Gauge theories on quantum spaces Kilian Hersenta Philippe Mathieub Jean-Christophe Walleta aIJCLab Universit e Paris-Saclay CNRSIN2P3 91405 Orsay France

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Gauge theories on quantum spaces

Kilian Hersenta, Philippe Mathieub, Jean-Christophe Walleta

aIJCLab, Universit´e Paris-Saclay, CNRS/IN2P3, 91405 Orsay, France

bInstitut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich.

e-mail: kilian.hersent@universite-paris-saclay.fr,

philippe.mathieu@math.uzh.ch,

jean-christophe.wallet@universite-paris-saclay.fr

Abstract

We review the present status of gauge theories built on various quantum space-

times described by noncommutative space-times. The mathematical tools and notions

underlying their construction are given. Diﬀerent formulations of gauge theory models

on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra

are covered, with particular emphasis on some explored quantum properties. Recent

attempts aiming to include gravity dynamics within a noncommutative framework are

also considered.

Contents

1 Introduction 4

2 Building the star-products. 8

2.1 Basicproperties. ................................. 8

2.2 Convolutionalgebras. .............................. 9

2.3 Twistdeformations. ............................... 10

2.3.1 Hopfalgebras. .............................. 10

2.3.2 Drinfeldtwists............................... 12

2.3.3 Constructions of twists. . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Derivations and noncommutative diﬀerential calculus. 14

3.1 Trading vectors ﬁelds for derivations. . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Connections on a right-module. . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Twopracticalschemes.............................. 17

3.3.1 The module as a copy of the algebra . . . . . . . . . . . . . . . . . . 17

3.3.2 The module as Ncopies of the algebra . . . . . . . . . . . . . . . . . 19

3.4 Connections on central bimodules . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Diﬀerential calculus from a twist deformation . . . . . . . . . . . . . . . . . 22

arXiv:2210.11890v1 [hep-th] 21 Oct 2022

4 Gauge theories on Moyal spaces R2n

θ. 24

4.1 The Moyal product and the Moyal space R2n

θ.................. 24

4.1.1 Thestar-product.............................. 24

4.1.2 Thematrixbase.............................. 26

4.2 Diﬀerential calculus, connections and curvatures on R4

θ. ........... 28

4.3 Gauge theories on R4

θas Yang-Mills type models . . . . . . . . . . . . . . . 33

4.3.1 The simplest noncommutative Yang-Mills model . . . . . . . . . . . 33

4.3.2 BF terms as a cure to the UV/IR mixing. . . . . . . . . . . . . . . . 36

4.3.3 Harmonic term, IR damping and gauge invariance. . . . . . . . . . . 38

4.3.4 1

p2gauge models and IR damping. . . . . . . . . . . . . . . . . . . . 40

4.3.5 Braided L∞-algebra gauge theory . . . . . . . . . . . . . . . . . . . . 44

4.4 From θ-expanded gauge models to phenomenological predictions . . . . . . 45

4.4.1 The Seiberg-Witten map . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4.2 Modiﬁed gauge and curvature . . . . . . . . . . . . . . . . . . . . . . 46

4.4.3 Phenomenological consequences . . . . . . . . . . . . . . . . . . . . . 46

4.5 Gauge theories on R4

θas matrix models: “Induced” gauge theory. . . . . . . 47

4.5.1 General properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5.2 Vacuum conﬁgurations. . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5.3 Quantum instability of the vacuum? . . . . . . . . . . . . . . . . . . 52

5 Gauge theories on deformations of R3. 55

5.1 From star-products to the matrix base for R3.................. 55

5.2 Noncommutative diﬀerential calculus on R3

λ................... 59

5.3 Gauge theories on R3

λ. .............................. 62

5.3.1 Gauge theories on R3

λas Yang-Mills type models. . . . . . . . . . . . 63

5.3.2 Gauge theories on R3

λas matrix models. . . . . . . . . . . . . . . . . 67

5.4 Relations with models of brane dynamics and group ﬁeld theory. . . . . . . 71

6 Gauge theories on κ-Minkowski space-time 75

6.1 Gauge theories with κ-Lorentz invariant calculus . . . . . . . . . . . . . . . 76

6.1.1 Star-product and symmetries . . . . . . . . . . . . . . . . . . . . . . 77

6.1.2 Diﬀerential calculus and integration . . . . . . . . . . . . . . . . . . 78

6.1.3 Action functional and gauge transformation . . . . . . . . . . . . . . 79

6.2 Gauge theories on κ-Minkowski via twists . . . . . . . . . . . . . . . . . . . 80

6.2.1 Star-product from twists . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2.2 Action functional from Hodge duality . . . . . . . . . . . . . . . . . 82

6.3 κ-Poincar´e invariant gauge theories . . . . . . . . . . . . . . . . . . . . . . . 83

6.3.1 A star-product from the aﬃne group algebra. . . . . . . . . . . . . . 84

6.3.2 Reconciling twisted trace and gauge invariance . . . . . . . . . . . . 86

6.3.3 Twisted diﬀerential calculus, curvature and connection. . . . . . . . 87

6.3.4 From classical gauge invariant action to BRST gauge-ﬁxing . . . . . 89

6.3.5 Some physical properties . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.4 Otherapproaches................................. 92

7 Gravity on quantum spaces: beyond Yang-Mills 94

7.1 Centralbimodules ................................ 95

7.2 Tame diﬀerential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.3 Pseudo-Riemannian calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.4 Quantum tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.5 Braidedgeometry.................................101

7.5.1 The new settings: braidings and symmetry . . . . . . . . . . . . . . 101

7.5.2 The previous setting: deformed vector ﬁelds . . . . . . . . . . . . . . 103

7.5.3 Braided L∞-algebras...........................104

7.6 Quantum principal ﬁber bundle . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.6.1 Frameresolution .............................107

7.6.2 Principal comodule algebras . . . . . . . . . . . . . . . . . . . . . . . 109

7.7 Fuzzyspaces ...................................109

7.8 Star-product incorporation . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.8.1 U(N)gaugestheories ..........................112

7.8.2 Gauge theories on real gauge groups . . . . . . . . . . . . . . . . . . 112

7.8.3 SL2(C)gauge...............................113

7.8.4 Other star-products . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.8.5 Teleparallel gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.9 Emergentgravity.................................115

7.9.1 Gauge theory on the Moyal space . . . . . . . . . . . . . . . . . . . . 116

7.9.2 Gravityemergence ............................117

7.9.3 Otherapproaches.............................118

7.10 Quantum Poincar´e gauge theory . . . . . . . . . . . . . . . . . . . . . . . . 119

7.11 Gravity on almost-commutative spaces . . . . . . . . . . . . . . . . . . . . . 120

8 Conclusion 121

1 Introduction

Quantum space-times, which are intended to substitute to the ordinary space-time at

an energy scale currently identiﬁed with the Planck mass, where the notion of (diﬀer-

entiable) manifold becomes meaningless, can be related to coordinates which no longer

commute. As such, quantum space-times are conveniently described by exploiting the

tools and concepts of noncommutative geometry [1], so that they are often called non-

commutative spaces. In what follows, we will use indiﬀerently both terminologies. Recall

that a quantum (noncommutative) space-time is modeled by a noncommutative but still

associative algebra, its derivations appearing to be natural noncommutative analogs of the

usual vector ﬁelds. Other natural correspondences between usual (commutative) objects

of diﬀerential geometry and noncommutative entities will be detailed through this paper

and in particular in section 3.

In quantum space-times, the space-time coordinates can be identiﬁed with operators

whose spectra (eigenvalues) provide the possible space-time localizations of a particle

(event). This natural identiﬁcation even permits one to get a pictorial ﬂavor of the quan-

tum space whenever the above spectra are discrete.

An instructive example is provided by the noncommutative space R3

λ[112] (covered in

section 5) which, up to some technical details, can be identiﬁed with R3

λ=⊕∞

n=1Mn(C)

where Mn(C) is the algebra of n×ncomplex matrices. Its corresponding coordinate

algebra is [xµ, xν] = iλεµνρxρ(where λis a constant with length dimension) , that is the

Lie algebra su(2). Each of the xµ’s has a spectrum involving only integers which is also the

case for the operator X=Xµxµwhen XµXµ= 1. Assuming that a quantum mechanical

framework holds, the operator Xpermits one to measure one coordinate in the direction

deﬁned by the vector Xµ. According to the property of its spectrum, the result of any

measurement will be integer, thus exhibiting the discrete nature of this quantum space,

which also inherits a continuous SO(3) symmetry stemming from the action of SU (2) on

su(2) by inner automorphisms as SU(2)/Z2'SO(3).

While it is not possible to measure a position in diﬀerent directions simultaneously,

since the above operators do not commute, it is however possible to perform simultaneously

another measurement by using the central operator x0=P∞

n=1(n−1)Pn, equation (5.27),

where Pnis the orthogonal projector on Mn(C). Its spectrum involves only positive

integers. This latter operator, called the radius operator, permits one to measure the

distance from the origin. In this description, Mn(C) looks like a kind of sphere of radius

n−1, which actually is the algebra modeling the so called fuzzy sphere, while, by using

the eigenvalues of the operator Xin the representation of Mn(C), one infers that the

coordinate in the direction deﬁned by Xµcan only take the nvalues n−1, n−3, . . . ,

−n+ 1.

There is a kind of consensus, for a rather long time, that the notion of manifold should

not be valid at ultra short distance when the strength of the gravitational interaction

becomes at the same order of magnitude than the one of the other interactions. In this

regime, it is known that an immediate problem arises for the exact localization of events

when combining quantum mechanics to general relativity. Indeed, according to the uncer-

tainity principle in quantum mechanics, any measurement of the position of a coordinate

xto a given accuracy ∆xcan be done provided an amount of energy ∆E∼ O(1

∆x). This

energy is then brought to the volume element (∆x)3, resulting in a huge energy density

whenever ∆xis required to be very small. But according to the Einstein equations, this

triggers the appearence of a black hole with Schwarzschild radius R∼E

MP`P, where `P

and MPare respectively the Planck length (`P∼ O(10−35) cm) and the Planck mass

(MP∼ O(1019) GeV). This radius therefore corresponds to the smallest length scale

which can be probed.

To escape this diﬃculty, the authors of [2] postulated modiﬁed uncertainity relations

∆xµ∆xν>1

2|θµν |, therefore generating a minimal length scale. This uncertainity relation

stems from the following noncommutativity of the coordinates: [xµ, xν] = iθµν

. This is

the commutation relation characterizing the by-now popular Moyal space R4

θ, see e.g. [28],

which will be considered in section 4. Further considerations on D-branes in [81] gave rise

to extended commutation relations of the form

[xµ, xν] = iθµν(x)

θµν +i

θµν

ρxρ+. . . . (1.1)

The second term in the right hand side of (1.1) corresponds to what is sometimes called in

the physics literature a “noncommutativity of Lie algebra type”. By-now popular related

quantum spaces are R3

λ, a deformation of R3mentioned above and considered in section

5 and the deformation of the usual Minkowski space-time, called κ-Minkowski space-time

introduced a long ago [149, 150, 151].

The latter space will be considered in section 6. The corresponding coordinate alge-

bra is [x0, xj] = i

κxj, [xj, xk] = 0 (x0,xjare respectively time and space coordinates)

where κis usually identiﬁed with the Planck mass. In this quantum space, the spatial

part stays commutative while the time becomes “noncommutative” generating modiﬁed

Lorentz boosts as well as modiﬁed relativistic dynamics, which produce sizable changes as

energy scale approaches the Planck mass scale. This noncommutative space is regarded

as phenomenologicaly promising.

A few years after, noncommutative structures have also shown up within Group Field

Theory, see [144] for a short review. These discoveries, performed within these diﬀerent

approaches to quantum gravity, have triggered a large interest for the study of ﬁeld the-

ories built on noncommutative (quantum) spaces, either as kind of prototypes or claimed

to describe eﬀective regimes of a more fundamental theory of quantum gravity. These ap-

proaches are attempts to characterize potentially observable eﬀects from quantum gravity.

For a recent review on the phenomenology of quantum gravity, see [155].

These mostly non local ﬁeld theories are generically called noncommutative ﬁeld the-

ories. For reviews related to the early time of noncommutative ﬁeld theories see [3].

Compared to the usual (commutative) ﬁeld theories, noncommutative ﬁeld theories have

somehow diﬀerent features. Many eﬀorts have been focused on the exploration of their

quantum and renormalisation properties. This latter aspect is known to be often a diﬃ-

cult task in noncommutative ﬁeld theories, due in particular to their non-local character,

thus precluding the use of the standard (perturbative) machinery used for the ordinary

local ﬁeld theories. This may even be complicated by the possible appearance of the

UV/IR mixing [46], a typical phenomenon of the noncommutative ﬁeld theories spoiling

renormalisability.

Gauge theories on quantum (noncommutative) spaces have also been the subject of an

intense activity. These inherit of all the diﬃculties inherent to the noncommutative ﬁeld

theories. These diﬃculties are even supplemented by additional ones of the gauge the-

ory context, including at least the construction of a suitable noncommutative diﬀerential

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GaugetheoriesonquantumspacesKilianHersenta,PhilippeMathieub,Jean-ChristopheWalletaaIJCLab,UniversiteParis-Saclay,CNRS/IN2P3,91405Orsay,FrancebInstitutfurMathematik,UniversitatZurich,Winterthurerstrasse190,CH-8057Zurich.e-mail:kilian.hersent@universite-paris-saclay.fr,philippe.mathieu@math.uzh.c...

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Gauge theories on quantum spaces Kilian Hersenta Philippe Mathieub Jean-Christophe Walleta aIJCLab Universit e Paris-Saclay CNRSIN2P3 91405 Orsay France

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