Local conserved charges in principal chiral models

Evans, J. M.; Hassan, M.; MacKay, Niall; Mountain, A. J.

doi:10.1016/s0550-3213(99)00489-7

Cited by 94 publications

(247 citation statements)

References 40 publications

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“…For a recent discussion and references to the older literature on local and non-local conservation laws for sigma models based on groups see [27,28]. The connection between the method described here and other methods is not clear.…”

Section: Infinite Number Of Conservation Lawsmentioning

confidence: 99%

Pseudoduality in sigma models

Alvarez¹

2002

Nuclear Physics B

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We revisit classical "on shell" duality, i.e., pseudoduality, in two dimensional conformally invariant classical sigma models and find some new interesting results. We show that any two sigma models that are "on shell" duals have opposite 1-loop renormalization group beta functions because of the integrability conditions for the pseudoduality transformation. A new result states for any two compact Lie groups of the same dimension there is a natural pseudoduality transformation that maps classical solutions of the WZW model on the first group into solutions of the WZW model on the second group. This transformation preserves the stress-energy tensor. The two groups can be non-isomorphic such as B l and C l in the Cartan notation. This transformation can be used for a new construction of non-local conserved currents. The new non-local currents on G depend on the choice of dual group G.

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Section: Infinite Number Of Conservation Lawsmentioning

confidence: 99%

Pseudoduality in sigma models

Alvarez¹

2002

Nuclear Physics B

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“…

where the generators t a belong to the defining representation of g. Commutation of the resulting charges depends upon some intricate algebraic identities satisfied by the k-tensors which we shall refer to as the commutation conditions.

…”

mentioning

confidence: 99%

“…This is a summary of progress made [1][2][3][4] in understanding the occurrence and properties of local, conserved, commuting charges in non-linear sigma-models, including principal chiral models (PCMs) and WZW models. Initial investigations [1] focussed on PCMs based on classical groups G and established that currents and symmetric tensors with the required properties could be defined by a formula…”

mentioning

confidence: 99%

“…Toda field theory [5] and these similarities offer a natural explanation of some otherwise mysterious common properties of PCM and affine Toda S-matrices [1].…”

mentioning

confidence: 99%

“…Initial investigations [1] focussed on PCMs based on classical groups G and established that currents and symmetric tensors with the required properties could be defined by a formula…”

mentioning

confidence: 99%

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Integrable Hierarchies and Modern Physical Theories

Aratyn¹,

Sorin²,

Theories³

2001

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This is a summary of progress made [1][2][3][4] in understanding the occurrence and properties of local, conserved, commuting charges in non-linear sigma-models, including principal chiral models (PCMs) and WZW models. Initial investigations [1] focussed on PCMs based on classical groups G and established that currents and symmetric tensors with the required properties could be defined by a formulawhere the generators t a belong to the defining representation of g. Commutation of the resulting charges depends upon some intricate algebraic identities satisfied by the k-tensors which we shall refer to as the commutation conditions. These findings were subsequently extended to incorporate the effects of WZ terms and supersymmetry [2] and analogous results were also established for sigma-models whose target manifolds are symmetric spaces G/H [3], but again only for G and H classical groups. The general case, including exceptional groups, has been treated quite recently [4] by making use of a direct link between the k-tensors and the Drinfeld-Sokolov modified KdV (DS/mKdV) hierarchies [6].The first step in establishing this link is to show that all relevant attributes of the k-tensors can be understood as properties of the restricted tensors k i1i2...im defined just on the Cartan subalgebra (CSA) of g (indices i refer to the CSA). It is well-known that any symmetric invariant tensor is completely determined by its restriction and that the remnant of G which fixes the CSA is the Weyl group; moreover, any Weylinvariant tensor on the CSA can be extended uniquely to a G-invariant tensor on g. It is less obvious that the key commutation conditions of specific interest to us can be simply expressed in terms of restricted tensors alone, but this is in fact the case [4].

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From PCM to KDV and Back

Evans

2001

Integrable Hierarchies and Modern Physical Theories

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This is a summary of progress made [1-4] in understanding the occurrence and properties of local, conserved, commuting charges in non-linear sigma-models, including principal chiral models (PCMs) and WZW models. In each PCM or WZW model with target manifold a compact Lie group G, there are infinitely many commuting charges whose currents have the form K m = k a1a2...am j a1 j a2. .. j am. The underlying fields j a take values in the Lie algebra g (indices a refer to some basis) and each tensor k a1a2...am is totally symmetric and G-invariant. The spins of the corresponding conserved charges s = m − 1 run over the exponents of the algebra modulo its Coxeter number h. Such patterns of spins are familiar from the context of affine Toda field theory [5] and these similarities offer a natural explanation of some otherwise mysterious common properties of PCM and affine Toda S-matrices [1]. Initial investigations [1] focussed on PCMs based on classical groups G and established that currents and symmetric tensors with the required properties could be defined by a formula K m = det(1 − µj a t a) s/h | µ s+1 , where the generators t a belong to the defining representation of g. Commutation of the resulting charges depends upon some intricate algebraic identities satisfied by the k-tensors which we shall refer to as the commutation conditions. These findings were subsequently extended to incorporate the effects of WZ terms and supersymmetry [2] and analogous results were also established for sigma-models whose target manifolds are symmetric spaces G/H [3], but again only for G and H classical groups. The general case, including exceptional groups, has been treated quite recently [4] by making use of a direct link between the k-tensors and the Drinfeld-Sokolov modified KdV (DS/mKdV) hierarchies [6]. The first step in establishing this link is to show that all relevant attributes of the k-tensors can be understood as properties of the restricted tensors k i1i2...im defined just on the Cartan subalgebra (CSA) of g (indices i refer to the CSA). It is well-known that any symmetric invariant tensor is completely determined by its restriction and that the remnant of G which fixes the CSA is the Weyl group; moreover, any Weyl-invariant tensor on the CSA can be extended uniquely to a G-invariant tensor on g. It is less obvious that the key commutation conditions of specific interest to us can be simply expressed in terms of restricted tensors alone, but this is in fact the case [4].

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Local conserved charges in principal chiral models

Cited by 94 publications

References 40 publications

Pseudoduality in sigma models

Pseudoduality in sigma models

Integrable Hierarchies and Modern Physical Theories

From PCM to KDV and Back

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