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Pseudoduality and Conserved Currents in Sigma Models 9 0 0 2 Mustafa Sarisaman∗ n a J 7 2 Department of Physics ] University of Miami h t P.O. Box 248046 - p Coral Gables, FL 33124 USA e h [ 2 v Monday, April 23, 2007 5 0 6 3 . Abstract 4 0 7 We discuss the current conservation laws in sigma models based 0 on a compact Lie groups of the same dimensionality and connected to : v each other via pseudoduality transformations in two dimensions. We i X show that pseudoduality transformations inducean infinitenumber of r nonlocal conserved currents on the pseudodual manifold. a ∗[email protected] 1 1 Introduction In this article we investigate some properties of conserved currents on target space manifolds that are pseudodual to each other, following a method dis- cussedin[1]wefindaninfinitenumber ofconservationlawsinthepseudodual manifold. We work out the currents in case of pseudodualities [2, 3] between the sigma model on an abelian group and a strict WZW sigma model [4] on a compact Lie group [5, 6] of the same dimensionality. We specialize to the case of the abelian group U(1)×U(1)×U(1), and of the Lie group SU(2). The strict WZW model is the model with the Wess-Zumino term normal- ized so that the canonical equations of motion are given by ∂ (g−1∂ g) = 0, − + where g is a function on spacetime taking values in some compact Lie group G. Pseudoduality is defined as the on shell duality transformation mapping the solutions to the equations of motion for different sigma models. This transformation preserves the stress energy tensor though it is not a canonical transformation. We know [1, 7] that if we are given a sigma model on an abelian group, and a strict WZW sigma model on a compact Lie group, there is a duality transformation between these two manifolds that maps solutions of the equations of motion of the first manifold into the solutions of the equations of motion of the second manifold. Solutions of the equations of motions allow us to construct holomorphic [5] nonlocal conserved currents on these manifolds. Pseudoduality relations provide a way to form pseudodual currents, and we show that these currents are conserved. Let Σ be two dimensional Minkowski space, and σ± = τ ± σ be the standard lightcone coordinates. Using maps x : Σ → M and x˜ : Σ → M˜ we may write pseudoduality relations as x˜ (σ) = +T(σ)x (σ) (1.1) + + x˜ (σ) = −T(σ)x (σ) (1.2) − − where T(σ) belongs to SO(n), and is a function of σ. Let M = G be a compact Lie group of dimension n with an Ad(G)- invariant metric, and g : Σ → G. We define the basic nonlocal conserved currents J(L) = (g−1∂ g) and J(R) = (∂ g)g−1 on the tangent bundle of + + − − G. What we demonstrate is that we can take these currents, and using the pseudoduality relations (1) we obtain currents on G (not G˜) and these currents are conserved. We would like to search for infinitely many conservation laws[8, 9, 10] on pseudodual manifolds. We first concentrate on a simple case, where M = 2 G = U(1)×U(1)×U(1) is an abelian group and M˜ = G˜ is SU(2). We show that infinite number of conservation laws of free scalar currents on G enable us to construct infinite number of pseudodual current conservation on G˜ by means of isometry preserving orthogonal map T between tangent bundles of these manifolds. We next focus our attention on a more complicated case, where M = G is the Lie group SU(2) and M˜ = G˜ is U(1)× U(1) × U(1). We find nonlocal conserved currents on G and construct pseudodual free currents on G˜ using pseudoduality relations. We show that pseudodual free scalar currents on G˜ gives us infinite number of conservation laws. 2 Pseudodual Currents : Simple Case WetakeM asanabeliangroup,andtheequationsofmotionbecome∂2 φi = +− 0, where φ is free massless scalar field. Currents on the tangent bundle of M are hence given by J(L) = (∂ φi)X and J(R) = (∂ φi)X , where {X } is a + + i − − i i basisfortheabelianLiealgebra. Wenoticethatthesecurrents areconserved, ∂ J(L) = ∂ J(R) = 0. Now we take M˜ as a compact Lie group of the same − + + − dimensionality with an Ad(G)-invariant metric. {X˜ } is the orthonormal i basis for the Lie algebra of G˜ with bracket relations [X˜ ,X˜ ] = f˜kX˜ , where i j G˜ ij k the structure constants f˜ are totaly antisymmetric in ijk. Using the map ijk g˜ : Σ → M˜ we may write equations of motion as ∂ (g˜−1∂ g˜) = 0. Currents − + onthismanifoldaredefinedbyJ˜(L) = (g˜−1∂ g˜)iX˜ andJ˜(R) = [(∂ g˜)g˜−1]iX˜ . + + i − − i Again, by virtue of equations of motion we observe that these currents are conserved, ∂ J˜(L) = ∂ J˜(R) = 0. − + + − To construct pseudodual currents on the manifold M we make use of the pseudoduality conditions. The pseudoduality relations between the sigma model on an abelian group and a strict WZW sigma model on a compact Lie group of the same dimension are (g˜−1∂ g˜)i = +Ti∂ φj (2.1) + j + (g˜−1∂ g˜)i = −Ti∂ φj (2.2) − j − where T is an orthogonal matrix and g˜−1dg˜ = (g˜−1dg˜)iX˜ . i Taking ∂ of the first equation (2.1) we conclude that T is a function − of σ+ only. Taking ∂ of the second equation (2.2) gives us the differential + equation for T 3 [(∂ T)T−1]i = −f˜i Tk∂ φl (3) + j kj l + where we used the antisymmetricity of f˜ at right hand side of equation. ikj To get pseudodual currents on the manifold M, we first solve this differential equation for T, and then plug this into pseudoduality equations with an initially given ∂ φi and from the pseudodual currents we find that these ± currents are conserved. 2.1 An Example We consider the sigma model based on the product group U(1)×U(1)×U(1) for M and a strict WZW model based on group SU(2) for M˜. We may write a point on the sigma model to M as φiX , where i = 1,2,3 and {X } are i i basis. Equations of motions are ∂2 φi = 0. Currents may be written as +− J(L) = (∂ φi)X and J(R) = (∂ φi)X . We learn from equations of motions + + i − − i that these currents are conserved. We denote any element in G˜ as g˜ = eiθ˜kX˜k, where {θ˜k} = (θ˜1,θ˜2,θ˜3) and {X˜ } = (−iσ1,−iσ2,−iσ3) is a basis for the Lie algebra of SU(2). Struc- k 2 2 2 ture constants are ǫ . Equations of motion for the strict WZW model are ijk ∂ (g˜−1∂ g˜) = 0,whereg˜−1dg˜ = (g˜−1dg˜)kX˜ . CurrentsfortheLiealgebraare − + k J˜(L) = (g˜−1∂ g˜)kX˜ and J˜(R) = [(∂ g˜)g˜−1]kX˜ . Again equations of motion + + k − − k ensure that these currents are conserved. We first solve the ordinary differential equation for T to find the pseudodual currents. Multiplying (3) by Tj from right we get n ∂ Ti = −f˜i TkTj∂ φl (4) + n kj l n + We put in an order parameter ε to look for a perturbation solution, ∂ Ti = −εf˜i TkTj∂ φl (5) + n kj l n + Presumably the solution is in the form T = eεα1e12ε2α2(I +O(ε3)), where α and α are antisymmetric matrices. Since we know that T is only a 1 2 function of σ+, α and α are also functions of σ+. If we expand T 1 2 1 T = I +εα + ε2(α +α2)+O(ε3) (6) 1 2 2 1 then taking ∂ we end up with + 4 1 ∂ T = ε∂ α + ε2[α (∂ α )+(∂ α )α +∂ α ]+O(ε3) (7) + + 1 1 + 1 + 1 1 + 2 2 If we compare (5) to (7), the latter may be written in tensor product form as ∂ T = −εf˜⊗(T∂ φ)⊗T = −εf˜⊗[(I +εα )∂ φ]⊗(I +εα ) + + 1 + 1 = −εf˜⊗∂ φ⊗I −ε2f˜⊗α ∂ φ⊗I −ε2f˜⊗∂ φ⊗α (8) + 1 + + 1 we find ∂ α = −f˜⊗∂ φ⊗I (9) + 1 + 1 [∂ α +α (∂ α )+(∂ α )α ] = −f˜⊗α ∂ φ⊗I −f˜⊗∂ φ⊗α (10) + 2 1 + 1 + 1 1 1 + + 1 2 Solving (9) we get α as follows 1 σ+ (α )i = − f˜i ∂ φkdσ′+ = −f˜i (φk +Ck) (11) 1 n Z kn + kn 0 where Ck is a constant, and we choose it to be zero. Since α is a function 1 of σ+ only, φk in the expression of α should involve σ+, not σ−. From this 1 we understand that we need to separate φ as right moving wave φ (σ−) and R left moving wave φ (σ+), i.e. φ = φ (σ+)+φ (σ−). Hence (α )i = −f˜i φk, L L R 1 n kn L from which we find 0 φ3 −φ2 L L α =  −φ3 0 φ1  (12) 1 L L φ2 −φ1 0  L L  Solving (10) we obtain σ+ σ+ (α )i = − (α ∂ α )i dσ′+ − [(∂ α )α ]i dσ′+ 2 n Z 1 + 1 n Z + 1 1 n 0 0 σ+ σ+ −2 f˜i (α )k∂ φl dσ′+ −2 f˜i ∂ φm(α )r dσ′+ (13) Z kn 1 l + L Z mr + L 1 n 0 0 σ+ = − (φi ∂ φn −φn∂ φi )dσ′+ Z L + L L + L 0 which gives us the following entries of α with the help of (12) 2 5 (α )1 = 0 (14.1) 2 1 σ+ (α )1 = [φ2(∂ φ1)−φ1(∂ φ2)]dσ′+ (14.2) 2 2 Z L + L L + L 0 σ+ (α )1 = [φ3(∂ φ1)−φ1(∂ φ3)]dσ′+ (14.3) 2 3 Z L + L L + L 0 σ+ (α )2 = [φ1(∂ φ2)−φ2(∂ φ1)]dσ′+ (14.4) 2 1 Z L + L L + L 0 (α )2 = 0 (14.5) 2 2 σ+ (α )2 = [φ3(∂ φ2)−φ2(∂ φ3)]dσ′+ (14.6) 2 3 Z L + L L + L 0 σ+ (α )3 = [φ1(∂ φ3)−φ3(∂ φ1)]dσ′+ (14.7) 2 1 Z L + L L + L 0 σ+ (α )3 = [φ2(∂ φ3)−φ3(∂ φ2)]dσ′+ (14.8) 2 2 Z L + L L + L 0 (α )3 = 0 (14.9) 2 3 Plugging (12) and (14) into T and setting ε = 1 gives us 1 Ti = δi +(α )i + [(α )i +(α2)i]+O(φ3) (15) j j 1 j 2 2 j 1 j 1 σ+ ≈ δi −f˜i φk − [ (φi ∂ φj −φj∂ φi )dσ′+ −f˜i f˜mφkφn] j kj L 2 Z L + L L + L km nj L L 0 so the entries of T becomes 1 T1 = 1− [φ2φ2 +φ3φ3] (16.1) 1 2 L L L L σ+ T1 = φ3 + φ2(∂ φ1)dσ′+ (16.2) 2 L Z L + L 0 σ+ T1 = −φ2 + φ3(∂ φ1)dσ′+ (16.3) 3 L Z L + L 0 σ+ T2 = −φ3 + φ1(∂ φ2)dσ′+ (16.4) 1 L Z L + L 0 6 1 T2 = 1− [φ1φ1 +φ3φ3] (16.5) 2 2 L L L L σ+ T2 = φ1 + φ3(∂ φ2)dσ′+ (16.6) 3 L Z L + L 0 σ+ T3 = φ2 + φ1(∂ φ3)dσ′+ (16.7) 1 L Z L + L 0 σ+ T3 = −φ1 + φ2(∂ φ3)dσ′+ (16.8) 2 L Z L + L 0 1 T3 = 1− [φ1φ1 +φ2φ2] (16.9) 3 2 L L L L WenotethatT isanorthogonalmatrix. Thetypeofthefieldφ(σ+,σ−) = φ (σ+)+φ (σ−) puts pseudoduality relations into the forms L R (g˜−1∂ g˜)i = +Ti∂ φj (17.1) + j + L (g˜−1∂ g˜)i = −Ti∂ φj (17.2) − j − R Wenotethatequation(17.1)hasaninvarianceunderg˜(σ+,σ−)−→h(σ−)g˜(σ+,σ−). From this we can look for solution g˜(σ+,σ−) = g˜ (σ−)g˜ (σ+), so first pseu- R L doduality relation is reduced to (g˜−1∂ g˜ )i = +Ti∂ φj. This equation gives L + L j + L us the left current. Next we have to find g˜ (σ−) using second pseudodual- R ity equation to construct right current. Plugging g˜(σ+,σ−) = g˜ (σ−)g˜ (σ+) R L into (17.2) and arranging terms we obtain g˜−1(σ−)∂ g˜ (σ−) = −g˜ (σ+)(X˜ Ti∂ φj )g˜−1(σ+) (18) R − R L i j − R L where {X˜ } are the Lie algebra basis of ˜g, and (X˜ ) = ǫ . Since we want i i jk jik to construct pseudodual currents in the order of φn, we need T(σ+) to the order of φn−1 to get J˜(L)(σ+) to the order of φn. From equation (18) we see + that the knowledge of T to O(φn−1) and g˜ to O(φn−1) allows us to construct L g˜ to O(φn), so we can construct J˜(R)(σ−) to O(φn). R − First we construct J˜(L)(σ+) to the order of φ2, so we need T to O(φ) + Ti = δi −f˜i φk +O(φ2) (19) j j kj L Therefore, using first pseudoduality relation (17.1) J˜(L)(σ+) = g˜−1∂ g˜ = X˜ ∂ φi −X˜ f˜i φk∂ φj (20) + L + L i + L i kj L + L 7 All we need is g˜ to the order of φ, so we need to solve g˜−1∂ g˜ = X˜ ∂ φi L L + L i + L for g˜ (σ+). Choosing initial condition as g˜ (σ+ = 0) = I, we get L L g˜ (σ+) = I +X˜ φi +O(φ2) (21) L i L Its inverse is g˜−1(σ+) = I −X˜ φi +O(φ2) (22) L i L Plugging these into (18) we find g˜−1∂ g˜ = −(I +X˜ φl )X˜ (δi −f˜i φk)∂ φj (I −X˜ φk) R − R l L i j kj L − R k L = −X˜ ∂ φi +O(φ3) (23) i − R We notice that the order of φ2 terms are cancelled, and g˜ is a function of R σ− only. We let g˜R = e−φiRX˜ieξkX˜k, where ξ represents O(φ2). Expanding g˜R 1 g˜ = (I −φi X˜ + φi φj X˜ X˜ )(I +ξkX˜ ) R R i 2 R R i j k 1 = I −φi X˜ + φi φj X˜ X˜ +ξkX˜ +O(φ3) (24) R i 2 R R i j k the inverse g˜R−1 can be found from g˜R = e−ξkX˜keφiRX˜i 1 g˜−1 = (I −ξkX˜ )(I +φi X˜ + φi φj X˜ X˜ ) R k R i 2 R R i j 1 = I +φi X˜ + φi φj X˜ X˜ −ξkX˜ +O(φ3) (25) R i 2 R R i j k It follows then that equations (24) and (25) lead to g˜−1∂ g˜ = −∂ φi X˜ +∂ ξkX˜ +O(φ3) (26) R − R − R i − k andcomparisonwith(23)evaluates∂ ξk = 0,so ξk isconstant andwechoose − it to be zero. Therefore, right current can be constructed using (24) and (25) as J˜(R)(σ−) = (∂ g˜ )g˜−1 = −(∂ φi )X˜ (27) − − R R − R i we see that order of φ2 disappears in the expression of right current. If we explicitly write pseudodual currents on the manifold M up to the order of φ3 using equations (20) and (27) we get the following 1 σ+ J˜(L)(σ+) =X˜ [∂ φi −f˜i φk∂ φj − [ (φi ∂ φj −φj∂ φi )dσ′+ + i + L kj L + L 2 Z L + L L + L 0 −f˜i f˜mφkφn]∂ φj] (28.1) km nj L L + L 8 J˜(R)(σ−) = −X˜ (∂ φi ) (28.2) − i − R Therefore, our currents can be written as J˜(µ) = J˜(µ) +J˜(µ) +J˜(µ) +O(φ3) (29) [0] [1] [2] where {µ} = (R,L). We can organize all these terms as ∞ J˜(µ)(φ) = J˜(µ)(φ) (30) [n] X 0 It is easy to see that these currents are conserved, i.e. ∂ J˜(R) = ∂ J˜(L) + − − + = 0, by means of the equations of motion ∂2 φi = 0. Since each term +− satisfies ∂ J˜(R) = ∂ J˜(L) = 0 for all n separately, we have infinite number of + [n] − [n] conservation laws for each order of φ as pointed out in [1]. 3 Pseudodual Currents : Complicated Case In this case we consider the pseudoduality between two strict WZW models based on compact Lie groups of dimension n with Ad-invariant metrics. If {X } are the orthonormal basis for the Lie algebra of G with commutation i relations [X ,X ] = fkX , where f are totally antisymmetric in ijk, and i j G ij k ijk g : Σ → M is the map to the target space, we may write equations of motion on G as ∂ (g−1∂ g) = 0. Therefore, currents become J(L) = (g−1∂ g)iX − + + + i and J(R) = [(∂ g)g−1]iX . These currents are conserved. We make similar − − i assumptions for the Lie group G˜. The pseudoduality equations are (g˜−1∂ g˜)i = +Ti(g−1∂ g)j (31.1) + j + (g˜−1∂ g˜)i = −Ti(g−1∂ g)j (31.2) − j − where T is an orthogonal matrix. Taking ∂ of the first equation (31.1) we − learn thatT is a function of σ+ only. Taking ∂ of the second equation (31.2) + we get the differential equation for T [(∂ T)T−1]i = −f˜i Tl(g−1∂ g)l +fk TiTj(g−1∂ g)m (32) + j kj k + ml k l + We follow the same method as we did in the previous part to find pseudodual currents. We first solve differential equation (32) for T, then replace this into the pseudoduality relations, and finally build pseudodual currents. We will see that these currents are conserved. 9 3.1 An Example To illustrate all these steps in an example, we consider a strict WZW model based on Lie group SU(2) for G, and a sigma model based on abelian group U(1) × U(1) × U(1) for G˜. Using the map g : Σ → G, we may represent any element in G by g = eiφkXk, where {φk} = (φ1,φ2,φ3) are commuting fields and {X } are the orthonormal basis for the Lie algebra of G, and k {X } = (−iσ ,−iσ ,−iσ ) for the case of SU(2). Structure constants k 2 1 2 2 2 3 are ǫi , and commutation relations are the familiar form of Pauli matrices, jk [−iσi,−iσj] = ǫk(−iσk). Equations of motion are ∂ (g−1∂ g) = 0. Nonlocal 2 2 ij 2 − + currents for the Lie algebra of SU(2) are J(L) = (g−1∂ g)kX and J(R) = + + k − [(∂ g)g−1]kX . We want to construct currents up to the order of φ2. If we − k consider infinitesimal coefficients {φk}, keeping up to second orders we may expand g as 1 g = 1+iφkX − (φkφl)(X X )+... (33) k k l 2 Since we are looking for J(L) and J(R) up to the order of φ2, we need g to + − the order of φ, hence g = 1+iφkX (34.1) k g−1 = 1−iφkX (34.2) k To this order the solution to equations of motion ∂ (g−1∂ g) = 0 is g = − + g (σ−)g (σ+), which leads to φ(σ+,σ−) = φ (σ−)+φ (σ+). Thus equation R L R L (34) can be written as g = 1+iφkX (35.1) L L k g = 1+iφkX (35.2) R R k and hence left and right currents can readily be obtained as 1 J(L) = g−1∂ g = i∂ φmX + fmφk∂ φl X (36.2) + L + L + L m 2 kl L + L m 1 J(R) = (∂ g )g−1 = i∂ φmX + fmφl ∂ φkX (36.2) − − R R − R m 2 kl R − R m Therefore, we conclude that ∂ J(L) = ∂ J(R) = 0, i.e, currents are con- − + + − served on G. We first solve equation (32) to figure out the pseudodual currents. Since f˜i = 0, we have kj 10