The Holographic Dual of Strongly \gamma-deformed N=4 SYM Theory: Derivation, Generalization, Integrability and Discrete Reparametrization Symmetry

Recently, we constructed the first-principle derivation of the holographic dual of N=4 SYM in the double-scaled $\gamma$-deformed limit directly from the CFT side. The dual fishchain model is a novel integrable chain of particles in AdS5. It can be viewed as a discretized string and revives earlier string-bit approaches. The original derivation was restricted to the operators built out of one of two types of scalar fields. In this paper, we extend our results to the general operators having any number of scalars of both types, except for a very special case when their numbers are equal. Interestingly, the extended model reveals a new discrete reparametrization symmetry of the"world-sheet", preserving all integrals of motion. We use integrability to formulate a closed system of equations, which allows us to solve for the spectrum of the model in full generality, and present non-perturbative numerical results. We show that our results are in agreement with the Asymptotic Bethe Ansatz of the fishnet model up to the wrapping order at weak coupling.


Introduction
The strongly γ-deformed limit of N = 4 SYM, also known as the fishnet model [1][2][3], is an ideal playground for developing our exact computational methods and understanding holography. It is an interacting four-dimensional QFT with SU (N ) symmetry that one could hope is solvable exactly in the planar limit. Just like the planar limit of N = 4 SYM theory, this model is conformal for any value of the 't Hooft coupling ξ. It consists of two complex N × N matrix scalars φ 1 and φ 2 that are weakly coupled when ξ → 0. At strong coupling ξ → ∞, a new classical dual description emerges in terms of a chain of scalars in AdS 5 [4]. At finite ξ the theory becomes rather complicated. At the same time, it is much simpler than N = 4 SYM. For example, at each order of the perturbation theory, there are only very few Feynman diagrams that contribute.
There are several properties that make the fishnet theory particularly interesting and tractable. First, the model can be proven to be integrable at the quantum level [1][2][3]. Second, its Holographic dual fishchain theory can be derived rigorously and quantized exactly [4,5]. This contrasts with the current status of N = 4 theory, where the integrability is conjectured and the dual super-string in AdS 5 × S 5 can only be quantized semi-classically.
So far, our derivation and quantization of the fishchain model was restricted to a set of states/operators that belong to the so-called u(1) sector. These are operators of the type . . ], which can carry arbitrary spin and are only charged under the u(1) 1 subgroup of the total u(1) 1 × u(1) 2 symmetry that rotates the two types of scalars. The main objective of this paper is to extend the derivation to general states, carrying both quantum numbers. This involves introduction of the magnons i.e. φ 2 fields in the background of J φ 1 's. While most of the steps go through very similar to our previous work, we find that the inclusion of the magnons leads naturally to a new notion of a discrete reparametrization gauge symmetry. It is the string-bit counterpart of the smooth string reparametrization freedom, which relates different ways of parametrizing the same state. 1 Furthermore, extending the integrability construction to all operators allows us, in particular, to solve the spectrum with magnons included, which is a new result by itself. The solution takes the form of a Baxter TQ-relations, subjected to the quantization conditions of [7] and generalizing results of [8]. It allows us to compute the conformal dimension of almost all single trace operators 2 . Moreover, this construction gives the general framework for development of the integraiblity based separation of variables method (SoV) for more general observables such as correlation functions.
The paper is organized as follows. In section 2 we introduce our notations and review the main previous results, which are important for this paper. In particular, we recall the main steps from [5] for the quantization procedure of the fishchain. In section 3, we generalize the notion of the CFT wave function, initially introduced in [5], to operators with magnons. We then construct the corresponding graph building operator and derive its holographic counterpart. In section 4, we establish the integrability of the fishchain model  [2]. This structure can be resummed and leads to integrability [3]. b) For operators with also net φ 2 charge, (2.3), the diagrams are of iterative spider type [9]. with magnons. In section 5, we identify redundancy in the space of CFT wave functions that we associate to a discrete reparametrization symmetry of the fishchain model. In section 6, we solve the spectrum of the model with magnons and prove its reparametrization symmetry. We conclude in the section 7.

Notations and set-up
Here, we introduce the notations and give a short introduction to our results from [4,5].

Fishnet model
The fishnet model [1] is defined by the following action 3 L 4d = N tr |∂φ 1 | 2 + |∂φ 2 | 2 + (4π) 2 ξ 2 φ † 1 φ † 2 φ 1 φ 2 . (2.1) This model was obtained in [1] by taking a double scaling limit of γ-deformed N = 4 SYM theory. Hence, we will still impose the Gauss constraint and will only consider operators that are neutral under the SU (N ) part of the global symmetries. Our previous consideration was restricted to a subset of all operators, known as the u(1) sector containing any number of derivatives, J-scalar fields φ 1 and any neutral combination of φ 2 and φ † 2 . Correlation functions of u(1) operators are given by the sum of fishnet type The initial conditions are restricted to a 2D plane after projection on the boundary. The plane is parametrized in the radial parameters by the angle φ and the radius r. The third axis corresponds to the time. The drift in φ is due to a non-zero total angular momentum S and the exponential expansion in r is due to ∆ > 0.
diagrams. These are diagrams that are made of the iterative wheel structure as is demonstrated in figure 1.a. General single trace operators in the model can also carry u(1) 2 charge. They take the schematic form In this paper we allow for arbitrary values of J 1 and J 2 as long as |J 1 | = |J 2 |. Surprisingly, this case seems to be very different as one can see already in the explicit results of [14]. Correlation functions of these type of operators consist of a very specific type of planar Feynman diagrams that have the shape of a spider web. In figure 1.b we draw an example of such a diagram for the case where J 2 = 1. In general, the diagrams can be constructed by first contracting all the φ 1 fields in the trace, (the black lines in figure 1). Then, the φ 2 's have no other option but to spiral around the trace, crossing the φ 1 lines from the right to the left, (the red line in figure 1.b). Each time a φ 2 line crosses a φ 1 line we have the four scalars interaction of the fishnet model (2.1). The absence of the complex conjugate interaction vertex forbids any other diagram to contribute in the planar limit.

Fishchain model
The holographic fishchain dual of the u(1) operators of charge J 1 consists of a chain of J 1 point-like scalar particles in AdS 5 . In the classical/strong coupling limit, ξ → ∞, the theory is defined by the action where X i ∈ R 1,5 . While at the quantum level the particles live on AdS 5 space, at the classical limit its radius collapses to zero and the dynamics take place on the light cone, One can think about (2.4) as a discretized version of the gauge fixed Polyakov action for a string, which requires a Virasoro condition. An analog of the Virasoro condition in the case of the fishchain isẊ 2 i = 1. In addition to the constraints the Hamiltonian of the system should vanish to ensure the time reparametrization symmetry The classical Hamiltonian H q follows from the action (2.4) together with the constraints (2.5), see [5].
As can be seen from figure 2, the action (2.4) produces rather non-trivial dynamics. Even though this dynamic may look chaotic, the model is integrable.
Quantization. We now briefly go through the main steps of the canonical quantization procedure. We will follow the Dirac procedure for the quantization of a system with constraints. To begin with, we notice that in addition to the primary constraints (2.5) we also have to include the secondary constraintẊ i .X i = 0. Using that the canonically conjugate to X i variable is P i =Ẋ i we have the following constraints Among these, ϕ j and ϕ j,P are second class constraints as follows from their Poisson bracket. Hence, in the quantization of the model we first introduce corresponding Dirac brackets and only then promote them into quantum commutation relations. For example, the commutator between the operatorsX andP takes a non-standard form Similarly, the commutator between different components ofX M k is no longer zero. To resolve these commutation relations we introduce the operatorŶ N i = i ξ ∂ P i,N , which commutes in the standard way withP N k . Then we show that we automatically solve all commutation relations ifX M i =Ŷ M i − (Ŷ i .P i )P M i . Next, we have to impose the constraints (2.7) at the quantum level and resolve possible ordering ambiguities. What we find is that • The quantum constraintφ i =X i .P i is automatically satisfied. Here, we had to choose an ordering betweenX i andP i inφ i . To satisfy the constraint, we had to correlate this choice with the one in the definition ofX i in terms ofŶ i .
• The constraintφ i,X is proportional to tr q 2 i , which is nothing but the quadratic Casimir,Ĉ i,2 /ξ 2 . Instead of setting it to zero we identify it with the dimension 1 of the scalar field φ 1 , corresponding toĈ i,2 = −3.
• The constraintφ i,P fixed the dependence of the wave function on R 2 i ≡ −Y i .Y i , as a result all the dynamical information is contained in the reduced wave-function • Finally, we have to resolve the ordering ambiguity in the quantum version of H q . Since the matrix elements ofq N M i do not commute with each other, the quantum version of q 2 i suffers from ambiguities. As we explain below, the relevant definition is the symmetric and traceless combination : where the charge density in the Z-coordinates iŝ The corresponding quantum fishchain Hamiltonian takes the form At the end of the day we end with a quantum theory in AdS 5 that is defined by two relations that the fishchain wave function Ψ(Z 1 , . . . , Z J ) has to satisfŷ where ∆ i = 1 is the dimension of the scalar φ 1 .
The Holographic map. The key object which allows to establish the Holographic duality between the fishchain model and the fishnet CFT is the CFT wave function introduced in [5]. This is a J + 1 point correlator of a local operator O and J scalar fields The knowledge of this function is equivalent to specifing the local operator O. In order to compute ϕ O in perturbation theory one has to sum an infinite number of wheel diagrams like in figure 1(a). To sum these diagrams one can use the so-called graph building operator B , which is an operator acting on the wave function ϕ O by adding one more wheel to the (2.14) Physical non-protected operators in the CFT spectrum are in one to one correspondence with stationary wave functionŝ The map between the CFT and the fishchain wave functions reads [5] where X M =−1,...,4 are six dimensional embedding coordinates with (−, +, +, +, +, +) signature [15]. They realize flat four dimensional space as the protective lightcone of R 1,5 as In this parameterization we have for example 1/(x−y) 2 = X + Y + /(−2X.Y ). When rewriting the correlation functions in embedding space we are following the general prescription of [16]. According to it we have to introduce extra factors of X + in agreement with the scaling dimensions of the corresponding operators. In particular, the embedding space wave function in (2.16) is defined as The introduction of these extra factors ensures that the r.h.s. stays invariant under all conformal transformations. It was shown in [5] that if φ O obeys the condition (2.15), then the l.h.s. of (2.16) will be automatically annihilated byĤ q . At the same time the second condition in (2.12) is also satisfied just because the kernel 1 (Z.X) 3 solves this equation for any null vector X. In the next section we will show how this Holographic map naturally generalizes to the case where both quantum numbers J 1 = J and |J 2 | < J are non-zero.

CFT wave function and the graph building operator
In this section we generalize the fishchain model to include magnons i.e. insertions of the φ 2 fields into the "vacuum" of φ 1 's. We assume that the number of φ 2 fields is less the number of φ 1 's. The starting point for the construction is the CFT wave function, which we introduced in (2.13) for the u(1) sector. Like before it is given by the (J + 1)-point The new ingredient is that the local operators χ a are not just single scalars φ † 1 , but come in two types . (3.2) We use the multi-index notation I = (I 1 , . . . , I J ) where the index I j ∈ {0, 1}. For example, when I j = 1 it indicates the position of the χ 1 insertion, which is a composite of two scalars φ † 1 and φ † 2 . Like before, the knowledge of this correlator is equivalent to the knowledge of the initial operator. Any other planar correlators, such as those with φ 2 φ † 2 and φ † 1 φ 1 , can be expressed in terms of this one. In section 5 we show that the order of the magnons inside the correlator can be interchanged. Hence, in order to reconstruct the local operator it is sufficient to know the CFT wave function for only one of all possible magnon orderings. In this section we consider all magnon orderings on equal footing.
As for the case with no magnons, the Feynman diagrams which contribute to the correlator ϕ I O in (3.1) are of iterative structure and can be represented as a simple geometric series built from the corresponding graph building operator 1/(1 −B). The only difference in comparison to the previous section is that now the graph building operatorB is slightly more complicated and is given bŷ where f is a probe function. In (3) the integration kernel is built out of elementary building blocks b a , which are different for the sites with magnons and without .

(3.4)
We see that at the sites with a magnon, the φ 2 propagator connects to the external point x j instead of the integrated y j point as in a site without a magnon. This is illustrated in figure 3, where the sites with magnons are highlighted by the blue circles. As we explained in the previous section the key equation which the CFT wave function should satisfy is (2.15). The wave function is the correlator computed at some finite value of the coupling ξ. It is given by an infinite sum of planar Feynman diagrams. As such it should stay invariant under addition of an extra layer i.e. the wave function should be an eigenstate of the graph building operator with a unit eigenvalue, (2.15). For explicit examples we refer to [14]. As was demonstrated in [14] the condition (2.15) singles out the physical states in the theory and leads to the discrete spectrum of the scaling dimensions ∆ corresponding to the properly regularized and renormalized single trace operators of the type (2.3). Like in our previous paper [5], we will now rewrite (2.15) in a form suitable for uplifting to AdS 5 space and show that the dynamics induced by (2.15) is exactly this of the quantum fishchain (QFC).

Inverting the graph building operator
Following [5] in order to relate the graph building operator B to the dual fishchain model in AdS 5 , we have to invert it and rewrite (2.15) as a zero energy Hamiltonian constraint This constraint is then interpreted as the result of a time reparametrization symmetry of the fishchain. We show below that, defined in this way,B −1 andĤ are differential operators.
In the case without magnons the invertion ofB, given by (2.14), was straightforward. We simply act with j on all the external x k points in (2.14). This has the effect of eliminating the corresponding y k integrals and the 1/(4π 2 (x i − y i ) 2 ) factors. 4 In the case at hand we have an extra complication -there are two propagators that end at a site with a magnon. Acting with j on the magnon site does not remove the integration anymore, see figure 3.
Let us show that the inversion of B is still possible in the situation when there is at least one site without a magnon, i.e. for J 2 < J 1 . Without loss of generality, we assume that there is a magnon on the k'th site, but there is no magnon at the neighboring site to the right i.e. I k−1 = 0, I k = 1, see figure 4. In this case we can hit x k−1 with k−1 , removing the integration in y k−1 . After that the propagator connecting y k−1 (which is now equal to x k ) with x k can be removed by multiplication with x 2 k,k−1 . 5 After this we are left . 5 We use the standard notation Here, we demonstrate how the operator is inverted for the case where there is a magnon at the k'th site but no magnons at the neighboring sites.
with a single propagator connecting x k with an integration point y k , which now can be treated with k .
It is easy now to figure out the general procedure. Assuming again that at the site k − 1 there is no magnon we begin by acting with k−1 , removing integration in y k . Then, if at the site k there is a magnon, we act with the combination k x 2 k,k−1 , see figure 4. Otherwise, on the sites with no magnon we act with the operator x 2 k,k−1 k . Acting in this way we can invertB completely. For example for J = 3, M = 1 with I 3 = 0, I 2 = 1 and We conclude that it is always possible to writeB −1 as a differential operator. Below we rewriteB −1 in a simple and manifestly conformally invariant way, applicable for all cases. For that we will have to first change to embedding coordinates, introduced in section 2.2.
Embedding coordinates. When manipulating conformal integrals such as the ones that are generated by the graph building operator, it is always advantageous to work in the embedding coordinates. In the case of the correlator (3.1), we have the χ I operators in (3.2) at positions X 1 , . . . , X J and O ∆ at X 0 . Hence, the corresponding embedding space wave function is defined as The advantage of the covariant formalism is that the invertedB can be written in a very concise formB −1 = tr (β J,I J . . . . .β 1,I 1 ) where β k,I k is a local operator that only acts on the X k 'th coordinate. It is given by To understand the mechanism behind (3.8) we look at the same example as in (3.6). There we took I 3 = 0, I 2 = 1 and I 1 = 0. For this case (3.8) gives does not depend on X − , we can simply replace K 2 i by − 1 ξ 2 i when acting on the CFT wave function. In addition we use to reproduce precisely (3.6). It is equally easy to see that the equation (3.8) works in the general situation, provided that J 2 < J 1 .
Finally, let us rewrite the local site operators, β k,I k , in terms of the local generator of the conformal symmetry, namely in terms of q k defined aŝ We can use that the CFT wave function is a homogeneous function of X k with degree of homogeneity −(1 + I k ), as follows directly from its definition (3.7). This property allows us to make the replacementsX k .K k → i ξ (1 + I k ). Using that in analogy with (2.9), we find where the " " means equality when acting on the wave function (3.7).
The strong coupling classical limit. Before we proceed to the holographic interpretation of equation (3.8) at the quantum level, we can already make a nontrivial conclusion about the strong coupling limit of the theory, where the dual description becomes classical [4]. In order to understand the classical limit we recall that the operator (B −1 − 1) can be interpreted as a worldline Hamiltonian. The classical limit is obtained by simply replacing − i ξ ∂ x i by the classical momentum p i [4]. As one can see from e.g. (3.9) the only difference between the situation with magnon and without magnon is the ordering of the operators X i and K i , which becomes irrelevant in the classical limit. From this simple observation we conclude that the classical theory is not affected by the presence of magnons. But there will be a difference visible already at the first quasi-classical correction.
One way to think about this is that in the strong coupling limit the difference between a dimension one scalar and a dimension two composite operator made of two scalars (which is the magnon), becomes negligible. Namely, in units of 1/ξ, both of them are indistinguishable from zero.

Quantum fishchain with magnons
In the previous section we understood that the classical limit of the theory with or without magnons should be the same. This means that essentially all the quantization procedure developed in [5] and described shortly in section 2.2 still applies here. There are only two steps in our quantization procedure which allow for a modification. First, in (2.12) we assumed that the local Casimir operator trq 2 i , which is a c-number, should be associated with the dimension of the scalar, or more precisely its AdS 5 mass m 2 = −3. This assumption does not look natural anymore as the magnons are the bound-states of two scalars φ † 1 and φ † 2 of the total dimension ∆ i = 2, meaning that the AdS 5 mass becomes −4 or equivalently trq 2 i = 8 ξ 2 . Another subtle point in the quantization procedure, which needs to be revisited, is the ordering ambiguity in the quantum HamiltonianĤ q (see (2.11) and (2.9)). We have a freedom to add to the naive term ( where c I k and d I k are some numbers, which for the case with no magnons were set to 2 and 1 correspondingly i.e. c 0 = 2, d 0 = 1. In this section we fix this quantization ambiguity by requiring that this quantum fishchain model is exactly dual to the Fishnet model for the states with magnons. Map between the wave functions. In [5] we have built an explicit map between the CFT wave function in the u(1)-sector and the wave function of the quantum fishchain in AdS 5 . This construction was reviewed in section 2.2, see (2.16). We now extend this map to the case with magnons. The map (2.16) takes the 4D flat space CFT wave function and returns a function of J points Z 1 , . . . , Z J , parametrizing a unit radius AdS 5 space, Z 2 i = −1. These are the coordinates of the particles constituting the fishchain.
The factors 1/(Z i .X i ) 3 in (2.16) are bulk to boundary propagators for J scalars in AdS 5 of mass m 2 = −3. It was designed to solve the constraint trq 2 i = 6 ξ 2 . As for the sites with magnons we should modify this constraint to trq 2 i = 8 ξ 2 , so we can take 1/(Z i .X i ) 2 . Hence, the generalization of the map (2.16) to wavefunctions with both χ 0 and χ 1 operators is (3.14) As before, 1/(Z i .X i ) 2 is the bulk to boundary propagator for an AdS 5 scalar of m 2 = −4. It is the fastest decaying solution to the bulk equation of motion with a source at the boundary point X i . 6 It is not too hard to invert the map (3.14). One way is to go to the boundary of AdS 5 . While with no magnon excitations the bulk wave function exactly reduces to the CFT wave function [5], in the case with magnons there is an addition logarithmic factor Exact duality. Having the map between the wave functions extended to the general case (3.14), we can prove the duality and also fix the remaining quantization ambiguity manifested by the existence of two arbitrary constants in the QFC Hamiltonian (3.13).
Following [5] we can establish the map between the QFC HamiltonianĤ q and the CFT Hamiltonian H by requirinĝ where H = (B −1 − 1) as defined in (3.5). Exactly like in [5] we simply use the identity Here, : q k : I k is given by This derivation extends the proof [5] of the exact duality between the QFC and the Fishnet CFT models beyond the u(1) sector. Note that so far we only considered the CFT wave functions, which contained φ † 1 and φ † 1 φ † 2 under the correlator with a local operator O. Whereas this does indeed go beyond the u(1) sector one could still question if that is the most general case. In particular one can also introduce "anti-magnons" i.e. φ 2 φ † 1 , as we do in the next subsection. Later, however, we show that the CFT wave functions with the same magnon number J 2 (defined as a difference of the number of magnons and anti-magnons) are all equivalent to each other.
It is yet very fruitful to have both magnons and anti-magnons at the same time as we will see in the next section 3.3.

Adding anti-magnons to the QFC
We consider the most general situation, where at a site of the chain we can have either no magnon, one magnon, one anti-magnon or a magnon-anti-magnon pair. The CFT wave function definition (3.1) should be extended by allowing for χ j to take two additional values . (3.20) for anti-magnon and magnon-anti-magnon pair correspondingly. All the steps from the previous parts of this section are carried out in a similar way. We summarize in table 1 the key elements.
type As before we assume that |M −M | < J where M is the total number of magnons andM is the total number of anti-magnons. In this case we again haveB −1 = tr (β J,I J . . . . .β 1,I 1 ) where the β's are summarized in table 1.
Next, one can also map the CFT wave function with the QFC wave function in AdS 5 using the generalization of (3.14) where again ∆ k is the conformal dimensions of χ I k , given in the table 1. Finally, the QFC HamiltonianĤ q is obtained by uplifting the CFT HamiltonianĤ, which results in fixing the quantization ambiguity parameters c I and d I as shown in the table 1.
To summarize, we see that the quantum fishchain naturally generalizes to incorporate magnons. They are in one to one correspondence with all possible orderings of K 2 k and the X k 's in the site operator β k .

Integrability of the fishchain with magnons
In this section we extend the proof of the quantum integrability of the QFC [5] to the general case with magnons. In order to demonstrate the integrability of the QFC model we will follow the same steps as in [5]. The strategy is to identify the Hamiltonian as a part of a large family of mutually commuting operators, which can be obtained as an expansion in powers of u of the T-operatorsT(u).
There are 3 non-trivial T-operators, which should contain a complete set of integrals of motion. The building blocks for these operators are the Lax matrices in antisymmetric finite-dimensional representations of sl(4). They were computed in [5] in terms ofq k . We will see that this derivation is general enough to accommodate the current case as well. The main building blocks for the operatorsT(u) are the L-matriceŝ Where Σ M N are 6-dimensional σ-matrices. Their explicit representation can be found in [5]. The mutually commuting families of operators are constructed from (4.1) viâ Where G is a constant twist matrix and {θ j } are the so-called inhomogeneities which can be taken as arbitrary. For the original fishnet model the twist matrix is trivial, G = 1. However, here we consider the most general deformed case, dual to the twisted fishnet of [17], with G = diag(λ 1 , λ 2 , λ 3 , λ 4 ) and we assume λ 1 λ 2 λ 3 λ 4 = 1 for unimodularity of G. Next, we have to choose the values of {θ j } so that the QFC Hamiltonian appears in (4.2) for a special value of u. More precisely we require that in the same manner as in the case without magnons we haveT 6 (0) =Ĥ q + 1.
We now consider all 4 cases of insertions one by one -the site without magnon, with magnon, with anti-magnon, and with magnon anti-magnon pair using the data from table 1.
No-magnon case. The case without magnon corresponds to a single scalar φ 1 , which allows for the value of the quandatic Casimir to be −3, implying trq 2 k = 6 ξ 2 . As a result (4.1) reduces toL , which is the correct factor inĤ q for sites without a magnon, meaning that for these sites we have to set θ k = 0.
Magnon case. For the site with magnon the dimension ∆ k = 2 corresponds to the quadratic Casimir −4, implying that trq 2 k = 8 ξ 2 . Consequently from (4.1) we get This time the ambiguity constants are c I = 3 and d I = 2 i.e. we need to tune θ k to obtain Note that with only one parameter θ k it is not guaranteed that we manage to fit the two constants correctly. Fortunately, by setting θ k = i 2ξ 7 we indeed obtainL 6 k (0) = 1 2 :q 2 k : 1 .
Anti-magnon case. The building block for the anti-magnon isL 6 Magnon-anti-magnon case. The Lax matrix, in this case, is identical to the nomagnon case considered above. The reason is that ∆(χ0) = 4−∆(χ 0 ) = 3, so the quadratic Casimir, given by c = ∆(∆ − 4) = −3 is exactly the same as that for ∆ = 1. This implies that in both cases we have tr q 2 k = 6 ξ 2 . Moreover, the quantization ambiguity constants c I and d I are also exactly the same (see table 1) and therefore θ k = 0 in this case as well. As is explained in the next section, at the fishnet side both the magnon-anti-magnon case and no-magnon case describe different ways of cutting the same propagation line of φ 1 along the fishnet diagrams. Hence, the corresponding transfer matrix that encodes all physical charges should be the same.
To summarize, we see that after fixing the impurities to θ k = iI k 2ξ we get indeedT 6 (0) = H q + 1. This property implies the quantum integrability of our model, as we identified the Hamiltonian of the system within a large family of mutually commuting operators. 8 In the current formulation the ordering of the magnons is quite important. Different distributions of magnons within the chain will produce different Hamiltonian and Toperators. We will see in section 6 that the eigenvalues of all T-operators are in fact independent of the order of magnons. This invariance has a very nice physical interpretation -it can be associated with the discrete reparametrization symmetry of the fishchain as we discuss in the next section 5. Based on the results of this section, in section 6 we present the general integrability formalism for the spectrum of the theory. 7 N.G. is grateful to V.Kazakov and Z.Tsuboi for discussing the possibility of inclusion of inhomogeneities in relation to the magnons. 8 Of course one should verify that there are sufficiently many independent integrals of motion. This is usually rather hard to verify rigorously, but we strongly believe this is the case. The rough counting of the integrals of motions indeed gives ∼ 4J, which coincides with the number of degrees of freedom of the theory.

Discrete reparametrization symmetry
When introducing the magnons we have to face the following paradox. On the CFT side, the primary operators with several magnons will be given by a complicated linear combination of all possible magnon permutations. At the same time on the fishnet side, it looks like we have the freedom to choose any ordering for the magnons. This choice is made explicitly in our definition of the CFT wave function by the correlator (2.3). Moreover, the fishchain HamiltonianĤ q itself depends on a particular fixed order. The danger is that the spectrum of the Hamiltonian is also order dependent, which would result in an inconsistency of our approach. The physical quantities such as dimensions of local operators should not depend on an arbitrary choice of order of insertions of φ 2 fields in the CFT wave function, which probes this local operator.
In this section we demonstrate that, luckily, the spectrum does not depend on the ordering of the magnons. This is due to a novel symmetry of the fishchain which we interpret as the discrete reparametrization symmetry. We will show that redistribution of the magnons can be understood as a map between Hilbert spaces of the two chains, which differ by the magnon positions. We will then prove that this map preserves the eigenvalues ofĤ q and of all other conserved charges encoded intoT(u), introduced in the previous section.

Cutting and leveling
Consider a correlation function between two local operators of the type (2.3) that consists of J 1 = J fields φ 1 and J 2 = M insertions of φ 2 's fields in the trace (as before we assume M < J). The planar Feynman diagrams that contribute to this correlator have cylindrical topology and are of an iterative spider-net-like structure as discussed in the introduction, see figure 5. The fishchain state, as given by the correlator in (3.1), corresponds to a specific choice of cut around this cylinder. For example the blue cut γ 1 in figure 5 corresponds to the correlator (3.1) with M = 2, where one magnon (χ 1 ) is at the k'th site of the chain and the second magnon is at the neighboring k + 1 site. On the other hand, when the same diagram is cut as is indicated by the γ 2 cut in figure 5, the second magnon is at the k + 2 site.
In general, a cut is a choice of a decomposition of the fishnet diagrams. It is specified by a line γ that goes around the cylinder and crosses a set of propagators on the way. Here, we choose to restrict our consideration to space-like cuts. These are cuts that cross every φ 1 line and every φ 2 line exactly once. Moreover, every φ 2 cut must be followed by a φ 1 cut. These are the cuts that look space-like on the fishnet lattice. They are the ones that allow us to obtain all the higher loop fishnet diagrams by iterative action of the corresponding graph building operator. Such cuts are analogues to Cauchy surfaces on the string worldsheet. The graph building operator generates the "worldsheet" time propagation on such surfaces.
We notice that a choice of space-like cut corresponds to a choice of a basis of CFT wave functions of the type (2.3) as follows. A cut of a φ 1 -propagator is mapped to a φ † 1 field in the corresponding correlator (2.3) and a cut of a φ 2 -propagator is mapped to a φ † 2 field (see figure 5). We then merge φ † 1 and φ † 2 fields to be at the same point as a χ 1 or χ −1 in accordance with the vertex above the cut. We will give more explicit examples below.
Let us point out that the situation in which two different cuts result in a different order of magnons in the chain like in figure 5 is not the most general. In fact one can also generate an arbitrary number of magnon-anti-magnon pairs. For example, we can start with a chain with no magnons at all and deform the cut. After doing so, we end up with a magnon and an anti-magnon, see figure 6.
One way to think about the difference between different cuts is as a map between Hilbert spaces. We will now construct explicit operators that implement this map between the Hilbert spaces of two different deformations of the cut.

The cut deformation operators
Any deformation of the cut, such as the one in figure 4, can be realized as the action of a cut deformation operator on the corresponding CFT wave function. The aim of this section is to construct the CFT cut deformation operators and then lift them into an operator acting on fishchain wavefunctions, i.e. acting on the AdS 5 variables Z i . We will use these operators in section 6 and will prove that they generate a discrete symmetry of the quantum fishchain, leaving the spectrum invariant.
Any cut deformation of the CFT wave function can be decomposed into a combination of two elementary operations As one can see from the equation the v k operator adds a vertical φ 1 propagator, and h k adds a horizontal φ 2 propagator. The corresponding inverse of these elementary operations are We can build out of v and h the operators which move a magnon/anti-magnon to the right/left (moves anti-magnon left) .

(5.6)
Let us show that the operators v k , r k+1,k ,r k+1,k and h −1 k+1,k constitute the 3 elementary moves which allow us to map any CFT wave function to a standard form. When acting on a state in a fishchain Hilbert space, characterized by the indexes I, they transform it to a state in a different Hilbert space, characterized by a different set of indexesĨ as follows Indeed, we can use v k to replace all0 by 0's. Next we can move magnons and anti-magnons towards each other using r k+1,k andr k+1,k and then annihilate them in pairs using h −1 k+1,k . After that, depending on the sign of J 2 there will be only magnons or anti-magnons left, which we can group together with r k+1,k orr k+1,k .
In the next section we derive what these operators correspond to in the dual description.

The fishchain reparametrization operators
Using the map between the CFT and the fishchain wavefunctions (3.21) we now map v −1 k and h −1 k into fishchain operators. Consider first the operator v −1 k . This operator acts on a site with no magnon and maps it to a site with a magnon anti-magnon pair. Plugging it into the map (3.21) and taking into account that the conformal dimension at that site changes from ∆ k = 1 to ∆ k = 3, we get the equation for the corresponding operator (5.11) Using that v −1 (x) = − 1 4ξ 2 x we conclude that V −1 (Z) = 1 2ξ 2 , i.e. up to a multiplication by a constant this operator acts trivially in the dual picture. This is because on the fishchain side there is no real distinction between no-magnon and a magnon-anti-magnon pair at the same site, as one can see from the table 1.  k+1,k , removing the φ 2 propagator stretched horizontally between the k+1 and k sites. It can be interpreted as an operator annihilating a magnon and anti-magnon situated at two neighbouring sites. k r k+1,k k k Figure 9. The operator r k+1,k moves a magnon at the (k + 1) site to the empty site on its right, (k). Similarly, the reflection of this figure describes how the operatorr k+1,k moves an anti-magnon at the k'th site to the empty site on its left (k + 1).
Similarly, for h −1 k+1,k (which decreases ∆ i by one on two consecutive sites) we find its AdS 5 counterpart H from the following identity where ∆ j can be either 2 or 3. We find that (5.14) As a consequence, the lift of the operator r k+1,k = h −1 k • v −1 k , that moves a magnon to an empty site to the right reads, up to an irrelevant constant factor In the next section we will show that the magnon move operator H 3,2 and magnon annihilation operator H 2,2 generate a gauge symmetry of the fishchain that we interpret as discrete reparametrization symmetry.

Integrability, Exact Spectrum and Reparametrization Invariance
In section 4 we constructed a set of mutually commuting operators which also include the HamiltonianĤ q . Then, in section 5, we have constructed the fishnet cut deformation operators and lifted them to fishchain operators. In this section, we use the set of commuting operators to solve for the spectrum of dimensions ∆ of local operators. We then show that the cut deformation operators generate similarity transformations of the transfer matrices that can be interpreted as gauge transformations of the corresponding Baxter equation.
This section generalizes the results of [8], where the non-perturbative spectrum was obtained for the operators J = 3, M = 0 10 , and [7], where the general J, M = 0 case was considered. An important result of [7,8], which we are going to use here is the quantization condition, which is needed in addition to the Baxter TQ-relations. We will propose the generalization of this quantization condition and also present the most general form of the TQ-relations valid for any values of the charges J 1 , J 2 such that J 1 > |J 2 |.
Using these techniques we will be able to solve numerically for several examples of states, J = 3, M = 1 and J = 3, M = 2. At finite ξ we explore a very rich analytic structure of the function ∆(ξ). At weak coupling we are able to compare our numerical results with the asymptotic Bethe ansatz, valid only up to the wrapping order ξ 2J+2|M |−2 .
Our Baxter TQ-relation, which we present in this section, together with the generalization of the quantization condition of [7,8] constitute the complete set of equations, analogous to the Quantum Spectral Curve equations for N=4 SYM [18] or ABJM [19] theory.

T-functions and Baxter TQ relation
We start by formulating the TQ-relations. In order to compare with the previous partial results of [7,8] it is convenient to rescale the spectral parameter by ξ. To avoid confusion we denote the rescaled spectral parameter by v, which is related to the initial one by u = v/ξ. 11 We also denote by T(u) the eigenvalue (or T-functions) of the simultaneously diagonalizable operatorsT(u).
The T-functions corresponding to the trivial representations can be computed explicitly Also we notice from (4.2)-(4.4) that T4 and T 4 are closely related where * indicates that one should inverse the order of the particles and interchange magnons and anti-magnons. This transformation is an obvious symmetry of the system, but of course not all states are invariant under this transformation.
Baxter TQ-relations. One of the key quantities in quantum integrability are the Qfunctions [20,21]. They can be defined as T-functions corresponding to a rather complicated auxiliary space representation. As a simple alternative to defining them as eigenvalues of these complicated Q-operators, one can find them from a finite difference TQ-relation, which in our case reads 12 11 When writing T(v), strictly speaking, what we mean is T(v/ξ). 12 We use the standard shorthand notations We notice that the coefficients in this equation are polynomials of different degree. However, as we can see from (6.2) even though T4(v) is a polynomial of degree 3J, it has a trivial part of degree 2J, which factorises. To remove the trivial factors we introduce the following gauge transformation This transformation is designed so that the equation (6.3) becomes more symmetric where we defined the degree-L polynomials P L in the following way The index a = 1, . . . , 4 in q a labels 4 independent solutions of (6.5). We specify below exactly how these solutions are defined by their large v asymptotic.
One can see that in the form (6.5) the Baxter TQ-equation generalizes the one derived in the CFT context in [7,8] for the case M = 0. Furthermore, in [8] it is in addition assumed that the state is invariant under the inversion of the order of the sites, which we do not assume here. Our equation is thus a generalization of the results of [7,8] to the most general states with an arbitrary number of magnons, as long as |J 2 | < |J 1 |.
Asymptotics of q. The global conformal group charges ∆, S 1 and S 2 , enter into the TQ-relations through the large v asymptotic of T-functions and consequently Q-functions. Indeed, at large v the product (4.2) becomes a sum ofq k , which is the global so (1,5) charge. This observation is reflected in the following expressions for the highest powers of the polynomials P L : whereλ a ≡ 1 λa . We remind that λ i 's are the diagonal elements of the twist matrix G. Next, we plug the asymptotics (6.7) into the TQ-relation (6.5) to deduce the following four possible asymptotics of the four linearly independent solutions of (6.5) Another important asymptotic is the value of P 6 2J (v) at the origin. Due to the relation (6.6) and the observation that T 6 (0) = H q + 1 we have the following relation This is central for how the fishchain condition H q = 0 injects the information about the coupling ξ into the TQ-relations. Except for (6.9) there is no other explicit dependence on ξ appearing in (6.5).

Discrete reparametrization symmetry and integrability
The general expectation from the discrete reparametrization symmetry, described in section 5, is that the spectrum and other integrals of motion should remain invariant. In this section we will prove this is indeed the case based on the integrability construction described in the previous section.
Magnon reordering symmetry. One can immediately notice that the Baxter equation (6.5) does not contain any explicit reference to the ordering of the particles. The equation only depends on the number of magnons M and anti-magnonsM , but not on their particular ordering along the fishchain. Furthermore, we will see below that the Baxter TQ-relations depends non-trivially only on the difference J 2 = M −M . At the level of the eigenvector this symmetry is less obvious. In section 5 we have shown that wave functions with different magnon positions corresponds to different ways of cutting the same Feynman diagrams. Using this picture, we have argued that a shift in the magnon position should be thought of as a discrete reparametrization symmetry of the fishchain model. This shift is generated by the fishchain operator H 3,2 in (5.15) (6.10) We will now prove that indeed, this operator generates a symmetry of the transfer matrix. The operator R 21 has the following key property At first sight it looks quite complicated, however, one can see easily that on the states with a magnon on site 1 and no magnon on 2 the r.h.s. become zero and the operator R become the interchange operator which exchanges the roles between the magnons and no-magnons.
The existence of such operators guarantees that the spectrum of all integrals of motion does not depend on the initial order of the magnons. This ensures the self-consistency of our approach. More precisely we can see that the magnon move operation results in the following similarity transformation where the set I is such that I k = 1 and I k+1 = 0, whereas the transformed set ↔ I corresponds to the magnon at k + 1 i.e. I k = 0 and I k+1 = 1. The equation (6.12) ensures that the spectrum of all conserved charges before and after the magnon move operation stays unchanged. Similarly, we can shift an anti-magnon to the left and all conserved charges are unaffected. Since states are uniquely characterize by the charges, this is a gauge symmetry that is associated with a redundancy in our description.
Magnon-anti-magnon annihilation. First, let us demonstrate that the spectrum does not change when we simultaneously remove one magnon and one anti-magnon. We will establish a simple relation between the solutions of the Baxter equation with different numbers of magnons. Consider the transformation M → M − n,M →M − n. Under this transformation (6.5) becomes First we see that the overall factor 1/v n cancels. Furthermore, by denoting q(v) =q (v) v n we return back to the initial form (6.5).
At the level of the fishchain wave function, we can always shift the positions of the magnon and anti-magnon until the anti-magnon is at the first site and the magnon is at the second site. To Annihilate them, we act with (6.14) This operator has the following property Similar to (6.11) the r.h.s. of the above equation vanishes on the states with magnon and anti-magnon. This relation leads to an analog of (6.12) where the initial set I has I k+1 = 1 and I k = −1, whereas the resulting set ↔ I has no magnons at these positions i.e. ↔ I k = ↔ I k+1 = 0. We see again that the magnon annihilation operation does not affect the spectrum or any of the the conserved charges.

Quantization condition and exact spectrum
In order to adapt the quantization condition of [7] to the general case with any number of magnons, we introduce two sets of solutions of the TQ-relations (6.5). One set, denoted by q ↓ a (v), is regular in the upper half-plane and satisfy (6.8) at large v. The other set, q ↑ a (v), is regular in the lower half-plane. It is easy to see that such solutions exist and that they can be constructed numerically, using the methods of [22].
Since there are only 4 independent solutions of the 4th order finite difference equation (6.5), these two sets q ↓ a (v) and q ↑ a (v) should be linearly dependent with i-periodic coefficients, i.e. , The matrix Ω a b (v) itself can be constructed from q-functions as the simple ratio of two determinants as follows from (6.16). In [7] it is shown that this matrix Ω a b (v) should satisfy the following quantization which singles out physical solutions of the TQ-relations. Furthermore, one can show [7] that the i-periodic function Ω a b (u) has simple poles at u = in of order J. This implies that each matrix element of Ω a b (u) can be parametrized by J + 1 constants C (n) as (1 − e 2πu ) J , (6.19) where in addition for off-diagonal elements only J constants C (n) are non-zero. This implies that we have 4J equations in (6.18). One can show that only 4J − 3 of these conditions are linearly independent (for large enough J), which is exactly how many free parameters we have in the polynomial coefficients P L in (6.5). Thus, using the quantization condition (6.18) we should be able to fix all coefficients in the Baxter TQ-relations, which also includes ∆.
In the next section we report on the numerical tests of this procedure for the case with magnons.

Numerical tests
In this section we apply the procedure described in section 6.3 for two cases, J = 3, M = 1 and J = 3, M = 2. In order to solve the Baxter TQ-relations (6.5) we use the method of [8,22]. Then we use the generalized gluing conditions (6.18) to fix the parameters in (6.5). In both cases we found perfect agreement with the Asymptotic Bethe Ansatz (ABA) of [9][10][11] at weak coupling. In this section we fix the twist parameters as λ 1 = e i/3 , λ 2 = e −i/3 , λ 3 = e i/2 , λ 4 = e −i/2 . We generated to a very high precision numerical values for a wide range of couplings using the method described in 6.3 (see figure 10). In order to check our results we have to recall that in the ABA approach for one magnon one only needs to know the dispersion relation ∆ ABA = J + 1 − 4ξ 2 , which is expected to be accurate up to the wrapping order ξ 2J+2M = ξ 8 .
Indeed, by making a fit of our numerical values with even powers of ξ we found the following expansion ∆ T Q = 4. − 2.ξ 2 − 2.ξ 4 − 4.ξ 6 + 0.155845ξ 8 + . . . . (6.20) We see that the first 4 coefficients are integers, and agree precisely with the magnon dispersion relation. At order ξ 8 we get some real number, which is expected to be a combination of two polylogarithms Li 3 and Li 5 with the arguments depending on the twists. What is very interesting is the analytic structure of our result in ξ. At small enough ξ the function has a regular (convergent) expansion in powers of ξ 2 . At ξ ∼ 0.7 the dimension ∆ tends to 0 and after that point become purely imaginary (see figure 10). The value ξ ∼ 0.7 is thus a branch point. This is not the only branch cut of the function ∆(ξ). The function ∆(ξ) is in fact a multivalued function with infinitely many sheets (see figure 11), containing not just one but infinitely many local operators, which can be identified at weak coupling as tr φ 3 1 φ 2 (φ 2 φ † 2 ) n (up to permutations). Figure 11. Single magnon operator tr φ 3 1 φ 2 has a dimension ∆, which is a multivalued function of ξ. At weak coupling the bare dimension of the state is J 1 + J 2 = 4. Analytically continuing in ξ one can discover an infinite tower of states with the bare dimensions 4 + 2n, n = 0, 1, . . . (n = 0, 1 are indicated by the red dots on the picture).

Length three, two magnons
We also found a numerical solution in the two magnon case for the chain of length J = 3. Again our results are in complete agreement with the ABA prediction at weak coupling. Note that this time, the ABA calculation involves a rather non-trivial dressing phase, which gives rise to the ζ values in the result ∆ ABA = 5 + √ 5 − 1 ξ 2 + 3 √ 5 − 1 ξ 4 − 2 + 2 5 √ 5 ξ 6 (6.21) Our numerical fit of the result of solution of the TQ-relations gives ∆ T Q = 5 + 1.23607ξ 2 + 0.341641ξ 4 − 2.17889ξ 6 − 2.50956ξ 8 − 0.575805ξ 10 + . . . (6.22) which agrees perfectly with the first 4-loops predicted by ABA (6.21). At the wrapping order ξ 2J+2M = ξ 10 , as expected, the ABA result is no longer valid and deviates from our numerical prediction (6.22).
In conclusion, the fact that we managed to reproduce the ABA prediction from the TQ-relations provides a very non-trivial test of our approach.

Conclusion
In this paper, we completed the derivation of the Holographic dual of the fishnet theory. Introducing the magnons, i.e. φ 2 and/or φ † 2 fields, into the consideration brought some surprises. Firstly, we found that the classical limit ξ → ∞ remain unchanged. Namely, we still find the same classically integrable fishchain model, which is a classical chain of particles on a light-cone with nearest-neighbor interaction. The essence of the magnons is hiding in the details of the quantization procedure, which also modifies the light-cone into the AdS 5 space. 14 This is, in fact, in complete agreement with the exact spectrum and 4-point function computation in [14] for the length-two operators. Secondly, the introduction of the magnons revealed a new nice feature, namely the discrete reparametrization symmetry -an invariance of the physics w.r.t. different sectioning of the discretized world-sheet of the fishchain into space-like slices. The magnons in this picture indicate bends of the curve cutting the world-sheet. Mathematically this symmetry manifests as a map between Hilbert spaces, preserving the integrability and conserved charges that uniquely characterize the state.
Another important output of our analysis is the universal integrability picture, which includes all possible insertions on equal footing. We found the quantum Baxter TQrelations, which together with the quantization condition [7,8], give us direct access to the quantum spectrum of the model in complete generality. 15 This also opens ways to the separation of variables (SoV) construction, along the lines of the recent works [24][25][26][27].
Among open questions remains a particular configuration when the number of magnons saturates the number of states, i.e. when both u(1) charges are equal |J 1 | = |J 2 |. In this case, surprisingly, the general strategy outlined in this paper fails. On some particular examples it is known that this case produces a more complicated result for the spectrum even for short operators, tractable with the conformal symmetry alone [14]. Nevertheless, the integrability should be present in this case too [7], but the strong coupling behavior is very different, indicating that it is mysteriously missing in the current picture. This problem could be related to another large part of states which happened to be protected, and decoupled from the dynamical part of the spectrum studied in this paper. The reason why there is a large number of states with protected dimension is also unclear, there is no manifest symmetry, like supersymmetry, which can help to explain this phenomenon.