PPrepared for submission to JHEP
Integrable Lambda Models And Chern-Simons
Theories
David M. Schmidtt Departamento de F´ısica, Universidade Federal de S˜ao Carlos,Caixa Postal 676, CEP 13565-905, S˜ao Carlos-SP, Brazil
Abstract:
In this note we reveal a connection between the phase space of lambdamodels on S × R and the phase space of double Chern-Simons theories on D × R and explain in the process the origin of the non-ultralocality of the Maillet bracket,which emerges as a boundary algebra. In particular, this means that the (classical) AdS × S lambda model can be understood as a double Chern-Simons theory definedon the Lie superalgebra psu (2 , |
4) after a proper dependence of the spectral parameteris introduced. This offers a possibility for avoiding the use of the problematic non-ultralocal Poisson algebras that preclude the introduction of lattice regularizations andthe application of the QISM to string sigma models. The utility of the equivalence atthe quantum level is, however, still to be explored. [email protected] a r X i v : . [ h e p - t h ] M a y ontents It is by now widely recognized that integrability plays a fundamental role on theAdS/CFT correspondence and that a way to explore the duality more efficiently isto study its underlying integrable structure in a systematic way. One logical strategyto do so is to implement deformations in a consistent mathematical way and then learnmore about the physical system from its response to the deformation. Recently, twodifferent but complementary kinds of deformations defined on the gravity side of theduality have been introduced [4, 7]. Both preserve the integrability of (super)-stringsigma models and are currently known as the eta models [1, 3–5] and the lambda models[2, 6–8]. The works [3–5] and [6–8], respectively, came as generalizations of the originalideas for deforming sigma models introduced by Klimˇc´ık in [1] for the eta models andby Sfetsos in [2] for the lambda models. In the particular case of the
AdS × S Green-Schwarz (GS) superstring, the main property is that its eta/lambda model realize aquantum group deformation of their parent sigma model S-matrix with a q that is realand a root-of-unity [9–11], respectively.Most of the physically interesting integrable field theories (including the ones men-tioned above) are of the so-called non-ultralocal type, a property that poses a majorobstacle to the use of powerful techniques like the algebraic Bethe ansatz and this iswhy a great amount effort has been invested along the years in trying to eliminatethis “pathological” behavior e.g. see [12–19] for several different approaches concern-ing this issue. The most important work dealing successfully with this problem is the– 1 –986 seminal paper by Faddeev and Reshetikhin (FR) [12], in which a (rather ad hoc)ultralocalization method for the SU (2) principal chiral model (PCM) was introducedallowing to exactly quantize the theory within the QISM scheme. Unfortunately, themethod only seemed to work with this case and not with the more interesting PCM’son any Lie group G or the more general sigma models on symmetric spaces F/G . It wasonly in 2012 where real progress was made by Delduc, Magro and Vicedo [18], in whichthe underlying algebraic mechanism behind the ultralocalization method of FR wasdiscovered, generalized and applied to any PCM and sigma model on (semi)-symmetricspaces . Unfortunately, in the case of sigma models on (semi)-symmetric spaces thenon ultralocality is still present albeit in an alleviated way and the introduction of alattice regularization (at quantum level) for the alleviated theories is still not knownbecause of the non-ultralocality persists .One of the main characteristics of the lambda deformation is that it implements theFR mechanism of [18] directly at the Lagrangian level [24] and this is the best we can do(to present knowledge) in handling analytically the non ultra-locality of the integrablefield theory from a world-sheet theory point of view. This means, in particular, thatthe problem is still present so apparently nothing seems to be gained by deforming theoriginal theory in this particular way. However, it is the same deformed theory thatsuggests there is a way out if we give up the world-sheet description.In this work we offer a new approach to deal with the non ultralocality of all knownlambda models, which have recently attracted a lot of attention. The idea is not totackle the problem in 1+1 dimensions, as customary, but rather from a 2+1 dimensionalpoint of view. As we shall see, by changing the dimensionality the problem ceases toexist (for any value of the deformation parameter λ ) and the strategy to do it is toexploit the natural relationship that exist between WZW models and Chern-Simons(CS) theories. We are also able to introduce the spectral parameter in the 2+1 theorygiving it a more prominent role. We expect this approach will provide a novel way totreat the 1+1 integrable field theories that fit within the formulation of lambda modelsbut one of the hopes is to leave open the possibility of generalizing the constructionso that more general theories can be treated in a similar way. For a new but differentapproach to non-ultralocal integrable field theories, see the very recent work [22]. Seealso [23] for another recent application of the QISM to the lambda model of the PCM.The lambda models have two important characteristic properties that are analoguesof similar relations present on ordinary chiral WZW models. They are summarized in The
AdS × S superstring was considered in [20, 21]. To the present, it has been only possible to construct a lattice Poisson algebra that is related tothe Pohlmeyer reduction of the string sigma models [18, 20, 21]. – 2 –he following pair of (on-shell) results [24, 25] m ( z ± ) = P exp (cid:104) ± πk (cid:90) S dσ J ∓ ( σ ) (cid:105) and F = Ψ( z + )Ψ( z − ) − . (1.1)In the first equation, m ( z ) is the monodromy matrix of the 2d theory, z ± = λ ± / ∈ R are two special values of the spectral parameter z and J ± are two currents satisfyingthe algebra of two mutually commuting Kac-Moody algebras. This relation have beenstudied in [24] for bosonic sigma models and after the use of a KM lattice regularizationresults in the presence of a quantum group symmetry with a deformation parameter q that is a root of unity.In the second equation, we have that F is the Lagrangian matrix field entering thedefinition of the lambda model action and Ψ( z ) is the wave function that appears asthe compatibility equation for the Lax pair representation of the equations of motion[25, 41]. A similar decomposition appears for ordinary chiral WZW models but with thevery important difference that for the lambda models the elements Ψ( z ± ) are far frombeing chiral . As it is well known [44–46], conventional WZW models are deeply relatedto 3d Chern-Simons gauge theories and under this connection, the non ultralocality ofthe Kac-Moody chiral algebras of the WZW model rises as a boundary effect afterthe impositions of certain constraints on the phase space. We will see below that thissituation persist also for lambda models but with the added advantage that a spectralparameter can be naturally introduced and that this time it is the Maillet bracket[43] that emerges as boundary algebra. Hopefully, this remarkable relation will revealunexpected connections between integrable string sigma models and gauge theories ofthe CS type that might assist in the quantization of the former theories.The paper is organized as follows. In section 1, we introduce the lambda modelsand emphasize the properties that are important for the topic of the present study. Insection 2, we elaborate on the version of the Chern-Simon theory that, after introductionof the spectral parameter, turns out to be equivalent to the lambda models at theclassical level. We finish with some remarks concerning our approach and mention onproblems to be considered in the near future. This paper is strongly inspired by the results of [26]. Precisely, this decomposition is used in [50] to construct the deformed giant magnon solutions oflambda models. – 3 –
Integrable Lambda Models
In this section we briefly review the most important aspects of the integrable deforma-tions that are of relevance for the present paper. We will restrict the discussion to thespecific example of the lambda model of the Green-Schwarz (GS) superstring on thecoset superspace
AdS × S but also make contact with similar lambda models whenuseful for clarifying purposes.Consider the Lie superalgebra f = psu (2 , |
4) of F = P SU (2 , , |
4) and its Z de-composition induced by the automorphism ΦΦ( f ( m ) ) = i m f ( m ) , f = (cid:77) i =0 f ( i ) , [ f ( m ) , f ( n ) ] ⊂ f ( m + n ) mod 4 , (2.1)where m, n = 0 , , ,
3. From this decomposition we associate the following twistedloop superagebra (cid:98) f = (cid:77) n ∈ Z (cid:16)(cid:77) i =0 f ( i ) ⊗ z n + i (cid:17) = (cid:77) n ∈ Z (cid:98) f ( n ) , (2.2)which is required to exhibit the integrable properties of the theory in terms of thespectral parameter z . Denote by G the bosonic Lie group associated to f (0) = su (2 , × su (4).The lambda model on the semi-symmetric space F/G is defined by the followingaction functional [7] S = S F/F ( F , A µ ) − kπ (cid:90) Σ d σ (cid:104) A + (Ω − A − (cid:105) , k ∈ Z , (2.3)where (cid:104)∗ , ∗(cid:105) = ST r ( ∗ , ∗ ) is the supertrace in some faithful representation of the Liesuperalgebra f , Σ = S × R is the world-sheet manifold parameterized by ( σ, τ ) andΩ ≡ Ω( λ ), where Ω( z ) = P (0) + zP (1) + z − P (2) + z − P (3) (2.4)is the omega projector characteristic of the GS superstring. The P ( m ) are projectorsalong the graded components f ( m ) of f . Above, we have that S F/F ( F , A µ ) = S W ZW ( F ) − kπ (cid:90) Σ d σ (cid:10) A + ∂ − F F − − A − F − ∂ + F − A + F A − F − + A + A − (cid:11) , (2.5) The 1+1 notation used in this paper is: σ ± = τ ± σ, ∂ ± = ( ∂ τ ± ∂ σ ) , η µν = diag (1 , − (cid:15) = 1, δ σσ (cid:48) = δ ( σ − σ (cid:48) ) and δ (cid:48) σσ (cid:48) = ∂ σ δ ( σ − σ (cid:48) ). We also have that a ± = ( a τ ± a σ ) . – 4 –here S W ZW ( F ) is the usual level k WZW model action. The original GS superstringcoupling constant is κ and it is related to k through the relation λ − = 1 + κ /k .From (2.3) we realize that the λ -deformation can be seen as a continuation of the GSsuperstring into a topological field theory defined by the gauged F/F
WZW model.The gauge field equations of motion are given by A + = (cid:0) Ω T − D T (cid:1) − F − ∂ + F , A − = − (Ω − D ) − ∂ − F F − , D = Ad F . (2.6)After putting them back into the action (2.3), a deformation of the non-Abelian T-dual of the GS superstring with respect to the global left action of the supergroup F is produced. A dilaton is generated in the process but we will not consider its effectshere as we are only concerned with the classical aspects of the theory.The F equations of motion, when combined with (2.6) can be written in twodifferent by equivalent ways[ ∂ + + L + ( z ± ) , ∂ − + L − ( z ± )] = 0 , (2.7)where L ± ( z ) = I (0) ± + zI (1) ± + z ± I (2) ± + z − I (3) ± (2.8)is the GS superstring Lax pair that besides satisfy the conditionΦ( L ± ( z )) = L ± ( iz ) . (2.9)Then, the lambda model equations of motion follow from zero curvature condition of L ± ( z ). Above, the I ( m ) ± , are the components of the deformed dual currents defined by I + = Ω T ( z + ) A + , I − = Ω − ( z − ) A − , z ± = λ ± / . (2.10)The flatness of the Lax pair is equivalent to the compatibility condition( ∂ µ + L µ ( z ))Ψ( z ) = 0 , (2.11)where Ψ( z ) is the so-called wave function. This last equation together with (2.6) and(2.8) allow to relate (on-shell) the Lagrangian fields of the lambda model to the wavefunction [25, 41]. For example, F = Ψ( z + )Ψ( z − ) − , A ± = − ∂ ± Ψ( z ± )Ψ( z ± ) − . (2.12) To match with the notation of [25, 50], take κ = 4 πg . – 5 –he spatial component of the Lax pair L σ ( z ) ≡ L ( z ) satisfy L ( z ± ) = ∓ πk J ∓ , (2.13)where the currents J ± obey the relations of two mutually commuting Kac-Moodyalgebras { J ± ( σ ) , J ± ( σ (cid:48) ) } = − [ C , J ± ( σ (cid:48) )] δ σσ (cid:48) ∓ k π C δ (cid:48) σσ (cid:48) . (2.14)Equation (2.13) is valid for all lambda models and as a consequence of this the firstrelation in (1.1) provide conserved Lie-Poisson charges [24]. On the constrained surfacedefined by (2.6) the KM currents take the form J + = k π (Ω T A + − A − ) , J − = − k π ( A + − Ω A − ) (2.15)and are used to relate J ± with the deformed dual currents I ± . This is a particularlyuseful relation because it means the current algebra for I ± follows from the algebra(2.14).By adding to the Lax operator arbitrary z -dependent terms proportional to theHamiltonian constraints (bosonic and fermionic) of the theory and by demanding thatthe condition (2.9) and the equation (2.13) are still valid, we obtain the Hamiltonianor extended Lax operator [8, 25, 41] L (cid:48) ( z ) = − πk ( z − z )( z − z − ) (cid:26) J (0)+ + z − z J (1)+ + z − z J (2)+ + z − z J (3)+ (cid:27) − πk ( z − z − )( z − z − ) (cid:26) J (0) − + z z J (1) − + z z J (2) − + z + z J (3) − (cid:27) . (2.16)Then, as a consequence of the Kac-Moody algebra structure of the theory (2.14), theHamiltonian Lax operator obeys the Maillet algebra { L (cid:48) ( σ, z ) , L (cid:48) ( σ (cid:48) , w ) } = − [ r , L (cid:48) ( σ, z )+ L (cid:48) ( σ (cid:48) , w )] δ σσ (cid:48) +[ s , L (cid:48) ( σ, z ) − L (cid:48) ( σ (cid:48) , w )] δ σσ (cid:48) − s δ (cid:48) σσ (cid:48) , (2.17)which reduce to the two mutually commuting Kac-Moody algebras at the special points z ± . We will deduce this bracket from a Chern-Simons theory point of view and writedown the explicit form of the r / s operators below. It is important to mention that bothGS and hybrid superstring formulations share the same extended Lax operator [8] butdefined in terms of the Lie superalgebras psu (2 , |
4) and psu (1 , | The Kac-Moody algebras are protected and does not change under the Dirac procedure [6] meaningwe can use them on the constrained surface defined by (2.6). – 6 –he last piece of information is related to the imposition of the Virasoro con-straints T ±± ≈
0, which renders the lambda model a string theory . The stress-tensorcomponents of the action (2.3) are given by T ±± = − k π (cid:10) ( F − D ± F ) + 2 A ± (Ω − A ± (cid:11) , (2.18)where D ± ( ∗ ) = ∂ ± ( ∗ ) + [ A ± , ∗ ]. On the surface defined by the gauge field equations ofmotion they reduce to the usual quadratic form albeit in terms of the deformed dualcurrents T ±± = k π ( z − z − ) (cid:10) I (2) ± I (2) ± (cid:11) , (2.19)that in terms of the Lax pair become T ±± = ± k π (cid:10) L ± ( z + ) − L ± ( z − ) (cid:11) . (2.20)From this last expression we can extract the Hamiltonian and momentum densities H = k π (cid:10) L τ ( z + ) L σ ( z + ) − L τ ( z − ) L σ ( z − ) (cid:11) ,P = k π (cid:10) ( L τ ( z + ) + L σ ( z + )) − ( L τ ( z − ) + L σ ( z − )) (cid:11) . (2.21)The expression (2.20) is not unique to the GS superstring and could be considered asa starting point. Indeed, if we take for example the Lax pair for the hybrid superstringon AdS × S given by [8] L + ( z ) = I (0)+ + zI (1)+ + z I (2)+ + z I (3)+ , L − ( z ) = I (0) − + z − I (1) − + z − I (2) − + z − I (3) − , (2.22)which also satisfy (2.9) and make use of (2.20), we do recover the known expressionsfor the stress-tensor T ±± = k π ( z − z − ) (cid:10) I (2) ± I (2) ± + 2 I (1) ± I (3) ± (cid:11) (2.23)but in terms of a different set of deformed dual currents I ± written down in [8]. Thisresult also applies to the PCM lambda model but with a different set of points z ± defined in [24].As we saw above, the lambda models are naturally equipped with two decoupledKac-moody algebras and the Lagrangian field decompose in a rather similar way asthe Lagrangian field in conventional chiral WZW models. This suggest that the known[44–46] relation between WZW models and CS theories could be present for lambdamodels as well and we now proceed to make this connection more precise. The lambda models are also consistent superstring theories at the quantum level, as has beenrecently shown in [51–53] for
AdS n × S n , n = 2 , , Use H = T ++ + T −− and P = T ++ − T −− – 7 – Double Chern-Simons theory
Consider the following double Chern-Simons action functional defined by S CS = S (+) + S ( − ) , (3.1)where S ( ± ) = ± k π (cid:90) M (cid:10) A ( ± ) ∧ ˆ d A ( ± ) + 23 A ( ± ) ∧A ( ± ) ∧ A ( ± ) (cid:11) . (3.2)The ( ± ) sub-index is just a label whose significance will emerge later on, M is a 3-dimensional manifold and A ( ± ) are two different 3-dimensional gauge fields valued inthe Lie superalgebra f . In what follows we will study the generic action S = k π (cid:90) M (cid:10) A∧ ˆ d A + 23 A ∧ A ∧ A (cid:11) , k = ± k for ( ± ) (3.3)to avoid a duplicated analysis.In order to define the Hamiltonian theory of our interest we consider the actionon the manifold M = D × R , where D is a 2-dimensional disc parameterized by x i ,i = 1 , R is the time direction parameterized by τ . It is useful to use radius-angle coordinates ( r, σ ) to describe D as well . In particular, we use σ as a coordinateof ∂D = S that is identified with the S entering the definition of the world-sheetΣ = S × R of the lambda model action in (2.3).Using the decomposition A = dτ A τ + A, ˆ d = dτ ∂ τ + d, (3.4)we end up with the following action functional S = k π (cid:90) D × R dτ (cid:104)− A∂ τ A + 2 A τ F (cid:105) − k π (cid:90) ∂D × R dτ (cid:104) A τ A (cid:105) , (3.5)where F = dA + A is the curvature of the 2-dimensional gauge field A = A i dx i notto be confused with the world-sheet gauge field entering the definition of the action(2.3). Notice that we have omitted the wedge product symbol ∧ in order to simplifythe notation but we can put it back if required. It is also useful to work in terms ofdifferential forms rather than in terms of components.The Lagrangian is then given by L = k π (cid:90) D (cid:104)− A∂ τ A + 2 A τ F (cid:105) − k π (cid:90) ∂D (cid:104) A τ A (cid:105) , (3.6) We do not know if there is a standard name in the literature for this type of action. – 8 –hose arbitrary variation is as follows δL = k π (cid:90) D (cid:104) δA τ F + δA ( DA τ − ∂ τ A ) (cid:105) + k π (cid:90) ∂D dσ (cid:104) δA σ A τ − δA τ A σ (cid:105) , (3.7)where D ( ∗ ) = d ( ∗ ) + [ A, ∗ ] is a covariant derivative. From this we find the bulkequations of motion F = 0 , ∂ τ A − DA τ = 0 , on D (3.8)stating that the 3-dimensional gauge field A is flat, as well as the boundary equationsof motion (cid:104) δA σ A τ − δA τ A σ (cid:105) = 0 on ∂D, (3.9)which must be solved consistently in order to obtain the field configurations minimiz-ing the action. A possible useful solution to the boundary equations of motion is todemand that A τ = ξA σ , for some constant factor ξ or the more general boundary con-ditions considered in [68]. However, as we shall see, for the lambda models they areautomatically satisfied.The Lagrangian (3.6) is already written in Hamiltonian form. The Hamiltonianincludes a boundary term and it is given by H = − k π (cid:90) D (cid:104) A τ F (cid:105) + k π (cid:90) ∂D (cid:104) A τ A (cid:105) . (3.10)The fundamental Poisson brackets extracted from (3.6) are found to be { A i ( x ) , A j ( y ) } = 2 πk (cid:15) ij C δ xy , { A τ ( x ) , P τ ( y ) } = C δ xy (3.11)and for arbitrary functions of A i , they generalize to { F ( A ) , G ( A ) } = 2 πk (cid:15) ij (cid:90) D d x δF ( A ) δA Ai ( x ) η AB δG ( A ) δA Bj ( x ) . (3.12)The definition of the functional derivatives δF/δA to be used in the bracket above issubtle because of the presence of boundaries [56, 58]. To find them, we start with thevariation δF ( A ) and subsequently find a way to write the result as an integral over thedisc D only. For example, for the Hamiltonian we find that δH = − k π (cid:90) D (cid:104) δA τ F + δADA τ (cid:105) − k π (cid:90) ∂D dσ (cid:104) δA σ A τ − δA τ A σ (cid:105) . (3.13) The 2+1 notation used in this paper is: (cid:15) = 1 and δ xy = δ (2) ( x − y ). For the Lie (super)-algebrawe define η AB = (cid:104) T A , T B (cid:105) , C = η AB T A ⊗ T B and u = u ⊗ I, u = I ⊗ u , etc. For arbitrary functions of A τ , the Poisson bracket is obvious and will not be written. – 9 –hen, to cancel the boundary term we must use the boundary equations of motion(3.9) in order to obtain the desired well-behaved result δH = − k π (cid:90) D (cid:104) δA τ F + δADA τ (cid:105) . (3.14)Now we are ready to consider the Dirac procedure. There is a primary constraint P τ ≈ , (3.15)whose stability condition leads to a secondary constraint F ≈ , (3.16)which is nothing but the first bulk equation of motion in (3.8). To study the secondaryconstraints we better introduce the general quantity G ( η ) = k π (cid:90) D (cid:104) ηF (cid:105) (3.17)and compute its variation assuming that the test functions η are independent of thephase space variables { A τ , A i } . We find that δG ( η ) = k π (cid:90) D (cid:104) δADη (cid:105) + k π δ (cid:90) ∂D (cid:104) ηA (cid:105) . (3.18)Then, the constraint with a well-defined functional derivative is actually the shiftedone G ( η ) = G ( η ) + G ( η ) , G ( η ) = − k π (cid:90) ∂D (cid:104) ηA (cid:105) . (3.19)Using this we can show that the action of the shifted constraint is a gauge transforma-tion δ η A ≡ { A, G ( η ) } = − Dη (3.20)and that the second equation of motion in (3.8) can be written as a special gaugetransformation ∂ τ A = δ ( − A τ ) A, (3.21)because δH = δG ( − A τ ). Then, despite of the fact that A τ is a phase space coordinateboth quantities turn out to generate the same action.The constraint algebra is now given by the bracket { G ( η ) , G ( η ) } = k π (cid:90) D (cid:104) DηDη (cid:105) = G ([ η, η ]) + k π (cid:90) ∂D (cid:104) ηdη (cid:105) (3.22)– 10 –fter some standard manipulations, showing that when the test functions or theirderivatives do not vanish on ∂D , the shifted constraints are actually second class be-cause of the presence of the boundary. On the other hand, the former constraints G ( η )are also second class [56] for the same kind of test functions and this means that noextra gauge-fixing conditions are required allowing to introduce a Dirac bracket forthe constraints F ≈ { G ( η ) , H } = − G ([ η, A τ ]) − k π (cid:90) ∂D (cid:104) ηdA τ (cid:105)≈ − k π (cid:90) ∂D (cid:104) ηDA τ (cid:105) . (3.23)Using this result we can find the time evolution of the secondary constraints. We obtain dG ( η ) dτ ≈ { G ( η ) , H } + (cid:90) ∂D δG ( η ) δη A ∂ τ η A ≈ k π (cid:90) ∂D dσ (cid:104) ηF τσ (cid:105) − k π (cid:90) ∂D dσ∂ τ (cid:104) ηA σ (cid:105) (3.24)and after pulling ∂ τ outside the integral as ddτ , we get the final result dG ( η ) dτ ≈ k π (cid:90) ∂D dσ (cid:104) ηF τσ (cid:105) , (3.25)which vanish if we demand that F τσ ≈ ∂D. (3.26)This is the second bulk equation of motion in (3.8) (or constraint) with i = σ extendedto ∂D . We will come back to this important boundary constraint later on. There areno tertiary constraints.Following [56], we now write down the non-zero Poisson algebra for the quantities G , G , G on the constraint surface F ≈
0. It is given by { G ( η ) , G ( η ) } ≈ − k π (cid:90) ∂D dσ (cid:104) η, D σ η (cid:105) , { G ( η ) , G ( η ) } ≈ k π (cid:90) ∂D dσ (cid:104) η, D σ η (cid:105) , { G ( η ) , G ( η ) } ≈ k π (cid:90) ∂D dσ (cid:104) η, D σ η (cid:105) . (3.27) It is important to mention at this point that G and G separately also have well-defined functionalvariations, as shown in [56] after a careful treatment of boundary terms. – 11 –n order to impose F ≈ { G ( η ) , G ( η ) } ∗ = k π (cid:90) ∂D dσ (cid:104) η, D σ η (cid:105) , { G ( η ) , G ( η ) } ∗ = k π (cid:90) ∂D dσ (cid:104) η, D σ η (cid:105) , (3.28)which are consistent with setting F = 0 strongly. From this follows that only theboundary contribution G ( η ) remains. Then, on the constrained surface, (3.20) takesthe form [56] δ η A ≡ { A, G ( η ) } ∗ = − Dη, (3.29)which in turn imply that the second set of equations of motion in (3.8) can be writtenas ∂ τ A = δ ( − A τ ) A = { A, H ∗ } ∗ , (3.30)because δH ∗ = δG ( − A τ ). In showing this last result we have used the boundary equa-tions of motion (3.9) as required before and the restriction of (3.10) to the constrainedsurface given by H ∗ = k π (cid:90) ∂D dσ (cid:104) A τ A σ (cid:105) . (3.31)Then, the time evolution on the constraint surface takes the correct form under theDirac bracket but now in terms of the boundary Hamiltonian.To identify the reduced phase space coordinates we replace (3.11) by its Diracbracket , but this is equivalent to pulling the symplectic form associated to (the firstbracket in) (3.11) back to the constraint surface [45, 46]. Namely, ω ∗ = k π (cid:90) D (cid:104) δA ∧ δA (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) A = − d ΨΨ − = − k π (cid:90) ∂D dσ (cid:10) δA σ ∧ D − σ δA σ (cid:11) . (3.32)The Poisson bracket that follows from this reduced symplectic form is equivalentto (3.28) and after eliminating the test functions, we reveal the Kac-Moody algebrastructure { A σ ( σ ) , A σ ( σ (cid:48) ) } ∗ = 2 πk (cid:0) [ C , A σ ( σ (cid:48) )] δ σσ (cid:48) + C δ (cid:48) σσ (cid:48) (cid:1) (3.33)of the associated WZW model on ∂D . In this sense we say that the phase space infor-mation of the CS theory is now completely stored on its boundary theory. Indeed, the We will not consider phase space functionals depending on P τ . The symplectic form operator is formally identified as ˆ ω = D − σ with an associated Poissonoperator given by ˆ θ = D σ that is responsible for the Kac-Moody algebra structure. – 12 –educed phase space is described by the data ( A σ | ∂D , H ∗ , {· , ·} ∗ ). The time evolution of A σ can be put in Hamiltonian form and it is given precisely by the boundary constraints(3.26), which as we shall see below are equivalent to the string lambda model equationsof motion. What we have presented here is nothing but the Hamiltonian version of the(well-known) relation that exists between the CS and the WZW theories that is foundin the literature but in a different guise.We are now ready to introduce the dependence of the spectral parameter z . Notsurprisingly, the ( ± ) sub-index introduced above make reference to the two specialpoints z ± = λ ± / in the complex plane parameterized by z and introduced in thelast section. We now make use of the twisted loop superalgebra structure (2.2) andconsider the problem of finding a z -dependent 2-dimensional gauge field A ( z ) on thedisc D satisfying the following two conditions . A ( z ± ) = A ( ± ) and Φ( A ( z )) = A ( iz ) . (3.34)The answer we will consider here (recall that A ( z ) = A i ( z ) dx i , i = 1 ,
2) is given by A ( z ) = f − ( z )Ω( z/z + ) A (+) − f + ( z )Ω( z/z − ) A ( − ) , f ± ( z ) = ( z − z ± )( z − z − ) , (3.35)where Ω( z ) = P (0) + z − P (1) + z − P (2) + z − P (3) (3.36)is another projector not to be confused with the one defining the GS lambda model(2.4) above. Actually, this same projector appears for both the GS [25] and the hybridsuperstring [8] and leads to the same Maillet bracket when (2.16) or (3.33) is used.Then, both superstring formulations are equivalent at this level of analysis.Using this z -dependent gauge field we can gather both Poisson brackets on the LHSof (3.11) into a single interpolating one { A i ( x, z ) , A j ( y, w ) } = − s ( z, w ) (cid:15) ij δ xy , (3.37)which, as we shall see, is the precursor of the Maillet bracket. From here we can appreci-ate that it is the Chern-Simons Poisson structure ( { A i , A j } ≈ (cid:15) ij and not { A i , A j } ≈ δ ij )the one responsible for the non skew-symmetry of the R -matrix entering the Mailletbracket and the very source of its non ultralocality. In this calculation we face exactlythe same situation of [8] and find that s ( z, w ) = − z − w (cid:80) j =0 { z j w − j C ( j, − j )12 ϕ − λ ( w ) − z − j w j C (4 − j,j )12 ϕ − λ ( z ) } , (3.38) Here we are considering only the horizontal fields A ( ± ) , but it works exactly the same way for A ( ± ) so we can consider A ( z ) instead from the beginning and then restrict it to the disc. – 13 –here ϕ λ ( z ) = kπ ( λ − − λ )( z − − z ) − ( λ − − λ ) (3.39)is the lambda deformed twisting function. Notice that the two special points z ± = λ ± / are poles of ϕ λ ( z ). In retrospective, we realize that our theory (3.2) actually consist oftwo Chern-Simons theories with opposite levels attached to the poles z ± of (3.39) inthe complex plane or the Riemann sphere after compactification.The symmetric operator s ( z, w ) satisfy s ( z ± , z ± ) = ∓ πk C = − πk C , s ( z ± , z ∓ ) = 0 (3.40)as required for the Poisson algebra (3.37) to reduce to (3.11) at the poles. It also satisfylim λ → s ( z, w ) = − πk C (00)12 (3.41)but we still do not have a proper interpretation for this limit which corresponds tothe ultra-localization limit of the lambda models and that is deeply related to thePohlmeyer reduction of the AdS × S GS superstring [42]. As customary, we will referto the limits λ → λ → F ( A ) = ( F, A ) ϕ λ , lim t → ddt F ( A + tX ) = ( dF, X ) ϕ λ , (3.42)the Poisson bracket (3.37) generalize to { F ( A ) , G ( A ) } = ( R ( dF ) , dG ) ϕ λ + ( dF, R ( dG )) ϕ λ , (3.43)where R = ± (Π ≥ − Π < ) is the usual AKS R -operator defined in terms of the projectorsΠ that act on elements of ˆ f . The definitions are as follows: For a z -dependent 2-formon D constructed from the pair X and Y , we have( X, Y ) ϕ λ = (cid:90) D (cid:104) X ( x ) , Y ( x ) (cid:105) ϕ λ , (3.44)where (cid:104) X ( x ) , Y ( x ) (cid:105) ϕ λ = (cid:73) dz πiz ϕ λ ( z ) (cid:104) X ( x, z ) , Y ( x, z ) (cid:105) (3.45)is the twisted inner product on the loop superalgebra (cid:98) f for fixed x . For example, thefunctions F ( ± ) ( A ) = (cid:90) D (cid:10) A ( ± ) ∧ η ( ± ) (cid:11) = (cid:90) D d x (cid:10) A i ( ± ) η j ( ± ) (cid:11) (cid:15) ij , (3.46)– 14 –here η ( ± ) are two test 1-forms can be written in terms of (3.44) as F ( ± ) ( A ) = ( η ( ± ) , A ) ϕ λ , (3.47)with η ( ± ) ( z ) = ϕ − λ ( z ) z ± ( z − + z ∓ ) { η (0)( ± ) + zz ∓ η (1)( ± ) + z z ∓ η (2)( ± ) + z z ∓ η (3)( ± ) } . (3.48)Now, because of the functions F ( ± ) ( A ) are linear in A ( ± ) , their differentials are dF ( ± ) ( z ) = η ( ± ) ( z ) . Using these expressions in (3.43), we recover the first expression in (3.11).Now we can compute the z -dependent boundary algebra after imposing the con-straints (3.16) strongly. We take i = σ in (3.35) and use (3.33). Explicitly, A σ ( z ) = ( z − z − )( z − z − ) (cid:26) A (0) σ (+) + z z A (1) σ (+) + z z A (2) σ (+) + z + z A (3) σ (+) (cid:27) − ( z − z )( z − z − ) (cid:26) A (0) σ ( − ) + z − z A (1) σ ( − ) + z − z A (2) σ ( − ) + z − z A (3) σ ( − ) (cid:27) . (3.49)As a consequence of the Kac-Moody algebra structure (3.33) and once more following[8], we obtain the Maillet bracket { A σ ( σ, z ) , A σ ( σ (cid:48) , w ) } ∗ = − [ r , A σ ( σ, z )+ A σ ( σ (cid:48) , w )] δ σσ (cid:48) +[ s , A σ ( σ, z ) − A σ ( σ (cid:48) , w )] δ σσ (cid:48) − s δ (cid:48) σσ (cid:48) , (3.50)where r ( z, w ) = 1 z − w (cid:80) j =0 { z j w − j C ( j, − j )12 ϕ − λ ( w ) + z − j w j C (4 − j,j )12 ϕ − λ ( z ) } , (3.51)is the anti-symmetric part of the R -matrix. This is the same algebra obtained with theextended Lax operator (2.16) and we now identify A σ ( z ) = L (cid:48) ( z ). The bracket satisfythe Jacobi identity and reduce to (3.33) at the poles. The sine-Gordon limit (3.41)applied to (3.50) is a continuous version of the alleviating mechanism introduced in[18] so the non ultralocality of the Maillet algebra is still present for (semi)-symmetriccosets. Then, in order to suppress the δ (cid:48) σσ (cid:48) completely for any value of λ , we must goto a higher dimension.Alternatively, by setting r = 12 ( R − R ∗ ) , s = −
12 ( R + R ∗ ) , (3.52) The R -operator with the minus sign is the one that reproduce the first Poisson bracket expressionin (3.11) explicitly. – 15 –e can write { A σ ( σ, z ) , A σ ( σ (cid:48) , w ) } ∗ = − [ R , A σ ( σ, z )] δ σσ (cid:48) + [ R ∗ , A σ ( σ (cid:48) , w )] δ σσ (cid:48) + ( R + R ∗ ) δ (cid:48) σσ (cid:48) , (3.53)where R ( z, w ) = 2 z − w (cid:80) j =0 z j w − j C ( j, − j )12 ϕ − λ ( w ) , R ∗ ( z, w ) = R ( w, z ) . (3.54)For arbitrary functions of A σ , (3.50) generalize to { F, G } ∗ ( A σ ) = − ( A σ , [ dF, dG ] R ) ϕ λ + ω ( R ( dF ) , dG ) ϕ λ + ω ( dF, R ( dG )) ϕ λ , (3.55)where [ ∗ , ∗ ] R is the R -bracket on (cid:98) f and ω ( X, Y ) ϕλ = (cid:90) ∂D dσ (cid:104) X ( σ ) , ∂ σ Y ( σ ) (cid:105) ϕ λ (3.56)is the co-cycle. The only difference when compared to (3.44) is that the inner productintegration is now performed on ∂D , the dσ is written explicitly and the X, Y areordinary functions on ∂D . Namely,( X, Y ) ϕ λ = (cid:90) ∂D dσ (cid:104) X ( σ ) , Y ( σ ) (cid:105) ϕ λ . (3.57)The bracket (3.50) can, alternatively, be written in the form { F, G } ∗ ( A σ ) = ( R ( dF ) , ˆ D σ dG ) ϕ λ + ( dF, ˆ D σ R ( dG )) ϕ λ , (3.58)where ˆ D σ ( ∗ ) = ∂ σ ( ∗ ) + [ A σ ( z ) , ∗ ]. From this we identify ˆ θ ( z ) = ˆ D σ ◦ R + R ∗ ◦ ˆ D σ ,which is the analogue of the ˆ θ in (3.33) (see footnote (15)). Following the same stepswe realize that (3.55) is the z -dependent extension of the Dirac bracket associated to(3.43) after imposing the constraints F ( z ± ) = 0. A comment is in order. Notice that weare referring to (3.55) as an extension of the Dirac bracket because F ( z ) (the curvatureof A ( z )) reproduce the correct Hamiltonian constraints only when it reach the poles z ± . In order to find a proper z -dependent constraint (if any), we probably would needto lift the action (3.1) to the loop superalgebra ˆ f and run the Dirac procedure againbut as we have seen, the introduction of the spectral parameter is rather innocuous anddoes not introduce new degrees of freedom or fields so no new constraints are expectedbeyond those attached to the points z ± . However, by an abuse of notation we will keepthe ∗ on (3.55). Up to a global minus sign this is the same Maillet braket constructed in [41]. – 16 –ow we are in the position to interpret the boundary equations of motion (3.9).First we note that the link between the Kac-Moody algebras (3.33) and (2.14) is throughthe relation A σ ( ± ) = ∓ πk J ∓ = L σ ( z ± ) , (3.59)where we have used (2.13) in the last equality. Now, the obvious solution to theboundary equations of motion is to identify A τ ( ± ) = L τ ( z ± ) . (3.60)To see why, we rewrite (3.9) in the form (cid:15) µν (cid:104) δ L µ ( z + ) L ν ( z + ) − δ L µ ( z − ) L ν ( z − ) (cid:105) = 0 (3.61)and use the fact that the product (cid:104) δ L ± ( z ) L ∓ ( z ) (cid:105) is independent of the spectral pa-rameter either for the Green-Schwarz or the hybrid superstring Lax Pairs (2.8) and(2.22), respectively. The PCM Lax pair also satisfy (3.61) trivially. Thus, for thelambda models the CS boundary equation of motion (3.61) is just an identity. Theexplicit transformation between the components of the CS gauge field on ∂D × R andthe lambda model gauge field on Σ is A σ ( ± ) = (cid:26) A + − Ω A − Ω T A + − A − , A τ ( ± ) = (cid:26) A + + Ω A − Ω T A + + A − . (3.62)The relations (3.59) and (3.60) allow to extract an important piece of informationfrom the boundary constraints (3.26) if we write them in the form[ ∂ + + L + ( z ± ) , ∂ − + L − ( z ± )] = 0 . (3.63)The first conclusion is that they are precisely the lambda model Euler-Lagrange equa-tions of motion (2.7) of the action (2.3), which also follows from a Lax pair zero cur-vature condition. The second conclusion is that they imply the conservation of thePoisson-Lie charges m ( z ± ) in (1.1) (see [24]). The boundary degrees of freedom of thedouble CS theory on psu (2 , |
4) are described by the
AdS × S lambda model action(2.3). This conclusion also apply to the other models we have considered so far.It is important to realize that the identification between (3.50) and (2.17) holds forthe extended Lax operator. At this point the result is rather generic (recall we onlyrequired (3.34) in the construction) and in order to consider a particular lambda modelthe omega projector Ω must be specified as it determines the Hamiltonian constraint– 17 –tructure of the theory under the Dirac procedure. In other words, it determines thedecomposition L (cid:48) ( z ) = L ( z ) + constraints , (3.64)from where the current algebra of the deformed dual currents I ± can be computed.It matches the one found by using the direct relation (2.15). See [60, 61] for the theconventional GS superstring and [8] for the lambda model of the hybrid superstring. Itis remarkable that the CS theory reproduce the Hamiltonian Lax connection as it hasinteresting properties, see [61] for the specific case GS formulation in relation to theinvolution properties of the charges extracted from the monodromy matrix. Notice thatthe twisted loop and the Kac-moody superalgebras combined are the ones responsiblefor introducing the R -matrix with spectral parameter.There is a certain amount of freedom in the construction of the current (3.35).For instance, one could consider several copies of the Chern-Simons actions (3.2) inthe definition of the theory involving different WZW levels k (cid:48) s , which would alter thefirst condition in (3.34) or we can also consider a consistent algebraic modification tothe second condition in (3.34). Either case, this could lead to more general twistingfunctions and r / s tensors and to novel multiparametric lambda deformations of stringsigma models. An example of a consistent modification of the second condition, when f is a bosonic Lie algebra, is the PCM. For this case, the loop superalgebra has Φ = I ,i.e. no Z grading and L (cid:48) ( z ) = L ( z ) = f − ( z ) J + + f + ( z ) J − , (3.65)i.e. no first class constraints. The explicit form of the functions f ± ( z ) for this casecan be found by working out explicitly the Lax pair representation. The r / s tensorsas well as the twisting function ϕ λ ( z ) and their poles z ± become those of the lambdadeformation of the PCM [24].On the constrained surface, the complete Hamiltonian (3.31) is given by (we dropthe ∗ ) H = k π (cid:90) ∂D dσ (cid:10) A τ ( z + ) A σ ( z + ) − A τ ( z − ) A σ ( z − ) (cid:11) , (3.66)which should be compared with the lambda model Hamiltonian in (2.21). Mimicking(2.21), we define the momentum generator P = k π (cid:90) ∂D dσ (cid:10) ( A τ ( z + ) + A σ ( z + )) − ( A τ ( z − ) + A σ ( z − )) (cid:11) , (3.67)which commutes with H under the bracket (3.33). The opposite signs of the levels at( ± ) are important in showing this. We can recover the Virasoro algebra structure ofthe lambda models if we define the usual T ±± components as in (2.20).– 18 –inally, the boundary equations of motion (3.9) can be understood as a conditiondictating the form of the Lax pair and the boundary constraints (3.26) as a conditiondictating the dynamics of the system because of their equivalence to the lambda modelEuler-Lagrange equations of motion. This is precisely the content of equation (2.11)which summarizes the integrability properties of the system. The flatness condition aswell as the analytic properties of L ± ( z ) are known to be preserved by the action ofthe group of dressing transformations [25, 62, 63] that can be seen now as an infinitedimensional symmetry group of the boundary theory. The main goal of this paper was to show how the lambda model (2.3) can be refor-mulated as a double Chern-Simons theory (3.1) and how the Lax pair representationand the Maillet bracket structure of the lambda model phase space emerge from theCS theory point of view. The strategy is to trade the non-ultralocal Maillet bracket(3.50) by the ultralocal CS bracket (3.37) at the expense of introducing a couple ofnew constraints (3.16), so the price to pay for the elimination of the problematic δ (cid:48) σσ (cid:48) term is to deal with the quadratic second class constraints F ( z ± ) ≈ T ±± ≈ λ -deformed BMN vacuum solution under theaction of the dressing group Ψ( z ) = χ ( z )Ψ ( z ).One may wonder what is to be gained in complicating even more the phase spacestructure of the string lambda models by introducing the constraints F ( z ± ) ≈ λ -models (i.e. PCM λ , F/G λ , GS on AdS × λ S and hybrid on AdS × λ S ) but also to do it for any (generic) value of the deformationparameter λ . On the other hand, the new theory being of the CS type can, in prin-ciple, be quantized in a number of ways. In particular, by employing a (disc) latticealgebra regularization [30–34] or by a path integral approach [44, 47–49]. However,for superstrings we have the added complication that the CS theories are defined onLie superalgebras, which is not a common feature of conventional CS theories . Theproblem of quantizing our Hamiltonian double CS theory in the presence of the spec- Superalgebra CS theories have been considered before in the literature albeit in a different context.See e.g. [36, 37]. – 19 –ral parameter, i.e. the quantization of the Poisson bracket (3.37), is currently underinvestigation [35] based on the combinatorial quantization approach of [30–34].Several natural questions raise from these first steps and in what follows we mentionsome of them:One potential application of this approach would be to study finite size effects. For r → ∞ , the boundary decompactifies and Σ = S × R → R , . In this situation, thelambda model action must be carefully modified along the lines of [54, 55] in order toaccommodate the new boundary conditions. In this limit though, asymptotic statesand their S-matrix can be defined but for finite r (to our knowledge) not much is known.It would be enlightening to study what the CS theory could tell us about the quantumintegrability of the 1+1 theory for any value of r . A strategy for quantization would beto quantize the ultralocal theory on the disc and afterwards project the quantum theoryto the boundary in some sort of holographic way (by imposing all the constraints). Thisis opposite to the usual symplectic reduction approach of first enforcing the constraints F = 0 classically and then quantizing the remaining degrees of freedom.The fundamental objects of our double CS theory would be the Wilson loops withspectral parameter W R ( C ; z ) = (cid:10) P exp (cid:0) − (cid:73) C A ( z ) (cid:1)(cid:11) R , (4.1)for C a knot in M = D × R and R a particular representation. If C is a horizontalcurve constrained to ∂D and wrapping it once, then we obtain W R ( S ; z ) = (cid:10) P exp (cid:0) − (cid:90) S dσ L (cid:48) ( σ ; z ) (cid:1)(cid:11) R (4.2)that is related to the monodromy of the extended Lax operator (2.16) and if we take z = z ± , then we get W R ( S ; z ± ) = (cid:10) P exp (cid:0) ± πk (cid:90) S dσ J ∓ ( σ ) (cid:1)(cid:11) R (4.3)that is related to the monodromy of the two Kac-Moody currents J ± as in (1.1),leading to quantum groups [24, 26]. We should then expect a natural affine quantumgroup symmetry enhancement in our theory in terms of a quantum R ( z )-matrix that(hopefully) is related to (3.54) in the classical limit. Recently, in [64] it is shownthat this enhancement do occur for the eta models and this is done by expanding themonodromy matrix around the poles of the eta-deformed twisting function ϕ η ( z ) so it Other important objects would be the vertical z -dependent Wilson lines but at this moment theirmeaning is less clear. – 20 –s reasonable to expect that this must be true for the lambda models as well as boththeories are, in a sense, complementary. Another issue is related to the computation ofthe (classical/quantum) algebra of z -dependent Wilson loops (4.1) defined on horizontalcurves corresponding to the continuation of (4.2) into the interior of D . At this point itis too premature to make strong statements about its properties or even its existence butthe results of [40], where a slightly similar situation is considered, suggest this algebracan be found precisely by exploiting the lattice simulation approach of [30–34].From (3.37) and (3.43), we realize that the Poisson structure is related to thesymmetric operator R + R ∗ . This suggest a possible lift of the action (3.5) to thetwisted loop superalgebra ˆ f in terms of the inner product (3.44). The kinetic term inthe Lagrangian (3.6) should, in principle, be replaced by something of the form L ∼ (cid:90) D (cid:10) A ∧ ( R + R ∗ ) − ∂ τ A (cid:11) ϕ λ (4.4)but we have not succeeded in showing it. In any case, it would be interesting to study ifthere is a connection of our CS formulation with the CS construction of lattice modelspresented in [65–67]. In particular, if the action (2.8) of the paper [65] could be relatedto our action (3.1) for a 1-form ω ∼ dzϕ λ ( z ) /z .A natural variation of our construction would be to investigate if the formulationin which the actions S (+) and S ( − ) are complex conjugated could be related to the etamodels and if the results of [68] could be applied to relate both formulations.The last question is how enters the lambda model dilaton field in the CS formulationafter quantization is performed.We will report on some of these questions elsewhere. Acknowledgements
The author would like thank T. J. Hollowood and J. L. Miramontes for valuable dis-cussions and collaboration.
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