New boundary monodromy matrices for classical sigma models

The 2d principal models without boundaries have $G\times G$ symmetry. The already known integrable boundaries have either $H\times H$ or $G_{D}$ symmetries, where $H$ is such a subgroup of $G$ for which $G/H$ is a symmetric space while $G_{D}$ is the diagonal subgroup of $G\times G$. These boundary conditions have a common feature: they do not contain free parameters. We have found new integrable boundary conditions for which the remaining symmetry groups are either $G\times H$ or $H\times G$ and they contain one free parameter. The related boundary monodromy matrices are also described.


Introduction
In this paper we investigate 1 + 1 dimensional O(N ) sigma and principal chiral models (PCMs). These are integrable at the quantum level i.e. infinite many conserved charges survive the quantization [1,2]. The scattering matrices (S-matrices) are factorized and they can be constructed from the two particle S-matrices which satisfy the Yang-Baxter equation (YBE). Thus, integrable theories at infinite volume can be defined by the solutions of the YBE. For example, it has been verified that the minimum solution of the O(N ) symmetric YBE is the S-matrix of the O(N ) sigma model [3].
In this paper we are interested in boundary conditions for these systems. There are three interesting type of boundary conditions which are: I Classically conformal -which means that the boundary condition does not break the classical conformal symmetry, which guaranties infinitely many conserved charges II Boundary conditions with zero curvature representation which means that there exists a κ-matrix (or classical reflection matrix) from which double row monodromy matrices can be constructed III Quantum integrable, which means that there exist a higher spin conserved charge even on the half line. These boundary conditions were investigated in [4,5,6,7,8,9] and was shown that all of them are conformal. What can we say about the quantum integrability of these boundary conditions? In some of these cases, one can also use the Goldschmidt-Witten argument [5,7] which is a sufficient condition for quantum integrability. With this argument it can be shown that boundary conditions 1b and ib are integrable at the quantum level. There is also a necessary condition for quantum integrability which comes from the boundary bootstrap. As we know, quantum integrable theories with boundary can be defined with the ? × iib ? ?  Thus we can infer that if the center of the residual symmetry algebra is u(1) then the reflection matrix contains a free parameter [11]. We can also classify the residual symmetries of PCMs. The bulk theory has G L × G R symmetry and the particles transform with respect to some representations of this symmetry. If the reflection matrix has a factorized form (R = R L ⊗ R R ), then the bYBE can be separated into equation for left and right reflection matrices. Thus, in principle, arbitrarily combined solutions R L and R R can be used to construct the full reflection matrix R. This implies that the remaining left and right symmetries can be different.
From the classification of the quantum reflection matrices [7,6,10,11,12] we can extract the possible residual symmetries therefore we can conclude that 1a, 2a, ia and iia can not be quantum integrable because their residual symmetries are different.
The zero curvature description is also known for some boundary conditions [4,8]. Their classical reflection matrices are constant matrices without any parameters.
The state of the art about boundary conditions and their integrability can be summarized in Table 1. With question marks we indicated the open questions. For example, 1b is quantum integrable (Goldschmidt-Witten argument) and it has O(k) × O(N − k) symmetry so it can be matched to the reflection matrix (coming from the bootstrap) with the same symmetry. Contrary, we have a U(N/2) symmetric reflection matrix with a free parameter and one can ask which boundary condition belongs to it. The boundary condition 2b is a natural candidate because it has a free parameter and the same symmetry. Indeed, in this paper we show that it has a zero curvature representation which may indicate the quantum integrability in view of the fact that a restricted boundary condition preserved the integrability at the quantum level if and only if there exists a zero curvature representation (see the table above).
In the PCM the remaining symmetries for the known classical integrable boundary conditions are H L × H R where H L ∼ = H R which means R L ∼ = R R (or the residual symmetry is G D which is the diagonal subgroup of G L × G R but in this case the reflection matrix is not factorized) [5,6]. This paper also provides a zero curvature representation for boundary condition iib where only the left or the right symmetries are broken therefore these can be candidates for reflection matrices where R L ∼ = R R .
We also derive that the traces of these new monodromy matrices Poisson commute therefore there are infinitely many conserved charges in involution. This Poisson algebra of the one and double row monodromy matrices are consistent if the r-matrix and classical reflection matrix (κ-matrix) satisfy the classical Yang-Baxter (cYBE) and the classical boundary Yang-Baxter equations (cbYBE). In [4] and [8] the Poisson algebra was investigated for non-ultralocal theories with constant κ-matrix. In [13] this was done for ultralocal theories with dynamical κ-matrix when the Poisson bracket of the κ-matrix and the Lax-connection vanished. In this paper we derive the Poisson algebra of non-ultralocal theories with κ-matrix whose Poisson-bracket with the Lax-connection does not vanish. However, the possible solutions of this equation have only been examined in a few cases. In this paper we classify the solutions of the field independent cbYBE and check that the new field dependent κ-matrix is satisfies the cbYBE for O(N ) sigma models.
The paper is structured as follows. In the next section, we start with the Lax formalism of the PCMs where we construct classical reflection matrices and use them to build double row transfer matrices. The conservation of these matrices (which is equivalent to the existence of infinite many conserved charges) provides the boundary conditions of the theories which belong to these boundary Lax representations. Using these results, we derive new double row monodromy matrices for the O(2n) sigma models and the corresponding boundary conditions will be determined too. In Section 4 we derive the Poisson algebra of the double row monodromy matrices and the cbYBE which is satisfied for the new κ-matrices.

Principal Chiral Models on the half line
In this section the new boundary monodromy matrix will be introduced. In the first subsection we will overview the Lax formalism of PCMs. After that the new reflection matrix and the related boundary condition will be derived. Finally we will show the corresponding Lagrangian descriptions and the unbroken symmetries of these models.

Lax formalism for PCMs
Let g be a semi-simple Lie algebra and G = exp(g). We use only matrix Lie-algebra and we work in the defining representation. The field variable is a map g : Σ → G where the space-time Σ = R × (−∞, 0] is parameterized with (x 0 , x 1 ) = (t, x). We can define two currents J R = g −1 dg and We will also use the following notations The ordinary letters denote forms and the italic letters denote the local coordinate functions of these.
Using these, the zero curvature condition can be written as The usefulness of the Lax connection lies in the fact that one can generate from it an infinite family of conserved charges. At first we define the one row monodromy matrix These monodromy matrices have an inversion property The monodromy matrix in the boundary case takes a double row type form where the κ L (λ), κ R (λ) ∈ G are the reflection matrices which will be specified later. In the following we use the right currents therefore we introduce the following notation J(λ) = J R (λ), The existence of infinitely many conserved quantities requires that the time derivative of the monodromy matrix has to vanishΩ(λ) = 0, which is equivalent to: where we assumed that the currents vanish at −∞. This is the boundary flatness condition. This equation can be translated to boundary conditions for the J R current. The consistency of the theory requires that the number of boundary conditions have to be equal to dim(g). Based on these, we call κ(λ) a consistent solution of (5) if it leads to exactly dim(g) boundary conditions.
The consistency of the definitions of double row monodromy matrices Ω L and Ω R (the boundary flatness condition implies the same boundary conditions with Ω L and Ω R ) implies that Using this equation, the double row monodromy matrices also have an inversion property: next subsection, we will try to find new consistent solutions with non-trivial spectral parameter dependency. Before that, we note that there is another possibility for the definition of the double row monodromy matrix, namely: This leads to the following boundary conditions Let us calculate the number of boundary conditions. For this, let us use the relation between the left and right currents.
We saw previously that this type of boundary condition is consistent if the operator Ad U −1 g is an involution on g which is equivalent to Clearly this restricted boundary condition is invariant under the transformation g → U g −1 0 U −1 gg 0 therefore it has the diagonal symmetry G D .
Finally, let us note that there is an other representation of this boundary condition. Using the inversion property (3) we can obtain an equivalent double row monodromy matrix: The conservation of this double row monodromy matrix requires that the following boundary flatness condition has to vanish.
Multiplying this by g from the right, we obtain which leads to the equations (8) and (9).

Spectral parameter dependent κ-matrices
In the previous subsection we summarized the spectral parameter independent κ-matrices. In this subsection, we try to find new spectral parameter dependent κs.

Solution of the boundary flatness equation
Let us use the following ansatz: where k(z) is a scalar and M ∈ g. Using this ansatz the equation (5) takes the following form: Which leads to the following system of equations: where [, ] + is the anti-commutator i.e. [X, Y ] + = XY + Y X. Since equation (12) provides already dim(g) boundary conditions, the consistency requires that the equations (13) and (14) should follow from (12). In the following, we look for constraints on M and N which ensure this.
Taking the anti-commutator of equation (12) with M gives where c is a constant. From this we can see that M commutes with N . Using this and the equation (14) we can obtain: Therefore, by taking the anti-commutator of equation (13) with N , we get Since J 0 spans the whole defining representation of g therefore N 2 has to be proportional to 1 so the automorphism Ad N has +1 and −1 eigenvalues and we denote the corresponding eigenspaces by h and f. Therefore N defines a Z 2 graded decomposition g = h ⊕ f. Equation (14) means that J 1 ∈ f i.e Π h (J 1 ) = 0 where Π h is the projection operator of h subspace. Putting this into (13): where we used that [M, N ] = 0 which implies M ∈ h. We can see from the last equation that equation (14) follows from (12) if M commutes with h. Summarizing, consistency of the solutions requires the following conditions These implies that Ad N generates a Z 2 graded decomposition and M is an element of h and also commutes with h. Therefore h has a non-trivial center which is generated by M . It follows that every Z 2 graded decomposition where hs are not semi-simple belong to these type of reflection matrices and boundary conditions. There are two classes of these κ matrices. The first is N = 0. The second case is N = 0, which implies that M 2 ∼ 1. In this case M defines the Z 2 graded decomposition. The projection operators to the h and f are: where U = N when N = 0 otherwise U = M . The classification of these κ-matrices for classical Lie-algebras are shown in the following.

Examples
We saw that the integrable boundary conditions described above belongs to a (g, h) symmetric pair for which G/H is a symmetric spaces (G = exp(g), H = exp(h)). The symmetric spaces are classified [14]. The spectral parameter dependent solutions belongs to not semi-simple h therefore there are three types of spectral parameter dependent κ-matrices.
These matrices are the classical counterparts of the h = u(1)⊕su(m)⊕su(n−m), h = u(1)⊕su(n) and h = so(2) ⊕ so(n − 2) symmetric solutions of the quantum boundary Yang-Baxter equation [7][6] [9]. The quantum reflection matrices are where ν i (θ) are some dressing phases and For the classical limit we define a scaling variable h for which The classical limit is h → 0. In this limit the R-matrices are proportional to the κ matrices:

Lagrangian and symmetries
In the previous subsection we found reflection matrices parameterized as (11) which leads to the following boundary condition: Using the left currents this condition takes the form: One can obtain the same boundary condition in the Lagrangian description. The Lagrangian density of the bulk theory is Thus if we add a boundary Lagrangian function as we get the boundary condition (17). This boundary condition was already investigated in [7] and [9]. It was shown that this is a conformal boundary condition for all M ∈ g. Now we have just shown that it has a zero curvature representation too for some special M s which satisfy the conditions (16). Now let us continue with the residual symmetries. The bulk Lagrangian has G L × G R symmetries which are the left/right multiplications with a constant group element: g(x) → g L g(x) and g(x) → g(x)g R . The transformations of the currents are the following: We can see that the boundary Lagrangian breaks the G R symmetry. The remaining symmetry is One can derive the Noether charges by the variation of the action but there is an easier way. We know that the J L and J R are the Noether currents of the bulk G L and G R symmetries. Let us define the following charges: By taking their time derivatives we obtaiṅ We can see thatQ and (20) are conserved charges. Finally we note that we could have used the left current J L with the κ-matrix This implies that the right reflection matrix, the boundary condition and the boundary Lagrangian are Therefore, in this case the residual symmetry is H L × G R .

O(N ) sigma model on the half line
The new reflection matrices of the PCM can be used to find new ones for the O(N ) sigma model. In particular, using the equivalence between SU(2) PCM and the O(4) sigma model we have immediately new reflection matrices for the O(N ) sigma model when N = 4. This solution then can be generalized for even N .

Lax formalism for the O(N ) sigma model
The field variables are n : Σ → R N with the n T n = 1 constrain. The bulk Lagrangian is from which equation of motion follows: We can define an O(N ) group element as: h = 1 − 2nn T which satisfies the following identities: h T h = 1 and h = h T . Using this, one can define a current:Ĵ = hdh = 2ndn T − 2dnn T which is the Noether current of the bulk global SO(N ) symmetry. The e.o.m with this current is d * Ĵ = 0 and the Lagrangian is The Lax connection is very similar to the PCM but here the current is constrained.
The double row monodromy matrix can be defined similarly as it was in PCMs. In the following we look for solutions of the boundary flatness equation Let us start with the constant κ-matrices i.e. κ(λ) = U where U ∈ O(N ) therefore the boundary flatness equation looks like which implies the following: In this subsection, we assume that U 2 = ±1 but we do not derive that. We will return to this at the next section. There are two kinds of U s: Let us start with the first case. Let the number of +1s and −1s be N − k and k respectively. Let us use the notation: n =ñ +n, with n = (n 1 , . . . , n N −k , 0, . . . , 0) ,n = (0, . . . , 0, n N −k+1 , . . . , n N ).
Using this, the equation (22) is equivalent tõ Multiplying byn from the right andñ T from the left, we can obtain the following two equations Similarly, from (23) we can get Let us assume thatn Tn = 0 which is equivalent toñ Tñ = 1 andn = 0. From this, the equations (24) and (26) where we used that 0 = n T n =n Tn′ +ñ Tñ′ =ñ Tñ′ . We can see that this is the restricted boundary condition to a sphere S k with maximal radius. Analogously, if we assume thatñ Tñ = 0 thenṅ = 0, which is the restricted bc to S N −k with maximal radius.
What happens whenn Tn = 0 andñ Tñ = 0. Let us multiply (24) withñ T form the left: n Tn ñ Tṅ = n Tṅ ñ Tñ , Using that n Tṅ + ñ Tṅ = 0 0 = n Tn +ñ Tñ ñ Tṅ = ñ Tṅ Let us continue with the second case i.e. U T = −U . Let us start with equation (25): Let us multiply this with n from the right: From this we can obtain the following two equations Let us multiply this with U from the left and U T from the right.
Using this and the original equation (25) we can obtain that J 1 = 0 which is equivalent to n ′ = 0. But we also have equation (24) therefore we have too many boundary condition which means thatĴ 0 = UĴ 0 U −1 andĴ 1 = −UĴ 1 U −1 are not consistent boundary conditions at the second case. We will also see at Subsection (4.3) that the κ-matrix of the second case do not satisfy the classical boundary Yang-Baxter equation.

Spectral parameter dependent solution for N = 4
In the last section, we found a new spectral parameter dependent reflection matrix for the SU(2) PCM. Since this model is equivalent to the O(4) sigma model we can obtain a new nonconstant κ-matrix for the O(4) sigma model by changing the notation to the O(4) sigma model language. We will see that this is a spectral parameter and field (!) dependent reflection matrix.
Thus we need to develop a dictionary between the SU(2) PCM and the O(4) sigma model. Let us introduce the following tensor: which satisfies the following relations: whereσ αα i is the complex conjugate of σ i αα . Using this we can change the basis in which the group element g 4 = SO(4) is factorized.
We can also find the relation between the variables of the O(4) model (h,Ĵ) and the SU(2) PCM (g, J L/R ). Using n = g 4 n 0 and h = 1 − 2nn T we obtain that h = g 4 jg T 4 where j = 1 − 2n 0 n T 0 = diag(1, 1, 1, −1) ∈ O(4). Since det(j) = −1, j is not factorized in the new basis: The group element h in the new basis takes the form: (g was defined in (28)) In the last line we used the following property: σ 2 gσ † 2 =ḡ andḡ denotes the complex conjugate of g. We can see that h is not factorized. This is because h is not an element of SO(4). It is convenient to introduce a new notation: Let us calculateĴ in the new basis.
whereJ R denotes the complex conjugate of J R . The Lax connection in the new basis is: Therefore the monodromy matrix of the O(4) sigma model factorized in the following way: The double row monodromy matrix in the new basis reads: Before we calculate the new κ-matrix let us apply the formula above to the known constant reflection matrices. The simplest known κ 4 is the identity matrix. This is factorized in the spinor basis: κ L = κ R = 1. Another known reflection matrix is κ = diag(−1, −1, 1, 1) in the vector basis. If we change the basis we get: −1). These two reflection factors are consistent if they satisfy the inversion property (6) i.e.
which means that g has to commute with them therefore g is restricted to H = U(1) at the boundary.
There is another known reflection matrix: κ = diag(1, 1, 1, −1) in the vector basis. If we change the basis we get: We can see this matrix is not factorized. Using this formula for the monodromy matrix, we obtain that This theory is consistent in the principal model language if g = g † at the boundary which is the boundary conditions (10). These were the relations of the well known reflection matrices of the SU(2) PCM and the O(4) sigma model. Let us continue with the new one. In the last section we found new reflection matrices for the PCM model which for g = su(2) simplifies to where M R is an arbitrary element of su (2). Without loss of generality one can choose M R = aσ 2 . We have seen that κ L (λ) = gκ R (1/λ)g † so we have Let us denote 1 ⊗M R in the vector representation by M . In the spinor basis hM h looks like Based on the above formulas, the new κ-matrix for O(4) takes the following form: where the matrix M looks like We can see that this κ is spectral parameter and field dependent too. We can give the boundary condition which correspond to this κ from the boundary conditions of SU(2) PCM (17), (18) and (29).Ĵ Using the definition of M and using (31) Therefore the boundary condition in language of the O(4) model is: This boundary condition was investigated in [9]. Using the definitionĴ = hdh = 2ndn T −2dnn T , we can get an equivalent form : From the boundary Lagrangian of the SU(2) PCM we get which agrees with [9]. Using the variables n: Finally, we can see that the residual symmetry is U(2) ∼ = SU(2) L × U(1) R which is a subgroup of SU(2) L × SU(2) R ∼ = SO(4). We saw in the PCMs that we have conserved chargesQ L andQ R . The conserved charge in the SO(4) language are: which is equivalent toQ where h = su(2) L ⊕ u(1) R , and Q is the bulk part of the charge:

Generalization for N = 2n
The result for N = 4 can be generalized for any even N . We assume that equation (32) can be used as κ matrix for N = 2n i.e.
where M = a 0 n×n 1 n×n −1 n×n 0 n×n .
We have to prove that the time derivative of the double row monodromy matrix is zero when the boundary condition is satisfied. The quantity ∂ 0Ω is zero when the boundary flatness condition is satisfied Now the RHS is not zero since the κ has field dependence.
Using this, equation (39) leads to the following three equations: If we take the anti-commutator of the boundary condition (33) with M then we will see that the third equation is satisfied. If we use the following identity This is also follows from the boundary condition.
Only the second equation remained. We have to prove that the following term vanish: Using the definition of h, we obtain that for any X ∈ so(2n).
For conserved charges, we can generalize the formula (37).
We can check the conservation of these charges.
where we used (41). The boundary Lagrangian can be written in the same form as we had for the case N = 4 (35) or (36): These have been studied earlier in [9] where it was showed that this is a conform boundary condition for any M ∈ so(2n) but in this paper we showed more, namely that it has a zero curvature representation only when M 2 ∼1.

Poisson algebra of double row monodromy matrices
In the previous sections we found new zero curvature representation of PCMs and O(N ) sigma models on a half line. This implies the existence of infinitely many conserved charges. In this section we want to prove that these conserved charges are in involution. For this we determine the Poisson algebra of the double row monodromy matrices (whose trace is the generating function of these charges). In the first subsection we summarize the formulas of general "bulk" non-ultralocal theories based on [15]. After that we derive the Poisson-algebra of the double row monodromy matrices and their consistency condition (which is the classical boundary Yang-Baxter equation) when the Poisson-bracket of the reflection matrix and the Lax-connection is not zero. This is a new result because, so far Poisson-algebras of non-ultralocal theories with boundaries were investigated only when the κ-matrix was field independent [4,8].
In the second and the third subsection we apply these general formulas for PCMs and non linear sigma models. We will use the following notations: where X ∈ End(V ) and Y ∈ End(V ) ⊗ End(V ) for a vector space V .
We can generalize the one row monodromy matrix for general paths from y to x: Let x 1 , x 2 , y 1 , y 2 be different positions and x 1,2 > y 1,2 then the general non-ultralocal Poissonbrackets of the monodromy matrices are the following [15]: where x 0 = min(x 1 , x 2 ), y 0 = max(y 1 , y 2 ) and This Poisson-bracket satisfies the Jacobi identity (for not coinciding points) if the generalized classical Yang-Baxter equation is satisfied: . For the calculation of the Poisson bracket of the global monodromy matrices (2) we have to take the limits x 1 → x 2 and y 1 → y 2 . However, the Poisson bracket (43) is not continuous due to the non ultra-locality. It is obvious that the equal intervals limit of the canonical brackets does not exist in a strong sense. More precisely, any strong definition implies the breakdown of the Jacobi identity for the canonical brackets of the global monodromy matrices (2).
However, it is possible to define this limit in a weak sense with respect to the canonical brackets based on a split-point procedure and a generalized symmetric limit. We consider canonical brackets of several monodromy matrices defined on intervals having coinciding end points. In order to compute them, let us first split the coinciding points and use (43) which then gives a completely consistent expression. Then if we symmetrize on all the possible splittings and go to the limit of equal points we get the "weak" algebras e.g. the weak algebra of the global monodromy matrices: The formulas above can be found in [15] but in this paper we use a different conventions for the Lax-pair i.e. we have to change L → −L to get the formulas in [15]. In the following we derive the Poisson-algebra. For this we need the κ-matrices which were derived in the previous sections. We saw that these matrices can depend on the fields but do not on the derivative of the fields therefore we assume that Let us continue with the generalized double row monodromy matrix: The Poisson bracket of Ω(x|λ) and Ω(y|µ) are not well defined even when x = y therefore we have to use the split-point procedure. For this, we can define a shifted double row monodromy matrix: where ∆ < 0. A general κ-matrix depends on the boundary value of the fields φ a (0) (i.e. κ(λ) = κ(φ a (0)|λ)) but we can extend this to arbitrary space coordinate: Using these the Poisson bracket of monodromy matrices are In the following we assume that Now we can calculate the symmetric limit: where x 0 = max(x 1 , x 2 ) and The existence of infinitely many conserved charges in involution requires that the following expression has to vanish.

This is the classical boundary Yang-Baxter equation (cbYBE). If the κ-matrix fulfill this equation then the Poisson-bracket of the double row monodromy matrix is
This Poisson-bracket satisfies the Jacobi identity (this can be derived by a straightforward but very long calculation). Using the split-point procedure and the symmetric limit we can calculate the "weak" Poisson algebra of the global double row monodromy matrix (4).

Poisson bracket in PCMs
Let us specify now the previous findings for the PCMs. The Poisson-algebra of the currents is the following [16,17]: In the following we will need the Poisson-bracket of the group element g and the current J L/R 0 . For this, we can use the following formula where we used the definition: and (48): Therefore The Poisson brackets of the space-like component of the Lax operator is [17]: In [17] a different convention is used which can be obtained by the following changes: L → −L,λ → −λ , γ → −1. This Poisson-bracket is the same as (42) but in this special case the r-and smatrices are space independent. Furthermore, we can find a consistency check for the classical boundary Yang-Baxter equation (cbYBE) in Appendix C where we prove that if κ R (λ) satisfies the cbYBE then κ L (λ) = gκ R (1/λ)g −1 also does which has to follow from the inversion property of the reflection matrices. In this derivation we have to use a non-trivial identity of the r-matrix In Appendix C we also show that this identity is a consequence of the inversion property and the s-matrix has a similar property: In the following we solve the classical boundary Yang-Baxter equation for constant κ-matrices.

Constant κ-matrices
Let κ(λ) = U where U ∈ G is a constant matrix. The cbYBE can be written as This equation has to be satisfied for every λ 1 , λ 2 ∈ C therefore The first equation is satisfied trivially because C 12 is invariant i.e. C 12 = U 1 U 2 C 12 U −1 1 U −2 2 . Let us multiply the second by U 1 from the left and by U −1 2 from the right Using the explicit form of C 12 we obtain that for all X ∈ g. Because we work with the defining representation (which is irreducible), U 2 has to be proportional to the identity. This is the same solution which we obtained from the analysis of the boundary flatness equation. Therefore we can conclude that the consistent solution of the flatness condition and the cbYBE are the same for the constant κ-matrix.
There is another consequence of the fact that we had to modify the equation (47)

Spectral parameter dependent κ-matrix
The κ-matrices described in Section 2 fulfill the classical boundary Yang-Baxter equation (45). The derivation can be found in Appendix B.
In [12] the following theorem was proven.
Theorem. Let U ∈ G for which Ad U defines a Lie-algebra involution and h := X ∈ g|U XU −1 = X . If κ(λ) is a solutions of the following cbYBE for reductive h where X 0 is a central element of h. The κ-matrix κ(λ) is unique for a given U (up to normalization) if we fix the norm of X 0 .
Previously we showed that these solutions exist therefore we classified the field independent solutions of the cbYBE.
We close this subsection with the Poisson-algebra of the Noether charges of the global symmetries. Let us start with the right charges Using the Poisson-algebra of the current we can obtain that We can decompose the basis {T A } into {T a ∈ h} and {T α ∈ f} . Using these, the equation above can be written as therefore they form the Lie-algebra h as expected. Let us continue with the Noether charges of the left multiplicatioñ is not well defined because it contains the following expression therefore we have to use the symmetric limit (∆ < 0): Using this, we can obtain the following equation which can be written as Clearly these charges form the Lie-algebra g as expected. This calculation shows the importance of the symmetric limit because if we do not use it properly then we cannot get the proper Poisson-algebra of the Noether charges of the symmetry G L .

Poisson bracket in O(N ) sigma models
The Poisson-algebra of the fields n i is the following From this one can calculate the Poisson-algebra of the currents [18]: and (P ) ij,kl = δ il δ jk , (K) ij,kl = δ ik δ jl are the permutation and the trace operators and (Z) ij = n i n j . Using this, one can obtain the non-ultralocal Poisson-algebra of the space-like component of the Lax-connection (42) where the r-and s-matrices are At first, we solve the cbYBE for constant κ-matrices and after that we check the spectral parameter and field dependent κ-matrix.

Constant κ-matrix
For κ(λ) = U ∈ O(N ), the cbYBE looks like After substitution, we obtain the following four equations: The first equation follows from the fact that U ∈ O(N ). From the second equation if follows that U 2 = ±1 i.e. U = ±U T . Multiplying the fourth one by U 1 from the left and right, we can see that the third one comes from the fourth. Let us write the third one explicitly.
Multiplying by U T 1 U T 2 from the left, we obtain the following Using the explicit form of C 12 , we can obtain that ). Let us multiply by P 12 from the left.
Taking the trace on the first site: Using thatZ T 2 =Z 2 , Tr Z = 0 and N > 2, we obtain that Since U can be U = ±U T , there are two cases.
1. U = U T . Using a global symmetry transformation U can be diagonalized as and Z in the same block diagonal form looks like From this explicit form we can see thatZ = 0 if and only ifñ = 0 orn = 0.
2. U = −U T . Using a global symmetry transformation U can be diagonalized as U = 0 n×n −1 n 1 n 0 n×n where n = N/2 andZ looks likẽ Multiplying the off-diagonal terms byn form the right, we obtaiñ n n Tn = −n ñ Tn and multiplying this byn T form the left, we obtain n Tn ñ Tn = 0.
At first, let us assume thatn = 0 thereforeñ Tn = 0. Substituting this to the previous equation, we obtain thatñ = 0. Using this in the diagonal term, we obtain thatnn T = 0 which contradicts ton = 0. Thereforen = 0. From n T n = 1 and from the diagonal, we obtain thatñ Tñ = 1 andññ T which is a contradiction. Therefore anti-symmetric U cannot be a solution of the cbYBE.
We can conclude that we have obtained the same constant κ-matrices from the cbYBE as we got from the boundary flatness condition.

Spectral parameter and field dependent κ-matrix
If we want to check that the new κ-matrix (32) satisfy the classical boundary Yang-Baxter equation (45) then we have to compute G 12 (λ 1 , λ 2 ). For this, we will need the following Poisson brackets: We checked the cbYBE for O(4) and O(6) sigma models with explicit calculations using Wolfram Mathematica. For this, we parameterized the sphere with stereo-graphic coordinates: Using this parameterization we can calculate explicitly the matrices r(λ, µ), G(λ, µ), κ(λ) and we can substitute these into the cbYBE. Using Mathematica we have checked that the cbYBE is satisfied for O(4) and O(6) sigma models.

Conclusion
In this paper new double row monodromy matrices have been determined for the principal chiral models. The corresponding integrable boundary conditions break one chiral half of the symmetry to G L × H R where H R was not arbitrary but G/H R had to be a symmetric space and the Lie algebra of H R was not semi-simple. We determined the boundary conditions which correspond to these monodromy matrices. Both the monodromy matrices and boundary conditions contain free parameters.
We used these results for finding new monodromy matrices for the O(N ) sigma models. At first, the SO(4) ∼ = SU(2) L × SU(2) R isometry was used to determine the SU(2) L × U(1) R symmetric κ matrices for SO(4) sigma models. These new spectral parameter dependent κ matrices were then generalized for O(2n) sigma models. They corresponds to U(n) symmetric boundary conditions.
We also showed that these κ-matrices satisfy the classical boundary Yang-Baxter equation therefore there exist infinitely many conserved charges in involution i.e. the boundary conditions proportional to these κs are classically integrable.
There exist quantum O(4) sigma models which have reflection matrix with two free parameters and the residual symmetry is O(2) × O(2) [10]. Therefore one interesting direction to pursue would be to find the classical field theoretical description of these quantum theories i.e. κ matrices and boundary conditions which have two independent parameters and residual symmetry O(2) × O(2). In the language of the SU(2) PCM, this means boundary conditions which independently break left and right symmetries. These results could be then generalized to general PCMs.
As a last remark, it would be interesting to check that the quantum version of the κ matrices determined in the paper are really the known reflection matrices. This could be done in the large-N limit. Recently, the large-N limit was studied for the CP N sigma models on finite intervals e.g. [19] [20]. These methods may also be applicable to the models studied in this paper.

Acknowledgment
I thank Zoltán Bajnok and László Palla for the useful discussions and for reading the manuscript. The work was supported by the NKFIH 116505 Grant.

Appendix A. Non-local conserved charges
If we expand the monodromy matrix around λ = λ 0 we get infinitely many conserved charges which are generally non-local. In this section we will deal with the expansions around λ = ∞ and λ = 0 and we will give the first two terms of these series.

Appendix A.1. Expansion around λ = ∞
We will start with the expansion of the one row monodromy matrix the expansion leads to which gives the first two charges In order to calculate the expansion of the monodromy matrix we will also need the following series: where M = aU . The conserved charges come from the expansion of the double row monodromy matrix.
R + UQ From this the first two conserved charges are the following: The first charge is equivalent to the charge (21) (up to a constant).Q (1) R is very similar to the charge for the g ∈ H restricted boundary condition but there is an extra term: U, Q (0) R [21].
These charges also satisfy the relations:Q (0) For a crosscheck we can take the time derivative of these charges and we will see that they all vanish.

Appendix A.2. Expansion around λ = 0
For the expansion around λ = 0, we can use the inversion property of the double row monodromy matrix (7): We can do the same calculation as before: We can see that the first conserved charge is equal to the Noether charge of the left multiplication symmetry (20):Q (0) L =Q L . The second set of charges vanish. This is similar to the case of the free boundary condition (g = h) in [21].
The derivation of (B.4) is similar.