Produtos multiplamente torcidos

Gueuvoghlanian, E. P.

doi:10.11606/d.45.2006.tde-20210729-144853

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“…The relativistic invariant abelian Conformal Toda field theories (3.9) can be called as open Toda field chains since the equations of motion of these theories reduce to the open Toda chain equations of motion when the fields do not depend on x. To the Toda field theories it is possible in general to associate a bicomplex [39], which is a special case of zero curvature formulation. Noncommutative extensions of integrable models can be obtained deforming the associated bicomplex by replacing the ordinary products of functions with the Moyal product [2].…”

Section: The Nc Abelian Toda Field Theories As Open Toda Field Chainsmentioning

confidence: 99%

Abelian Toda field theories on the noncommutative plane

Cabrera-Carnero

2005

J. High Energy Phys.

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Generalizations of GL(n) abelian Toda and GL(n) abelian affine Toda field theories to the noncommutative plane are constructed. Our proposal relies on the noncommutative extension of a zero-curvature condition satisfied by algebravalued gauge potentials dependent on the fields. This condition can be expressed as noncommutative Leznov-Saveliev equations which make possible to define the noncommutative generalizations as systems of second order differential equations, with an infinite chain of conserved currents. The actions corresponding to these field theories are also provided. The special cases of GL(2) Liouville and GL(2) sinh/sine-Gordon are explicitly studied. It is also shown that from the noncommutative (anti-)self-dual Yang-Mills equations in four dimensions it is possible to obtain by dimensional reduction the equations of motion of the two-dimensional models constructed. This fact supports the validity of the noncommutative version of the Ward conjecture. The relation of our proposal to previous versions of some specific Toda field theories reported in the literature is presented as well. on the manifold B remains commutative, i.e. [z,z] = θ, [y, z] = [y,z] = 0. The Euler-Lagrange equations of motion corresponding to (2.2) arē ∂J = ∂J = 0, (2.3)where J andJ represent the conserved chiral currents(2.4)The fields α a parameterize the group element g ∈ G through g = e αaTa ⋆ , where T a are the generators of the corresponding Lie algebra G. It is our interest to define the this model taking n = 2 in (3.8), saywhere we have called φ + = φ 1 , φ − = φ 2 and µ 1 = µ. One can now also compute the sum and difference of that equations to find(3.23)for SL(2). In this way one find that the equations that define the nc extension of Liouville model in this alternative parameterization are(3.30)One can also combine the previous equations to find the system of coupled second order equations,which are more suitable for applying the commutative limit. When θ → 0 that system easily reduced to (3.24).One can write an action for the nc Liouville model (3.30) using the general expression (2.19) and making use of the nc generalization of the Polyakov-Wiegmann identity (3.13). For this purpose, introduce (3.26) and (3.27) in (2.19) to obtain the corresponding action of nc Liouville (3.30)where using the notation of [8] we have defined). (3.33) Looking at the action (3.32) it is noticeable the presence of the topological term of W ZNW ⋆ , in this case shifted to the ϕ 0 field. This situation contrasts with the ordinary commutative case, where for Liouville and more generally for any abelian subspace this term equals zero. The parameterizations (3.26) and (3.24) for the element B belonging to the zero grade subgroup in the usual commutative case are identical, but in a nc space-time they lead to equivalent models. Taking e ϕ 1 ⋆ ⋆ e ϕ 0 ⋆ → e φ + ⋆ and e −ϕ 1 ⋆ from (3.36) through a dimensional reduction process some nc extensions of twodimensional integrable models have been obtained [4, 7, 8]. In this part of the section we w...

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Section: The Nc Abelian Toda Field Theories As Open Toda Field Chainsmentioning

confidence: 99%