Deformed Chern-Simons theories

Bimonte, Giuseppe; Musto, R.; Stern, A.; Vitale, Patrizia

doi:10.1016/s0370-2693(97)00689-8

Cited by 4 publications

(5 citation statements)

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“…Finally, the most immediate and physically relevant follow up of our work is the search for hidden deformed structures for 3 + 1 dimensions Einstein general relativity. As we have already mentioned, this search has been successful, and we will report on it in a forthcoming article [4].…”

Section: Inclusion Of a Cosmological Constantmentioning

confidence: 89%

“…While the space-time metric does not commute with the dreibeins and the spin connections, the Christoffel symbols and the Riemann curvature tensor, being in the center of the algebra, can be thought to be ordinary numbers and hence a sensible gravity theory can be defined. In a future article [4], we shall exhibit a similar hidden quantum group gauge symmetry in four dimensional gravity, although it will obviously not be described by a CS theory.…”

Section: Introductionmentioning

confidence: 98%

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2 + 1 Einstein Gravity as a Deformed Chern-Simons Theory

Bimonte

Musto

Vitale

et al. 1998

Int. J. Mod. Phys. A

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The usual description of 2 + 1 dimensional Einstein gravity as a Chern-Simons (CS) theory is extended to a one parameter family of descriptions of 2 + 1 Einstein gravity. This is done by replacing the Poincaré gauge group symmetry by a q−deformed Poincaré gauge group symmetry, with the former recovered when q → 1. As a result, we obtain a one parameter family of Hamiltonian formulations for 2 + 1 gravity. Although formulated in terms of noncommuting dreibeins and spin-connection fields, our expression for the action and our field equations, appropriately ordered, are identical in form to the ordinary ones. Moreover, starting with a properly defined metric tensor, the usual metric theory can be built; the Christoffel symbols and space-time curvature having the usual expressions in terms of the metric tensor, and being represented by c−numbers. In this article, we also couple the theory to particle sources, and find that these sources carry exotic angular momentum. Finally, problems related to the introduction of a cosmological constant are discussed.

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Section: Inclusion Of a Cosmological Constantmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 98%

2 + 1 Einstein Gravity as a Deformed Chern-Simons Theory

Bimonte

Musto

Vitale

et al. 1998

Int. J. Mod. Phys. A

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“…This result was first achieved in three space-time dimensions [2] using a deformed Chern-Simons action [5] and the quantum Poincaré group ISO q (2, 1) [6]. In the resulting description of 2+1 gravity, the dreibeins and spin-connections are not c-numbers, but instead, obey nontrivial braiding relations.…”

mentioning

confidence: 99%

“…In a couple of recent papers [2,3] we obtained a generalization of the gauge theory description of general relativity where the gauge group is replaced by a q-gauge group [4]. This result was first achieved in three space-time dimensions [2] using a deformed Chern-Simons action [5] and the quantum Poincaré group ISO q (2, 1) [6]. In the resulting description of 2+1 gravity, the dreibeins and spin-connections are not c-numbers, but instead, obey nontrivial braiding relations.…”

mentioning

confidence: 99%

Comments on the non-commutative description of classical gravity

et al. 1998

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We find a one-parameter family of Lagrangian descriptions for classical general relativity in terms of tetrads which are not c-numbers. Rather, they obey exotic commutation relations. These noncommutative properties drop out in the metric sector of the theory, where the Christoffel symbols and the Riemann tensor are ordinary commuting objects and they are given by the usual expression in terms of the metric tensor. Although the metric tensor is not a c-number, we argue that all measurements one can make in this theory are associated with c-numbers, and thus that the common invariant sector of our one-parameter family of deformed gauge theories (for the case of zero torsion) is physically equivalent to Einstein's general relativity.It is well known that 3 + 1 gravity admits a gauge theory description [1]. In this description, the connection one forms correspond to the tetrads and spin connections, while the dynamics is given by the Palatini action. The gauge group is the Poincaré group, although the action is only invariant under local Lorentz transformations.In a couple of recent papers [2,3] we obtained a generalization of the gauge theory description of general relativity where the gauge group is replaced by a q-gauge group [4].

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“…e à teoria de solubilidade local de operadores diferenciais lineares [16]. Recentemente, as profíquas idéias matemáticas de quantização algébrica [8,9,19,55], juntamente com as idéias da Geometria Não-comutativa [14,27,42], vêm promovendo importantes avanços no estudo de uma versão algébrica para o Teorema do Índice [48]. Também recentemente, uma versão do Teorema de Riemann-Roch foi obtida por meio de quantização por deformações [12].…”

Section: Formalismo Hamiltoneano Na Linguagem Dos Operadores Diferenc...unclassified

O papel algebrico dos operadores diferenciais no formalismo variacional

Carvalho¹,

Rodrigues²

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O propósito desta tese é estudar, sob o ponto de vista algébrico, o papel desempenhado pelos operadores diferenciais nos formalismos variacionais Lagrangeano e Hamiltoneano. Apresentamos uma aplicação simples das idéias e resultados básicos da teoria dos operadores diferenciais às álgebras de Clifford, obtendo uma relação entre os operadores diferenciais e o operador de Dirac. Introduzimos um formalismo Hamiltoneano, com base nos módulos de símbolos dos operadores diferenciais, generalizando os resultados para anéis comutativos. Nesse formalismo. encontramos importantes propriedades algébricas para a Hamiltoneana, e destacamos o colchete de Poisson como uma estrutura mais básica que a forma simplética canônica. Introduzimos o conceito de adjunta de um operador diferencial e, por meio dela, caracterizamos as formas integrais em termos das formas de Berezin. Obtemos uma seqüência espectral relacionando a cohomologia das formas integrais com a cohomologia de De Rham, tanto para variedades quanto para supervariedades. Introduzimos o conceito de Lagrangeana, e analisamos sua relação com as formas de Berezin. Nesse contexto, estudamos as leis de conservação, e obtemos um equivalente algébrico para o Teorema de Noether. Finalmente, essas construções nos encaminham rumo a uma versão algébrica para o teorema do índice.

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Deformed Chern-Simons theories

Cited by 4 publications

References 61 publications

2 + 1 Einstein Gravity as a Deformed Chern-Simons Theory

2 + 1 Einstein Gravity as a Deformed Chern-Simons Theory

Comments on the non-commutative description of classical gravity

O papel algebrico dos operadores diferenciais no formalismo variacional

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