Non-commutative solitons and strong-weak duality

Blas, H.; Carrion, H. L.; Rojas, M.

doi:10.1088/1126-6708/2005/03/037

Cited by 10 publications

(51 citation statements)

References 41 publications

(105 reference statements)

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“…This choice will lead to a nc sine-Gordon defined as a system of two coupled second order equations for two scalar fields that reduced to sine-Gordon model and a free scalar field in the commutative limit. The nc extensions of the Leznov-Saveliev equations (2.15) for GL(2) were also obtained in [42] from the nc generalization of the SL(2) affine Toda model coupled to matter (Dirac) fields. As shown in [6], the equations of motion (2.15) can be expressed as a generalized ⋆-zero-curvature condition…”

Section: Constrained W Zn W ⋆ Modelmentioning

confidence: 96%

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: Constrained W Zn W ⋆ Modelmentioning

confidence: 96%

Abelian Toda field theories on the noncommutative plane

Cabrera-Carnero

2005

J. High Energy Phys.

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“…(5.11)-(5.14) of ref. [2]). In fact, the later system is contained in the The four field interaction terms in the action (1.1) can be re-written as a sum of Dirac type current-current terms for the various flavors (j = 1, 2, 3).…”

Section: A Version Of the Noncommutative Generalized Massive Thirringmentioning

confidence: 99%

“…In fact, the later system is contained in the The four field interaction terms in the action (1.1) can be re-written as a sum of Dirac type current-current terms for the various flavors (j = 1, 2, 3). In the constructions of the relevant currents the double-gauging of a U (1) symmetry in the star-localized Noether procedure deserve a careful treatment [2]. So, one has two types of currents for each flavor [2]…”

Section: A Version Of the Noncommutative Generalized Massive Thirringmentioning

confidence: 99%

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Some comments on the integrability of the noncommutative generalized massive Thirring model

Blas¹,

Carrion²

2009

Proceedings of 5th International School on Field Theory and Gravitation — PoS(ISFTG)

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Some properties of a non-commutative version of the generalized massive Thirring theory (NCGMT) are studied. We develop explicit calculations for the affine Lie algebra gl(3) case. The NCGMT model is written in terms of Dirac type fields corresponding to the Moyal product extension of the ordinary multi-field massive Thirring model. We discuss the Lagrangian formulation, its zero-curvature representation and integrability property of certain submodels.

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“…Inside the context of Noncommutative Field theories (NCFT), noncommutative (nc) extensions of two-dimensional Integrable Field Theories have been investigated [1,2,3,4,5,6,9,7,8,10,11,12] in the last few years. Particularly in [12] nc extensions of Toda and affine Toda theories were proposed.…”

Section: Introductionmentioning

confidence: 99%

About the self-dual Chern–Simons system and Toda field theories on the noncommutative plane

Cabrera-Carnero¹

2006

J. Phys. A: Math. Gen.

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The relation of the noncommutative self-dual Chern-Simons (NCS-DCS) system to the noncommutative generalizations of Toda and of affine Toda field theories is investigated more deeply. This paper continues the programme initiated in JHEP 0510(2005)071, where it was presented how it is possible to define Toda field theories through second order differential equation systems starting from the NCSDCS system. Here we show that using the connection of the NCSDCS to the noncommutative chiral model, exact solutions of the Toda field theories can be also constructed by means of the noncommutative extension of the uniton method proposed in JHEP 0408 (2004)054 by Ki-Myeong Lee. Particularly some specific solutions of the nc Liouville model are explicit constructed. 1 email:cabrera@cpd.ufmt.br With the constant generators of grade ±1 aswhere E ±α i are the step generators associated to the positive/negative simple roots of the algebra and µ i are constant parameters, the equations of motion that define the nc extension of abelian Toda models area system of n-coupled equations (k = 1 . . . n) where we have changed variables to:Notice that for the first and last equation µ 0 = µ n = 0 and φ 0 = φ n+1 = 0. A particular example of these field theories is the nc extension of the Liouville model:

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Non-commutative solitons and strong-weak duality

Cited by 10 publications

References 41 publications

Abelian Toda field theories on the noncommutative plane

Abelian Toda field theories on the noncommutative plane

Some comments on the integrability of the noncommutative generalized massive Thirring model

About the self-dual Chern–Simons system and Toda field theories on the noncommutative plane

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