Nonlinear Sigma Models in (1 + 2)-Dimensions and An Infinite Number of Conserved Currents

We treat in this paper non-linear sigma models such as CP -model, QP -model and etc, in 1 + 2 dimensions. For submodels of such ones we definitely construct an infinite number of nontrivial conserved currents. Our result is a generalization of that of authors (Alvarez, Ferreira and Guillen). 0 Introduction Integrable models in (1 + 1)-dimensions are good toy models to understand (1+3)-dimensional field theories such as Yang-Mills one or Yang-Mills-Higgs one. See, for example, [1]. They have an infinite number of nontrivial conserved currents, which usually correspond to an infinite dimensional Lie algebra such as Kac-Moody algebra, Virasoro algebra or W-algebra. See [2]. For many such models the equations of motion are formulated in terms of the zero-curvature condition F01 = 0. Under this condition a holonomy operator doesn’t depend on paths whose end points are kept fixed. Then we can construct the conserved charges from this operator. See [3]. This method Department of Mathematics, Yokohama City University, Yokohama 236, Japan, E-mail address: [email protected] Department of Mathematics, Waseda University, Tokyo 169, Japan, E-mail address: [email protected] 1 is very powerful and useful. But this is typical of one or two-dimensional cases. To generalize the method to higher dimensional cases beyond the two dimensions is not easy task. But recently Alvarez, Ferreira and Guillen in the interesting paper [3] proposed a new idea to generalize the method in two dimensions. In particular they defined a three dimensional integrability and applied their method to the CP -model in (1 + 2)-dimensions to obtain an infinite number of nontrivial conserved currents. But their results (calculation) are not complete. The aim of this paper is to give explicit forms to conserved currents of a submodel of CP -model and also apply their ! method to other non-linear sigma models in (1 + 2)-dimensions to obtain an infinite number of nontrivial conserved currents. 1 Review of [3] We first make a review of [3] within our necessity. Let M be a (1 + d)-dimensional differential manifold and x0 ∈ M a fixed point. We denote by G a Lie group and by L0M a path space on M starting from x0, L0M ≡ {x : [0, 2π] → M |x(σ) ∈ M,x(0) = x0}. (1) Let W be a holonomy operator W : L0M → G. That is, W is defined by the differential equation dW dσ + Aμ dx dσ W = 0 on Γ ∈ L0M, (2) where Aμ(μ = 0, 1, · · · , d) is a connection on a principal G-bundle on M . We set the initial condition W (0) = I. We note that W is not a local function of x = (x) and is usually given by the path-ordered integral. We investigate the conditions under which W becomes local. For this aim we study a variation of W under deformations of Γ keeping the initial and end points (the boundary). The result is W(2π)δW (2π) = ∫ 2π 0 dσWFμνW dx dσ δx , (3) where Fμν is the curvature of Aμ, Fμν ≡ ∂μAν − ∂νAμ + [Aμ, Aν ]. (4) 2 If this curvature vanishes Fμν = 0, (5) then δW (2π) = 0 from (3), so W becomes path-independent. Namely W is a local function on Γ (W = W (x(σ))). From (2) Aμ is written as Aμ = −∂μWW. (6) Aμ is a pure gauge. Next we consider a two-dimensional “holonomy” operator. Let Γ be a fixed loop at x0. We denote by SΓM a space of surfaces on M with the boundary Γ, SΓM ≡ {Σ : a surface on M |∂Σ = Γ}. (7) Γ is parametrized by σ ∈ [0, 2π], so we parametrized Σ as follows. We scan Σ with loops passing through x0 and being parametrized by τ ∈ [0, 2π] such that τ = 0 is the infinitesimal loop around x0 and τ = 2π is Γ. We want to identify a surface Σ in SΓM with such a parametrization by τ ∈ [0, 2π]. Here we introduce a gauge field Aμ and an anti-symmetric tensor field Bμν . Then a two-dimensional “holonomy” operator V : SΓM → G is defined by the differential equation dV dτ − V T (B,A; τ) = 0 on Σ ∈ SΓM (8) where T (B,A; τ) ≡ ∫ 2π 0 dσWBμνW ∂x ∂σ ∂x ∂τ = 1 2 ∫ 2π 0 dσWBμνW ∂(x, x) ∂(σ, τ) (9) and W is given by (2) and x = x(σ, τ). We set the initial condition V (0) = I. In general there are an infinite number of methods to scan Σ in SΓM . The quantity V in (8) should not depend on such methods. Therefore we assume Aμ be flat (Fμν = 0). Then W becomes local by the arguments below (5), so T = T (B,A; τ) in (9) becomes also local. But V itself is still non-local. We research the conditions under which V becomes local. For this 3 aim we study a variation of V under deformations of Σ keeping the boundary Γ. The result is δV (2π)V (2π) = ∫ 2π 0 dτ V (τ) {∫ 2π 0 dσW(DλBμν +DμBνλ +DνBλμ)W × μ ∂σ ∂x ∂τ δx −[T (B,A; τ), ∫ 2π 0 dσWBμνW dx dσ δx ] } V (τ), (10) where DλBμν ≡ ∂λBμν + [Aλ, Bμν ]. If the R.H.S. of (10) vanishes, then δV (2π) = 0. Namely V becomes surfaceindependent or V becomes a local function on Σ. A comment is in order. The R.H.S. of (10) is identified with the curvature of a principal G-bundle on the loop space ΩG. See the appendix in [3]. Now we can construct conserved charges from V . This is the main story of [3]. Next we must study the vanishing conditions of the R.H.S. of (10). First we restrict a Lie group G. Let g be its Lie algebra. Here we assume g is non-semisimple. If we set p be the radical of g, then g is decomposed into g = h ⊕ p by the Levi’s theorem [4]. Now we assume Aμ ∈ g and Bμν ∈ p. (11) Since p is the radical (abelian) and WBμνW ∈ p, so we have [T (B,A; τ), ∫ 2π 0 dσWBμνW dx dσ δx ] = 0. (12) Second we restrict gauge fields Aμ and Bμν . We have choosed Aμ a flat connection (Fμν = 0). Therefore if we have Gμνλ ≡ DλBμν +DμBνλ +DνBλμ = 0, (13) the R.H.S. of (10) vanishes. That is to say, we call Fμν = 0 and Gμνλ = 0 (14) 4 local integrability conditions under (11). Up to now our arguments are based on any (1+ d)-dimensional manifold M . From here let us restrict to (1+2)-dimensional case. For Bμν we consider its dual B̃ ≡ 1 2 ǫBνλ, (15) where ǫ = 1 = −ǫ012. Then we have Gμνλ = 0 ⇔ DμB̃ = 0. Therefore we can write local integrability conditions (14) as Fμν = 0 and DμB̃ μ = 0. (16) This is a typical feature of 3-dimensions. Next let us construct a non-semisimple Lie algebra ĝ starting from g. Let R be a representation of g, R : g → gl(P ), where P is a representation space (abelian ideal). The construction of ĝ is as follows: 0 → P → ĝ → g → 0. (17) Let {Ta} be a basis of g and {Pi} of P . The commutation relations in ĝ are [Ta, Tb] = f c abTc, [Ta, Pi] = PjRji(Ta), (18) [Pi, Pj] = 0. Now we choose Aμ and Bμν as Aμ ∈ g and Bμν ∈ P (19) satisfying (16). Then the current Jμ ≡ WB̃μW (20) is conserved ∂μJ μ = WDμB̃ W = 0. (21) On the other hand since Jμ ∈ P Jμ = dimP