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Algebra of Lax Connection for T-Dual Models

We study relation between T-duality and integrability. We develop the Hamiltonian formalism for principal chiral model on general group manifold and on its T-dual image. We calculate the Poisson bracket of Lax connections in T-dual model and we show that they are non-local as opposite to the Poisson brackets of Lax connection in original model. We demonstrate these calculations on two specific examples: Sigma model on S(2) and sigma model on AdS(2).


Introduction and Summary
One of the most remarkable achievements in string theory that were reached in past few years is a discovery of integrability of N = 4 superconformal Yang-Mills theory in its planar limit together with integrability of AdS 5 × S 5 superstring [1, 2, 3] 1 . Indeed, the wellknown classical integrability of bosonic string on AdS 5 × S 5 was extended to κ-symmetric Green-Schwarz superstring [11,12], or to the pure spinor formulation of superstring as well [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] 2 . For example, it was found that the classical superstring possesses an infinite number of conserved non-local charges. These charges have their counterpart in planar gauge theory at weak coupling in the spin-chain formulation for the dilatation operator [13,14].
It is also believed that the integrability of the N = 4 SYM should have an impact on the spectrum of other observables in the theory, for example on the structure of expectation values of certain Wilson loops. The dual formulation of these objects in AdS/CF T correspondence are partition functions of open AdS 5 ×S 5 strings that end on some contours at the boundary of AdS 5 [35,36]. Moreover, it turned out that the open string description of Wilson loops is directly related to the T-duality in AdS 5 × S 5 [37,38] 3 , where T-dual formulation appears to be important in discovery of connection between maximally helicity-violating (MHV) gluon scattering amplitudes and special Wilson loops (defined on contours formed by light-like gluon momentum vectors). The classical SO(4, 2) conformal symmetry of the T-dual AdS 5 geometry seems to be related to a mysterious "dual" conformal symmetry that was observed in the momentum-space integrands of loop integrals for planar gluon scattering amplitudes [42,43,44,45]. From the string theory point of view this dual conformal symmetry of AdS 5 ×S 5 sigma model could be related to the presence of hidden symmetries in T-dual string theory. In fact, since T-duality is an on-shell symmetry, or in other words it maps classical solutions to classical one and since the T-dual geometry is again AdS 5 × S 5 we can expect that T-dual model is also integrable and consequently possesses an infinite number of conserved charges that should correspond to generators of some symmetries of dual Wilson lines in N = 4 SYM.
In phase space formulation of string theory the statement that T-duality is on-shell symmetry is that T-duality is canonical transformation [63,64]. This fact has a significant consequence for the calculation of the Poisson brackets of Lax connection in T-dual theory. In fact, it is an important question how the Lax connection in original integrable theory is mapped to its T-dual counterpart 4 . These problems were discussed recently in very nice papers [46,47,48]. In particular, the paper [46] discussed the interplay between T-duality and integrability on two examples: the two-sphere S 2 and the two-dimensional anti-de Sitter space AdS 2 . It was argued there that in order to perform T-duality explicitly we have to express T-dual coordinates in terms of the original ones. Further, it is also important that this relation is non-local and hence it is a non-trivial task to find the T-duality image of Lax connection since Lax connection-as opposite to the sigma model action-depends explicitly on coordinate that parameterizes T-duality direction. On the other hand it was shown in [46] that it is possible to eliminate the explicit dependence of Lax connection on the T-duality direction coordinate with the help of some special field redefinition that preserves the flatness of Lax connection of original theory. The new Lax connection depends on the derivatives of the isometric coordinate only. Then T-duality on these flat currents can be easily implemented and then it is easy to find the T-dual flat currents.
The goal of this paper is to study properties of the Lax connection in T-dual background further. We are mainly interested in the calculation of the Poisson bracket of the T-dual Lax connections. We find explicit form of these Poisson brackets and we argue that they have exactly the same form as the brackets introduced in [59,60]. Then, following discussion presented in these papers and reviewed in appendix we can argue that T-dual theory possesses an infinite number of conserved charges that are in involution 5 in the sense that their Poisson brackets commute. On the other hand it is desirable to find an explicit form of matrices r, s that appear in these Poisson brackets and that are crucial for the study of classical or quantum mechanical integrability of given theory [59,60]. We show that even if it is rather straightforward to find the form of these matrices in case of principal chiral model it is very difficult to find their form in T-dual model. In fact, we will argue that the original constant matrices r, s map under T-duality to non-local expressions. We claim that this is in accord with the observation that local charges map to non-local ones under T-duality.
The organization of this paper is as follows. In the next section (2) we review the calculation of Poisson bracket of Lax connection in case of principal chiral model and we show that in this case the matrices r, s are constant. Then in section (3) we perform calculation of Poisson bracket of Lax connection for the sigma model that is T-dual to the sigma model on S 2 . We review the construction of Lax connection performed in [46] and express it as function of canonical variables. Then we calculate the Poisson brackets of Lax connection for T-dual action and argue for integrability of given theory. In section (4) we study another example of T-dual sigma model that arises by performing T-duality along non-compact direction of AdS 2 . We again calculate the Poisson brackets of Lax connection that is an image of original Lax connection under T-duality. We stress however that T-duality in case of AdS 2 background is special since now T-duality along non-compact direction leads to background that it is again AdS 2 . Then it is natural to presume that this background possesses Lax connection that has the same form as the original one now expressed as a function of T-dual variables. One can then expect that these two Lax connections are related and it was shown for example in [47] this is really true. We plan to discuss the Hamiltonian formulation of these Lax connections and relations between them in forthcoming publication. Finally in section (5) we present the general analysis of the calculation of Poisson bracket between Lax connections in sigma model that is related to the original one by duality that is canonical transformation in phase space. We define the Lax connection in dual model in two steps. We firstly perform the gauge transformation that preserves the flatness of given Lax connection. In the second step we express this Lax connection as a functions of phase space variables that are related to the original ones by canonical transformations. In fact, it is well known that T-duality can be considered as some kind of canonical transformations. We find that the Poisson brackets of dual Lax connections are non-local however their forms again imply that the new theory possesses an infinite number of conserved charges that are in involution.
Let us outline our result. We derive Poisson brackets of Lax connections in sigma models that are T-dual to original integrable models. We argue that the new integrable models contain an infinite number of conserved charges that are in involutions even if the Poisson brackets of Lax connections are non-local and the matrices r, s are functions of phase space variables. We hope that these calculations can be useful for further study of relations between T-duality and integrability.

Review of Principal Chiral Model S 2 and Poisson Bracket of its Lax Connection
In this section we give a brief review of calculation of Poisson bracket of Lax connection of the sigma model on S 2 . The main goal is to demonstrate the difference between rather straightforward calculations given here with respect to the analysis of the Poisson bracket of Lax connection in case of T-dual model that will be presented in next section. We start with the action that governs the dynamics of the string on S 2 where γ αβ , α, β = 0, 1 is world-sheet metric and where σ α , σ 0 = τ , σ 1 = σ are world-sheet coordinates.
We start with observation that this theory possesses Noether currents in the form where df ≡ ∂ α f dσ α . Then it is easy to see that where A, B, . . . = 1, 2, 3 and where the Cartan-Killing form is given by Then using this result we can rewrite the action (2.1) in the form that clearly demonstrates that the dynamics of string on S 2 is governed by principal model action. Further, we observe that the currents (2.3) are flat where the structure constants are defined as Here ǫ ABC is totally antisymmetric with ǫ 123 = −1. In what follows we will be more general and introduce the general coordinates x M on manifold M that in the particular case of M = S 2 are x M = (Θ, Φ). As the next step we introduce E A M in order to write the current Then the conjugate momenta p M defined as δS δ∂τ x M take the form Further we define the current j A P as Then it is easy to calculate the Poisson bracket and also the relations that follow from the fact that the current j A = E A M dx M is flat. In the same way we obtain (2.14) Now we are ready to determine the Poisson brackets of Lax connection for principal chiral model. Note that the Lax connection is defined as where the Hodge dual is defined as for any function f . Further, a, b given in (2.14) depend on spectral parameter Λ as Then using (2.11) and (2.14) we determine the Poisson brackets between the spatial components of Lax connections for two different spectral parameters Λ, Λ ′ Comparing with the equation (A.16) we find Fortunately in this particular case we can rather easily guess the form of the matrices r AB , s AB . In fact, let us presume that the right side of the equation (2.19) can be written in the form Then comparing expressions proportional to ∂ σ δ(σ − σ ′ ) in (2.19) and in (2.21) we obtain where A, B are constants. Inserting these expressions into (2.21) and comparing with right side of (2.19) we find and then using (2.22) we finally obtain It is important that in case of principal chiral model the objects r AB , s AB are constants and depend on spectral parameters Λ, Λ ′ only. On the other hand the situation is much more involved in case of T-dual sigma model.

Poisson Brackets of Lax Connection in T-dual theory on S 2
In this section we determine the Poisson bracket of Lax connections in the sigma model that is related to the sigma model on S 2 by T-duality along the compact U (1) isometry cycle. Since the procedure how to derive T-dual action from sigma model action on S 2 is nicely described in the paper [46] we use results derived there and immediately write the T-dual action where the dual variableΦ is related to Φ through the relation While the original action was SO(3) invariant the manifest symmetry of the T-dual action (3.1) is simply U (1) shift ofΦ. As was argued in [46] the full SO(3) symmetry group is hidden and it is realized non-locally.
As usual in Hamiltonian formalism we firstly determine from the action (3.1) the momenta P Θ , PΦ conjugate to Θ,Φ with corresponding Poisson brackets To proceed further we have to say few words considering the problem how Lax connection behaves under T-duality transformation. It is well known that T-duality transformation (3.2) cannot be directly performed on the currents (2.15) and (2.18) since they depend not only on dΦ but also explicitly on coordinate Φ. Solution of this problem is based on the observation that for any g ∈ G the new current is again flat 6 . Then there exists an element g ∈ SO(3) that transforms the original currents into new ones that depend on Φ only through its derivatives. The forms of the matrix g and corresponding current J ′ were found in [46] with the result (3.6) Then using (3.2) we define T-dual flat currents asĴ A = J ′A (Φ →Φ). Explicitly, their spatial components take the form 6 We will discuss the general procedure in section (5).
or alternatively, as functions of phase space variables (Θ, P Θ ,Φ, PΦ) (3.8) Then using (3.4) we calculate the Poisson brackets of spatial components of Lax connections for two spectral parameters Λ, Λ ′ . After some straightforward calculations we obtain (3.12) As a check, note that A AB (σ, Λ, Λ ′ ), C AB (σ, σ ′ , Λ, Λ ′ ) and B AB (σ, σ ′ , Λ, Λ ′ ) obey the consistency relations (A.10). Using these results we can partially determine the matrices r, s wherer AB is solution of the differential equation (A.13). Unfortunately, due to the fact that A, B, C explicitly depend on the phase space variables it is very difficult to solve this differential equation (A.13) and we were not able to find explicit form ofr AB . On the other hand it is important to stress that the Poisson brackets of Lax connections take the form as in (A.9) and hence following arguments given in appendix we can argue that T-dual sigma model contains an infinite number of conserved charges that are in involution in the sense that their Poisson brackets vanish. In summary, T-dual theory is classically integrable as well in spite of the fact that the Poisson bracket structure is intricate.

Second Example: T-dual AdS 2 String
As the second example of T-dual theory we consider the case of bosonic sigma model on AdS 2 and its T-dual version. Recall that the dynamics of bosonic string on AdS 2 is governed by an action (4.1) With analogy with previous section we introduce three currents Further, it can be shown that these currents are flat where f A BC = − 1 √ 2 ǫ BCD K DA and where the Cartan-Killing form K AB is equal to K AB = diag(−1, 1, 1). Then it is easy to see that the sigma model action (4.1) can be expressed as principal chiral model with corresponding Lax connection Our goal is to develop Hamiltonian formalism for T-dual theory where T-duality is performed along X direction [46] so that T-dual action takes the form where we also introducedỸ defined asỸ It is clear that the action (4.6) again describes dynamics of string on AdS 2 background and hence the Lax connection for given theory is the same as the original one (4.5) when we replace X, Y withX andỸ . On the other hand there exists Lax connection in T-dual background that is related to the original Lax connection by gauge transformations and then by substitutions X, Y →X,Ỹ . This Lax connection was derived in [46] and takes the formĴ (4.8) To proceed further we derive from (4.6) the conjugate momenta (4.9) Then the spatial components of Lax connection expressed as functions of canonical variables are equal toĴ (4.10) Now we are ready to determine Poisson bracket of spatial components of Lax connection. Again, after some calculations we derive the Poisson brackets that have the same form as in (A.5) where the matrices A, B and C are equal to and (4.12) As a check note that the matrices (4.11),(4.12) obey the consistency relations (A.6). Further, we can also determine the matrix s AB however we are not able to fully determine r AB due to the fact that the matrices A, B, C are functions of phase space variables. It is clear that the theory is classically integrable since we can in principle find an infinite number of charges that are in involutions. However the consequence of the non-local nature of T-dual Lax connection is that the matrices r, s now explicitly depend on phase space variables and are non-local. On the other hand the case of AdS 2 is exceptional since we know that its T-dual image is again AdS 2 so that we can find Lax connection corresponding standard principal chiral with constant r and s matrices. We are not going to study the relations between these two Lax connections in this paper. We hope to return to the study of this problem in future publication.

General procedure
In this section we consider general situation when we have principal chiral model with a field g(σ) that maps the string world-sheet into some group G with Lie algebra g. Further we presume that the Lie algebra g has generators T A , A = 1, . . . , dim(g) that obey the relation we can construct a current j in the form and where we introduced sigma model coordinates x M . Then the dynamics of the theory is governed by the action where K AB = Tr(T A T B ). As we reviewed in section (2) the principal chiral model possesses Lax connection J = aj + b * j that is flat where Λ is a spectral parameter. The principal chiral model has an important property that when we perform the gauge transformation from g ∈ G on the original Lax connection we obtain that the new one is again flat (5.7) To proceed further we write the gauge transformation (5.6) in component formalism. Since where generally Ω A C , e A are functions of phase space variables. Our goal is to determine the Poisson bracket of Lax connection in T-dual theory. The first step in this direction is to determine the Poisson bracket of Lax connection J ′ . Using (5.8) we obtain Then, using the arguments above and due to the fact that J C σ is linear in momenta we can presume that , Further, let us presume that Poisson brackets between J A σ (σ, Λ) and J B σ (σ ′ , Γ) take the form Then if we insert this expression into (5.10) we obtain that the Poisson bracket of Lax connections J ′ has the same form as (5.13) where Now we are ready to calculate the Poisson brackets of Lax connection in T-dual theory. Let us denote η I (σ) = x 1 (σ), . . . , x n (σ), p 1 (σ), . . . , p n (σ) , I = 1, . . . , 2n where n is dimension of M and introduce symplectic structure J IJ (σ, σ ′ ) defined as Then the Poisson bracket of two functions F (η), G(η) can be written as .
An important property of T-duality is that it is a sort of canonical transformation. More precisely, let us denote variables in T-dual theory asη I = x M (σ),p M (σ) and presume that they can be expressed as functions of original variables Now if the transformation from η toη is canonical, we have that the matrix preserves the symplectic structure in the sense that Using this fact we immediately obtain η I (σ),η J (σ) η = J IJ (σ, σ) . where η andη are related by cannonical transformation. Then if we apply these considerations to the case of Lax connections in original and T-dual theories we obtain whereÃ AB (η(σ), Λ, Γ) ≡ A ′AB (η(η(σ)), Λ, Γ) , 26) and where in the first step we used (5.14) and in the second one we expressed (5.15) as functions ofη. This result implies that we can express the Poisson bracket of Lax connections in T-dual theory using the known form of original Lax connection J A σ and the known form of e A σ , Ω B A that are in the final step expressed as functions of T-dual variables. Then is it clear from (5.15) that the dual theory is again classically integrable in the sense that there are infinite number of integrals of motion that are in involution. On the other hand the complicated form of matricesÃ,B,C implies that generally the matricess and r are functions of phase space variables. Moreover we can also expect thatr is non-local due to the fact thatr is a solution of the differential equation
(A. 22) In other words we obtain that the theory contains infinite number of conserved charges that are in involution. This fact implies a classical integrability of given theory. 7 For very nice recent discussion, see [61].