Open Fishchain in N=4 Supersymmetric Yang-Mills Theory
PPrepared for submission to JHEP
Open Fishchain in N=4 SupersymmetricYang-Mills Theory
Nikolay Gromov, a,b
Julius Julius, a Nicolò Primi a a Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, UK b St.Petersburg INP, Gatchina, 188 300, St.Petersburg, Russia
E-mail: nikolay.gromov • kcl.ac.uk, julius.julius • kcl.ac.uk,nicolo.primi • kcl.ac.uk Abstract:
We consider a cusped Wilson line with J insertions of scalar fields in N = 4 SYM and prove that in a certain limit the Feynman graphs are integrable to all loop orders.We identify the integrable system as a quantum fishchain with open boundary conditions.The existence of the boundary degrees of freedom results in the boundary reflection operatoracting non-trivially on the physical space. We derive the Baxter equation for Q-functionsand provide the quantisation condition for the spectrum. This allows us to find the non-perturbative spectrum numerically. a r X i v : . [ h e p - t h ] J a n ontents R -matrix 225.4 Transfer matrix 235.4.1 Transfer matrix in fundamental representation 235.4.2 Ingredients of the transfer matrix in vector representation 245.4.3 Hamiltonian from the transfer matrices 265.4.4 Ingredients of the transfer matrix in the anti-fundamental represen-tation 275.4.5 Ingredients of the quantum determinant 285.4.6 J = 0 example 295.4.7 Eigenvalues of the transfer matrices 30 v asymptotic of Q-functions 33 T ( u )
37B Parity of quantum transfer matrices 40C Explicit form of transfer matrices 42 – 1 –
Generalisation: addition of impurities 44
Integrability in
D > quantum field theories takes its roots in quantum chromodynamicswhere it was first observed that in the BFKL limit the evolution kernel admits integra-bility [1, 2]. Later on it was found in N = 4 Supersymmetric Yang-Mills Theory (SYM)in [3] in a different regime, where the one-loop mixing matrix of a single trace of scalarswas identified with the Hamiltonian of a closed integrable SO (6) Heisenberg spin-chain inthe large- N limit (see [4–6] for recent reviews). Even though this observation was furthergeneralised to two loops and a number of tests was done at higher loop orders and alsonon-perturbatively, there is still no direct proof of integrability of N = 4 SYM. Even ifthere is very little doubt about the integrability of this theory, having a rigorous proof of itwould give us new tools and may also allow us to go beyond the spectrum in applicationsof integrability to non-perturbative gauge theories .Recently, the so-called fishnet limit of N = 4 SYM attracted much attention [8]. Inits simplest version this is a limit where only two scalar fields remain coupled and have aYukawa-type interaction. This theory has much simpler Feynman diagrams in the planarlimit and thus provides perfect playground to test various non-perturbative techniquesincluding integrability. In [8–13] it was shown how the integrability emerges directly fromthe Feynman graphs and the connection with the integrability structure of N = 4 SYMsuch as Quantum Spectral Curve was established. This gives a number of clues of how theintegrability realises itself in more complicated theories such as N = 4 SYM. The maindrawback of the fishnet theory is that it is not a unitary CFT and it is not known whetherthe conformal symmetry persists beyond the planar limit.In this paper we consider a Wilson loop with local operator insertions in undeformed N = 4 SYM and then take the so-called ladders limit. We will use the methods developedfor the fishnet theories [10, 13, 14] to develop an integrability based description of theseobservables and obtain a solution for the spectrum. The solution takes the form of a Baxterfinite difference equation supplemented with a particular quantisation condition.The setup that we will consider in this paper is the following: we have a cuspedMaldacena-Wilson line [15, 16] with internal cusp angle ϕ , as in figure 1. The scalar Φ couples to the left ray and Φ cos θ + Φ sin θ couples to the right ray. J scalars Z = √ (Φ + i Φ ) , orthogonal to the ones that couple to the rays, are inserted at the cusp.Here Φ i , i = 1 , · · · , are the six scalars of N = 4 SYM theory. In addition, we caninclude in our description the excited states, in analogy with [17–19], which corresponds toinsertions of linear combinations of the scalars coupled to the lines. This observable hasa well defined anomalous dimension, which was studied in [20–22] by means of TBA andthen QSC methods. In this paper we will start from scratch, carrying out a first principles For long operators the integrability based
Hexagon approach [7] works very well in some regimes. – 2 – igure 1 : The CFT wavefunction for J = 2 is a sum of the fishnet diagrams with anynumber of bridges. This figure shows one such diagram with l = 4 bridges. The graphbuilding operator is highlighted.derivation in the so-called ladders limit , which we describe below. The only insight weborrow from the QSC approach is a simple quantisation condition, which would requirefurther efforts to derive from first principles.Another motivation behind the work we present in this paper is due to the recentstudy of structure constants by the separation of variables (SoV) method [14, 17, 23, 24],where the explicit form of the Baxter equation was shown to be at the heart of the SoVapproach. An alternative to the approach of this paper would be to derive the Baxterequation from the QSC, which has a number of technical complications. Whereas at leastnumerically QSC [25] would give us a full control over this observable for a very wide rangeof parameters, extracting the closed system of equations in the ladders limit analyticallyhas proven to be quite a challenging task (which was performed for J = 0 case in [26]).The ladders limit which we study in this paper was first introduced for the case J = 0 in [27] and then used in [28]. This is obtained by taking the coupling g → and θ → i ∞ , insuch a way that ˆ g ≡ g (cid:16) exp( − iθ/ (cid:17) J +1 is kept constant. For the case J = 0 it was noticedin [27, 28] that only the ladder graphs contribute to the anomalous dimensions and thecorrelation functions. In this paper we show that for the general J > case the diagramswhich survive are those of the fishnet type with a boundary corresponding to the two raysof the Wilson line (see figure 1). This drastic simplification in Feynman graphs allows us to– 3 –onstruct the resummation procedure involving a graph-building operator. Such an operatorwas first constructed in the case of a Wilson-Maldacena loop with no scalar insertions in [17]and for the fishnet theory in [10]. A new ingredient in the construction is the boundary ofthe fishnet, which itself carries a D dynamics. We had to adapt the boundary integrabilitymethods for spin chains, developed by Sklyanin in [29]. In our case, however, the boundaryreflection matrix itself is a nontrivial operator in the physical space. In this paper we first explore the integrability in the classical (strong coupling) limit ˆ g → ∞ and then quantise this system and develop the full quantum integrability. Like in thecase of the fishnet theory the integrability description comes from a chain of particles livingon AdS (with radius going to zero at strong coupling) also known as “fish-chain” [13, 31, 32].This time, however, we have two particles with zero conformal weight at the ends of the chainwhose motion is additionally restricted to the Wilson lines. In the quantum construction weidentify explicitly the conserved charges of the system in the commuting family of operators,and prove that the graph-building operator of the Feynman graph in the perturbation theoryis one of them. In this way we obtain a full quantum non-perturbative description for thespectrum.We also briefly discuss an interesting limit when the cusp becomes a straight line. Inthis limit the insertion becomes an operator in 1D defect CFT, where one could make aconnection with the bootstrap methods of [33].This paper is organised as follows. In section 2, we derive the graph building opera-tor starting from Feynman diagrams in the ladders limit. In section 3, we construct theLagrangian of the open fishchain and solve the equations of motion. In section 4 we showthat our model is classically integrable and construct the Lax and the boundary reflectionoperators. Then in section 5 we show that integrability carries forward to the quantum case.In section 6 we construct the Baxter equation for arbitrary number of scalar insertions of Z at the cusp. In section 7 we present the numerical non-perturbative spectrum for variousinsertions and geometric parameters. Finally we conclude in section 8. In this section we will describe the Feynman diagrams contributing to the expectation valueof the cusped Wilson line. We show that in the ladders limit it gets an iterative Dyson-Schwinger structure, governed by a graph building operator. The graph building operatoris a hybrid between that obtained for J = 0 in [27, 28] for the cusp without insertion andthe one for the fishnet theory [8, 9]. In the rest of the paper we develop the integrabilitybased method to diagonalise this operator.– 4 – igure 2 : We only need a subset of all Feynman diagrams. Above are the conventions forthe scalar propagators and the interaction vertex between Φ and Z = Φ + i Φ √ . We use thestandard definition g = √ λ π with the ‘t Hooft coupling λ = g Y M N . The Maldacena-Wilson Loop with J insertions of scalar fields is given by: W = 1 N tr P exp (cid:90) ∞ dt πg (cid:0) i A · x (cid:48) ( t ) + Φ | x (cid:48) ( t ) | (cid:1) × Z (0) J × P exp (cid:90) ∞ ds πg (cid:2) − i A · x (cid:48) ( s ) + (Φ cos θ + Φ sin θ ) | x (cid:48) ( s ) | (cid:3) , (2.1)where x (cid:48) ( t ) ≡ ∂x ( t ) ∂t and x (cid:48) ( s ) ≡ ∂x ( s ) ∂s . The two scalars that couple to the individual Wilsonrays form an angle θ between each other. The expectation value of this quantity is divergentin both the IR and the UV, with the divergence controlled by the dimension ∆ : (cid:104) W (cid:105) ∼ (cid:18) R IR (cid:15) UV (cid:19) − ∆ , (2.2)where ∆ corresponds to the overall scaling dimension of W and is also known as the cuspanomalous dimension in the J = 0 case. In this paper we will study a more generalobservable which is the expectation value of W with J additional insertions (under thetrace) of complex scalar fields ¯ Z = √ (Φ − i Φ ) at points y i which lie outside of thecountour, and also truncate the upper limit in Wilson lines at some finite t and s . Inanalogy with [13] we call this object the CFT wavefunction ψ ( t, s, y i ) . At first sight thisobject is not gauge invariant, however in the ladders limit it is well defined. In fact, it canbe made gauge invariant by closing the Wilson loop by introducing additional segments ofnon-supersymmetric Wilson lines running through the ¯ Z insertions, which will decouple inthe ladders limit, as in fig. 1. As we will see, the role of the effective coupling in the ladderslimit is played by ˆ g ≡ g (cid:18) exp ( − iθ/ (cid:19) J +1 , (2.3)which we will assume finite while g → and θ → i ∞ . In this limit we will get the followingsimplifications: Similar situation can be found e.g. in [30]. Strictly speaking for
J > it is only divergent for large enough coupling as at tree level we have ∆ = J .For J = 0 it is divergent for any g > . – 5 – First of all, since we are taking the ’t Hooft coupling to zero, the gluons and fermionsdecouple, and we are left with a theory of interacting scalar fields. Hence, we candrop out the gauge field A from the definition (2.1).• In a Feynman diagram expansion, only the contributions with the highest power of cos θ will survive. For J = 0 , the only diagrams at l -loop order correspond to ladderdiagrams, that is, diagrams that contain l scalar propagators beginning on one of theWilson lines and ending on the other [28].• For J > , the scalars at the cusp Z can only contract with the external insertions of ¯ Z . This means that only one type of vertex allowed, i.e. the one in figure 2. This isanalogous to what one finds in the simplest fishnet CFT. Consequently, only “fishnet”diagrams contribute.Using these simplifications, we can define the CFT wavefunction in the ladders limit as: ψ ( t, y , . . . , y J , s ) ≡ N (cid:42) tr J (cid:89) j =1 ¯ Z ( y j ) × P exp (cid:90) t dt (cid:48) (4 πg ) | x (cid:48) ( t (cid:48) ) | Φ × Z (0) J × P exp (cid:90) s ds (cid:48) (4 πg ) | x (cid:48) ( s (cid:48) ) | Φ cos θ (cid:43) . (2.4)The CFT wavefunction is obtained expanding the path-ordered exponentials ψ ( t, y , . . . , y J , s ) = ∞ (cid:88) l =0 ψ l ( t, y , . . . , y J , s ) = ∞ (cid:88) l =0 tr (cid:90) t dt l | x (cid:48) ( t l ) | (cid:90) t l dt l − | x (cid:48) ( t l − ) | · · · (cid:90) t dt | x (cid:48) ( t ) | (cid:90) s ds l | x (cid:48) ( s l ) | (cid:90) s l ds l − | x (cid:48) ( s l − ) | · · · (cid:90) s ds | x (cid:48) ( s ) | F l ( y j , t i , s i ) . (2.5)Here, ψ l ( y j , t i , s i ) represents the contribution of the l -bridge fishnet Feynman graph, wherea bridge is defined as a series of J + 1 propagators connecting the left and right Wilsonrays, as can be seen in figure 1. Note that the sum goes in the number of bridges l . Theintegrand in s i , t i is given by: F l ( y j , t i , s i ) = 1 N (cid:18) π N (cid:19) l ( J +1)+ J ( l +1) (64 π N g ) l J (16 π g cos θ ) l N ( l +1)( J +1)+1 (cid:90) J (cid:89) i =1 l (cid:89) j =1 d x i,j (cid:32) J (cid:89) r =0 l (cid:89) k =1 x r +1 ,k − x r,k ) (cid:33) (cid:32) J (cid:89) m =1 l (cid:89) n =0 x m,n +1 − x m,n ) (cid:33) . (2.6)Here we have defined x k, ≡ y ≡ (cid:0) e t k , , , (cid:1) , x k,J +1 ≡ y J +1 ≡ ( e s k cos ϕ, e s k sin ϕ, , ∀ k = 1 . . . l , and x l +1 ,j ≡ y j , x ,j ≡ ∀ j = 1 . . . J . In the formula (2.6), the second factor inthe first line of the r.h.s contains the contribution from the propagators, the third the onefrom the vertices, the fourth comes from the expansion of the path-ordered exponentials,– 6 –hile the fifth represents the contribution from the closed index loops of the planar dia-gram. In the second line we first integrate over all positions of the vertices, the second termis a collection of all vertical propagators, while the third contains that of the horizontalones (see figure 1 for the case of J = 2 and l = 4 ). Notice that at any loop order thesegraphs have the same order in N , coherent with the fact that we are using a planar diagramexpansion. Instead of computing this integral we notice that we can define it recursively interms of the inverse of a graph building operator as we illustrate below. First, notice that (cid:3) y i acts on scalar propagators as: (cid:3) y j y j − x j,l ) = − π δ ( y j − x j,l ) . (2.7)Moreover, acting with ∂ t ∂ s on the contour of a Wilson line brings down the expansion ofthe path ordered exponential by one step, at the cost of a factor | y (cid:48) ( t ) || y (cid:48) J +1 ( s ) | . Thereforeacting on ψ with a string of (cid:3) y j , followed by ∂ t ∂ s , we get back ψ expanded to one lessbridge, up to a multiplicative factor: ∂ t ∂ s J (cid:89) j =1 (cid:3) y j ψ l = ( − J (4ˆ g ) J +1 | y (cid:48) || y (cid:48) J +1 | (cid:81) Ji =0 ( y i − y i +1 ) ψ l − , (2.8)where we use the definition of ˆ g from (2.3). From this we can extract an operator annihi-lating the CFT wavefunction: ( ˆ B − − ψ = 0 , ˆ B − ≡ ( − J (4ˆ g ) J +1 (cid:81) Ji =0 ( y i − y i +1 ) | y (cid:48) || y (cid:48) J +1 | ∂ t ∂ s J (cid:89) j =1 (cid:3) y j . (2.9)We refer to ˆ B − as an inverted graph-building operator. The role of ˆ B − − was realisedin [31] to be the analogue of the world-sheet Hamiltonian of a string theory. We will explorethis further in the next section.The Wilson loop with insertion W is invariant under dilatations, which stretches thespace around the origin (which we take to be the position of the cusp). Thus we can usethe following dilatation operator, acting on the CFT wavefunction ˆ D = − i (cid:32) ∂ t + ∂ s + J (cid:88) i =1 ( y i · ∂ y i + 1) (cid:33) , (2.10)to measure the dimension ∆ of the initial cusped Wilson line. More precisely, the eigenvalueof ˆ D is i ∆ . This operator commutes with ˆ B as it is easy to see. Another operator whichcommutes with ˆ B is the generator of rotations in the orthogonal plane to the Wilson line: ˆ S = i J (cid:88) i =1 (cid:16) y i ∂ y i − y i ∂ y i (cid:17) . (2.11)This operator measures the spin of W . For Z J scalar insertions S = 0 , but one can alsostudy more general insertions with derivatives in the orthogonal plane, corresponding to S (cid:54) = 0 , which are also described by our construction.– 7 –n analogy with the fishnet [32] one should diagonalise both ˆ S and ˆ D . After doing so,the equation ( ˆ B − − ψ = 0 should restrict us to the discrete spectrum of eigenvalues of thedilatation operator, which would give us all the anomalous dimensions of the operators withgiven quantum numbers. Indeed we will find that there are infinitely many (but a discreteset) of such ψ ’s diagonalising all the operators. In analogy with [17–19] we expect each ofthem to correspond to a particular insertion of operators, which could include derivativesand extra Φ , Φ fields in addition to Z J , whose number is fixed by the R-charge. Thesetype of insertions at the cusp will not modify the iterative structure of the diagrams, insteadjust adding a finite number of propagators close to the cusp (cf. figure 3). Therefore, allthese states should be governed by the same equation (2.8). Figure 3 : An example of an “excited state” for J = 2 . Here, propagators from theextra insertion of Φ at the cusp contract with the Wilson line without crossing any otherpropagators of Φ (shown in red), as such diagrams would be subleading in the ladderslimit.In the next sections we will explore how the integrability arises explicitly in this system.In particular, we will show first in the classical (strong coupling) limit and then in generalthat the operators ˆ B, ˆ D and ˆ S are part of a bigger commuting family of operators. In this section, following [31], we interpret the inverse of the graph-building operator asa Hamiltonian of a quantum system of particles. Then we take the quasi-classical strongcoupling limit of the system, deriving the classical fishchain with specific open boundaryconditions. We analyse in detail the classical system and find some of the solutions of theequations of motion.
The starting point for the strong coupling ˆ g → ∞ analysis is equation (2.8). By re-writing(2.9) in terms of the conjugate momenta: p i = − i∂ y i , π t = − i∂ t , π s = − i∂ s , (3.1)– 8 –e obtain the Hamiltonian ˆ H , governing a system with J + 2 degrees of freedom, givenby: ˆ H = π t π s J (cid:89) i =1 p i + (4ˆ g ) J +1 | y (cid:48) ( t ) || y (cid:48) J +1 ( s ) | (cid:81) Ji =0 ( y i − y i +1 ) . (3.2)In this section we will be treating this Hamiltonian as the one of a classical system. Inanalogy with [31] we will see that the classical limit corresponds to the strong coupling ˆ g → ∞ limit of the original quantum system (2.9). We will now demonstrate the classicalintegrability of this system and then describe its quantisation in section 5.We remark that y i , i = 1 . . . J are 4D vectors with four bulk degrees of freedom foreach one, while y and y J +1 are 4D vectors having one boundary degree of freedom each.Therefore, without loss of generality, we parametrise the latter as: y ( t ) = (cid:0) e t , , , (cid:1) , y J +1 ( s ) = ( e s cos ϕ, e s sin ϕ, , . (3.3)We will find it beneficial to embed the system in D space, which will allow to makethe conformal symmetry of the system manifest, but will also result in a local action withnearest neighbour interaction.In the rest of this section we will deduce the classical equations of motion of this system,using the Lagrangian formalism. First, by performing a Legendre transformation on (3.2),we find the Lagrangian to be: L = 2 − J J +1 (2 J + 1) (cid:32) ˙ t ˙ s J (cid:89) i =1 ˙ y i (cid:33) J +1 − (4ˆ g ) J +1 | y (cid:48) ( t ) || y (cid:48) J +1 ( s ) | (cid:81) Ji =0 ( y i − y i +1 ) , (3.4)where ˙ f ≡ ddτ f , with τ being a “world-sheet” time variable (conjugate to the Hamiltonian(3.2)). The action S = (cid:82) L dτ is not invariant under time reparametrisation symmetry τ → f ( τ ) , which is needed to ensure ˆ Hψ = 0 . In order to enforce this symmetry weintroduce an auxiliary field γ transforming as γ → γ ˙ f when τ → f ( τ ) which gives L = 2 − J J +1 (2 J + 1) (cid:32) γ ˙ t ˙ s J (cid:89) i =1 ˙ y i (cid:33) J +1 − γ (4ˆ g ) J +1 | y (cid:48) ( t ) || y (cid:48) J +1 ( s ) | (cid:81) Ji =0 ( y i − y i +1 ) . (3.5)This is now time-reparametrisation invariant. We then eliminate the auxiliary field settingit to its extremum (by a suitable time reparametrisation). We have to remember that y (and y J +1 ) is not itself a canonical coordinate, but depends on the world-sheet time through t ( τ ) (and s ( τ ) respectively). Thus we can use ˙ y = y (cid:48) ˙ t and similarly ˙ y J +1 = y (cid:48) J +1 ˙ s . Afterthat we get: L = (2 J + 2)(2 i ) J +1 ˆ g (cid:34) | ˙ y || ˙ y J +1 | (cid:81) Ji =1 ˙ x i (cid:81) Ji =0 | y i − y i +1 | (cid:35) J +1) . (3.6)– 9 –e now embed the system in D Minkowksi spacetime, using lightcone coordinates in thePoincare’ slice: y µi = X µi X + i , X i = 0 , X + i = X i + X − i . (3.7)Hence we get: L = (2 J + 2)(2 i ) J +1 ˆ g (cid:34) | ˙ X || ˙ X J +1 | (cid:81) Ji =1 ˙ X i (cid:81) Ji =0 ( − X i .X i +1 ) (cid:35) J +1) . (3.8)Furthermore, we can disentangle this action to bring it to a Polyakov-like form, by intro-ducing auxiliary fields α i , getting: L = ξ (cid:32) α | ˙ X || ˙ X J +1 | J (cid:88) i =1 (cid:32) α i ˙ X i η i X i (cid:33) + ( J + 1) J (cid:89) k =0 ( − α k X k .X k +1 ) − J +1 (cid:33) , (3.9)where ξ ≡ (2 i ) J +1 ˆ g . (3.10)In (3.9) we also introduced the light-cone constraint X i = 0 via the Lagrange multiplier η i . In order to get back the Nambu-Goto-like form (3.8), we have to extremise the fields α i and plug these values back into (3.9). It is possible to do this due to the new re-scalingsymmetry of X i . More precisely, the Lagrangian (3.9) has J + 3 gauge symmetries: time-dependent rescaling X i → g i ( τ ) X i , α i → α i g − / i ( τ ) , η i → η i g − / i ( τ ) , i = 0 . . . J + 1 and time reparametrisation τ → f ( τ ) , under which fields transform as X i → X i ˙ f , α i → ˙ f α i , η i → η i ˙ f . Instead of setting α i ’s to their extreme values we can use the symmetries toimpose α i = 1 , ∀ i = 0 , . . . , J . This would lead to the following constraints (the same wayas one gets Virasoro constraints): ˙ X k = L , (3.11)where L ≡ J (cid:89) i =0 ( − X i · X i +1 ) − J +1 , (3.12)with k = 1 , . . . , J in the first equation. Furthermore, from the equation of motion for α we get | ˙ X || ˙ X J +1 | = L : this still leaves us with the freedom to rescale X → h ( τ ) X and simultaneously X J +1 → h ( τ ) X J +1 , which we can fix by imposing additionally | ˙ X | = | ˙ X J +1 | . Hence, we can just extend the range of k in (3.11) to k = 0 , . . . , J + 1 . Finally, tofix the remaining time-reparametrisation gauge freedom we can set: L = 1 , (3.13)which is a convenient gauge to work with. We have imposed J + 3 conditions, so all gaugedegrees of freedom are fixed. The gauge fixed Lagrangian is then: L = ξ (cid:32) | ˙ X || ˙ X J +1 | J (cid:88) i =1 ˙ X i J + 1) J (cid:89) k =0 ( − X k .X k +1 ) − J +1 (cid:33) , (3.14)– 10 –inally, by noticing that | ˙ X || ˙ X J +1 | = ˙ X + ˙ X J +1 − ( | ˙ X | − | ˙ X J +1 | ) we can replace | ˙ X || ˙ X J +1 | → ˙ X + ˙ X J +1 in (3.14), modulo terms quadratic in constraints. Similarly,defining y = 2 (cid:81) Ji =0 ( − X i · X i +1 ) − J +1 (cid:39) on constraints, we have y = e log y = 1 + log y + O (log y ) , which allows us to replace the potential term by (cid:80) Jk =0 12 log − X k .X k +1 e . Thereforewe get the gauge fixed Lagrangian: L = ξ (cid:32) ˙ X J (cid:88) i =1 ˙ X i X J +1 − J (cid:88) i =0
12 log − X i .X i +1 e (cid:33) , (3.15)with constraints given by: J (cid:89) i =0 − X i .X i +1 , (3.16) X i = 0 , ˙ X i = 1 , i = 0 , . . . , J + 1 . (3.17)Note that on the constraints we also have L (cid:39) ξ ( J + 1) . In this form the Lagrangian isexplicitly local and the interaction is only between the nearest neighbours. It may appeara bit strange that the boundary particles has mass / w.r.t. to the particles in the bulk,however, we will see in the next section that in this way the equations of motion are moreuniform. The reason is that the bulk particles has to be split in two and reflected, unlikethose at the boundaries. In (3.14) the D variables X i , i = 1 , . . . , J are independentcanonical coordinates, constrained by (3.11) and (3.13). At the same time the boundaryparticles X and X J +1 are encoded in terms of one variable each t ( τ ) and s ( τ ) , due to (3.3).Explicitly: X i ( τ ) = r i ( τ ) (cosh w i ( τ ) , − sinh w i ( τ ) , cos φ i , sin φ i , , i = 0 , J + 1 , (3.18)where φ = 0 , φ J +1 = ϕ , w ( τ ) = t ( τ ) and w J +1 ( τ ) = s ( τ ) . On the constraint ˙ X i = 1 wealso have r i ( t ) = w i ( τ ) . Apart from this, the Lagrangian (3.14) is very similar to the onefound in the classical limit of the fishnet graphs in [31]. It can be interpreted as the one of adiscretised string with string-bits having a nearest neighbour interaction. However, due tothe difference in the boundary DOFs it still remains to see whether the system is classicallyintegrable, as it was in the original case [31]. We now compute the Euler-Lagrange equations starting from (3.15). The J equations forbulk variables are interpreted as equations of motions for J bulk particles, while the equa-tions for X and X J +1 are interpreted as equations of motion for two particles constrainedon the two Wilson lines. Since the Lagrangian (3.15) has nearest neighbour interactions,only the first and last bulk particles in the spin chain will feel the presence of the particleson the Wilson lines. For example, for the bulk particle j we have [31]: ¨ X j = 2 η j X j − (cid:18) X j +1 X j +1 .X j + X j − X j .X j − (cid:19) , j = 1 , . . . , J (3.19)– 11 –or the particles on the Wilson lines, we only have one physical degree of freedom for each, t ( τ ) and s ( τ ) . The relative equations of motion are given by: ¨ t ˙ t = X .∂ t ( τ ) X X .X , ¨ s ˙ s = X J .∂ s ( τ ) X J +1 X J .X J +1 . (3.20)These two equations can be written in the form (3.19) by introducing the reflected particles X − and X J +2 as the reflection of the particles X and X J w.r.t. the ray parametrised by t and s respectively. More precisely we introduce the reflection matrix and rotation matrices: C MN = − − − MN , G M N = ϕ − sin ϕ ϕ cos ϕ MN . (3.21)Then we define the images of the particles and J by the reflection about the ray parametrisedby t and s respectively as X − = C.X and X J +2 = G.C.G − .X J = G .C.X J . With thesedefinitions the equations (3.20) coincide with (3.19) for j = 0 and j = J +1 correspondingly.Thus, we conclude that at the level of the classical equations of motion the open versionof the fishchain we consider here is identical to the double-size closed fishchain of [31], withlength J + 2 and quasi-periodic boundary condition twisted by a ϕ rotation (see figure 4).However, there are some important differences in the Poisson structure and consequentlythe quantisation is different. The presence of boundaries in the open fishchain breaks the SO (1 , symmetry that itsclosed counterpart enjoyed to the subgroup SO (2) × SO (1 , . Nevertheless it is useful todefine q MNj ≡ ˙ X Mj X Nj − ˙ X Nj X Nj = 2 ˙ X [ Mj X N ] j , (3.22)for j = 0 , . . . J + 1 , which are the local SO (1 , generators for j = 1 , . . . , J . We also definethe total charge Q MN = ξ (cid:32) q MN J (cid:88) i =0 q MNj + q MNJ +1 (cid:33) . (3.23)As the SO (1 , symmetry is broken, only the components of Q MN corresponding to the un-broken symmetry subgroup will remain conserved in time. Thus we only have two Noethercharges, corresponding to the SO (2) angular momentum and to the anomalous dimension: S = Q , , D = Q − , = i ∆ . (3.24)– 12 – igure 4 : Schematic representation of our method of images construction Now we proceed to the numerical solution of the system (3.19). To do so, we introduce thefollowing parametrisation for the bulk particles, which is similar to the one used for theboundary particles (3.18): X a ( τ ) = 1 (cid:113) ˙ w a ( τ ) + ˙ φ a ( τ ) (cid:16) cosh w a ( τ ) , − sinh w a ( τ ) , cos φ a ( τ ) , sin φ a ( τ ) , , (cid:17) . (3.25)where a = 1 , . . . , J . This resolves the X = 1 and ˙ X = 1 constraints. For the ansatz(3.25) the particles are all in the same plane. The boundary particles are constrained tomove on the Wilson rays, so their angular position is fixed φ ( τ ) = 0 , (3.26) φ J +1 ( τ ) = ϕ . (3.27)We can imagine the simplest solution would be when these particles move along straightlines. For that we need to compensate the interaction with neighbours, which could other-wise bend the trajectory, so we require φ k ( τ ) = kJ + 1 ϕ . (3.28)To simplify our ansatz further we can assume that w k ( τ ) = W ( τ ) . Plugging this ansatzinto the EOMs (3.19) we obtain w k ( τ ) = β τ . (3.29)– 13 –inally, constraint (3.16) gives sin (cid:16) ϕ J +2 (cid:17) β J +2 = 1 , (3.30)which has J + 2 different solutions β = e πi n J +2 sin (cid:18) ϕ J + 2 (cid:19) , n = 1 . . . J + 2 . (3.31)To get an interpretation of this, we also compute the anomalous dimension, using (3.24) ∆ = − ( J + 1) iβ ξ . (3.32)We see that the ambiguous factor can be absorbed into ξ . In fact, the initial graph buildingoperator only depends on ξ J +2 , thus this type of ambiguity is expected. In fact this is thesame as in the case of the closed fishchain [31], where the solutions were found to multiplyin a similar way and were responsible for the different asymptotics of a point correlator.We can check our classical solution by comparing with some known results for J = 0 case.From (3.32) for J = 0 we obtain: ∆ = ± g sin ϕ (3.33)which agrees perfectly with the equation (E.6) in [17]. We note that for ˆ g > only theminus sign solution appears in the spectrum whereas the plus sign solution corresponds tolarge and negative ˆ g .More general solutions can be obtained numerically. We generated a couple of solutions,obtained by perturbing the analytic solution we just presented. These can be found infigure 5a and figure 5b. In this section we prove that the dual model is integrable at the classical level by studyingits Poisson structure. We will construct the Lax matrices, corresponding to the particles inthe bulk, and the dynamical reflection matrix will represent the boundary particles. Usingthese building blocks we will construct a family of mutually Poisson-commuting objects.The main purpose of this section is to establish the grounds for quantisation. For thisreason we will only build here a subset of all commuting integrals of motion, as they willanyway appear in the quantum case in full generality.
In this section we discuss the Poisson structure following from the Lagrangian (3.15). Forthe bulk DOFs the Poisson structure is identical to the closed fishchain case already studiedin [32]. One can find the conjugate momenta and define the Poisson bracket in the standardway. In particular, for the bulk particles the momentum conjugate to X i,M is P Mi = ξ ˙ X Mi , (4.1)– 14 – a) J = 2 — Notice that the boundary particles runaway to infinity in a finite amount of time, whilethe particles in the bulk have only moved a finiteamount. (b) J = 3 — Notice that one of the bulk particlesproceeds to make a complete loop. Figure 5 : Plot of the motion of particles obtained by a numerical solution of the classicalequations of motion. In these solutions, motion is restricted to the plane of the Wilson loop.As expected, the boundary particles are confined to fixed rays whereas the bulk particlesare free to move anywhere in the plane.and the Poisson bracket is defined as { X i,M , P Nj } = δ ij δ NM . Due to the constraints thePoisson brackets is ambiguous, and we could define a Dirac bracket. Alternatively, onecan work with gauge invariant quantities. The gauge invariant combination of phase spacecoordinates for the bulk particles are the local symmetry generators q MNi = 1 ξ (cid:0) X Ni P Mi − X Mi P Ni (cid:1) = X Ni ˙ X Mi − X Mi ˙ X Ni , (4.2)which form the SO (1 , algebra under the Poisson bracket: { q MNk , q
KLk } = 1 ξ (cid:0) − η MK q NLk + η NK q MLk + η ML q NKk − η NL q MKk (cid:1) , k = 1 , . . . , J . (4.3)Similarly, one can proceed with the boundary degrees of freedom t and s . The canonicallyconjugate momenta to t ( τ ) and s ( τ ) are Π t = ξ t (cid:48) ( τ ) , Π s = ξ s (cid:48) ( τ ) . (4.4)Even though the boundaries explicitly break down the SO (1 , symmetry, it is still usefulto define q MN and q MNJ +1 in a similar to (4.2) way q NM = 2 ξ (cid:16) Y M Y (cid:48) N − Y N Y (cid:48) M (cid:17) Π t , q NMJ +1 = 2 ξ (cid:16) Y MJ +1 Y (cid:48) NJ +1 − Y NJ +1 Y (cid:48) MJ +1 (cid:17) Π s , (4.5)– 15 –here Y = { cosh t, − sinh t, , , , } , Y J +1 = { cosh s, − sinh s, cos ϕ, sin ϕ, , } . (4.6)Since the Wilson lines explicitly break conformal symmetry, the Poisson bracket of q ismodified to { q MN , q KL } = 1 ξ (cid:0) − ˜ η MK q NL + ˜ η NK q ML + ˜ η ML q NK − ˜ η NL q MK (cid:1) , ˜ η = η ( + C ) , (4.7)where C is the reflection matrix defined in (3.21). Similarly for the right boundary we get { q MNJ +1 , q KLJ +1 } = 1 ξ (cid:0) − ˜ η MKφ q NLJ +1 + ˜ η NKφ q MLJ +1 + ˜ η MLφ q NKJ +1 − ˜ η NLφ q MKJ +1 (cid:1) , ˜ η φ = η ( + G.G.C ) , (4.8)where G is the rotation matrix defined in (3.21).Finally, let us write the Hamiltonian H , corresponding to the lagrangian (3.15) in termsof the local symmetry generators q i . For this we introduce H q ≡ J +2 tr (cid:0) q .q . . . . q J .q J +1 .G.G.C.q J . . . . .q .C (cid:1) − . (4.9)Then we find that H q is proportional to the Hamiltonian H up to a constant multiplier andup to second order in constraints H q = exp (cid:16) ξ H (cid:17) − (cid:39) ξ H + O ( H ) . (4.10)As our constraint implies H = 0 we can equivalently use ξ H q instead. The advantage of H q over H is that it is written explicitly in terms of q i ’s. At it is explained in [32] in thecase of q i ’s there is no difference between the Poisson and Dirac brackets and so they aremore convenient for the quantization.Next, we will build the Lax representation based on the Poisson structure explainedhere and develop the integrability construction. In this section we will build the classical integrals of motion. As at the classical level theequations of motion mostly coincide with the closed fishchain case of [31, 32], we will reviewthe construction from there adapting for our notations.In order to build the Lax representation, it is useful to introduce the local current j MNi = − X [ Mi − X N ] i X i − .X i , satisfying ˙ q MNi = { q MNi , H } = − (cid:0) j MNi +1 − j MNi (cid:1) , i = 0 , . . . , J + 1 . (4.11)This allows us to define the Lax pair of matrices L i and V i : L i = u I x + i q MNi Σ MN , V i = − i u j MNi Σ MN , (4.12)– 16 –here Σ MN are the D σ matrices, giving a D representation of SO (1 , . The explicitform we are using can be found in [32]. One can show [31] from (3.19) that L i and V i satisfy the flatness condition ˙ L i = L i . V i +1 − V i . L i = V i +1 . L i − L i . V i . (4.13)From that it follows immediately that the combination T ( u ) = tr L − J ( u ) . L − ( u ) . L ( u ) . L ( u ) · · · L J ( u ) . L J +1 ( u ) .G .G , (4.14)is conserved in time for any value of u , i.e. { T ( u ) , H } = 0 . In the above expression we havedefined L − i ( u ) = C . L ti ( − u ) .C , (4.15)and also the reflection matrix and the twist matrix in irrep. : C ab = C ab = ab , G a f = e i ϕ e − i ϕ e − i ϕ
00 0 0 e i ϕ af . (4.16)As each coefficient in the polynomial in u , T ( u ) , is an integral of motion we get ≤ J +2 integrals of motion. As we have J + 2 degrees of freedom in our system, some integrals ofmotion are still missing. The remaining ones are hiding in T ( u ) and T ¯4 ( u ) – the transfermatrices in vector and anti-fundamental representations. We will discuss them in detailin the quantum case in the next section. The classical construction for T ( u ) can be alsodeduced from [32]. To prove that the model is classically integrable, we also need to showthat the integrals of motion are in convolution, i.e. that { T ( u ) , T ( v ) } = 0 . This in turnis less trivial and cannot be obtained from the closed fishchain case immediately, as thePoisson structure is modified due to the presence of the boundary particles.In order to prove the convolution property of integrals of motion one can use (4.3) and(4.12) to show that, for ≤ n, m ≤ J : ξ { ( L n ) ab ( u ) , ( L m ) cd ( v ) } = ( L n ) ad ( u )( L n ) cb ( v ) − ( L n ) cb ( u )( L n ) ad ( v ) u − v δ nm (4.17)and analogously for − J ≤ n, m ≤ − . This relation can also be written by defining thedynamical r -matrix r ( u, v ) = P u − v , where P is the × permutation matrix: ξ { L n ( u ) , L m ( v ) } = [ r ( u, v ) , L n ( u ) ⊗ L m ( v )] δ nm . (4.18)For the boundary particles we have a different relation due to the modifications in thePoisson brackets (4.7) and (4.8). Denoting K ( u ) ≡ C. L ( u ) , ¯ K ( u ) ≡ G − . L J +1 ( u ) .G.C , (4.19)we have: ξ { K ab ( u ) , K cd ( v ) } = K ad ( u ) K cb ( v ) − K ad ( v ) K cb ( u ) u − v − K db ( u ) K ca ( v ) − K bd ( v ) K ac ( u ) u + v . (4.20)– 17 –nd the same for ¯ K . In Appendix A we use these identities to show that indeed { T ( u ) , T ( v ) } = 0 . (4.21)In the next section we show how this consideration extend to the quantum case. In order to demonstrate the integrability at the quantum level we will have to embed thegraph building operator into a family of commuting operators. To first approximation, onecan replace the local SO (1 , generators q i by the operators ˆ q i . However, there are somequantum corrections to work out due to non-commutativity of various components of ˆ q MNi ,and this is what we will do in this section.We will define the ˆ L and ˆ K operators as a quantum version of the classical ones. Theywill continue to be × matrices, but now each component will become a differentialoperator. Thus we will treat them as tensors acting on a tensor product of a D vectorspace and a functional space. We will refer to these spaces as auxiliary and physical spacesas usual.Our strategy to fix the quantisation ambiguities is to make sure that ˆ L and ˆ K satisfy theYang-Baxter equation and the Boundary Yang-Baxter equation correspondingly, which aregeneralisations of the classical Poisson brackets (4.17) and (4.20). After that we will buildexplicitly the quantum integrals of motion, demonstrate that the graph building operator(2.9) is one of them and prove that they mutually commute with each other. In order toget the complete set of integrals of motion, we will have to construct the transfer matricesin all antisymmetric representations of sl (4) . We do this via the fusion procedure [34].Next we will use integrability to compute the quantum spectrum. For that we willconstruct the Baxter equation [35] and use it to determine the Q-functions. Then imposinga suitable quantisation condition on the Q-functions we will demonstrate how to obtain thespectrum non-perturbatively. We need to build the quantum analogue of (4.18), which is the Yang-Baxter equation, andof (4.20), which is given by the boundary Yang-Baxter equation.
Quantum Lax matrix.
The quantum version of the Lax matrix is ˆ L ai b ( u ) = u δ ab + i q MNi Σ aMN b , (5.1)where ˆ q MNi is the local generator of SO (1 , , obtained as a quantisation of (4.2), i.e. byreplacing P Kj → ˆ P Kj = − i∂ X j,K : ˆ q MNj = − iξ (cid:18) X Nj ∂∂X j,M − X Mj ∂∂X j,N (cid:19) . (5.2) Note that our conventions differ by sign in comparison with [32]. – 18 –t satisfies the SO (1 , commutation relation: (cid:2) ˆ q MNk , ˆ q KLk (cid:3) = iξ (cid:0) − η MK ˆ q NLk + η NK ˆ q MLk + η ML ˆ q NKk − η NL ˆ q MKk (cid:1) , k = 1 , . . . , J . (5.3)As explained in [32] ˆ q i can be understood as acting on the functions of -dimensionalvariables y i (e.g. CFT wave function) as if it was the corresponding conformal generator in D . In other words one can use the following map between the functions of D coordinates y m and functions of D coordinates X M f ( y , . . . , y m ) → X − + X f (cid:18) X X − + X , . . . , X X − + X (cid:19) (5.4)as q i preserves the interval X M X M we can set it to zero consistently. Note the actionon the D is only well defined for observables build out of q i ’s. In particular ˆ P j and ˆ X j themselves are operators living in AdS [32].The Yang-Baxter (YB) equation can also be obtained by replacing {· , ·} by i [ · , · ] in(4.17). One should, however, pay attention to the order of terms in the r.h.s. of (4.17) as atthe quantum level the order does matter. So the correct generalisation of (4.17), followingfrom (5.1) is: ˆ L be ( u )ˆ L df ( v ) R efac ( v − u ) = R bdfe ( v − u )ˆ L ec ( v )ˆ L fa ( u ) , (5.5)where we introduced the R matrix, a quantum version of the r matrix seen in (4.18), whichacts on two copies of the D auxiliary space and is defined as: R b ca d ( u ) = I ( aux × aux ) + iξ u P = δ ba δ cd + iξ u δ bd δ ca . (5.6)Here P is the permutation operator, acting on vectors in the direct product of two spacesby interchanging them.It will be also convenient to introduce the Lax operator for the reflected particles: ˆ¯ L ai b ( u ) = − ˆ L ai b ( u ) , (5.7)and the corresponding ¯ R -matrix: ¯ R b ca d ( u ) = R b ca d ( − u ) . (5.8) Diagrammatic notation.
In what follows it is extremely convenient to use the followingnotations: we denote the physical space by a double green line, the boundary spaces bythick black lines; the auxiliary space will be denoted by a solid line, and for the reflectedauxiliary space we use a dotted line; the auxiliary space is equipped with a direction anda spectral parameter; then various tensors correspond to intersection vertices following therules depicted in figure 6.For example, the YB equation (5.5) can be easily expressed using this notation in thefollowing way: – 19 – igure 6 : Diagrammatic rulesIn addition we have two other YB equations involving reflected auxiliary space (see figure 7).
Figure 7 : Additional Yang-Baxter equations, which follow from (5.5), but look a slightlydifferently diagrammatically.
In the classical case at the boundary we found that q and q J +1 satisfied the modifiedPoisson brackets (4.7). This in turn results in a different relation (4.20) for the boundaryLax-type operator denoted by K . In order for integrability to persist at the quantum level(4.20) should become the boundary Yang-Baxter equation (BYBE) [29].– 20 –he quantum version of q and q J +1 are again obtained by replacing Π t → − i∂ t and Π s → − i∂ s , and read: ˆ q NM ≡ − i ξ ( Y M ˙ Y N − Y N ˙ Y M ) ∂ t , ˆ q NMJ +1 ≡ − i ξ ( Y MJ +1 ˙ Y NJ +1 − Y NJ +1 ˙ Y MJ +1 ) ∂ s , (5.9)where Y (cid:48) s are explicit functions of s and t , parameterising the Wilson rays defined in (4.6).Following the classical case, we also introduce: ˆ L a b ( u ) = u δ ab + i q MN Σ aMN b , ˆ L aJ +1 b ( u ) = u δ ab + i q MNJ +1 Σ aMN b . (5.10)Next we need to identify the quantisation of K (4.19), such that (4.20) becomes the BYBE,which for the left boundary is: ˆ K d c ( v ) ¯ R c c a d ( u + v ) ˆ K c a ( u ) R d d b b ( v − u ) = R c c a a ( v − u ) ˆ K b d ( u ) ¯ R d d c b ( u + v ) ˆ K d c ( v ) . (5.11)Diagrammatically this equation becomes =We find that at the quantum level there is a quantum correction to the spectral parameter,invisible in the classical ξ → ∞ limit. Namely the equation (5.11) is solved by ˆ K ( u ) = C. ˆ L ( u − i ξ ) , (5.12)where C is the same reflection matrix as the classical case (4.16). Similarly for the rightboundary we have the following BYBE: R b b c c ( u − v ) ˆ¯ K d c ( v ) ¯ R a c d d ( − u − v ) ˆ¯ K a d ( u ) = ˆ¯ K c b ( u ) ¯ R d b c c ( − u − v ) ˆ¯ K d c ( v ) R a a d d ( u − v ) , (5.13)which can be expressed diagrammatically as follows– 21 –One can easily verify that the solution to this equation has the following form: ˆ¯ K ( u ) = G − . L J +1 ( u + i ξ ) .G.C , (5.14)where G is the twist matrix defined in (4.16). This expression is again identical to theclassical expression up to a quantum correction in the spectral parameter.In the rest of this section we will use the building blocks K , L and R to build a completesystem of conserved, mutually commuting operators and also show that the inverse graph-building operator is part of this family. R -matrix The R -matrix itself is defined up to an arbitrary scalar factor, which does not affect any ofthe previous relations. However, in the next sections we will be using the fusion procedurefor the boundary reflection matrix which is sensitive to the normalisation.In order to fix the normalisation one can think about the R -matrix as an S-matrix andimpose unitarity. Therefore we denote the normalised R-matrix by S : S ( u ) = a ( u ) R ( u ) , ¯ S ( u ) = a ( − u ) ¯ R ( u ) , (5.15)where a ( u ) is the normalisation factor which we fix by the unitarity condition S ( u ) S ( − u ) = I , (5.16)which diagrammatically an be expressed as follows:and takes the following form when written explicitly with all indices: S a db e ( u ) S b ec f ( − u ) = ¯ S a db e ( − u ) ¯ S b ec f ( u ) = δ ac δ df . (5.17)– 22 –o satisfy this, we must have that: A ( u ) ≡ a ( u ) a ( − u ) = u ξ − u ξ . (5.18)Below we will only need the combination of the scalar factors A ( u ) , so we do not need todecode individual a ( u ) by imposing additional analyticity conditions. In this section we will build a family of mutually commuting operators out of the buildingblocks discussed above. First we will build the transfer matrix in fundamental representa-tion, following the discussion in the classical case. As we know already, it cannot encode allthe integrals of motion. In order to complete our system of IMs we will also have to buildthe transfer matrix in vector and anti-fundamental representations. For that we will followthe fusion procedure.
Figure 8 : Transfer matrix in fundamental representation for J = 3 . For the transfer matrix in fundamental representation we can mainly mimic the classicaltransfer matrix (4.14). From that we can deduce the diagrammatic representation as infigure 8 and follow the rules outlined in section 5.1 to deduce the quantum counterpart.We will use the index to indicate the fundamental representation, and define the transfermatrix as: ˆ T ( u ) = tr (cid:2) ¯ L J ( − u ) · · · ¯ L ( − u )¯ L ( − u ) K ( u ) L ( u ) . . . L J − ( u ) L J ( u ) G ¯ K ( u ) G t (cid:3) . (5.19)We show now that the transfer matrices form a family of mutually commuting operators: [ˆ T ( u ) , ˆ T ( v )] = 0 . (5.20)This is particularly easy to see using the diagrammatic representation as we do in figure 9for the particular case J = 0 for simplicity. The step 1 represents ˆ T ( u )ˆ T ( v ) . In step 2,we use unitarity of S -matrix (5.16) to insert 4 scattering matrices. In step 3 we use BYBE(5.11) and (5.13). In step 4 we cancel the S -matrices using unitarity again, obtaining ˆ T ( v )ˆ T ( u ) . – 23 –n order to be able to conclude that the quantum system is integrable we need todemonstrate that the Hamiltonian is part of the system of commuting operators. Forthat end in the next section we will build the transfer matrix in vector representation anddemonstrate that it does contain the Hamiltonian. (a) Step 1 — We start off with T ( u ) T ( v ) ,which acts as a differential operator onthe quantum space. (b) Step 2 — By introducing the identityas a product of S -matrices from (5.16),we can pass the particle line of T ( u ) through T ( v ) .(c) Step 3 — Now we apply the bound-ary Yang-Baxter equations (5.11) and(5.13). (d) Step 4 — Finally we resolve theidentity again using (5.16), to obtain T ( v ) T ( u ) hence proving that indeed thetransfer matrices commute with eachother for arbitrary values of the spec-tral parameter. Figure 9 : Diagrammatic proof of [ˆ T ( u ) , ˆ T ( v )] = 0 In order to build T – the transfer matrix in vector representation – we will need thecorresponding building blocks, which are K and L in vector representation. The simplestway to obtain those is by applying the fusion procedure [34]. Roughly speaking, we willneed two copies of L (or K ) in the fundamental representation with spectral parameters u ± i ξ combined together in an antisymmetrised way. This procedure was already appliedto L in [32] where L , L ¯ and L ¯ – i.e. L in all anti-symmetric representations – werecomputed.The L needed in this section has auxiliary space being the -dimensional Minkowskispace with metric η MN . The Lax operator is now a quadratic polynomial with coefficients– 24 – a) ˆ K (b) ˆ¯ K Figure 10 : Boundary reflection matrices in vector representation via fusion procedure.built out of local charge operators ˆ q i as follows: ˆ L MNi ( u ) = (cid:18) u −
18 tr ˆ q i (cid:19) η MN − u ˆ q MNi + (cid:18)
12 ˆ q MNi − iξ ˆ q MNi + 14 ξ η MN (cid:19) . (5.21)We also define the reflected operators ˆ¯ L i by ˆ¯ L i ( u ) ≡ ˆ L i ( − u ) . (5.22)Having ¯ L defined allows us to maintain the same definition for T as the other T -operatorslike (5.19). Boundary reflection operator.
What remains to be done is fusing the reflection op-erators. In order to keep them covariant and ensure the structure of the BYBE, we haveto insert an additional R -matrix between the tensor product of two reflection matrices, asshown in figure 10. In terms of Y = Y , defined in (4.6), we get: ˆ K MN ( u ) = C MN u (cid:18) u − iξ (cid:19) + u iξ ( Y N ˙ Y M − Y M ˙ Y N ) ∂ t (5.23) + 2 ξ Y N ˆ ∂ t Y M ˆ ∂ t − iξ u Y M ˆ ∂ t Y N ˆ ∂ t . As we can see, it is a second order differential operator in t and a second order polynomialin the spectral parameter u . Similarly for the right boundary we get (replacing t by s ): ˆ¯ K MN ( u ) = C MN u (cid:18) u + iξ (cid:19) + u iξ ( Y N ˙ Y M − Y M ˙ Y N ) ∂ s (5.24) + 2 ξ Y M ˆ ∂ s Y N ˆ ∂ s + 2 iξ u Y N ˆ ∂ s Y M ˆ ∂ s . Due to the sign difference in ˆ L i ( u ) in comparison with [32], the linear term in u also has a differentsign. – 25 –n the equations above we are using the reflection matrix C in vector representation is thematrix C we introduced in (3.21). An important property, which follows directly from thedefinitions (5.23) and (5.24), is that K (+ i/ξ ) = 0 and ¯ K ( − i/ξ ) = 0 .Using ˆ q (5.9) we can write ˆ K ( u ) = C u (cid:18) u − iξ (cid:19) − u ˆ q + 12 ˆ q − i ξu (ˆ q ) T . (5.25)For the right boundary we have very similar expression ˆ¯ K ( u ) = ( G ) − . (cid:18) u (cid:18) u + iξ (cid:19) − u ˆ q J +1 + 12 (ˆ q J +1 ) T + i ξu ˆ q J +1 (cid:19) .G .C . (5.26)where G in vector representation is the twist matrix G from (3.21). The twist matrix G appears in the expression (5.26) for the right boundary reflection operator, as it is definedin a way that does not depend on ϕ .Having all the needed ingredients we can compute T by replacing in the r.h.s. of (5.19)all the operators and matrices by their vector representation counterpart. In addition, oneshould multiply the result by A (2 u ) in order to account for the correct normalisation ofthe extra R (2 u ) and ¯ R (2 u ) appearing in the fusion procedure of the boundary reflectionoperators (see figure 10).We will present some explicit examples of the transfer matrices later in section 5.4.6and Appendix C. In the next sections we will first prove that the transfer matrix in thevector representation contains the Hamiltonian, and then proceed with the antifundamentalrepresentation. In this section we will show that the Hamiltonian of the system is a part of the commutingfamily of operators. For that consider: T (0) = 4 lim u → u ξ tr (cid:2) ¯ L J (0) · · · ¯ L (0)¯ L (0) K ( u ) L ( u ) . . . L J − (0) L J (0) G ¯ K ( u ) G t (cid:3) . (5.27)First we can use that: ˆ L MNi (0) = ˆ q MNi − iξ ˆ q MNi − η MN q i + η MN ξ = : ˆ q i : MN ξ X Mi X Ni ∂ X Ki . (5.28)where in the last equality we used the identity from [32]. Also from (5.23) and (5.24) wehave u ξ ˆ K MN ( u ) (cid:12)(cid:12)(cid:12) u =0 = − i q ) NM = − iξ Y M ˆ ∂ t Y N ˆ ∂ t , (5.29) uξ (cid:16) G ˆ¯ K ( u )( G ) − (cid:17) MN (cid:12)(cid:12)(cid:12)(cid:12) u =0 = + i q J +1 ) MN = + 2 iξ Y NJ +1 ˆ ∂ s Y MJ +1 ˆ ∂ s . (5.30)Combining all parts together, up to sub-leading terms in /ξ we get the quantum versionof H q + 1 , where H q is defined in (4.9). In order to check that this produces the correct– 26 –uantisation of H q , i.e. the one related to the graph building operator, we have to analysethe expression (5.28) more carefully. Paying attention to the order of the operators we get T (0) = 4 42 J ξ J +4 η NM X MJ X J .X J − . . . X .Y ∂ t J (cid:89) i =1 (cid:3) (6) i (5.31) × Y .X X .X . . . X J .Y J +1 ∂ s Y NJ +1 ∂ s ∂ t J (cid:89) i =1 (cid:3) (6) i , where all derivatives are understood as operators acting on the CFT wavefunction embeddedin the lightcone of D Minkowski spacetime. In order to relate the above expression with thegraph building operator (2.9), which is expressed in terms of derivatives acting on functionsin D Euclidean spacetime, we recall that T is built out of q i ’s and as such we can actwith it, in a consistent way, on functions of D coordinates, following the prescription (5.4).Furthermore, one can just replace the D d”Alembertian operator in D d”Alembertian dueto the identity (cid:3) (6) = (cid:3) (4) + ∂ X + ∂ X − , (5.32)and the fact that there is no dependence on X − in the D functions, by construction (5.4).Therefore, Y .X X .X . . . X J .Y J +1 ∂ s ∂ t J (cid:89) i =1 (cid:3) (6) i = (5.33) (cid:0) − (cid:1) J +1 ∂ s ∂ t | y (cid:48) || y (cid:48) J +1 | J (cid:89) i =0 ( y i − y i +1 ) J (cid:89) i =1 (cid:3) (4) i = (cid:0) (cid:1) J +1 (4ˆ g ) J +1 ˆ B − , (5.34)where we used that X .X = − ( x − x ) and Y .X = − e − t ( x − x ) . We use theexpression for the inverse of the graph-building operator ˆ B − from (2.9). Then for T (0) one gets precisely T (0) = 4 ˆ B − . (5.35)Where we used (3.10) to relate ξ and ˆ g . We see that all factors cancel exactly, implyingthat at the quantum level we also have T (0) ψ = 4 ψ as it follows from (2.9). At the sametime we see that the quantum graph building operator ˆ B is indeed a part of the commutingfamily of operators, which demonstrates the integrability of the initial system of Feynmangraphs.Now in order to find the spectrum ∆( ξ ) we will have to build two remaining transfermatrices in the two sections below. Here we compute the ¯ transfer matrix, corresponding to the antisymmetrisation of thetensor product of three copies of irrep. ingredients with the corresponding shifts in thespectral parameters, dictated by the fusion procedure. The calculation for L was done in– 27 –32]. The result for L ¯ can be re-expressed in terms of one L times a scalar polynomialfactor ˆ L ¯ bk a ( u ) = (cid:18) u − tr ˆ q k ξ (cid:19) ˆ¯ L bk a ( − u ) , (5.36)and ˆ¯ L ¯ ak b ( u ) = ˆ L ¯ ak b ( u ) . The fusion of the boundary reflection operator is done in analogywith representation. For that one follows the diagram in figure 11 to obtain: ˆ K ¯ ab ( u ) = − (cid:18) u − iuξ + 34 ξ (cid:19) ˆ¯ K ba ( − u ) , ˆ¯ K ¯ ab ( u ) = − (cid:18) u + iuξ + 34 ξ (cid:19) ¯ K ba ( − u ) . (5.37)Finally, the twist matrix is the inverse of the one for the irrep. (4.16). The polynomialfactors in (5.36) and (5.37) will play an important role below, when we derive the TQ-relations. (a) ˆ K ¯4 (b) ˆ¯ K ¯4 Figure 11 : Fusion of the boundary reflection operators to anti-fundamental irrep. ¯4 Here we compute the ingredients of the transfer matrix in the representation ¯1 , also knownas the quantum determinant. Like in the previous sections, this can be computed as anantisymmetrisation of the tensor product of four copies of L and K in the irrep. For both L ¯1 and K ¯1 we find that they are just fourth order polynomials in u acting trivially on thephysical space. Again the calculation of L ¯1 was already performed in [32] and the resultreads: ˆ L ¯ i ( u ) = (cid:18) u − tr ˆ q i ξ (cid:19) + tr ˆ q i ξ − ξ . (5.38)– 28 –he expression for ˆ¯ L ¯ i is the same. For the boundary reflection operator we follow thediagram on figure 12 to obtain: K ¯ ( u ) = (cid:18) u − iξ (cid:19) (cid:18) u − iξ (cid:19) u (cid:18) u + iξ (cid:19) , ¯ K ¯ ( u ) = (cid:18) u + 2 iξ (cid:19) (cid:18) u + iξ (cid:19) u (cid:18) u − iξ (cid:19) . (5.39) (a) K ¯1 (b) ¯ K ¯1 Figure 12 : Fusion of the boundary reflection operators to obtain the quantum determinant.In the next section we will discuss the implications of these polynomial factors for theanalytical properties of the transfer matrices and then derive the TQ-relations. J = 0 example Before discussing the general case we first give the explicit result for the simplest case ofa chain of zero length. This means that we are only left with the boundary reflectionoperators. Furthermore, the graph-building operator is a second order differential operatorin s and t , as it should commute with the dilatation operator only one variable remains.If we further impose T (0) = 4 , we will automatically diagonalise all transfer matrices– 29 –btaining the following results for their eigenvalues: T ( v ) = 1 ξ (cid:0) v cos ϕ + cos ϕ + 8 ξ (cid:1) , T ( v ) = A (2 v ) v + 1 ξ v (cid:0) v (2 cos(2 ϕ ) + 4) + v (cid:0) ξ cos ϕ − sin ϕ (cid:1) + 16 ξ (cid:1) , T ¯4 ( v ) = A (2 v ) A (2 v + i ) A (2 v − i ) (cid:0) v + 1 (cid:1) (cid:0) v + 9 (cid:1) ξ (cid:0) v cos ϕ + cos ϕ + 8 ξ (cid:1) , T ¯ ( v ) = A (2 v ) A (2 v + i ) A (2 v − i ) A (2 v + 2 i ) A (2 v − i ) v (cid:0) v + 1 (cid:1) (cid:0) v + 4 (cid:1) ξ , (5.40)where we introduced the rescaled spectral parameter v = u ξ . The factors A , where A ( v ) = v v , are due to the R-matrix normalisation as discussed in section 5.3. We also workedout the form of the transfer-matrix eigenvalues for J = 1 case in Appendix C in terms of afew unknown constants. We have explicitly verified that all the T -operators for J = 0 and J = 1 commute between themselves and with the charges ∆ , S, H as expected.In the next section we will extend these results to the general J case. Here we deduce the general form of the eigenvalues of the transfer matrices. First, one cannotice explicitly that for J = 0 and J = 1 case they are even functions of the spectralparameter. In Appendix B we prove that this is true for any J . Some other properties ofthe transfer matrices are:• T is a polynomial of degree J + 2 in v , as it follows from its definition (5.19).• T is a rational function with two poles at v = ± i/ , coming from the normalisationfactor A (2 v ) . Another potential pole at v = 0 , coming from the boundary reflectionoperators, is cancelled by the same A (2 v ) . At large v it behaves as ∼ v J +4 .• Previously we noticed that K ( u = + i/ξ ) = 0 and ¯ K ( u = − i/ξ ) = 0 , therefore wecan see that T should have a prefactor of v + 1 = ξ u + 1 .• Finally, in section 5.4.3 we have showed that T (0) = 4 due to (5.35).• The properties of T ¯ are very similar to those of T , apart from the trivial factors of A ’s and additional trivial factors coming from L ¯4 and K ¯4 .• Finally, T ¯ (the quantum determinant) contains only trivial factors and can be com-puted explicitly for any J . – 30 –asing on these observations we can write the transfer matrices in terms of the polynomials P λk ( v ) as: T ( v ) = 1 , T ( v ) ≡ P J +1 ( v ) ξ J +2 , T ( v ) ≡ A (2 v ) v + 1 v P J +2 ( v ) ξ J +4 , T ¯ ( v ) = A (2 v ) A (2 v + i ) A (2 v − i ) ( v + )( v + ) J +1 P ¯4 J +1 ( v ) ξ J +6 , T ¯ ( v ) = A (2 v ) A (2 v + i ) A (2 v − i ) A (2 v + 2 i ) A (2 v − i ) ( v + 4)( v + 1) J +2 v J +2 ξ J +8 . (5.41)Here, P λk is a polynomial of degree k , labelled by the representation λ in the auxiliary space.The eigenvalues of the conserved charges of the system are the coefficients of the powers of v in these polynomials. We will denote them as (defining w ≡ v ): P J +1 ( w ) = J +1 (cid:88) i =0 a i w − i + J +1 ,P J +2 ( w ) = J +2 (cid:88) i =0 b i w − i +2 J +2 ,P ¯4 J +1 ( w ) = J +1 (cid:88) i =0 c i w − i + J +1 . (5.42)In our definition, a will represent the coefficient of the highest power in v in P J +1 , a thesecond highest etc. The leading coefficients are easy to compute explicitly directly from thedefinition a = c = 4 cos ϕ , b = 2 cos 2 ϕ + 4 . (5.43)They just give the twisted (or q-)dimension of the corresponding representations. Sincethe leading coefficients are trivial, in total we get J + 4 non-trivial coefficients in thepolynomials P . As our system has J + 2 degrees of freedom it may suggest that thereare more relations between the coefficients of the polynomials P . Indeed, in the J = 0 and J = 1 cases we found them by computing the differential operators explicitly, but it israther hard to deduce the general relations. In J = 1 case we found exactly independentoperators guarantees integrability of the system.We found that the global charges ∆ and S are encoded into the sub-leading coefficientsin the following way: c − a = 8 i S ∆ sin ϕ , (5.44) ( a + c ) cos ϕ − b = 2 (cid:0) S + 2 ∆ + J (cid:1) sin ϕ + 2 cos ϕ . – 31 –hese relations are also quite hard to derive in general, but we explicitly verified the firstrelation up to J = 3 and the second up to J = 2 .Finally, the condition T (0) = 4 implies: b J +2 = ξ J +4 = 16ˆ g J +4 . (5.45)In order to find the eigenvalues of all coefficients of the transfer matrices we will haveto develop a numerical procedure. For that we will first build the TQ-relations in the nextsection. In this section we follow the derivation of [13] to deduce the general simplified form ofthe TQ-relations and deduce asymptotic of the Q-functions. The starting point is theTQ-relation [35–37]: Q ( v + 2 i ) + T ( v + i/ Q ( v + i ) + T ( v ) Q ( v ) + T ¯ ( v − i/ Q ( v − i ) + T ¯ ( v − i ) Q ( v − i ) = 0 . (6.1)As we discussed above the transfer matrices have a number of trivial factors. In order toremove these fixed, non-dynamical factors, we perform the following gauge transformationof the Q-function Q ( v ) = q ( v ) e π ( J +1) v Γ( − iv ) ξ i ( J +1) v Γ( iv + 1) − J − Γ (cid:0) − iv − (cid:1) Γ( iv + 2) , (6.2)which brings (6.1) to a simpler and more symmetric form: P J +2 ( v ) v J +3 q ( v ) = − ( v + i ) J +1 q ( v + 2 i ) − v + i v ( v + i ) P J +1 (cid:0) ( v + i ) (cid:1) q ( v + i ) (6.3) − ( v − i ) J +1 q ( v − i ) − v − i v ( v − i ) P ¯4 J +1 (cid:0) ( v − i ) (cid:1) q ( v − i ) . As a test of this equation we can compare with the case J = 0 , studied as a ladder limit ofQSC in N = 4 SYM. For J = 0 , by plugging in the explicit form of the polynomials (5.40)into (6.3), we obtain: q ( v ) (cid:32) (cid:0) g v cos( ϕ ) + 8ˆ g + v (cos(2 ϕ ) + 2) (cid:1) v − sin ( ϕ ) v (cid:33) + 2(2 v − i ) q ( v − i ) (cid:0) g + v ( v − i ) cos( ϕ ) (cid:1) v ( v − i ) + 2(2 v + i ) q ( v + i ) (cid:0) g + v ( v + i ) cos( ϕ ) (cid:1) v ( v + i )+ ( v − i ) q ( v − i ) + ( v + i ) q ( v + 2 i ) = 0 . (6.4)This is the same as what was found in [17, 26] for a cusped Wilson line in the ladders limit,as expected (see detailed comparison in Appendix C).The equation (6.3) for general J is one of our main result. As we show in section 7 itlets us evaluate numerically the spectrum.– 32 – .1 Large v asymptotic of Q-functions For the numerical evaluation, which we describe in the next section, it is important to havethe large v asymptotics under control. As the leading and partially subleading coefficientsin the polynomials P are known from (5.43) and (5.44), we can deduce that the linearlyindependent solutions of the equation (6.3) should have the following large v asymptoticexpansion: q = e + φv v +∆+ S − J (cid:16) c , v + . . . (cid:17) ,q = e − φv v +∆ − S − J (cid:16) c , v + . . . (cid:17) ,q = e + φv v − ∆ − S − J (cid:16) c , v + . . . (cid:17) ,q = e − φv v − ∆+ S − J (cid:16) c , v + . . . (cid:17) . (6.5)where φ = π − ϕ . The above asymptotics suggest the following relation to the QSC Q-functions of [22]: q i ( v ) ∼ Q i ( v ) v J +1 / , (6.6)which is similar to the relations found in the fishnet model [10]. Subleading coefficients in /v can be found systematically in terms of the coefficients of the polynomials P , i.e. a i , b i and c i , by plugging the expansion (6.5) into (6.3). In order to fix the coefficients of thepolynomials P one has to use the gluing (or quantisation) condition, which we describe inthe next section. After having established the key properties of the Baxter equation we can solve themnumerically and fix the remaining coefficients a i , b i and c i . The method we implement isessentially the one of [25] which was adopted and simplified to the current type of problemsin [10, 13, 14, 26]. The 4th order finite difference equation (6.3) has linearly independentsolutions with the asymptotic (6.5). The way to find them numerically is first finding theasymptotic solution at large v , where (6.3) reduces to a linear problem for the asymptoticexpansion coefficients. The truncated asymptotic series gives a very good approximationat sufficiently large | Im v | . In order to bring Im v to a finite value, we can simply use (6.3)itself, as it allows to find q ( v ) in terms of q ( v + in ) , n = 1 , . . . , (or q ( v − in ) , n = 1 , . . . , ).Using (6.3) as a recursion relation, we can gradually decrease | Im v | . By doing this thereare two options: starting from + i ∞ or from − i ∞ . Correspondingly, we will find analyticsolutions in the upper-half plane, q ↓ i , and other analytic in the lower-half plane, q ↑ i .Since the Baxter equation is a fourth order equation, we can have only four independentsolutions, meaning that the q ↑ i and q ↓ i should be related by a linear transformation. Weshould therefore have: q ↑ i ( v ) = Ω ji ( v ) q ↓ j ( v ) , Ω ji ( v + i ) = Ω ji ( v ) , (7.1)– 33 –here Ω ji ( v ) = (cid:15) j j j j
3! det n =0 ,..., (cid:110) q ↑ i ( v − in ) , q ↓ j ( v − in ) , q ↓ j ( v − in ) , q ↓ j ( v − in ) (cid:111) det n =0 ,..., (cid:110) q ↓ ( v − in ) , q ↓ ( v − in ) , q ↓ ( v − in ) , q ↓ ( v − in ) (cid:111) . (7.2) Ω ji ( v ) is an i -periodic function which can have poles at v = in of order no higher than q i ( v ) ’s themselves. From the Baxter equation (6.3) it is easy to see that q i ( v ) only haspoles at v = in of maximal order J + 2 , which implies that Ω ji ( v ) is a trigonometricrational function of the form: Ω ji ( v ) = J +2 (cid:80) n =0 C ( n ) ji e π n u (1 − e π u ) J +2 , (7.3)The quantisation condition can be obtained by comparing with the QSC description of thecusped Wilson line [22], where one defines an antisymmetric matrix ω ik , related to Ω ji inthe following way: ω ik = Ω ji Γ jk , (7.4)where the so-called gluing matrix is: Γ jk = γ sinh(2 πv ) 0 γ γ sinh(2 πv ) 0 γ γ γ , (7.5)where γ i are some constants. All we need to know, from QSC, is that ω in (7.4) is anti-symmetric i.e. Ω ji Γ jk = − Ω jk Γ ji , which, in particular, implies: Ω = Ω = 0 . (7.6)As each component of Ω( u ) is a nontrivial function parametrised in terms of J +3 constants C ( n ) ij , imposing (7.6) is usually sufficient to fix J + 4 unknown constants, contained in theBaxter equations. Tests.
By applying the numerical method we studied the spectrum for J = 1 and J = 2 cases. For J = 1 we also found a large number of excited states (see figure 13), correspondingto additional insertions of Φ and Φ fields at the cusp, as discussed in [17]. We tested ourresults against the weak coupling result of [21], which in our notations reads ∆ = J + ˆ g J +2 ( − J J +3 π J +1 csc( ϕ ) B J +1 (cid:0) ϕ π (cid:1) Γ(2 J + 2) + O (ˆ g J +4 ) , (7.7)which agreed with high precision (of more than digits) with our numerical data for J = 0 , J = 1 and J = 2 . For example for J = 2 and ϕ = 2 π/ we get the following fit forthe numerical data on figure 14: ∆ = 2 − . g + 23271 . g + . . . (7.8)– 34 – ull Figure 13 : Numerical spectrum with excited states for J = 1 and S = 0 . The lowest curve(starting at ∆ = 1 at zero coupling) corresponds to the case with a single insertion of Z atthe cusp. The curves which begin at higher integers at zero coupling correspond to excitedstates of the solution of the Baxter equation, which correspond to additional insertions of Φ and Φ at the cusp (see [17] for some explicit examples). Whereas for the ground statethe dimension ∆ is real, excited states could appear in complex conjugate pairs.in agreement with (7.7), which for J = 2 gives g ϕ (cid:0) ϕ − πϕ + 20 π ϕ − π (cid:1) csc ϕ = 2 − ˆ g π √ . (7.9)The states Z J for the cases J = 1 and J = 2 do not behave classically at large ξ , i.e. ∆ decreases faster than linear. Like in [10] we expect the classical regime to describe thehighly excited states. – 35 – .2 0.4 0.6 0.8 1.0 g - - - Δ J = φ = π / Figure 14 : Numerical data (dots) for the ground state of length-2 chain with ϕ = 2 π/ .Red solid line shows the Lüscher formula prediction of [21]. In this paper we show that the cusped Wilson-Maldacena loop with insertions of J orthog-onal scalars is an integrable system, and also describe its strong coupling dual descriptionin terms of a classical chain of particles with nearest neighbour interactions, sitting on thelightcone in -dimensional Minkowski spacetime — the open fishchain. We compute thetransfer matrices of this model, which gives us all the conserved charges of the system.Moreover, we obtain a Baxter equation which can be solved numerically for any J and findthe spectrum of dimensions ∆ non-perturbatively (for J = 0 , , ). This lets us find the Q -functions of the system, which are a crucial quantity in a quantum integrable systems.There is a number of future directions of our work. We outline some of them below.Since we have the spectrum under control, the natural next step is to compute correla-tion functions. In [17, 38], the first steps towards this is taken, where the authors calculatethe three point functions of three cusped Wilson lines in the ladders limit. Remarkably,they observe that the structure constants can be expressed as overlaps between the same q -functions that appear in the QSC that describes the system. This is one of many results[14, 23, 24, 39–47] in the rapidly developing separation of variables (SoV) program. Anotherpossibility is to first probe the SoV structure at strong coupling, using the dual fishchaindescription which becomes classical in this limit.In general it would be interesting to get away from the ladders/fishnet limit both inour set-up and in deformed N = 4 SYM. Such an exploration could give some clues as tohow to develop a first principles holographic derivation for the full theory.It would be also interesting to establish links with the bootstrap program, in particularin the straight line limit, where the Wilson line defines a defect CFT. Our Baxter equationallows to more easily generate the spectrum here than in the case of the full N = 4 SYM,– 36 –hich could give a better understanding on how the conformal bootstrap works in this case.
Acknowledgements
We are especially grateful to A. Cavaglia for numerous discussions and for carefully readingthe manuscript prior to its publication. N.G. is also grateful to F. Levkovich-Maslyuk,A. Sever, E. Sobko and A. Tumanov for discussions on various stages of this project. Thework of N.G. was supported by European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme (grant agreement No. 865075)EXACTC.
A Proof of Poisson-commutativity of T ( u ) In this appendix, we prove that the classical transfer matrix (4.14) forms a family of mu-tually Poisson-commuting functions for any value of the spectral parameter and for any J .We start by the J = 0 case. We have that: { T ( u ) , T ( v ) } = { K ab ( u ) , K αβ ( v ) } ¯ K ba ( u ) ¯ K βα ( v ) + K ab ( u ) K αβ ( v ) { ¯ K ba ( u ) , ¯ K βα ( v ) } . (A.1)Using (4.20) and its analog for ¯ K : { T ( u ) , T ( v ) } = − ¯ K ba ( u ) ¯ K βα ( v ) ξ ( u + v ) [ K βb ( u ) K αa ( v ) − K bβ ( v ) K aα ( u )] ++ ¯ K ba ( u ) ¯ K βα ( v ) ξ ( u − v ) [ K aβ ( u ) K αb ( v ) − K aβ ( v ) K αb ( u )] −− K ab ( u ) K αβ ( v ) ξ ( u + v ) (cid:104) ¯ K αa ( u ) ¯ K βb ( v ) − ¯ K aα ( v ) ¯ K bβ ( u ) (cid:105) ++ K ab ( u ) K αβ ( v ) ξ ( u − v ) (cid:104) ¯ K bα ( u ) ¯ K βa ( v ) − ¯ K bα ( v ) ¯ K βa ( u ) (cid:105) . (A.2)By relabelling indices appropriately, all terms cancel as expected.For the rest of this section, we will use the shorthand notation: ( L ) ab ( u ) ≡ ( L ( u ) . L ( u ) . . . L J ( u )) ab , (¯ L ) ba ( u ) ≡ ( L − J ( u ) . L − ( J − ( u ) . . . L − ( u )) ba . (A.3)It is easy to see that these matrices follow the same Poisson brackets as individual L -– 37 –atrices, i.e. (4.17). Thus, for general J we have that: { T ( u ) , T ( v ) } = { K ab ( u ) , K αβ ( v ) } (¯ L ) ac ( u )(¯ L ) αγ ( v ) ¯ K dc ( u ) ¯ K δγ ( v )( L ) bd ( u )( L ) βδ ( v )+ K ab ( u ) K αβ ( v )(¯ L ) ac ( u )(¯ L ) αγ ( v ) { ¯ K dc ( u ) , ¯ K δγ ( v ) } ( L ) bd ( u )( L ) βδ ( v )++ ( Poisson brackets between L ) == (¯ L ) ac ( u )(¯ L ) αγ ( v )( L ) bd ( u )( L ) βδ ( v ) (cid:40) − ξ ( u + v ) (cid:20) K βb ( u ) K αa ( v ) ¯ K dc ( u ) ¯ K δγ ( v ) − K bβ ( v ) K aα ( u ) ¯ K dc ( u ) ¯ K δγ ( v )++ K ab ( u ) K αβ ( v ) ¯ K γc ( u ) ¯ K δd ( v ) − K ab ( u ) K αβ ( v ) ¯ K cγ ( v ) ¯ K dδ ( u ) (cid:21) ++ 1 ξ ( u − v ) (cid:20) K aβ ( u ) K αb ( v ) ¯ K dc ( u ) ¯ K δγ ( v ) − K aβ ( v ) K αb ( u ) ¯ K dc ( u ) ¯ K δγ ( v )++ K ab ( u ) K αβ ( v ) ¯ K dγ ( u ) ¯ K δc ( v ) − K ab ( u ) K αβ ( v ) ¯ K dγ ( v ) ¯ K δc ( u ) (cid:21)(cid:41) ++ ( Poisson brackets between L ) . (A.4)We now relabel indices in order to collect the boundary reflection matrices as: { T ( u ) , T ( v ) } = K ab ( u ) K αβ ( v ) ¯ K dc ( u ) ¯ K δγ ( v ) (cid:40) − ξ ( u + v ) (cid:20) (¯ L ) βc ( u )(¯ L ) αγ ( v )( L ) bd ( u )( L ) aδ ( v ) −− (¯ L ) aδ ( u )(¯ L ) αγ ( v )( L ) bd ( u )( L ) βc ( v )++ (¯ L ) ac ( u )(¯ L ) αd ( v )( L ) bγ ( u )( L ) βδ ( v ) −− (¯ L ) ac ( u )(¯ L ) bγ ( v )( L ) αd ( u )( L ) βδ ( v ) (cid:21) ++ 1 ξ ( u − v ) (cid:20) (¯ L ) ac ( u )(¯ L ) αγ ( v )( L ) βd ( u )( L ) bδ ( v ) −− (¯ L ) αc ( u )(¯ L ) aγ ( v )( L ) bd ( u )( L ) βδ ( v )++ (¯ L ) aγ ( u )(¯ L ) αc ( v )( L ) bd ( u )( L ) βδ ( v ) −− (¯ L ) ac ( u )(¯ L ) αγ ( v )( L ) bδ ( u )( L ) βd ( v ) (cid:21)(cid:41) ++ ( Poisson brackets between L ) . (A.5)– 38 –he Poisson Brackets between L -matrices give: { T ( u ) , T ( v ) } = (Poisson brackets between K )++ K ab ( u ) K αβ ( v ) ¯ K dc ( u ) ¯ K δγ ( v ) (cid:34) { (¯ L ) ac ( u ) , (¯ L ) αγ ( v ) } ( L ) bd ( u )( L ) βδ ( v )++ (¯ L ) αγ ( v ) { (¯ L ) ac ( u ) , ( L ) βδ ( v ) } ( L ) bd ( u )++ (¯ L ) ac ( u )(¯ L ) αγ ( v ) { ( L ) bd ( u ) , ( L ) βδ ( v ) } ++ (¯ L ) ac ( u ) { ( L ) bd ( u ) , (¯ L ) αγ ( v ) } ( L ) βδ ( v ) (cid:35) . (A.6)Using the Poisson brackets (4.17) and their analogues: ξ { ( L − n ) ba ( u ) , ( L − m ) dc ( v ) } = ( L − n ) da ( u )( L − n ) bc ( v ) − ( L − n ) bc ( u )( L − n ) da ( v ) u − v δ nm , (A.7) ξ { ( L − n ) ba ( u ) , ( L m ) c d ( v ) } = ( L − n ) ca ( u )( L n ) b d ( v ) − ( L − n ) bd ( u )( L n ) c a ( v ) u + v δ m n , (A.8)we get: { T ( u ) , T ( v ) } = (Poisson brackets between K )++ K ab ( u ) K αβ ( v ) ¯ K dc ( u ) ¯ K δγ ( v ) (cid:40) ξ ( u − v ) (cid:20) (¯ L ) αc ( u )(¯ L ) aγ ( v )( L ) bd ( u )( L ) βδ ( v ) −− (¯ L ) aγ ( u )(¯ L ) αc ( v )( L ) bd ( u )( L ) βδ ( v )++ (¯ L ) ac ( u )(¯ L ) αγ ( v )( L ) bδ ( u )( L ) βd ( v ) −− (¯ L ) ac ( u )(¯ L ) αγ ( v )( L ) βd ( u )( L ) bδ ( v ) (cid:21) ++ 1 ξ ( u + v ) (cid:20) (¯ L ) βc ( u )(¯ L ) αγ ( v )( L ) bd ( u )( L ) aδ ( v ) −− (¯ L ) aδ ( u )(¯ L ) αγ ( v )( L ) bd ( u )( L ) βc ( v ) −− (¯ L ) ac ( u )(¯ L ) bγ ( v )( L ) αd ( u )( L ) βδ ( v )++ (¯ L ) ac ( u )(¯ L ) αd ( v )( L ) bγ ( u )( L ) βδ ( v ) (cid:21)(cid:41) . (A.9)It is easy to verify that the terms from the Poisson Brackets of L -matrices cancel exactlythe ones from the Poisson brackets of K -matrices. Therefore, the transfer matrices form afamily of functions in convolution between themselves: { T ( u ) , T ( v ) } = 0 . (A.10)– 39 – Parity of quantum transfer matrices
In this appendix we will prove explicitly the parity of the quantum transfer matrices in allthe antisymmetric representations of the auxiliary space.
Parity of T We need to evaluate: ˆ T ( − u ) = Tr (ˆ¯ L J ( u ) . ˆ¯ L J − ( u ) . . . ˆ¯ L ( u ) . ˆ K ( − u ) . ˆ L ( − u ) . ˆ L ( − u ) . . . ˆ L J ( − u ) .G . ˆ¯ K ( − u ) .G T ) . (B.1)Transposing inside of the trace: ˆ T ( − u ) = Tr ( G . ˆ¯ K T ( − u ) .G T . ˆ L TJ ( − u ) . ˆ L TJ − ( − u ) . . . ˆ L T ( − u ) .. ˆ K T ( − u ) . ˆ¯ L T ( u ) . ˆ¯ L T ( u ) . . . ˆ¯ L TJ ( u )) . (B.2)Now since ˆ L Tj ( − u ) = − ˆ¯ L j ( − u ) and ˆ¯ L Tj ( u ) = − ˆ L ( u ) we get: ˆ T ( − u ) = Tr (ˆ¯ L J ( − u ) . ˆ¯ L J − ( − u ) . . . ˆ¯ L ( − u ) . ˆ K T ( − u ) . ˆ L ( u ) . ˆ L ( u ) . . . ˆ L J ( u ) .G . ˆ¯ K T ( − u ) .G T ) . (B.3)We can now insert a pair of ¯ S -matrices near the ˆ K -operator using the unitarity condition ¯ S (2 u ) ¯ S ( − u ) = I and then commute ¯ S (2 u ) through the ˆ L -operators using the Yang-Baxterequation, obtaining: ˆ T ( − u ) = Tr (ˆ¯ L J ( − u ) . ˆ¯ L J − ( − u ) . . . ˆ¯ L ( − u ) . ˆ K T ( − u ) . ¯ S ( − u ) .. ˆ L ( u ) . ˆ L ( u ) . . . ˆ L J ( u ) .G . ¯ S (2 u ) . ˆ¯ K T ( − u ) .G T ) . (B.4)Using the following identities: ¯ α (2 u ) ¯ R c da b (2 u ) ˆ K cd ( − u ) = ˆ K ba ( u ) , (B.5) ¯ α ( − u ) ¯ R a bd c ( − u )( ˆ¯ K ) dc ( − u ) = ( ˆ¯ K ) ba ( u ) , (B.6)which in matrix notation are: ˆ K T ( − u ) . ¯ S (2 u ) = ˆ K ( u ) , (B.7) ¯ S ( − u ) . ˆ¯ K T ( − u ) = ˆ¯ K ( u ) , (B.8)we obtain that ˆ T ( − u ) = ˆ T ( u ) , thus seeing that ˆ T is even for any J .We will now give a more detailed proof of the last passage above. We will use the YBEin figure 7 (substituting R ( − u ) with ¯ S ( u ) ), the unitarity condition (5.16) and the identities(B.5) and (B.6). Starting from (B.3) we insert an identity and we use (5.17) to get: ˆ T ( − u ) =(ˆ¯ L J ( − u ) . ˆ¯ L J − ( − u ) . . . ˆ¯ L ( − u )) γa ¯ S δ αζ η (2 u ) ¯ S ζ ηγ β ( − u ) ˆ K αδ ( − u )(ˆ L ( u ) . ˆ L ( u ) . . . ˆ L J ( u )) βh ( G . ˆ¯ K T ( − u ) .G T ) ha . (B.9)– 40 –e can now use (B.5) to get: ˆ T ( − u ) =(ˆ¯ L J ( − u ) . ˆ¯ L J − ( − u ) . . . ˆ¯ L ( − u )) γa ¯ S ζ ηγ β ( − u ) ˆ K ζη ( u )(ˆ L ( u ) . ˆ L ( u ) . . . ˆ L J ( u )) βh ( G . ˆ¯ K T ( − u ) .G T ) ha . (B.10)We now use YBE in figure 7 (substituting R ( − u ) with ¯ S ( u ) ) to commute the remainingS-matrix through all the ˆ L i and ˆ¯ L i , obtaining for i = 1 : ˆ T ( − u ) =(ˆ¯ L J ( − u ) . ˆ¯ L J − ( − u ) . . . ˆ¯ L ( − u )) ωa ¯ S γ β(cid:15) ω ( − u )ˆ L η γ ( u )ˆ L ζ β ( − u ) ˆ K ζη ( u )(ˆ L ( u ) . . . ˆ L J ( u )) (cid:15)h ( G . ˆ¯ K T ( − u ) .G T ) ha . (B.11)Continuing this process for ∀ i = 2 ...J we obtain: ˆ T ( − u ) =(ˆ¯ L J ( − u ) . ˆ¯ L J − ( − u ) . . . ˆ¯ L ( − u ) . ˆ K ( u ) . ˆ L ( u ) . ˆ L ( u ) . . . ˆ L J ( u )) (cid:15) ω ¯ S (cid:15) ωh a ( − u )( G . ˆ¯ K T ( − u ) .G T ) ha . (B.12)Finally we use (B.6) and get: ˆ T ( − u ) =(ˆ¯ L J ( − u ) . ˆ¯ L J − ( − u ) . . . ˆ¯ L ( − u ) . ˆ K ( u ) . ˆ L ( u ) . ˆ L ( u ) . . . ˆ L J ( u )) (cid:15) ω ( G . ˆ¯ K ( u ) .G T ) ω (cid:15) = ˆ T ( u ) . (B.13) Parity of T Remembering that ¯ L i ( u ) = L i ( − u ) , we write: ˆ T ( u ) = Tr (cid:16) ˆ L J ( u ) . . . ˆ L ( u ) . ˆ K ( u ) . ˆ L ( u ) . . . ˆ L J ( u ) .G . ˆ¯ K ( u ) .G T (cid:17) . (B.14)Hence we have that: ˆ T ( − u ) = Tr (cid:16) ˆ L J ( − u ) . . . ˆ L ( − u ) . ˆ K ( − u ) . ˆ L ( − u ) . . . ˆ L J ( − u ) .G . ˆ¯ K ( − u ) .G T (cid:17) . (B.15)Taking a transpose inside the trace and noticing from definition (5.21) that ˆ L Ti ( − u ) =ˆ L i ( u ) : ˆ T ( − u ) = Tr (cid:16) ˆ L J ( u ) . . . ˆ L ( u ) . ˆ K T ( − u ) . ˆ L ( u ) . . . ˆ L J ( u ) .G . ˆ¯ K T ( − u ) .G T (cid:17) . (B.16)We now need identities analogous to (B.5) for irrep. First, we need the ¯ S -matrix.Making the ansatz that it is formed by all compatible indices structures, we can fix therelative coefficients by requiring that it satisfies the Yang-Baxter equation: ˆ L BE ( u )ˆ¯ L DF ( − v ) ¯ S E FA C ( u + v ) = ¯ S B DF E ( u + v )ˆ¯ L EC ( − v )ˆ L FA ( u ) . (B.17)We get that: ¯ S A CB D ( u ) = c ( u ) δ AB δ CD − iξ u δ AD δ CB + iξ (cid:16) u − iξ (cid:17) η AC η BD . (B.18)– 41 –he overall coefficient c ( u ) is fixed by unitarity, ¯ S A CB D ( u ) ¯ S B DE F ( − u ) = δ AE δ CF , as: c ( u ) = u (cid:16) u − iξ (cid:17) ξ + u (cid:16) u + iξ (cid:17) . (B.19)The identities we need are: ˆ K T ( − u ) . ¯ S (2 u ) = ˆ K ( u ) , (B.20) ¯ S ( − u ) . ˆ¯ K T ( − u ) = ˆ¯ K ( u ) . (B.21)Hence, inserting into (B.16) a pair of ¯ S -matrices via unitarity and repeating the passagesof the section above, it is easy to prove that: ˆ T ( − u ) = ˆ T ( u ) . (B.22) Parity of T ¯4 Using the definitions of section 5.4.4 one can rewrite ˆ T ¯4 ( u ) in terms of ˆ L -operators and ˆ K -operators as: ˆ T ¯4 ( u ) = ˆ β ( u ) Tr (ˆ¯ L ( u ) . . . ˆ¯ L J ( u ) . ( G ) − T . ˆ K ( − u ) . ( G ) − . ˆ L J ( − u ) . . . ˆ L ( − u ) . ˆ¯ K ( − u )) . (B.23)Where ˆ β ( u ) is an operator which is an even polynomial in u , composed by all the prefactorsappearing in the definitions of the ¯4 operators. Following the same passages used for theparity of T , it is then easy to prove that: T ¯4 ( − u ) = T ¯4 ( u ) . (B.24) Parity of T ¯1 From the definitions of section 5.4.5, it is evident that the L ¯1 -operators are even polynomialsin u . Also, since the K ¯1 -operators are proportional to the identity operator, we can movethem together through all L ¯1 . Their product is: K ¯1 ( u ) ¯ K ¯1 ( u ) = (cid:18) u + 4 ξ (cid:19) (cid:18) u + 1 ξ (cid:19) u . (B.25)Hence, T ¯1 ( u ) is even as it is a product of even functions. C Explicit form of transfer matrices J = 0 case The polynomials P that enter the transfer matrices for the J = 0 case in (5.40) are P = P ¯4 = 4 cos ϕ v − h + cos ϕ ,P = (2 cos 2 ϕ + 4) v − (4 ∆ sin ϕ + 16 h cos ϕ ) v + 16 h , (C.1)– 42 –here h = − ˆ g B − . The above expressions lead to the Baxter equation (6.4).In order to make a comparison with [17, 26] we introduce the notation for the finaldifference operators ˆ O ± ˆ O ± q ≡ q ( u ) (cid:0) g − u cos( φ ) ± u sin( φ ) (cid:1) + u q ( u − i ) + u q ( u + i ) . (C.2)The second order equations used in [17, 26] was of the form ˆ O ± q = 0 . At the same timethe forth order equation (6.4) can we written as − u ˆ O + u ˆ O − q = − u ˆ O − u ˆ O + q = 0 . (C.3)We see that the four independent solutions of the two second order equations ˆ O ± q = 0 arethe solutions of (6.4), which indeed demonstrates their equivalence. J = 1 case For the J = 1 case we explicitly built all transfer matrices as differential operators actingon the CFT wavefunction of variables s, t, (cid:126)x . We verified the general analytic propertiesoutlined in the text in section 5.4.7. Furthermore, we found some additional relationsbetween the coefficients as shown below P = 4 cos ϕ v + a v + a h −
14 cos ϕ ,P = (2 cos 2 ϕ + 4) v + 2 (cid:20) cos ϕ a + c ϕ ∆ − (∆ + 1) − S sin ϕ (cid:21) v + b v + b v + 16 h ,P ¯ = 4 cos ϕ v + c v + c h −
14 cos ϕ . (C.4)Here, h = − ˆ g B − (cid:39) − ˆ g so we can see that the relation (5.35) does hold indeed. Thecoefficients a , c , b , b and h are complicated differential operators whose explicit form canbe provided upon request. There are no further simple relations we found between themexcept for c − a = 8 i S ∆ sin ϕ , agreeing with (5.44). This implies that under spin flipping S → − S , T ( u ) interchanges with T ¯4 ( u ) up to the trivial explicit prefactor. We see thatin total we have independent commuting operators a , b , b , h, S, ∆ , which equates thenumber of degrees of freedom in J = 1 case.The limit of straight line ϕ → π is especially interesting as 1D Conformal symmetrygets restored. The space naturally decomposes into a D line and the D space orthogonalto it. The corresponding symmetry is thus SO (3) × SO (2 , , and its representations areparametrised by the spin S of SO (3) and the conformal weight ∆ of SO (2 , . Fixing ∆ and S removes two variables in our CFT wavefunction out of . Furthermore, we can restrictourselves to Highest Weight states w.r.t. to both subgroups, which imposes on the wavefunction more conditions Kψ = 0 and S + ψ = 0 , which can be used to further reduce thenumber of variables from to . In this reduced system B − and b remain two non-trivialdifferential operators, whereas all others can be expressed explicitly in terms of ∆ and S . In– 43 –articular, a becomes P K + ∆ − ∆ + 1) , where K µ is the special conformal transfor-mation generator and P µ is the generator of translations, and thus simplifies considerablyfor the primary operators in the 1D defect CFT, for which by definition K = 0 .In the simplified case S = 0 we get the following relations P = − v + (cid:0) −
2∆ + 2 (cid:1) v + 14 (cid:0) −
2∆ + 3 (cid:1) − g ,P = +6 v − (4(∆ − v + v (cid:0) (∆ − ∆ + 16ˆ g (cid:1) + b v + 16ˆ g ,P ¯ = − v + (cid:0) −
2∆ + 2 (cid:1) v + 14 (cid:0) −
2∆ + 3 (cid:1) − g , (C.5)so there are only two non-trivial functions ∆( g ) and b ( g ) , which can only be deducednumerically. D Generalisation: addition of impurities
To generalise our setup, it is possible to introduce impurities in the spin chain. This is doneby introducing a dependence on some parameters { θ i } , i = 1 . . . J in the rapidities of thebulk particles. To preserve parity in the argument u of the T -operators, the correct choice(up to a normalisation of the θ i ) amounts to: ˆ T λθ ( u ) = Tr (cid:16) ˆ¯ L λJ ( − u − θ J ) . . . ˆ¯ L λ ( − u − θ ) . ˆ K λ ( u ) . ˆ L λ ( u − θ ) . . . ˆ L λJ ( u − θ J ) .G λ . ˆ¯ K λ ( u ) .G λ t (cid:17) , (D.1)where λ = { , ¯4 , , ¯1 } . These transfer matrices form a family of mutually commutingoperators: this was verified explicitly up to the case J = 1 . However, they do not commutewith the original Hamiltonian H : this is expected, as introducing impurities changes thephysical system and thus the Hamiltonian as well.The next step is to introduce the polynomials P λk and to write the Baxter equation.In this case, the polynomials will acquire a { θ i } dependence. Moreover, the prefactorsappearing in equations (5.41) will also be modified. We obtain that: T θ ( v, { ζ i } ) = 1 , T θ ( v, { ζ i } ) ≡ P J +1 ( v , { ζ i } ) ξ J +2 , T θ ( v, { ζ i } ) ≡ A (2 v ) v + 1 v P J +2 ( v , { ζ i } ) ξ J +4 , T ¯ θ ( v, { ζ i } ) = A (2 v ) A (2 v + i ) A (2 v − i ) ( v + )( v + ) (cid:81) Ji =1 (cid:16) ζ i + (cid:0) v − ζ i + (cid:1) (cid:17) P ¯4 J +1 ( v , { ζ i } ) ξ J +6 , T ¯ θ ( v, { ζ i } ) = A (2 v ) A (2 v + i ) A (2 v − i ) A (2 v + 2 i ) A (2 v − i )( v + 4)( v + 1) v (cid:81) Ji =1 (cid:16)(cid:0) v − ζ i (cid:1) (cid:16) ζ i + (cid:0) v − ζ i + 1 (cid:1) (cid:17)(cid:17) ξ J +8 , (D.2)– 44 –here v ≡ u ξ and ζ i ≡ θ i ξ . We can now rewrite the Baxter equation (6.1). Defining ζ ≡ , ζ − i ≡ − ζ i and identifying: Q ( v ) → Γ( − iv ) exp( π ( J + 1) v ) q ( v ) ξ iv ( J +1) Γ (cid:0) − iv − (cid:1) Γ( iv + 2) J (cid:89) i = − J (cid:0) Γ( i ( v + ζ i ) + 1) − (cid:1) , (D.3)we obtain: P J +2 ( v ) v (cid:81) Ji = − J ( v − ζ i ) q ( v ) = − J (cid:89) i = − J ( v + i − ζ i ) q ( v + 2 i ) − v + i v ( v + i ) P J +1 (cid:0) ( v + i ) (cid:1) q ( v + i ) − J (cid:89) i = − J ( v − i − ζ i ) q ( v − i ) − v − i v ( v − i ) P ¯4 J +1 (cid:0) ( v − i ) (cid:1) q ( v − i ) . A similar construction with inhomogeneities in the closed fishchian is introduced in [48].
References [1] L. N. Lipatov,
Asymptotic behavior of multicolor QCD at high energies in connection withexactly solvable spin models , JETP Lett. (1994) 596 [ hep-th/9311037 ].[2] L. D. Faddeev and G. P. Korchemsky, High-energy QCD as a completely integrable model , Phys. Lett.
B342 (1995) 311 [ hep-th/9404173 ].[3] J. A. Minahan and K. Zarembo,
The Bethe ansatz for N=4 superYang-Mills , JHEP (2003) 013 [ hep-th/0212208 ].[4] N. Beisert et al., Review of AdS/CFT Integrability: An Overview , Lett. Math. Phys. (2012) 3 [ ].[5] P. Dorey, G. Korchemsky, N. Nekrasov, V. Schomerus, D. Serban and L. Cugliandolo, eds., Integrability: From Statistical Systems to Gauge Theory , vol. 106 of
Lecture Notes of the LesHouches Summer School . Oxford University Press, 2019.[6] N. Gromov,
Introduction to the Spectrum of N = 4 SYM and the Quantum Spectral Curve , .[7] B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in PlanarN=4 SYM Theory , .[8] O. Gürdoğan and V. Kazakov, New Integrable 4D Quantum Field Theories from StronglyDeformed Planar N = , Phys. Rev. Lett. (2016)201602 [ ].[9] A. Zamolodchikov, ’FISHNET’ DIAGRAMS AS A COMPLETELY INTEGRABLESYSTEM , Phys. Lett. B (1980) 63.[10] N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, Integrability of ConformalFishnet Theory , JHEP (2018) 095 [ ].[11] N. Gromov, V. Kazakov and G. Korchemsky, Exact Correlation Functions in ConformalFishnet Theory , JHEP (2019) 123 [ ].[12] D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly γ -Deformed N = 4 Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory , Phys. Rev.Lett. (2018) 111601 [ ]. – 45 –
13] N. Gromov and A. Sever,
The Holographic Dual of Strongly γ -deformed N=4 SYM Theory:Derivation, Generalization, Integrability and Discrete Reparametrization Symmetry , .[14] A. Cavaglià, D. Grabner, N. Gromov and A. Sever, Colour-Twist Operators I: Spectrum andWave Functions , .[15] J. M. Maldacena, Wilson loops in large N field theories , Phys. Rev. Lett. (1998) 4859[ hep-th/9803002 ].[16] J. K. Erickson, G. W. Semenoff and K. Zarembo, Wilson loops in N=4 supersymmetricYang-Mills theory , Nucl. Phys.
B582 (2000) 155 [ hep-th/0003055 ].[17] A. Cavaglia, N. Gromov and F. Levkovich-Maslyuk,
Quantum spectral curve and structureconstants in N = 4 SYM: cusps in the ladder limit , JHEP (2018) 060 [ ].[18] D. Grabner, N. Gromov and J. Julius, Excited States of One-Dimensional Defect CFTs fromthe Quantum Spectral Curve , JHEP (2020) 042 [ ].[19] J. Julius, “ Baxter Equation for One-Dimensional Defect CFT , to appear.”[20] N. Drukker,
Integrable Wilson loops , JHEP (2013) 135 [ ].[21] D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cuspanomalous dimension from a TBA equation , JHEP (2012) 134 [ ].[22] N. Gromov and F. Levkovich-Maslyuk, Quantum Spectral Curve for a cusped Wilson line in N = 4 SYM , JHEP (2016) 134 [ ].[23] N. Gromov, F. Levkovich-Maslyuk, P. Ryan and D. Volin, Dual Separated Variables andScalar Products , Phys. Lett. B (2020) 135494 [ ].[24] N. Gromov, F. Levkovich-Maslyuk and P. Ryan,
Determinant Form of Correlators in HighRank Integrable Spin Chains via Separation of Variables , .[25] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Quantum Spectral Curve and the NumericalSolution of the Spectral Problem in AdS5/CFT4 , JHEP (2016) 036 [ ].[26] N. Gromov and F. Levkovich-Maslyuk, Quark-anti-quark potential in N = , JHEP (2016) 122 [ ].[27] J. K. Erickson, G. W. Semenoff, R. J. Szabo and K. Zarembo, Static potential in N=4supersymmetric Yang-Mills theory , Phys. Rev.
D61 (2000) 105006 [ hep-th/9911088 ].[28] D. Correa, J. Henn, J. Maldacena and A. Sever,
The cusp anomalous dimension at threeloops and beyond , JHEP (2012) 098 [ ].[29] E. K. Sklyanin, Boundary conditions for integrable quantum systems , Journal of Physics A:Mathematical and General (1988) 2375.[30] T. Gombor and Z. Bajnok, Boundary states, overlaps, nesting and bootstrapping AdS/dCFT , .[31] N. Gromov and A. Sever, Derivation of the Holographic Dual of a Planar Conformal FieldTheory in 4D , Phys. Rev. Lett. (2019) 081602 [ ].[32] N. Gromov and A. Sever,
Quantum fishchain in AdS , JHEP (2019) 085 [ ].[33] P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect , JHEP (2018) 077 [ ]. – 46 –
34] O. Lipan, P. Wiegmann and A. Zabrodin,
Fusion rules for quantum transfer matrices as adynamical system on Grassmann manifolds , Mod. Phys. Lett. A (1997) 1369[ solv-int/9704015 ].[35] R. Baxter, Exactly solved models in statistical mechanics . 1982.[36] R. Baxter,
Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenbergchain. i. some fundamental eigenvectors , Annals of Physics (1973) 1 .[37] V. V. Bazhanov, S. L. Lukyanov and A. B. Zamolodchikov, Integrable structure of conformalfield theory. 2. Q operator and DDV equation , Commun. Math. Phys. (1997) 247[ hep-th/9604044 ].[38] J. McGovern,
Scalar Insertions in Cusped Wilson Loops in the Ladders Limit of Planar N =4SYM , .[39] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, New Construction of Eigenstates andSeparation of Variables for SU(N) Quantum Spin Chains , JHEP (2017) 111 [ ].[40] N. Gromov and F. Levkovich-Maslyuk, New Compact Construction of Eigenstates forSupersymmetric Spin Chains , JHEP (2018) 085 [ ].[41] J. Maillet and G. Niccoli, On quantum separation of variables , J. Math. Phys. (2018)091417 [ ].[42] P. Ryan and D. Volin, Separated variables and wave functions for rational gl(N) spin chainsin the companion twist frame , J. Math. Phys. (2019) 032701 [ ].[43] S. Giombi and S. Komatsu, More Exact Results in the Wilson Loop Defect CFT: Bulk-DefectOPE, Nonplanar Corrections and Quantum Spectral Curve , J. Phys.
A52 (2019) 125401[ ].[44] A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk,
Separation of variables and scalarproducts at any rank , JHEP (2019) 052 [ ].[45] S. Derkachov and E. Olivucci, Exactly solvable magnet of conformal spins in fourdimensions , Phys. Rev. Lett. (2020) 031603 [ ].[46] P. Ryan and D. Volin,
Separation of variables for rational gl(n) spin chains in any compactrepresentation, via fusion, embedding morphism and Backlund flow , .[47] S. Derkachov and E. Olivucci, Exactly solvable single-trace four point correlators in χ CFT , .[48] A. Cavaglia, N. Gromov and F. Levkovich-Maslyuk, “ Separation of Variables for the FishnetCFT 1: Functional Approach and Scalar Products , to appear.”, to appear.”