REVIEW
Symmetry, Gravity and Noncommutativity
High Energy Physics - Theory · Submitted 23 Jun 2006 · 122 references
Pith's verdict: A solid, well-organized review of noncommutative gravity that is honest about what it does not settle; the teleparallel section is the load-bearing piece and it is quoted, not re-derived.
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Referee report
The paper
The paper claims
The review argues that noncommutative Yang-Mills theory can be recast as a teleparallel theory of gravity, with gauge fields playing the role of frame fields and field strengths playing the role of torsion.
The reading
What carries the argument
The argument runs through the Moyal star-product and its star-gauge transformations. The covariant coordinates $C_i=B_{ij}x^j+A_i$ turn spacetime derivatives into star-commutators, and the large-$N$ limit of the resulting matrix model identifies the gauge group with a quantum deformation of the Poisson diffeomorphism group. For the gravitational reduction, the noncommutativity tensor is split into blocks $theta^{mu a}$, the gauge field is truncated to be linear in the auxiliary fibre coordinates $y^a$, and the surviving components $omega_{mu a}$ define a frame and a Weitzenböck connection. The same star-product technology, now twisted by a Drinfeld twist, gives deformed Poincaré and diffeomorphism Hopf algebras and a star-metric version of Einstein gravity.
Extended reading notes
Core claim
The central claim is that noncommutative Yang-Mills theory is not merely a gauge theory with a deformed product; after a consistent truncation it contains the kinematics of gravity. The truncation keeps gauge fields linear in the fibre coordinates, $A=omega_{mu a}(x)y^a dx^mu+xi_a(x)dy^a$, and identifies $e^nu{}_mu=delta^nu{}_mu-theta^{nu a}omega_{mu a}$ as a frame field. The noncommutative field strength becomes the torsion of the Weitzenböck connection, and the leading lagrangian (3.66) is equivalent, up to a total derivative, to the Einstein-Hilbert lagrangian (3.68). In this sense the open-string gauge theory supplies closed-string gravitational degrees of freedom, with $kappa=g,|mathrm{Pf}(theta)|^{1/2d}$.
Load-bearing premise
The load-bearing premise is that the truncation to gauge fields linear in the fibre coordinates $y^a$ is the physically relevant consistent subsector, so that its leading-order dynamics really captures the gravitational sector rather than only a special corner of noncommutative Yang-Mills theory.
Editorial extensions
If this is right
- Spacetime translations become part of the noncommutative gauge group, so the split between internal gauge symmetry and general coordinate symmetry is not sharp in these theories.
- The noncommutative gauge group is a quantum deformation of the Poisson diffeomorphism group, which makes the gravitational sector naturally volume-preserving or unimodular at the level of the full gauge symmetry.
- The induced Planck scale obeys $kappa=g,|mathrm{Pf}(theta)|^{1/2d}$, tying the strength of gravity to the Yang-Mills coupling and the noncommutativity scale.
- Twisted diffeomorphisms give a systematic star-product deformation of the Einstein-Hilbert action, with a star-metric, star-Christoffel symbols, and a star-Ricci tensor.
- D-branes in curved superstring backgrounds realize nonconstant $theta$, and the $AdS_3 times S^3$, Melvin, and Nappi-Witten examples show how twisted spacetime symmetries survive in those settings.
Reading between the lines
- If the teleparallel map is more than a subsector, the open-string gauge theory and the closed-string gravitational theory are two descriptions of the same degrees of freedom in the Seiberg-Witten limit, not two separate sectors coupled by holography.
- A natural test is to look for torsion-like rather than curvature-like signatures in the worldvolume effective action of D-branes with constant $B$-field; the teleparallel form predicts that the leading gravitational coupling is carried by the frame field $e^nu{}_mu$.
- The twisted-diffeomorphism construction has no clear string realization in the review; a useful next step would be to ask whether it can be embedded in open-closed string field theory or recovered as a particular Seiberg-Witten map.
- Because the derivation uses constant $theta$ and a linear fibre truncation, extending it to $theta(x)$ is the obvious way to test whether the correspondence survives when the noncommutativity itself is dynamical.
Original abstract
We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. The geometry of canonical noncommutative gauge transformations is analysed in detail and it is shown how noncommutative Yang-Mills theory can be related to a gravity theory. The construction of twisted spacetime symmetries and their role in constructing a noncommutative extension of general relativity is described. We also analyse certain generic features of noncommutative gauge theories on D-branes in curved spaces, treating several explicit examples of superstring backgrounds.
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.
The abstract, sentence by sentence
We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. The geometry of canonical noncommutative gauge transformations is analysed in detail and it is shown how noncommutative Yang-Mills theory can be related to a gravity theory. The construction of twisted spacetime symmetries and their role in constructing a noncommutative extension of general relativity is described. We also analyse certain generic features of noncommutative gauge theories on D-branes in curved spaces, treating several explicit examples of superstring backgrounds.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. | contribution 0.76 | 0.30 | 0.17 | hedged possibility 0.83 |
| s1 | The geometry of canonical noncommutative gauge transformations is analysed in detail and it is shown how noncommutative Yang-Mills theory can be related to a gravity theory. | contribution 0.90 | 0.84 | 0.25 | asserted 1.00 |
| s2 | The construction of twisted spacetime symmetries and their role in constructing a noncommutative extension of general relativity is described. | contribution 0.98 | 0.66 | 0.15 | asserted 0.90 |
| s3 | We also analyse certain generic features of noncommutative gauge theories on D-branes in curved spaces, treating several explicit examples of superstring backgrounds. | contribution 1.00 | 0.82 | 0.06 | asserted 0.98 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How the literature uses it
63 arXiv papers cite this in 87 sentences
Papers citing it · 28 sentences without a confident label
Uses its method or tool 1 paper
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Using [ref] we can define a new function $f$ for every function $f$ by [ref] f = f +(ħ)^2/2 U_3 (f,Θ,Θ) +∑_n=3^∞ (ħ)^n/n!
Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds · Derivation properties and associator · 0.89
Cites it as support 1 paper
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As in the case of string field theory, nonassociativity in this case is an off-shell phenomenon: In the integrated on-shell open string scattering amplitudes the effects of nonassociativity (but not noncommutativity) vanish due to appropriate cyclicity properties of the nonassociative star products [ref] , consistently with the requirements of crossing symmetry of the worldsheet conformal field theory of open strings (which still involves ordinary quantum fields that necessarily associate); see e.g. [ref] for a review of these and other developments along these lines.
An Introduction to Nonassociative Physics · Nonassociativity in string theory · 0.96
Cites it as background 50 papers
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This approach is in the spirit of noncommutative gauge theory (cf. [ref] and references therein), where target space has no direct physical meaning, and is not inhabited by propagating fields; it therefore does not need to be compactified.
Modified Einstein equations from the 1-loop effective action of the IKKT model · Introduction · 0.94
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Despite recent advances in noncommutative gravity [ref] there is still no unique and totally satisfactory way to realize the full diffeomorphism group on a noncommutative manifold.
Diffeomorphism covariant star products and noncommutative gravity · Introduction · 0.93
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In addition, several versions of the noncommutative theory of gravitation have been suggested in Refs. [ref] .
Constraining non-commutative space-time from GW150914 · Introduction · 0.96
Discussion
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