Noncommutative Geometry and Gravity

Frank Meyer; Julius Wess; Marija Dimitrijevic; Paolo Aschieri

REVIEW

Noncommutative Geometry and Gravity

Paolo Aschieri, Marija Dimitrijevic, Frank Meyer, Julius Wess

High Energy Physics - Theory · Submitted 6 Oct 2005 · 57 references

Pith's verdict: A solid general-twist framework paper whose Einstein equation rests on an unpublished companion paper for the one ingredient that makes it definite.

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arXiv IDhep-th/0510059 (version 2)

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CitationFrank Meyer, Julius Wess, Marija Dimitrijevic, Paolo Aschieri. "Noncommutative Geometry and Gravity." arXiv:hep-th/0510059 (2005). https://pith.science/paper/ANNDDXNN

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The thing to know: this is the general-twist version of the authors' own θ-constant gravity paper [14], and it is a real generalization, not a repackaging. They build the full coordinate-independent machinery — tensor fields as a ⋆-tensor algebra, vielbeins and dual frames, a ⋆-pairing with the right A⋆-linearity, covariant derivative with the deformed Leibniz rule, torsion and curvature as left A⋆-linear maps, Ricci, metric, and finally Ric − ½ g ⋆ R = 0 in Eq. 7.11. The Hopf algebra side is done properly: UΞ⋆ is constructed, shown isomorphic to UΞ_F, and the coproduct (3.12) is the one that actually has the Woronowicz form ∆⋆(Ξ⋆) ⊂ Ξ⋆ ⊗ 1 + UΞ⋆ ⊗ Ξ⋆. The superspace extension in §2.3.1 is a nice bonus and covers the non-anticommutative cases as special twists. The written-out derivations — associativity, the ⋆-Jacobi identity, the coproduct checks in Appendix A.1 — are careful and I did not find errors on inspection. The construction is not circular; it builds from standard Drinfeld twist machinery rather than fitting to a target. Credit where due: this is the cleanest statement of twist-based noncommutative Riemannian geometry I know of in this period, and the examples (Moyal-Weyl, quantum plane, Jordanian, extended Jordanian) are worked out concretely enough to be usable. Now the soft spot, and it is the load-bearing one. The scalar curvature in 7.9 is defined with respect to a connection introduced in §7 as 'the connection that has vanishing torsion and that is metric compatible... See [44].' Reference [44] is 'P. Aschieri et al. In preparation.' That is the one object that turns 7.11 from a formal expression with a free connection into a definite equation, and the manuscript never shows such a connection exists for a general twist. Torsion-freeness (6.1) plus ∇⋆g = 0 is a system of ⋆-equations for Γ whose solvability is not obvious given the R-matrix orderings in (5.3). I would push back on one part of the reader's diagnosis: left A⋆-linearity of torsion is not purely cited. §6 gives an explicit in-text derivation (6.8–6.9 and the paragraph joining them, using R⁻¹ as a permutation representation), with [44] invoked only for the full technical proof; curvature is then dispatched by 'and similarly,' which is thinner but not empty. So the linearity concern is weaker than stated, the connection concern stronger. Two smaller things worth recording: they prove a first Bianchi identity (6.18) but no contracted one, so divergence-freeness of the left side of 7.11 — the natural commutative consistency check — is not shown; and the reality condition (8.7) on F is imposed as an extra assumption and not verified for the Jordanian twist (2.37). Neither is fatal, both are the kind of thing a referee should ask for. Who should read it: anyone working on twist-based noncommutative gravity or on deformed differential calculi à la Woronowicz; the tensor/vielbein formalism is the part that will get reused. It deserves a serious referee — the framework is right and the gap is fixable, but the gap should be closed in this paper, not deferred to a preprint that does not yet exist.

Referee report

Significance

If the construction holds, this is a genuinely foundational framework paper: it gives a twist-generic, coordinate-independent deformation of Riemannian geometry that subsumes a large family of previously studied noncommutative spaces rather than treating them case by case. The technical core is solid where it is displayed: associativity of the star-product is proved from the twist cocycle condition (Appendix A.3), the star-Jacobi identity is proved (Appendix A.2), the Hopf algebra axioms for UΞF are verified (Appendix A.1), and the coproduct formula (3.12) is shown to satisfy the crucial closure property (3.23), Δ⋆(Ξ⋆) ⊂ Ξ⋆ ⊗ 1 + UΞ⋆ ⊗ Ξ⋆, which is what makes a Woronowicz-style deformed differential calculus possible. The observation that the transposition operator Λ⋆ on Ω⋆ ⊗⋆ Ω⋆ coincides with the classical transposition (Section 7) is a clean and useful result, and the explicit statement that noncommutativity does not single out a preferred metric is an honest and important qualification.

Major objections (3)

[§7, Eqs. (7.9)–(7.11)] The scalar curvature R entering the central equation (7.11) is defined via 'the connection that has vanishing torsion and that is metric compatible, ∇⋆_u g = 0', and this connection is asserted by citation: 'See [44], see also [14] for the case θ-const.' Reference [44] is listed as 'P. Aschieri et al. In preparation.' Since torsion-freeness (6.1) plus ∇⋆g = 0 is a system of star-equations for Γ^μ_{νσ} whose solvability is constrained by the deformed Leibniz rule (5.3) and by the R-matrix orderings, the existence of such a connection is not obvious and is exactly what turns (7.11) from a formal expression with a free connection into a definite equation. The manuscript should either include an existence argument (even an order-by-order one in λ, in the spirit of the proof of Lemma 1) or state explicitly that (7.11) is conditional on the results of [44].
[§6, Eqs. (6.5)–(6.7) and the paragraph following Eq. (6.9)] Left A⋆-linearity of the curvature, Eq. (6.5), is what makes R a tensor rather than an arbitrary trilinear map, and hence what makes Eq. (7.11) meaningful. The manuscript gives an explicit argument for the torsion case (Eqs. 6.8–6.9), but labels it 'an easy intuitive argument' and then dispatches the curvature case with 'and similarly.' Given that this linearity is load-bearing for the paper's main claim, the curvature case deserves the same explicit treatment as the torsion case, or a precise pointer to a published proof rather than to [44].
[§4.2 (statement after Lemma 3) and §3.3] Two structural results used downstream are cited only to [44]: the converse of Lemma 3 ('any left A⋆-linear map Φ : Ξ⋆ → A⋆ is of the form ⟨ , ω⟩⋆ for some ω'), which is used in §6 to construct the Ricci map and to prove its frame-independence, and the isomorphism UΞ⋆ ≅ UΞF of Hopf algebras, which underlies the triangular structure and the R-matrix properties (3.19)–(3.22). Since [44] is unpublished, the manuscript's self-containedness is reduced at precisely the points where the geometry is tied together. At minimum, the converse of Lemma 3 should be proved or its use in §6 made explicit as a hypothesis.

Minor objections (6)

[§2.2, Eq. (2.24)] The notation F = f^α ⊗ f_α and F^{-1} = f^α ⊗ f_α distinguishes the two expansions only by bars and underlines, which are easy to lose in typesetting and hard to track through Eqs. (2.25)–(2.30) and Appendix A.1. Distinct base letters for the two expansions would make the paper substantially easier to check.
[§4.2, Lemma 3] 'We leave it to the reader to prove the following Lemma 3' is acceptable for a routine computation, but the A⋆-linearity properties (4.29)–(4.30) are used repeatedly in §6 and §7; a two-line indication of the argument would help the reader.
[§2.3.1, Eq. (2.44)] The sign factor (−1)^{|f^α||g|} in the superspace star-product should be accompanied by a sentence fixing the grading convention relative to the multiplication rule (2.39), since the two conventions interact.
[§7, paragraph after Eq. (7.1)] The statement that 'there are infinitely many metrics compatible with a given noncommutative differential geometry' is an important qualification that distinguishes this approach from [8] and [55]; it would be worth flagging in the introduction, where a reader would otherwise expect noncommutativity to select a metric.
[Throughout] Typographical errors to correct: 'funtcions' and 'diﬀeomorhsisms' (p. 3), 'Expession' (§5), 'permutaion' and 'inital' (§6), 'Morover' (§4), 'Lengingrad' (ref. [19]), 'New Yourk' (ref. [47]), and the encoding of 'Birkh¨ auser' in refs. [56] and [57].
[§8, Eq. (8.7)] The reality condition (S ⊗ S)F^{-1}_{21} = F^{-1*⊗*} is described as 'a mild natural extra condition'; it would be useful to state explicitly which of the twists in §2.3 satisfy it and which do not, since the reality discussion is otherwise conditional.

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The paper

The paper claims

A general twist deformation of infinitesimal diffeomorphisms yields a coordinate-independent noncommutative Riemannian geometry and a candidate noncommutative Einstein equation.

The reading

Ordinary gravity is written in the language of smooth manifolds: points, coordinates, vector fields, a metric, a connection, and curvature. This paper asks what happens to that language if the algebra of functions on the manifold is deformed so that coordinates no longer commute, using a Drinfeld twist of the Lie algebra of vector fields. The twist is a general object, so the construction covers Moyal-Weyl noncommutativity, quantum-plane-like products, Jordanian deformations, and even non-anticommutative superspace as special cases. The authors build the deformed analogues of vector fields, one-forms, tensor fields, a covariant derivative, torsion, curvature, and a Ricci tensor, all as left module maps over the deformed algebra of functions. They then define a metric as a star-symmetric element of the star-tensor product of one-forms, define scalar curvature by contracting the Ricci map with the inverse metric, and write down a noncommutative Einstein equation in vacuum. The claim is that the whole package is covariant under deformed infinitesimal diffeomorphisms and does not depend on a choice of coordinates. The paper is a framework construction, not a test: it does not compare to data or produce a number that an experiment could check. Several of the supporting lemmas, notably the existence of a torsion-free metric-compatible connection and the left linearity of torsion and curvature, are cited to a companion paper listed as in preparation rather than proved here.

Extended reading notes

Core claim

The paper constructs a coordinate-independent differential geometry on a star-product algebra of functions, covariant under deformed infinitesimal diffeomorphisms, and derives a noncommutative vacuum Einstein equation Ric minus one half g star R equals zero (Eq. 7.11), with torsion, curvature, and Ricci defined as left module maps. If correct, this gives a general framework for gravity on noncommutative manifolds for any twist, covering Moyal-Weyl, quantum-plane, Jordanian, and superspace cases as special cases.

Load-bearing premise

The existence and uniqueness of a torsion-free, metric-compatible star-connection (the noncommutative Levi-Civita connection), which is required to define the scalar curvature R in Eq. 7.9 and hence the Einstein equation 7.11. Section 7 asserts it by citation: 'Consider now the connection that has vanishing torsion and that is metric compatible, nabla-star_u g = 0. See [44].' Reference [44] is listed as 'P. Aschieri et al. In preparation.' The same unpublished source is cited for the left A-star-linearity of torsion and curvature (Section 6), which is what makes them tensors rather than arbitrary maps.

Original abstract

We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The class of noncommutative spaces studied is very rich. Non-anticommutative superspaces are also briefly considered. The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein's equations for gravity on noncommutative manifolds.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The class of noncommutative spaces studied is very rich. Non-anticommutative superspaces are also briefly considered. The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein's equations for gravity on noncommutative manifolds.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study a deformation of infinitesimal diffeomorphisms of a smooth manifold.	contribution 0.98	0.91	0.02	asserted 0.95
s1	The deformation is based on a general twist.	method 0.92	0.54	0.02	asserted 1.00
s2	This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product.	method 0.52	0.78	0.05	asserted 1.00
s3	The class of noncommutative spaces studied is very rich.	result 0.51	0.43	0.02	asserted 1.00
s4	Non-anticommutative superspaces are also briefly considered.	contribution 0.75	0.69	0.02	asserted 0.93
s5	The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent.	result 0.74	0.88	0.07	asserted 1.00
s6	The main target of this work is the construction of Einstein's equations for gravity on noncommutative manifolds.	contribution 0.85	0.41	0.21	asserted 0.93

What kind of paper the classifier reports this is: theoretical result 0.74 · evidence from proof or derivation 0.99.

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levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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How the literature uses it

86 arXiv papers cite this in 199 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 114 sentences without a confident label

Contests it 1 paper

To understand why this improvement was necessary, we show explicitly where the original definition of [ref] breaks down.

Noncommutative Gravity and Quantum Field Theory on Noncommutative Curved Spacetimes · Covariant derivatives and their extension: · contest 0.90

Builds on it 3 papers

This section builds and expands on the nonassociative differential geometry machinery developed by [ref] , and on the noncommutative geometry techniques and results of [ref] .

Nonassociative differential geometry and gravity with non-geometric fluxes · Introduction · 0.97
We use a minor generalization of the formalism of [ref] and define the metric $g∈Ω⊗_⋆Ω$ to be a hermitian and nondegenerate tensor.

Field Theory on Curved Noncommutative Spacetimes · NC geometry from Drinfel'd twists · 0.81
This formalism generalizes the one previously studied by [ref] of Drinfeld twists applied to the canonical Hopf algebra of the universal enveloping algebra of vector fields.

Gauge theory models on $\kappa$-Minkowski space: Results and prospects · Outlook. · 0.92

Uses its method or tool 10 papers

Our work is based on the approach outlined in [ref] , where a noncommutative gravity theory based on an arbitrary twist deformation is established.

Symmetry Reduction in Twisted Noncommutative Gravity with Applications to Cosmology and Black Holes · Introduction · 0.87
=1 In this work we follow the formulation proposed by Julius Wess and his group [ref] to describe NC gravity and field theories.

Field Theory on Curved Noncommutative Spacetimes · Introduction · 1.00
In this paper we use the methods of noncommutative geometry and noncommutative gravity [ref] first to recapitulate a construction of the NC scalar and gauge field model from the reference [ref] and then to generalize it to the second order in the Seiberg-Witten map [ref] .

Noncommutative Reissner-Nordstr\"om black hole from noncommutative charged scalar field · Introduction · 0.83

Compares with it 6 papers

The expression [ref] for the curvature agrees with that of [ref] after taking into account their different conventions;[We are grateful to Michael Fuchs for pointing this out to us.] for trivial associator it reduces to the general expression in [ref] .

Nonassociative differential geometry and gravity with non-geometric fluxes · Curvature · 1.00
This is the reverse statement to the one found in [ref] , namely that the cocycle condition leads to associative star product.

Twists, realizations and Hopf algebroid structure of kappa-deformed phase space · Twist operator · 0.93
(3.7) (3.9), (3.10) in [ref] ; indeed the quantum Lie algebras presented here and there are isomorphic via $^-1$ (cf. also [ref] ).

Observables and Dispersion Relations in k-Minkowski Spacetime · Quantum Symmetries and Quantum Lie algebras · 0.93

Cites it as support 2 papers

Property iii) holds because it is equivalent to equation (4.6) in [ref] (just apply $$ to (4.6)).

Observables and Dispersion Relations in k-Minkowski Spacetime · Quantum Symmetries and Quantum Lie algebras · 0.96
Thirdly, as pointed out in [ref] , homothetic Killing vector fields play an important role in constructing exact solutions of the NC Einstein equations [ref] .

QFT on homothetic Killing twist deformed curved spacetimes · Introduction · 0.97

Cites it as background 41 papers

In section [ref] we briefly review the noncommutative Riemannian geometry and Einstein equations of [ref] .

Cosmological and Black Hole Spacetimes in Twisted Noncommutative Gravity · Introduction · 0.89
More details can be found in [ref] .

Symmetry Reduction in Twisted Noncommutative Gravity with Applications to Cosmology and Black Holes · Basics of Twisted Differential Geometry and Gravity · 0.98
We will give an introductory overview of the noncommutative gravity theory developed in [ref] .

Symmetry Reduction and Exact Solutions in Twisted Noncommutative Gravity · Introduction · 0.93

Discussion

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