Geometry of the Grosse-Wulkenhaar model

Burić, Maja; Wohlgenannt, Michael

doi:10.1007/jhep03(2010)053

Cited by 39 publications

(66 citation statements)

References 22 publications

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“…In recent paper [20] we proposed a geometric interpretation of the oscillator term: It was shown there that the Grosse-Wulkenhaar action can be interpreted as an action for the scalar field coupled to the curvature of a background noncommutative space. The coupling has the usual form, Rφ 2 , and in this term the oscillator potential is contained.…”

Section: Jhep07(2010)010mentioning

confidence: 99%

“…The plan of the paper is the following: in section 2 we recollect some results of [20] and also some steps of the construction of local symmetries in the noncommutative frame formalism, [1][2][3]21]. We apply the formalism to the truncated Heisenberg space and then we reduce to subspace z = 0 which gives the relevant two-dimensional theory in section 3.…”

Section: Jhep07(2010)010mentioning

confidence: 99%

“…The truncated Heisenberg space, [20], is a three-dimensional noncommutative space defined by coordinates x, y and z and the commutation relations…”

Section: Truncated Heisenberg Algebramentioning

confidence: 99%

“…Some geometric properties of (2.1) were analyzed in [20]; here we wish to define gauge fields and therefore we need to explore structure of the algebra of differential forms. We have established already that though algebra (2.5), defined as a limit of matrix truncations for n → ∞, is two-dimensional its cotangent space is three-dimensional.…”

Section: Jhep07(2010)010mentioning

confidence: 99%

“…Note that relations (2.27) give the Hodge dual uniquely only if, when multiplying forms with their duals the products are symmetrized. Geometric characteristics of the truncated Heisenberg space as the connection and the curvature were discussed in [20]. In order to define gauge fields we only need the Ricci rotation coefficients C α βγ , dθ α = − …”

Section: Jhep07(2010)010mentioning

confidence: 99%

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Gauge fields on noncommutative geometries with curvature

Burić

Grosse

Madore

2010

J. High Energ. Phys.

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It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can be written as lagrangian of the scalar field propagating in a curved noncommutative space. In this interpretation, renormalizability of the model is related to the interaction with the background curvature which introduces explicit coordinate dependence in the action. In this paper we construct the U 1 gauge field on the same noncommutative space: since covariant derivatives contain coordinates, the Yang-Mills action is again coordinate dependent. To obtain a two-dimensional model we reduce to a subspace, which results in splitting of the degrees of freedom into a gauge and a scalar. We define the gauge fixing and show the BRST invariance of the quantum action.

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Section: Jhep07(2010)010mentioning

confidence: 99%

Section: Jhep07(2010)010mentioning

confidence: 99%

“…The truncated Heisenberg space, [20], is a three-dimensional noncommutative space defined by coordinates x, y and z and the commutation relations…”

Section: Truncated Heisenberg Algebramentioning

confidence: 99%

Section: Jhep07(2010)010mentioning

confidence: 99%

Section: Jhep07(2010)010mentioning

confidence: 99%

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Gauge fields on noncommutative geometries with curvature

Burić

Grosse

Madore

2010

J. High Energ. Phys.

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Snyder-de Sitter Meets the Grosse-Wulkenhaar Model

Franchino-Viñas

Mignemi

2020

Progress and Visions in Quantum Theory in View of Gravity

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We study an interacting λ φ 4 ⋆ scalar field defined on Snyder-de Sitter space. Due to the noncommutativity as well as the curvature of this space, the renormalization of the two-point function differs from the commutative case. In particular, we show that the theory in the limit of small curvature and noncommutativity is described by a model similar to the Grosse-Wulkenhaar one. Moreover, very much akin to what happens in the Grosse-Wulkenhaar model, our computation demonstrates that there exists a fixed point in the renormalization group flow of the harmonic and mass terms.

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Approximate treatment of noncommutative curvature in quartic matrix model

Prekrat

Ranković

Todorović-Vasović

et al. 2023

J. High Energ. Phys.

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We study a Hermitian matrix model with the standard quartic potential amended by a tr(RΦ2) term for fixed external matrix R. This is motivated by a curvature term in the truncated Heisenberg algebra formulation of the Grosse-Wulkenhaar model — a renormalizable noncommutative field theory. The extra term breaks the unitary symmetry of the action and leads, after perturbative calculation of the unitary integral, to an effective multitrace matrix model. Accompanying the analytical treatment of this multitrace approximation, we also study the model numerically by Monte Carlo simulations. The phase structure of the model is investigated, and a modified phase diagram is identified. We observe a shift of the transition line between the 1-cut and 2-cut phases of the theory that is consistent with the previous numerical simulations and also with the removal of the noncommutative phase in the Grosse-Wulkenhaar model.

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Geometry of the Grosse-Wulkenhaar model

Cited by 39 publications

References 22 publications

Gauge fields on noncommutative geometries with curvature

Gauge fields on noncommutative geometries with curvature

Snyder-de Sitter Meets the Grosse-Wulkenhaar Model

Approximate treatment of noncommutative curvature in quartic matrix model

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