse diagram (i.e. and are Hodge dual to each other)11, this is an additional condition which replaces (17) for the super-spin chain.12 It actually reflects the projectivity and super-unimodularity of the symmetry of the system.

This gauge appears to be the most suitable for the formulation of analyticity properties of the whole -system. But these properties are more complicated then the polynomial ansatz (15) since we deal not with the rational super-spin chain (which however occurs to be the case in the weak-coupling, one-loop limit of SYM [66, 67, 68]) but with the integrable string sigma-model on coset.

We introduce special notations for the most useful “near-equator” -functions mentioned above:

(21)
(22)

where by we denote again the missing index from the set . For example, and . Hence have indices and has indices in the standard notations for functions, as in (5).

Their Hodge dual -functions have the same, but upper indices:

(23)
(24)

For example, and . The positions of these functions on Hasse diagram are pictorially presented on Fig 5:

Figure 5: Schematic presentation of positions of functions and functions (21),(22),(23),(24), within the Hasse diagram. They are neighboring the two “poles” of Hasse diagram corresponding to empty-set and full-set labels. Each pair of functions or , with the same label, are placed at the diametrally opposite vertices of Hasse diagram, i.e. they are Hodge dual to each other w.r.t. the Grassmannian structure of the -system. These 16 functions have the simplest analytic structure on the physical sheet. The gray “cloud” signifies all the elements of 8D hypercube missing on the picture.

The and functions are, roughly, responsible for the dynamics of string fields on (related to R-symmetry) and on (related to the conformal symmetry) projections of the dual string sigma-model, as will be seen from their large asymptotics.

Another useful set of 16 -functions and of their 16 Hodge duals deserves a special notation:

(25)
(26)

where by “hat” we denote again the missing member from consecutive integers. For example, and .

Due to the determinant flow (5), together with the gauge conditions (20), these functions satisfy a useful set of algebraic identities:

(27)
(28)

Notice that we can raise and lower the indices by the rules similar to the standard tensor algebra. A useful automatic consequence of the Grassmannian structure of -system and of the gauge (20) is the orthogonality relations: .

The QSC formalism is based on a set of 256 -functions, out of which only a few are algebraically independent. The rest of them can be deduced from the determinant flow or from the Plücker QQ relations (12). The choice of the most convenient algebraically independent subset of -functions depends on the problem being solved, i.e. on the type of studied operators and on the chosen approximations (weak coupling, strong coupling, numerics, etc). Thus there exist many useful forms of QSC equations. Let us mention one particularly important, especially for various weak coupling limits - the coupled system of 4th order Baxter equations on the functions (21),(22),(23),(24). Namely, excluding the functions from the relations (27)-(28)13 one gets the following linear 4th order finite difference Baxter equation[69]

(29)

where the coefficients are explicit functionals of -functions:

Four solutions of this equation give the functions . Of course, any independent linear combinations of these 4 -functions with -periodic coefficients 14 are also algebraically admissible -functions.

For a general state/operator, this Baxter equation should be supplemented by three similar equations. One of them, for -functions, uses the Hodge duality of the -system, obtained from the above equation by exchange of all upperlower indices of the -functions in the coefficients. Two other 4th order Baxter equations, on - and -functions, can be obtained from the previous two by simply exchanging all - and -functions. The existence of the last two equations is a simple consequence of the algebraic symmetry within the full -system between - and -functions.

Let us note that the most frequent cases of the SYM operators studied in the literature are those which obey the so called left-right (LR) symmetry w.r.t. to the exchange of two subgroups of the full superconformal group: . This symmetry has direct algebraic consequence for the underlying AdS/CFT -system. Namely, due this symmetry we can raise and lower the indices of -functions and -functions, i.e. -type or -type, by means of a “metric” whose role is played by a fixed constant matrix [70, 15]:

(31)

Obviously, in this case only two 4th order Baxter equations are algebraically independent: one for and one for , which significantly simplifies the problem. In addition, in various weak coupling limits, such as one-loop[71, 68] or BFKL[69] approximations, or the double scaling (DS) limit of -deformed SYM[30] described in the next section, the analytic properties of -functions simplify even further: they can have only finite order poles at and and thus they can be parameterized by a finite number of coefficients in the corresponding polynomials. Then the Baxter equation (3.1) on -functions starts to remind the one for the integrable spin chain reflecting the 4D conformal symmetry of the problem.

3.2 Analyticity: quantum spectral curve as a Riemann-Hilbert problem

The QSC formalism is based on two fundamental ingredients: the first is the algebraic structure of the underlying -system, entirely based on the superconformal symmetry of the model, and the second is the analyticity properties of the underlying -functions. The analyticity is greatly, but not completely dictated by the algebraic structure of -system. It was established in the original papers [14, 15]. It was extracted from the exact solution of the AdS/CFT spectral problem, first proposed in the form of the AdS/CFT Y-system [72] and then via the TBA approach [73, 74, 75]. The papers [76, 70] have been important steps towards the discovery of the QSC formalism.

In the rest of this section, we will describe the analytic properties of -functions. The main ingredients of their analytic structure are i) Infinitely branching Riemann surface due to branch cuts at fixed positions – “Zhukovsky cuts” 15; ii) Asymptotics at large values of spectral parameter fixing the representation of state/operator; iii) Riemann-Hilbert “sewing” conditions relating various -functions via monodromies around Zhukovsky cuts; iv) Absence of any other singularities anywhere on the Riemann surface of any -function, except mentioned above. Let us inspect these properties in detail.

Figure 6: Schematic depiction of analytic structure of Riemann surfaces of and functions defined by (21),(22),(23),(24). On the left, and have, each, a special, physical sheet of the Riemann surface where it has only one Zhukovsky cut for the range of spectral parameters , where is the ’t Hooft coupling. This cut is connected to the next sheet which has a ladder of equidistant Zhukovsky cuts spaced by , at positions , along the imaginary axis. On the right(up), the same picture of Riemann surface is true for the and functions, with an important difference: short Zhukovsky cuts should be replaced by long Zhukovsky cut, passing through , i.e. for . On the next sheets we have an infinite ladder of such cuts spaced by . On the right(down) we show the rearrangement of Riemann surface, by re-gluing the upper-half plane of the physical sheet with the lower-half-plane of the next sheet, and vice versa for the other two halves. This flips the long Zhukovsky cut on the real axis to short cut, but also creates a sequence of cuts in the lower half plane (which can be made short by the same re-gluing procedure for the next sheets).

Zhukovsky branch cuts and Riemann surface for -functions The main analyticity observation in QSC formalism is about the particular subset of 16 -functions, precisely the ones listed in (21),(22),(23),(24) and shown on Fig.5. Namely, the 8 functions and have, each, a special sheet of the Riemann surface (which will be called physical) where it has only one Zhukovsky cut for the range of spectral parameters and is the ’t Hooft coupling16. The physical sheet is depicted on the left of Fig 6(left). Similarly, the other 8 -functions, and , have a special, physical sheet where they have only one cut with the same branch-points but passing through , i.e. for  . The physical sheet is depicted on Fig 6(upper-right). It is natural call the first type of cuts as “short cuts” and the second one as “long cuts”. The positions of the branchpoints of these cuts are actually the only place in the QSC formalism where the ’t Hooft-Yang-Mills coupling constant is encoded.17

There are no other singularities on the physical sheets of these -functions except the one at , described in the next subsection.

Next, we want to know what happens under the cut, on the next sheet of the Riemann surface. In fact, the structure of the -system, and in particular of the QQ-relations (12), dictates that for and functions, apart from the same short cut , we find on the second sheet an infinite “ladder” of its periodically18 repeating replicas at , as shown on the right of the Fig 6(left). If we pass through any of these cuts we will encounter another sheet, with the same infinite ladder of short cuts repeating periodically along the whole imaginary axis. Passing through any of these cuts we discover the other sheets with the same infinite ladder of cuts. Consequently, each -function lives on an infinitely branching Riemann surface of the topology of sphere with a puncture at .19

As for the functions and , the picture of cuts on the sheets next to the physical one is exactly the same as for the functions and , except that all cuts are long, i.e. they are -periodic, at positions , as shown on the right of the Fig 6(upper-right). Of course the fact that these cuts are accumulated at leads to an infinite branching at infinity and allows for asymptotics with arbitrary power law w.r.t. spectral parameter. As we will see below, this is the way how the parameter – the anomalous dimension of the operator – arises in the QSC formalism as a power in the large asymptotics of -functions.

Large asymptotics Now we describe the behavior of - and -functions at on the physical sheet. To avoid complications with degeneracy of solutions we first consider the case of the totaly deformed superconformal symmetry of the model: . This is done by introduction of special twist on the CFT duality  parameterized by a fixed Cartan group element: , with the group constraint . This deformation is easy to perform directly in the SYM action [1, 6, 77, 8] for the case of so called -twist, when the conformal part of the superconformal symmetry is not twisted and the are parameters of -symmetry deformation.20

The picture for -functions is very simple: since the only singularity at the finite part of the -plane is a short Zhukovsky cut, we can approach the by any path, and the asymptotics is completely fixed by the global R-symmetry charges and the value of the twist:

(32)
(33)

So we see that these asymptotics can have only integer or half-integer powers21.

The situation with large asymptotics on the physical sheet of -functions is slightly more involved: due to the presence of the long Zhukovsky cut we should speak in principle separately of the large asymptotics in the upper-half plane (UHP) and in the lower-half plane (LHP). However one can easily argue that those two asymptotics can be different only by an overall constant (see [15] for the calculation of this constant). We can thus impose that, for example in the UHP, far away from the real axis (to avoid the vicinity of the long cut) their exponential and power-like parts are defined, respectively, by twists and the Cartan charges of conformal group , as follows

(34)
(35)

Here are integer conformal spins and is the dimension of the studied operator (energy of the state on the string side of duality) which is the main quantity under study in QSC formalism. Generically, is a complicated function of the ’t Hooft coupling , of conserved charges and of the twist parameters . With all these parameters fixed, we should have a finite or infinite discrete set of operators/states with different anomalous dimensions fixed by the values of the other conserved charges present in this integrable model. The presence of an arbitrary (if we vary ) power in the asymptotics means the presence of, in general, infinite branching at . This is a natural consequence of the presence of a long cut passing through point. Notice that on the next sheets of -functions it is hardly possible to speak about such power-likeexponential asymptotics due to the accumulation of long cuts forming an infinite ladder. On the contrary, one can define this kind of asymptotics at large for the -functions if we avoid approaching along the imaginary axis, in the vicinity of infinite ladder of short cuts.

Notice that we did not impose separately the asymptotics of Hodge dual and -functions since those are not independent of and . They are completely constrained by the structure of the -system (with an important role of the gauge condition (20)) and the leading asymptotics are inverse powers w.r.t. the original and , namely,

(36)

It is important to notice that all these asymptotics are multiplied by the expansion in integer powers w.r.t. . This is a special choice of the -functions, since any linear combination of them would spoil this property and mix up different combinations of twists and charges. We call our choice “pure” asymptotics, and this choice will be important for the rest of analyticity properties given below in the form of Riemann-Hilbert conditions.

We can partially remove the deformations by making some of the twist parameters equal to each other, i.e. restoring some subgroups of ) symmetry. Then we have to modify the asymptotics (32),(34) by shifting the exponent by certain integers because, asymptotically, certain determinant formulas for -functions will become ambiguous and will not render the right asymptotics. The whole classification of various twist configurations and of the corresponding asymptotics is given in [16]. We will discuss a couple of the most important cases. One of them, used in the next section, is the so called -deformation which preserves the entire conformal subgroup , i.e. , and leaves arbitrary twists , thus breaking -symmetry . Then the asymptotics (34), should be modified as follows

(37)

where

(38)

where as the leading asymptotics of remain as given by (32),(33).

Finally, the most studied case is of course the fully untwisted, completely symmetric SYM theory (or the equivalent dual superstring sigma-model on background). In this case, the above asymptotics of are the same as in (34), but for they look now as follows[14, 15]

(39)

with

(40)

Using these asymptotics and the Grassmannian structure of the QQ system we can even compute a few leading coefficients of all these asymptotics, which appear to depend only on the global charges, not on particular solutions [14, 15]. The classification of the coefficients of the leading asymptotics can be found in [16]. We don’t give here explicit formulas since we limit ourselves only to the formulation of basic rules of QSC construction, leaving aside its consequences.

Riemann-Hilbert sewing conditions Finally, we have to describe how one can move among the sheets of the Riemann surface for the and -functions. In other words, one should detail the properties of monodromy around the branch-points of Zhukovsky cuts of these functions.

It was noticed in[22] that, after having imposed the “purity” of asymptotics, as discussed after eq.(36), one can fix completely the system of spectral equations, by demanding within the QSC formalism the following Riemann-Hilbert sewing conditions [15]22

(41)

where means the complex conjugation, i.e. reflection of the main sheet w.r.t. the real axis where the long cut is present, see Fig.7(left),

Figure 7: Demonstration of the Riemann-Hilbert sewing relations: on the left, the complex conjugation relation (41) between a pair of functions on the physical sheet with a single long cut is presented. Notice that the path connecting them should go between the branch-points. On the right, the same relation is demonstrated on the sheet with short cuts. It takes the form . Dotted cuts, are situated on the second sheet. The conjugation path is now passing through the short cut at the real axis, i.e. the conjugation involves now the monodromy (denoted by tilde) as well.

and are constants non-trivially depending on the parameters of the operator/state, to be defined self-consistently in the process of solution of QSC equations. The origins of this sewing condition originate already from the properties of quasi-momenta of classical finite gap solution of the string dual – the sigma model on coset [78, 79, 80].23

These conditions mean that the -functions are not all independent but rather glued together into a smaller number of analytic functions. This sewing condition is the finite element of QSC construction which locks completely the QSC relations into a closed system of equations for spectrum. Their solution renders a discrete set of dimensions/energies of all the operators/states with the given ’t Hooft coupling , the global charges and twist parameters