ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation

We study the massive ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation. Based on the $\psi$-system for the solutions of the associated linear problem, we obtain the Bethe ansatz equations. We also discuss the T-Q relations, the T-system and the Y-system, which are shown to be related to those of the $A_3/{\bf Z}_2$ integrable system. We consider the case that the solution of the linear problem has a monodromy around the origin, which imposes nontrivial boundary conditions for the T-/Y-system. The high-temperature limit of the T- and Y-system and their monodromy dependence are studied numerically.


Introduction
It has been recognized that the relation between classical and quantum integrable systems is useful for studying non-perturbative properties of supersymmetric gauge theories and the AdS/CFT correspondence [1,2,3]. The ODE/IM correspondence [4,5,6] provides an interesting example of this classical/quantum correspondence, which relates the spectral determinants of certain ordinary differential equations (ODE) to the Bethe ansatz equations in the massless limit of certain integrable models (IM). It is an interesting problem to make a complete list of this ODE/IM correspondence. The ordinary differential equations for the integrable models related to classical Lie algebras have been proposed in [7]. The Wronskian of the solutions obeys the functional relations called the ψ-system, which leads to the Bethe ansatz equations of the related quantum integrable system. The ψ-system for classical Lie algebra has been reformulated in the form of the matrix valued linear differential equations [8], where the Bethe ansatz equations of the integrable models associated with the untwisted affine Lie algebra X (1) of a classical Lie algebra X are related to the linear differential equations associated with the Langlands dual (X (1) ) ∨ .
The ODE/IM correspondence has been generalized to massive integrable models. It was found that for the classical sinh-Gordon equation modified by a conformal transformation, the spectral problem for the associated linear problem leads to the functional relations of the quantum sine-Gordon model [9]. By taking the conformal limit, it reduces to the ODE/IM correspondence for the Schrödinger type differential equation [4,5].
Recently the massive ODE/IM correspondence has been generalized to a class of modified affine Toda field equations [10,11,12,13,14,15]. In particular, Locke and one of the present authors studied the modified affine Toda equations for affine Lie algebraĝ ∨ , whereĝ is an untwisted affine Lie algebra including exceptional type [13]. It has been shown that from their associated linear problems one obtains the ψ-system which leads to the Bethe ansatz equations for the affine Lie algebraĝ.
It would be an interesting problem to explore the modified affine Toda field equation with an affine Lie algebraĝ which is not of the form of the Langlands dual of an untwisted one, where the corresponding integrable models are not identified yet. In this paper we will work with the modified affine Toda field equation associated with the affine Lie algebra B (1) 2 (or C (1) 2 ), which provides the simplest and nontrivial example. This equation also appears in the study of the area of minimal surface with a null-polygonal boundary in AdS 4 spacetime [16,17,18,19], which is dual to the gluon scattering amplitudes with specific momentum configurations. The equation of motion of strings is described by the B (1) 2 affine Toda field equation modified by the conformal transformation. The Stokes problem of the associated linear system determines the functional equations for the crossrations of external momenta. These functional relations are known to be the same as the Y-system of the homogeneous sine-Gordon model [20,21] and the free energy of the Y-system determines the area of the minimal surface.
The purpose of this paper is to apply the massive ODE/IM correspondence to the modified B (1) 2 affine Toda field equation and to investigate the functional relations for the Stokes coefficients of the linear problem, which include the Bethe ansatz equations, the T-Q relations, the T-system and the Y-system. We study the boundary condition of the T-system arising from the nontrivial monodromy of the linear problem solution around the origin. This monodromy condition also appears in the study of the form factors via the AdS/CFT correspondence [22,23].
This paper is organized as follows: In sect. 2, we introduce modified B 2 affine Toda equation and the associated linear problem. In sect. 3, we discuss the ψ-system and derive the Bethe ansatz equations. In sect. 4, we discuss the spinor representation of B 2 in detail and study the quantum Wronskian and the T-Q relations. In sect. 5, we argue the T-system and Y-system and their boundary conditions which come from the monodromy of the solution of the linear system around the origin. In sect. 6, we investigate the hightemperature limit of the Y-system in the presence of monodromy. Sect. 7 is devoted for conclusions and discussion. In the appendix, we summarize the auxiliary T-functions and their functional relations used in this paper.
Let φ = (φ 1 , φ 2 ) be the two-component scalar field on the complex plane with coordinates (z,z). We define the modified affine Toda field equation for B where ∂ = ∂ ∂z ,∂ = ∂ ∂z , m is a mass parameter and β is a coupling parameter. p(z) is a holomorphic function of z and is chosen as with M > 1 3 and s is a complex parameter. This equation is obtained by the conformal transformation z → w with ∂w ∂z = p 1 4 and the field redefinition φ → φ − 1 4β ρ ∨ log(pp), where ρ ∨ = ω 1 + 2ω 2 is the co-Weyl vector. 1 Note that the Coxeter number of B 2 is 4. Eq. (2.1) can be written in the form of the compatibility condition [∂ + A z ,∂ + Az] = 0 of the linear differential equations defined in a B 2 -module: 1 Note that the modified equation in [17] is ∂∂φ − m 2 β (α 1 √ ppe βα1·φ + 2α 2 e βα2·φ + √ ppα 0 e βα0·φ ) = 0, which is obtained by the same conformal transformation but a different field redefinition φ → φ − 1 4β α 2 log(pp). This modified equation is related to (2.1) by a field redefinition.
where the connections are defined by Here λ is the spectral parameter. We are interested in the special class of solutions of (2.1), which satisfy the periodicity condition φ(ρ, θ + π 4M ) = φ(ρ, θ) and the boundary conditions at infinity and the origin of the complex plane: where we have introduced the polar coordinate (ρ, θ) by z = ρe iθ and g is a 2-vector satisfying βα a · g + 1 > 0 (a = 0, 1, 2). Due to the special form (2.2) of p(z), (2.1) and the linear problem are invariant under the Symanzik rotation for an integer k. This also acts on the solution Ψ(z,z), which is denoted as Ψ k (z,z) := Ω k Ψ(z,z). The linear problem is also invariant under the transformation: where S = exp( 2πi 4 ρ ∨ · H). We now consider the solutions of the linear differential equations (2.3) in the basic B j ) i , where i, j = 1, · · · , dimV (a) . For the Lie algebra B 2 , V (1) is 5-dimensional vector representation, whose matrix representation is given by and Here e ab denotes the matrix whose (i, j)-element is δ ia δ jb . Similarly, V (2) is a 4-dimensional spinor representation. Its matrix representation is given by and E −α i = E T α i . We are interested in the small (or subdominant) solution Ψ (a) , which decays fastest along the positive real axis. This was studied in [13] forĝ ∨ for an untwisted affine Lie algebra g. In general, the small solution Ψ (a) at large ρ is given by with C (a) being a normalization constant. Here µ (a) and µ (a) denote the eigenvector and its eigenvalue of the matrix Λ + = E α 0 + E α 1 + E α 2 with the eigenvalue of the largest real part. Applying the Symanzik rotationΩ k (k ∈ Z), one obtains the small solution Ψ We define the basis of the solutions around ρ = 0 behaves as ρ → 0: which are invariant underΩ k [13]. The small solution Ψ (1) 1 2 and Ψ (2) can be expanded in this basis as (2.14) We call Q i (λ, g) satisfy the quasi-periodicity condition: Note that we can rescale z andz such that the mass parameter m is fixed to be an arbitrary non-zero constant. Then the Q-functions depend on the mass parameter through s/m.

ψ-System and the Bethe ansatz equations
The linear problem in the basic B 2 -modules V (a) can be also defined in other B 2 -modules corresponding to the (anti-)symmetrized tensor product of V (a) 's. The inclusion maps between the modules induce the relation between the small solutions, which is called the ψ-system [7]. For example, we consider the inclusion map By these maps the highest weight state e 1 and e 1 . We use this map to relate the solutions of the linear problem defined on the different modules. Ψ (2) with the same asymptotic behavior at large ρ. In a similar way we can identify Ψ . Thus we obtain the ψ-system: Expanding the small solutions in the basis {X Denoting the zeros of the Q-functions Q Note that these differ from those of the integrable model based on the U q (A 3 ) [24,25], which is expected from the Langlands duality between A 3 ) do not include the squared Q-functions. It would be interesting to study the solutions of the Bethe ansatz equations (3.7) and (3.8) in the conformal limit and explore the corresponding integrable model.

Quantum Wronskian and T-Q relations 4.1 Spinor representation and discrete symmetries
The ψ-system in the previous section has been obtained by investigating the asymptotic solution in a single Stokes sector, S 0 for example. Now we consider the solutions of the linear problem in the whole complex plane. We focus on V (2) because this is the minimal dimensional representation and the solution in the vector representation can be constructed via the inclusion map ι 2 .
Since we are considering a SO(5) spinor, it is natural to introduce the charge conjugation. Associated with the linear problem (2.3) in the spinor representation, we define the transposed linear problem: The solutionΨ(z,z|λ, g) of these equations are related to Ψ(z,z|λ, g) by the charge conjugation:Ψ This is a Z 2 symmetry of the linear problem. Note thatΨ = −Ψ.
One can define the inner product Ψ , Ψ := 4 α=1Ψ α Ψ α between Ψ = (Ψ α ) and Ψ = (Ψ α ). The inner product is independent of z andz when Ψ (Ψ) is a solution of the (transposed) linear problem. The Wronskian of any four linearly independent solutions Ψ i (i = 1, 2, 3, 4) is also independent of z andz. We define the (−k)-rotated solution s k := Ψ −k in the module V (2) . This is the subdominant solution in the Stokes sector S k but it gives a divergent solution in the sectors S k−2 and S k+2 . One can choose {s k−1 , s k , s k+1 , s k+2 } as a basis of the solutions. We normalize the solution s k such that by choosing the normalization constant C (2) in (2.11) as (−16) − 1 4 . From the asymptotic behavior of s k ands k at large ρ, we find s k , s k = s k , s k±1 = 0 and s k , s k+2 = 1 16 .
Then from the condition (4.5) we find which simplifies the functional relations described below.

T-Q relation
Now we take {s −2 , s −1 , s 0 , s 1 } as the basis of the solutions of the linear system. We introduce a set of functions T a,m (λ) (a = 1, 2, 3, m ∈ Z) by  The coefficients of s −1 , s 0 and s 1 follow from the definition of T a,m directly. The coefficient of s −2 is evaluated as s k , s −1 , s 0 , s 1 . Using the identity: which follows from the Symanzik rotation, it is shown to be equal to s −2 , s −1 , s 0 , s k−1 We expand s −k in terms of the basis X i : where Q i := Q i . The exterior product is also expanded in the basis X (2) i . The coefficient of the highest weight vector is evaluated as 1 ∧ · · · ∧ X (2) p + · · · , (4.14) where we introduce the determinant . For p = 4, we obtain W (4) In particular, from the normalization condition (4.7) we find that W (4) k−1,k,k+1,k+2 = 1. (4.16) The relation (4.16) can be regarded as the quantum Wronskian relation [9]. Let us consider two more examples. For p = 2 with i 1 = −k and i 2 = −k + 1, using the ψ-system (3.4), we find k+1,k,k−1 , using (4.6), we have which becomes s −k , F T X by the formula (4.2). We then get We note that the determinants (4.15) satisfy the Plücker relations In particular one finds This is the T-Q relation for the A 3 /Z 2 -type. From this relation we obtain the Bethe equations, which was also derived in the previous section by using the ψ-system. One can also derive a set of the relations: 134 W 12 . (4.24) From these equations W At the zeros λ

T-system and Y-system
Now we study the functional relations which are satisfied by T a,m . First we calculate the product of T a,1 and T 1,m . From the Plücker relation s j 1 , s j 2 , s j 3 , s j 4 s i 1 , s i 2 , s i 3 , s i 4 − s i 1 , s j 2 , s j 3 , s j 4 s j 1 , s i 2 , s i 3 , s i 4 + s i 4 , s j 2 , s j 3 , s j 4 s j 1 , s i 2 , s i 3 , s i 1 = 0. 1,m T where the second term in the r.h.s. is not the form of the T a,m functions. We add this function to a member of the T-functions and define for m ∈ Z. The new function T 2,m satisfies the identity We then introduce But this is not new. Using (4.2) and (4.6), we can show that T 3,m = T 1,m . Finally we obtain the T-system of A 3 /Z 2 type: which is obtained by the reduction of A 3 T-system with the identification T 1,m = T 3,m . Other functions T 2,m , T 3,m can be expressed in terms of T a,m by using T 2,1 = T 3,1 and (5.2). They also satisfy the identities: We next introduce the Y-functions by They satisfy the Y-system of where a = 1, 2 and Y 3,m = Y 1,m . The T-system (5.8) and the Y-system (5.11) imply that the Langlands duality between the modified B (1) 2 affine Toda equation and the functional equations of the A 3 /Z 2 quantum integrable system.
We now discuss the boundary condition of the T-system and the Y-system. It is easy to see that T a,−1 = 0 and T a,0 = 1. In order to determine the boundary conditions T 1,m for large m, we need to study the small solutions s m in the whole complex plane. When 4(M + 1) is not a rational number, the Stokes sectors cover the complex plane infinitely many times. So the T-functions T a,m are defined independently for arbitrary positive integer m.
In this paper we will consider the case 4(M + 1) = n with n ≥ 6 being a positive integer in detail. 2 In this case there are n Stokes sectors in the complex plane. When we go around the origin, the solution s k (ze −2πi ) is defined in the sector S k+n , which is the same as S k . Then the small solution s k+n (z) is proportional to s k (ze −2πi ): For g = 0, the linear system has no simple pole at the origin. The solution has no monodromy around it. Then we have s k (ze −2πi ) = s k (z), which implies The condition (5.13) leads to the boundary conditions for the T-/Y-functions: T a,n−3 = 0 and Y a,n−4 = 0. The truncated T-/Y-system becomes the same as the one for the n-point gluon scattering amplitudes in AdS 4 at strong coupling [19]. For g = 0, the solutions of the linear system have monodromy around the origin. We introduce a monodromy matrix Ω(λ) by (5.14) From the normalization condition (4.5) we find detΩ(λ) = 1. We also introduce the proportionality factor B(λ) in (5.13) for k = 1 by Let us expand the solution s 0 (z, λ) in the basis X i (z,z|λ, g) whose coefficient has been defined as Q i (λ, g). Then we substitute its Symanzik rotation into (5.15). In the basis X i , the monodromy matrix becomes diagonal and takes the form diag(e 2πiβg·h (2) 1 , . . . , e 2πiβg·h (2) 4 ). Moreover from the quasi-periodicity condition (2.15) one finds that B(λ) = −1. Plugging (5.15) into (5.14), we get the relation which generalizes the condition (5.13) and determines the boundary condition for the T-system. It is convenient to use the (multi-)trace of the monodromy matrix Ω: trΩ and tr (2) Ω ≡ 1 2 ((trΩ) 2 − trΩ 2 ), which are basis independent quantities. These traces can be also expressed using the Wronskians: tr (2) Ω = s −2 , s −1 , s n , s n+1 + s 0 , s 1 , s n−2 , s n−1 + s −2 , s n−1 , s 0 , s n+1 + s n−2 , s −1 , s 0 , s n+1 + s −2 , s n−1 , s n , s 1 + s n−2 , s −1 , s n , s 1 .
The Y-system (5.11) for m = n−2 and (5.21) contains Y a,n−1 in the r.h.s. of the equations. Y a,n−1 are expressed as and T a,n are expressed in terms of the lower T-functions. For the n = 4ℓ (ℓ = 1, 2, · · · ) case, they are also expressed in terms of the lower Y-functions by solving (5.10). Then (5.11) and (5.21) with (5.22) become the closed functional relations. Note that the present T-and Y-systems are the same as those of form factors in AdS 4 [23]. However the function p(z) has different pole structure from the present one.

Conclusions and discussion
In this paper, we studied the massive ODE/IM correspondence for modified B the modified affine Toda equation, we derived the ψ-system. This leads to the Bethe ansatz equations corresponding to the integrable model which is not identified yet. We also derived the same Bethe ansatz equations from the T-Q relations. We constructed the T-system and Y-system from the Wronskians of the solutions of the linear problem. These systems have non-trivial boundary conditions due to the presence of monodromy around the origin. It would be interesting to generalize the present approach to modified affine Toda field equations associated with other affine Lie algebras which are not of the Langlands dual of an untwisted affine Lie algebra [30]. It is also interesting to study the massless limit of this linear problem and investigate the description by using the free field realization of conformal field theory [31,32,33,34]. For the linear system associated the null-polygonal minimal surface in AdS 4 , we have seen that the corresponding integrable system is the homogeneous sine-Gordon model [20,21]. When the solution has monodromy around the origin, we have seen the T-system and Y-system are extended and they take the form that appears in the strong-coupling limit of the form factor in N = 4 super Yang-Mills theory. For a general polynomial p(z) and the appropriate boundary conditions for the solutions of the linear problem, one can describe the minimal surface problem using the massive ODE/IM correspondence. In particular it is interesting to explore the ODE/IM correspondence for the minimal surface in AdS 5 , where the corresponding quantum integrable model is not known yet. (JSPS).
Auxiliary T-functions In this appendix we summarize the auxiliary T-functions and their recursion relations [23]. From these relations we express trΩ and tr (2) Ω in terms of T a,m (m ≤ n). Furthermore we can express Y a,n−1 in the lower Y-functions and get a closed Y-system. We define the functions U 1,m , V 1,m , W 1,m , W 2,mW2,m (m ∈ Z) by From these equations we can write T 1,n and T 2,n in terms of lower T-functions. In the case of n = 4ℓ with ℓ = 1, 2, · · · , the T-functions can be further expressed in terms of the Y-functions by solving (5.10). We then obtain a closed Y-system.