Liuba Mazzanti- Topics in Noncommutative Integrable Theories and Holographic Brane-World Cosmology

8/3/2019 Liuba Mazzanti- Topics in Noncommutative Integrable Theories and Holographic Brane-World Cosmology

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Universita di Milano-Bicocca - Dip. di Fisica G. OcchialiniEcole Polytechnique - CPhT

Topics in noncommutative integrable theories

and holographic braneworld cosmology

Doctoral Thesis of: Liuba Mazzanti(international Ph.D. project)

Supervisor at University of Milano-Bicocca: Prof. Silvia PenatiSupervisor at Ecole Polytechnique: Prof. Elias Kiritsis

External Referees: Prof. Gianluca Grignani, Prof. David Langlois and Prof. Boris Pioline

Dottorato di Ricerca in Fisica ed Astronomia (XIX ciclo)Doctorat en Physique Theorique


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Acknowledgements

I am grateful to all who somehow shared with me these 3-4 years of Ph. D.. . .

First of all, it is my duty and pleasure to thank my two advisors Silvia Penati and Elias Kiritsis.

Then the whole two string groups in Milano and Paris; in particular Roberto (if he can hear me,lost in babys cries), Angel, Francesco, Umut (Ill have to thank you two for some more time ahead)

and Alberto (MOA, for once), Gabri (see you in Perth), Marco, Ago, Alberto (a kiss).

Finally my family for supporting. Marcello (dad) and Claudia for constructive and healthycriticisms, appreciations and all their love. Muriel (mum) and Andrea for everything I could need

and more. A huge kiss to my sister Arianne, always giving me the sweetest encouragements.Alberto, for making sun shine, most of the times.


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Resume de la these

Ma these se deroule suivant deux principales lignes de recherche. Les deux arguments traites con-stituent une relation entre la theorie des cordes et les aspects phenomenologiques/cosmologiques.Dune part, la geometrie noncommutative (NC) est une consequence naturelle de la presence de branes

et flux dans la theorie des cordes. La noncommutativite deforme certaines proprietes fondamentalesdes theories ordinaires decrivant par exemple les interactions electrofaibles et fortes ou les modelesstatistiques. Cest dans ce sens que la geometrie NC represente une application a la phenomenologiedes cordes. Dautre part, les branes sont lingredient cle des modeles dunivers branaires. Le modelede RandallSundrum (RS) en particulier offre de nouvelles perspectives tant du point de vue de lacosmologie, ouvrant des scenarios devolution cosmologique non conventionnelle, que du point de vuede lholographie.

La premiere partie de la these est dediee a la geometrie NC et, en particulier, aux theories dechamps NC integrables. Le but principal du travail a ete detudier les consequences de la noncom-mutativite par rapport a lintegrabilite. Plus precisement, on a voulu verifier ou refuter dans uncontexte NC le theoreme qui lie, en deux dimensions, lintegrabilite a la factorisation de la matrice

S. Avec integrabilite on parle de lexistence dun nombre infini de courants locaux conserves, associesaux symetries de la theorie de champs. Le point de depart a donc ete de garantir la presence de telscourants, au moyen du formalisme du bicomplexe. Cette methode permet dobtenir les equations dumouvement en tant que conditions dintegrabilite dun systeme dequations differentielles lineaires. apartir des solutions du meme systeme lineaire suivent les courants conserves. En exploitant le for-malisme de Weyl, la procedure est immediatement generalisable a la geometrie NC. Une algebre defonctions (operateurs de Weyl) definie sur un espace NC est associee a une algebre NC de fonctionsou la multiplication est executee au moyen dun produit NC de Moyal: le produit . En introduisantle produit au niveau du systeme lineaire et en en deduisant les equations du mouvement NC, onobtient la generalisation NC du bicomplexe. On a infere le premier modele en generalisant le bi-complexe du modele de sineGordon (SG) a la geometrie NC. Nous avons deduit (en collaboration

avec Grisaru, Penati, Tamassia) laction corresp ondante aux equations du mouvement precedemmentetablies par Grisaru et Penati. Le calcul des amplitudes de diffusion et production a determine lescaracteristiques de la matrice S du modele. Des comportements acausaux ont ete releves pour lesprocessus de diffusion. En outre, les processus de production possedent une amplitudes non nulle:dou la non validite du theoreme dintegrabilite vs. factorisation pour cette version NC du modelede SG. Dautres proprietes ont ete mises en evidence, comme la relation avec la theorie des cordeset la bosonisation. Le deuxieme modele de SG NC a ete propose en collaboration avec Lechtenfeld,Penati, Popov, Tamassia. Les equations du mouvement ont ete tirees de la reduction dimensionnelledu modele sigma NC en 2+1 dimensions, qui a son tour est la reduction de la theorie de selfdual


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Resume de la these

YangMills NC en 2+2 dimensions (decrivant les supercordes N= 2 avec champs B). Laction a etecalculee de meme que les amplitudes. Les processus de production possedant des amplitudes nulles et

ceux de diffusion ne dependant pas du parametre de NC, entranent ainsi un comportement causal.Le deuxieme modele de SG NC semble donc obeir a lequivalence entre integrabilite et factorisationde la matrice S. La reduction de la theorie des cordes garde sa validite meme au niveau de laction,contrairement au modele precedent.

La deuxieme partie de ma these traite des modeles dunivers branaires, ou plus precisement desmodeles de RS. Le modele propose par Randall et Sundrum se situe dans un bulk 5dimensionnel, car-acterise per une symetrie dorbifold Z2 par rapport a la position de la brane 4dimensionnelle. Graceau facteur de warp qui multiplie le sousespace 4dimensionnel parallele a la brane, on obtient la local-isation des modes du graviton. Par consequent, le potentiel gravitationnel efficace est newtonien auxenergies inferieures a la masse de Planck. En introduisant en outre un terme de matiere dans le bulk eten considerant lechange denergie entre brane et bulk, une variete de nouvelles cosmologies en derive.

Dans la premiere partie de mon travail sur RS nous avons propose un modele analogue situe dans unbulk 7dimensionnel. La brane 6dimensionnelle ayant compactifie deux dimensions est placeeau point fixe de lorbifold Z2. Afin detudier levolution cosmologique en nous mettant en relation avecles observations, nous avons introduit lechange denergie entre brane et bulk. Les scenarios possiblessont nombreux et dependent de la forme explicite du parametre dechange denergie. Entre autres, lespoints fixes possedent une acceleration positive, pouvant ainsi representer la recente acceleration delunivers. Il sont egalement stables pour un large ensemble des valeurs des parametres. Finalement,on peut tracer des scenarios qui partent dune phase initiale acceleree, en passant successivement aune ere de deceleration, pour terminer sur un point fixe stable dinflation. Les modeles duniversbranaires a la RS possedent un dual holographique via AdS/CFT. La correspondance AdS/CFTetablit quune theorie de supergravite (ou, plus generalement, de cordes) dans un champ de fond

danti de Sitter (AdS) en d + 1 dimensions est duale a une theorie de champs conforme (CFT) end dimensions. Tenant compte des divergences presentes dans les deux descriptions, cette correspon-dance a ete rendue plus precise par la formulation de la renormalisation holographique. Si lespacede AdS est regularise au moyen dun cutoff infrarouge, la correspondante CFT resulte regularisee parun cutoff ultraviolet et couplee a la gravite ddimensionnelle. En analogie a lanalyse effectuee encinq dimensions par Kiritsis, nous avons construit la theorie duale au modele cosmologique de RS ensept dimensions. Pour capturer les dynamiques dictees par lechange denergie entre brane et bulk,la theorie holographique en six dimensions a ete generalisee au cas interagissant (entre matiere etCFT) et non conforme. Le resultat sont les relations entre les parametres de masse appartenant auxdeux descriptions et entre lechange denergie, dun cote, et le parametre dinteraction, de lautre. Deplus, le parametre de rupture conforme est associe au parametre dautointeraction du bulk dans la

description de supergravite 7dimensionnelle.Le travail de recherche inclut donc des resultats pouvant trouver leur application dans la pheno-

menologie et cosmologie des cordes. Dune part on a enqueter sur linfluence de la noncommutativiteliee a lintegrabilite du modele de SG. Dautre part, les consequences cosmologiques de lemplacementdu modele de RS en sept dimensions ont ete etudiees et la correspondance AdS/CFT a ete appliqueeafin den tirer des informations sur la theorie duale, couplee a la gravite.


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Riassunto della tesi

Il mio lavoro di tesi si sviluppa seguendo due principali linee di ricerca. Entrambi gli argomentiaffrontati costituiscono una relazione tra la teoria delle stringhe e aspetti fenomenologici/cosmologici.Da un lato, la geometria noncommutativa (NC) e una naturale conseguenza della presenza di brane e

flussi nella teoria di stringa. La noncommutativita deforma alcune proprieta fondamentali delle teorieordinarie che ad esempio descrivono le interazioni elettrodebole e forte o modelli statistici. In talsenso, la geometria NC rappresenta unapplicazione alla fenomenologia di stringa. Daltro canto, lebrane rappresentano un ingrediente chiave nei modelli di braneworld. Il modello di RandallSundrum(RS), in particolare, offre nuove prospettive sia dal punto di vista della cosmologia, aprendo scenaridi evoluzione cosmologica non convenzionale, sia dellolografia.

La prima parte della tesi e dedicata alla geometria NC ed, in particolare, a teorie di campo NCintegrabili. Il principale scopo del lavoro di ricerca e stato studiare le conseguenze della noncommu-tativita sullintegrabilita. Piu esplicitamente, si e voluto verificare o confutare in un contesto non-commutativo il teorema che lega, in due dimensioni, lintegrabilita alla fattorizzazione della matriceS. Per integrabilita si intende lesistenza di un infinito numero di correnti locali conservate, associate

alle simmetrie della teoria di campo. Il punto di partenza e stato dunque garantire la presenza di talicorrenti attraverso il formalismo del bicomplex. Questo metodo consente di ottenere le equazioni delmoto come condizioni di integrabilita di un sistema di equazioni differenziali lineari. Dalle soluzionidello stesso sistema lineare e possibile ricavare le infinite correnti conservate. Sfruttando il formalismodi Weyl, il procedimento e immediatamente generalizzabile alla geometria NC. Un algebra di funzioni(operatori di Weyl) definite sullo spazio NC viene associata ad un algebra NC di funzioni in cui lamoltiplicazione e implementata attraverso un prodotto NC di Moyal: il prodotto . Introducendonel sistema differenziale lineare il prodotto e deducendone le equazioni del moto NC, si ottiene lageneralizzazione NC del metodo del bicomplex. Il primo modello considerato e stato ricavato gen-eralizzando il bicomplex per il modello di sineGordon (SG) alla geometria NC. Dalle equazioni delmoto ottenute in precedenza da Grisaru e Penati abbiamo dedotto lazione corrispondente (in col-

laborazione con Grisaru, Penati, Tamassia). Il calcolo delle ampiezze di scattering e produzione hadeterminato le caratteristiche della matrice S del modello. Sono risultati comportamenti acausali peri processi di scattering. Inoltre, poiche i processi di produzione di particelle non possiedono ampiezzanulla, il teorema integrabilita vs. fattorizzazione non rimane valido per tale generalizzazione NC delmodello di SG. Altre proprieta sono state evidenziate, come la relazione con la teoria di stringa e conla bosonizzazione. Il secondo modello di SG NC e stato proposto in collaborazione con Lechtenfeld,Penati, Popov, Tamassia. Le equazioni del moto sono state derivate dalla riduzione dimensionaledel modello sigma NC in 2+1 dimensioni, che a sua volta e la riduzione dimensionale della teoriadi selfdual YangMills NC in 2+2 dimensioni (che descrive le superstringhe N= 2 con campo B).


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Contents

Contents i

List of figures v

Introduction and outline vii

I Noncommutative integrable theories 1

1 Integrable systems and the sine-Gordon model 3

1.1 SineGordon and relations to other models . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 The XY model as the sineGordon theory . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Massive Thirring model and bosonization . . . . . . . . . . . . . . . . . . . . . 5

1.2 SineGordon at classical and quantum level . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Gauging the bicomplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Quantum properties of ordinary sineGordon . . . . . . . . . . . . . . . . . . . 10

1.2.3 The Smatrix and its features for 2D integrable theories . . . . . . . . . . . . . 11

1.3 Generalities on solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Basics and origins of noncommutative field theories 15

2.1 Weyl formalism and Moyal product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Moyal product arising from Weyl transform . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Moyal product defined by translation covariance and associativity . . . . . . . 18

2.2 From quantum Hall to strings and branes . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Noncommutativity in strong magnetic field . . . . . . . . . . . . . . . . . . . . 20

2.2.2 String backgrounds and noncommutative geometry . . . . . . . . . . . . . . . . 21

2.3 How to deform quantum field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 The free theory, interactions and Feynmann rules . . . . . . . . . . . . . . . . . 30

2.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.3 Infinite conserved currents and noncommutative deformations . . . . . . . . . . 35

2.3.4 Phenomenological and field theoretical consequences of noncommutativity . . . 36

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CONTENTS

3 Noncommutative sineGordon with non factorized Smatrix 41

3.1 Noncommutative sineGordon from the bicomplex . . . . . . . . . . . . . . . . . . . . 41

3.1.1 The equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1.2 The action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Connections to strings and dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Equations of motion from noncommutative selfdual YangMills . . . . . . . . . 45

3.2.2 Noncommutative Thirring model and bosonization . . . . . . . . . . . . . . . . 46

3.3 Properties of Smatrix and integrability . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Remarks on the (non)integrable noncommutative sineGordon . . . . . . . . . . . . . 50

4 Integrable noncommutative sineGordon 53

4.1 Noncommutative integrable sigma model and the bicomplex . . . . . . . . . . . . . . . 54

4.2 The noncommutative sineGordon action from dimensional reduction . . . . . . . . . . 57

4.2.1 Connection to previous noncommutative sineGordon generalizations . . . . . . 614.3 Noncommutative solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.1 Dressing approach in (2 + 1) dimensions . . . . . . . . . . . . . . . . . . . . . . 62

4.3.2 Solitons in noncommutative integrable sineGordon . . . . . . . . . . . . . . . 65

4.4 Properties of Smatrix leading to integrability . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Remarks on the noncommutative integrable sineGordon model . . . . . . . . . . . . . 74

II Holography and cosmology 75

5 AdS/CFT correspondence and branes 77

5.1 The duality paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.1 String theory fields vs. field theory operators . . . . . . . . . . . . . . . . . . . 81

5.1.2 The duality in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Holographic renormalization: spacetimes with boundaries . . . . . . . . . . . . . . . . 88

5.2.1 Renormalized gravity action and boundary counterterms . . . . . . . . . . . . . 88

5.2.2 Renormalized scalar action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3 RandallSundrum and its holographic interpretation . . . . . . . . . . . . . . . . . . . 95

6 Cosmology fundamentals and braneworlds 99

6.1 The conventional scenario of cosmological evolution . . . . . . . . . . . . . . . . . . . . 99

6.1.1 Big Bang! The early Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.1.2 The importance of inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.1.3 Present era: dark energy and open questions . . . . . . . . . . . . . . . . . . . 112

6.2 Embedding cosmology in strings via braneworlds . . . . . . . . . . . . . . . . . . . . 114

6.2.1 Compatification of extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2.2 The RandallSundrum alternative . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Cosmologies in RandallSundrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3.1 Non conventional vs. conventional cosmology . . . . . . . . . . . . . . . . . . . 127

6.3.2 Branebulk energy exchange and late time acceleration . . . . . . . . . . . . . 134

6.4 RandallSundrum cosmology from the holographic point of view . . . . . . . . . . . . 142

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CONTENTS

6.4.1 The dual theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.4.2 Holographic cosmology in four dimensions . . . . . . . . . . . . . . . . . . . . . 144

6.4.3 Generalization in the holographic description . . . . . . . . . . . . . . . . . . . 147

7 RandallSundrum cosmology in seven dimensions 151

7.1 7D RS setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.2 Cosmological evolution in the bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.2.1 Equal pressures in 3D and 2D compact space . . . . . . . . . . . . . . . . . . . 156

7.2.2 Equal scale factors (generic pressures) . . . . . . . . . . . . . . . . . . . . . . . 157

7.2.3 Static compact extra dimensions (generic pressures) . . . . . . . . . . . . . . . 158

7.2.4 Proportional Hubble parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3 Bulk critical point analysis with energy exchange . . . . . . . . . . . . . . . . . . . . . 1617.3.1 Small energy density and flat compact extra dimensions . . . . . . . . . . . . . 162

7.3.2 Critical points with general energy density . . . . . . . . . . . . . . . . . . . . . 167

7.3.3 Small density and free radiation equation of state: an explicit solution . . . . . 174

7.4 Remarks on 7D RS cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8 Holography in the seven dimensional RandallSundrum background 179

8.1 Construction of the holographic dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.1.1 Renormalization on the gravitational side . . . . . . . . . . . . . . . . . . . . . 180

8.1.2 Gauge/gravity duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.2 Holographic cosmological evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.2.1 Simplifications and ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.3 Holographic critical point analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.3.1 Flat compact extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.3.2 Static compact extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.3.3 Equal scale factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.3.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.4 Brane/bulk correspondence at work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.4.1 Slowly scaling approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.5 Non conformal interacting generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 196

8.5.1 Generalized evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8.5.2 Critical points and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

8.5.3 Correspondence: interactions and conformal breaking vs. energy exchange andbulk selfinteraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8.6 Remarks on the 7D RS holographic description . . . . . . . . . . . . . . . . . . . . . . 201

Summary and further considerations 202

Appendices 207

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CONTENTS

A Hographic Weyl anomaly and critical point analysis 211A.1 Conformal anomaly and traces in six dimensions . . . . . . . . . . . . . . . . . . . . . 211

A.2 Fixed points in the holographic description . . . . . . . . . . . . . . . . . . . . . . . . 213A.2.1 Flat compact extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 214A.2.2 Static compact extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 217A.2.3 Equal scale factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

References 220

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List of Figures

7.1 Phase spaces q/ with influx of branebulk energy exchange in the small energy density approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.2 Phase spaces q/ with influx of branebulk energy exchange and general energy density 172

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Introduction and outline

String theory is a wide web of interlacing theories which encloses gauge theories and gravity insome low energy limits. By now, string theory in its supersymmetric version, provides a consistentdescription of quantum gravity. However, it is still not completely clear how to merge real (hence

non supersymmetric) fundamental interactions in the strings framework, despite the fact that muchwork has been recently devoted to this purpose. The unifying theories of strings (which in turnmay be argued to be incorporated in the larger M theory) contain degrees of freedom which cannotbe described by ordinary gauge theories. This is why new features arise in this context and newmathematical techniques as well as new objects must be studied. My thesis essentially tackles two ofthese stringy issues: noncommutative geometry and braneworlds.

Both topics deal with stringy effects on some aspects of (hopefully) realistic description of knownphysics. On the one hand, noncommutative (NC) geometry emerges in relation to particular stringconfigurations involving branes and fluxes. Gauge theories arise in the low energy limit of open stringdynamics, with string ends attached on the branes. When non trivial fluxes are turned on, ordinaryfield theories get deformed by noncommutativity. On the other hand, braneworlds originated from

the intuition that matter fields can be localized on branes, while gravity propagates in the wholestring target space. Braneworlds can thus yield effectively four dimensional gauge theories withobvious phenomenological implications, despite the existence of extra dimensions. Furthermore, nonstaticity of the brane worldvolume produces cosmological evolution, opening the issue of the braneworld cosmology. Branes turn out to be key ingredients for both topics. They indeed represent atpresent the main motivation to study noncommutative geometry and create a link between stringtheory and phenomenology/cosmology.

Noncommutative geometry

Independently of string theory successes, NC geometry was initially formulated with the hope that

it could milden ultraviolet divergences in quantum field theories [1]. However, noncommutativityusually does not qualitatively modify renormalization properties, except for the mixing of infraredand ultraviolet divergences IR/UV mixing , which on the contrary generally spoils renormaliza-tion. Noncommutative relation among spacetime coordinates may also be interpreted as a possibledeformation of geometry beyond the Planck scale. In fact, we can imagine that spacetime can be nomore endowed with a pointlike structure. Indeed, this is a consequence of noncommutativity. Pointswould be subtituted by space cells with Planck length dimension, so that ordinary geometry is recov-ered at energies lower than the Planck scale. It is also true that noncommutativity arises in the largemagnetic field limit of quantum Hall effect [2, 3]. There, space coordinates are forced not to commute

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due to the very large magnetic field, or equivalently to the very small particle mass. However, thestrongest motivation is string theory, since it naturally describes noncommutative embeddings.

Field theories in NC geometry represent the low energy limit of dynamics of open strings endingon branes with appropriate non zero fluxes. A paradigmatic configuration is that of D3branes in IIBstring theory with constant NeveuSchwarzNeveuSchwarz (NS-NS) form B, yielding noncom-mutative four dimensional super YangMills theory (SYM), with instanton solutions described byselfdual YangMills (SDYM) equations [47]. Noncommutative versions of well known field theorieshave been studied over the last ten years [58][62], which single out the interesting results ensuingfrom noncommutativity. For instance, noncommutative relation among spacetime coordinates implya correlation between infrared and ultraviolet divergences in the field theory [33, 34]. This followsintuitively from the uncertainty principle involving the coordinates, which connects small distances tolarge distances dynamics, just as quantum mechanic uncertainty principle connects large momenta tosmall distances and viceversa. Hence, field theory renormalization also depends on the IR behavior.

Although in most cases renormalizability doesnt change going to NC geometry except for UV/IRmixing , it can be explicitly destroyed in some particular models by noncommutativity, as I willshow.

Besides renormalization, there has been much interest in studying integrability properties of non-commutative generalizations. Integrable theories share very nice features, in particular restricting totwo dimensions [10]. Their Smatrix has to be factorized in simple two particle processes and can beexplicitly calculated in some cases. Furthermore, no particle production or annihilation occurs. Mo-menta of initial states must be mapped in the final states, precisely. Solitons, i.e. localized classicalsolutions preserving their shape and velocity in scattering processes, are usually present. The originof these nice properties is the presence of an infinite number of conserved currents, which are indeedresponsible for the integrability of the theory. It is interesting to note that most of the known inte-

grable bidimensional models come from dimensional reduction of four dimensional SDYM. In turn,(2 + 2)dimensional SDYM is the effective field theory for N= 2 open superstrings on D3branes,whose noncommutative version is obtained by turning on a constant NS-NS two form [9, 48]. Wemay now wonder if noncommutativity influences integrability of known models. This is basically thequestion I tried to answer with my collaborators, restricting to a special integrable model, namelythe two dimensional sineGordon theory.

SineGordon equations of motion are related to the integrability of a system with an infinitenumber of degrees of freedom, giving the infinite number of conserved currents. The gauged bicomplexapproach guarantees the existence of the local currents as solutions to an infinite chain of conservationequations, for any integrable theory. These come from solving a matrix valued equation, order byorder in an expansion parameter a Lax pair of differential operators (guaranteeing integrability)

can also be found in some cases related to the bicomplex formulation. Furthermore, the compatibilitycondition of the matrix equation yields the equations of motion, from which an action can in somecases be derived for sineGordon, for instance. Soliton solutions can also be constructed via thedressing method [72] in integrable theories, exploiting the solutions to the integrable linear system ofequations.

Using the gauged bicomplex formalism, S. Penati and M.T. Grisaru wrote the equations of motionfor a noncommutative version of sineGordon, introducing noncommutativity in the two dimensionalmatrix equation [68]. Successively, we found the corresponding noncommutative action and studiedproperties of the Smatrix at tree level [63]. Noncommutativity entailed acausal behaviors and non

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factorization of the Smatrix. Acausality is actually a typical problem in NC field theories whennoncommutativity involves the time coordinate. It has been shown that also unitarity is broken by

time/space noncommutativity. Indeed, in two dimensions, a noncommuting time is unavoidable.However, NC generalizations are not unique, since different deformations can yield the sameordinary theory in the commutative limit. Indeed, a second noncommutative sineGordon modelwas proposed in my publication [46] in collaboration with O. Lechtenfeld, S. Penati, A. D. Popov,L. Tamassia, where noncommutativity has been implemented at an intermediate step in the dimen-sional reduction from SDYM. The action and tree level Smatrix were computed. Scattering processesdisplayed the nice properties expected in integrable models and causality was not violated. More-over, we also provided a general method to calculate multisoliton solutions in this integrable NCsineGordon model.

As I anticipated, string theory suggests a deformation of spacetime, leading to noncommutativefield theories. The relation to specific string configurations pass through dimensional reduction of

higher dimensional integrable theories namely 4 or (2 + 2)dimensional SDYM describing theopen string dynamics. Phenomenological consequences other that integrability in two dimensionscan be investigated. Most of related literature focuses on Lorentz violation in Standard Modelnoncommutative generalizations [4]. In fact, in other than two dimensions, Lorentz invariance isbroken, due to the non tensorial nature of noncommutativity parameter (which I assume to beconstant non constant generalizations have been considered, though [6]). Cosmological issues,such as noncommutative inflation, have also been subjects of research [8, 5]. Thorough studies havebeen devoted to non(anti)commutative generalizations of supersymmetric theories, which imply a nontrivial extension of noncommutative relations to fermionic variables [7]. Summarizing, NC geometryhas its modern origin in string theory and its implications can be analyzed in the perspective offinding phenomenological indications of strings.

Braneworlds

Conversely, taking as an input the low energy physics as we know it Standard Model, GeneralRelativity we may wish to find its description inside the string theory framework. A very successfulintuition going in this direction is the braneworld idea. As I mentioned, the low energy effective fieldtheories living on the brane worldvolume are gauge theories. Thus, we may hope to describe elec-troweak interactions and QCD in a braneworld picture, allowing large and eventually non compactextra dimensions. However, going towards realistic theories implies for instance that supersymmetryand conformal invariance, as they appear in string theory, have to be broken. Some literature isdevoted to the search of branes configurations [86, 115, 117] (intersecting branes, for example) which

realize Standard Model features in string theory.A great improvement in the subject of braneworld models is represented by the AdS/CFT cor-

respondence. In early times, it was already pointed out that large N gauge theories N is the rankof the gauge group displayed stringy characteristics. The large N expansion can indeed be relatedto the closed string loop expansion if the string coupling is identified with 1/N [84]. Furthermore,gauge theories naturally arise in string theory in the presence of branes, more precisely Dbranes. Onthe other hand, so called black brane solutions in supergravity were argued to describe Dbranes inthe classical limit [91]. This was a hint going towards the formulation of a duality connecting gaugetheories on the Dbranes to supergravity in the black brane backgrounds. Stronger indications came

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from the counting of BPS states and absorption cross sections calculations in Dbranes configura-tions compared to entropy and absorption processes in the supergravity description. In particular,

the (1 + 1)dimensional CFT living in the intersection of the D1-D5 system was suggested to bedual to a charged black hole supergravity solution, whose near horizon geometry yields AdS3 S3[94, 92, 177]. Analogously, the D3 configuration in type IIB string theory, giving an effective SU(N)

N= 4 SYM theory (N is the number of coincident Dbranes), was compared to the black 3branesupergravity solution where the near horizon geometry is AdS5 S5 (N units of five form flux arepresent and N also determines the AdS5 and S

5 radii) [95, 177].Finally, Maldacena formulated his conjecture [174], stating that the large N field theory describing

the dynamics of opens strings on Dbranes (or Mbranes, if we consider M theory) is dual to the fullstring theory in the corresponding AdS background. Such a duality is holographic in the sense thatthe dynamics in the supergravity bulk is determined only in terms of boundary degrees of freedom.Furthermore, the boundary conditions are exactly identified with the sources of the CFT operators. In

this spirit, the gauge theory can be thought to live on the boundary of the AdS space. It is particularlyinteresting to note that the BekensteinHawking formula for entropy already suggested the existenceof an holographic principle, relating gravity solutions to the dynamics of the background boundary.The matching of global symmetries also supported Maldacenas idea. The highly non trivial contentof AdS/CFT correspondence is that perturbative approximations in the two descriptions hold inopposite regimes for the effective string coupling constant gsN. The correspondence instead relatesthe two full theories.

From the time the conjecture was formulated, many checks (mainly on protected quantities) andimprovements have been worked out. A rigorous treatment of the divergences that plague the two sidesof the duality is provided by holographic renormalization [189, 190, 193]. It has hence been used toperform correlation functions calculations on the gravity side, using a covariant regularization, and to

compare them with the CFT results. An important consequence of holographic renormalization is itsapplication to AdS/CFT duals of supergravity solutions with cutoff spacetimes. Such backgroundsappear in RandallSundrum models (RS) [113], where spacetime is a slice of AdS with a brane placedat the fixed point of a Z2 orbifold, playing the role of a IR cutoff. It has been argued [178, 179, 180, 181]that the holographic dual theory is a cutoff CFT living on the boundary of AdS, coupled to gravity andhigher order corrections. Indeed, it can be shown that EinsteinHilbert action and the higher ordercorrections are the boundary covariant counterterms appearing in the regularization procedure. Thepresence of gravity is intriguing since we expect the braneworld to display gravitational interactionif it has to describe real universe (using General Relativity as a theoretical instrument).

Braneworld cosmology is a rather broad subject, including applications of stringy models todifferent issues of cosmology. Among the mostly investigated scenarios, brane induced gravity

proposed by Dvali, Gabadadze and Porrati, also called the DGP model [129] and RandallSundrummodel [113] are two alternative ways to obtain 4D gravity in a background with an infinite extradimension. DGP and RS models display Newtonian gravity in opposite regimes. Namely, gravityinduced on the brane yields effective 4D behaviors at high energies, while in this same regime 5D effectsappear in RS. If the two models are merged, getting induced gravity on a RS brane [114], Newtonspotential can be recovered at all energies if the brane induced gravity term is strong compared to theRS scale. On the other hand, we may modify gravity to get non conventional cosmological features primordial inflation, late time acceleration including higher order corrections such as GaussBonnetterms. GaussBonnet braneworlds [138][140] admit a 4D gravity description in the low energy

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regime, as RS models. Moreover, the issue of primordial inflation has been addressed in further stringtheory contexts. Brane/antibrane inflation [143, 144, 145, 135, 136, 137], for instance, is a thoroughly

investigated subject embodying the initial inflationary era in string compactification. Related topicsinclude brane inflation [129], further works on brane induced gravity [130][134], particular exampleswith varying speed of light [141], cosmological evolution induced by the rolling tachyon [142] andrecent braneworld models [146][152]. Braneworld cosmology [119][125] can also be analyzed, in arather general way, following the mirage cosmology approach [127, 128], where evolution is driven bya mirage energy density, which encodes bulk effects no matter term is there from the beginning.

Non conventional cosmology [126] at late times can be obtained in RS braneworlds by consideringthe interaction between brane and bulk. Indeed, models with branebulk energy exchange have beendiscussed in literature [159][173]. It has been shown that a rich variety of cosmologies are producedin the original 5D RS model with the presence of energy exchange. The brane motion is driven bothby the matter energy density and by a mirage radiation, which takes account of the bulk dynamics.

Some of the features that these models exhibit can fit into the cosmological observational data.For instance, eternally accelerating solutions can be found. Furthermore, an holographic cosmologyanalysis has been performed in [125], exploiting gauge/gravity duality specified to 5D RS model.Non conventional cosmology results are found in the 4D picture and compared to the 5D description,yielding the matching of dimensionful parameters on the two sides.

My work is inspired to the cosmological analysis in RS braneworlds, both from the bulk grav-itational point of view and in the holographic description. A RS model in seven dimensions wasconsidered in [154], tracing it back to the M5-M2 configuration in M theory. Bao and Lykken con-centrated on the graviton mode spectrum analysis. They found new features with respect to the 5Dpicture. New KaluzaKlein (KK) and winding modes appear due to the additional compactificationon the internal two dimensional compact manifold. Whether the two additional extra dimensions also

lead to new properties for the cosmological evolution is the question I address in the second part ofmy thesis.

I proposed a 7D RS setup, with a codimensionone 5brane and matter on the brane, as well asin the bulk [96]. In order to make contact with our four dimensional universe, I further compactifyspacetime on a two dimensional internal manifold, without necessarily impose homogeneity inthe sense that evolution in the 3D and 2D spaces may in general be different. The detailed studyof the brane cosmological evolution from the 7D gravity viewpoint is carried, yielding acceleratingsolutions at late times, among the other possibilities. New features with respect to the 5D setupappear. I moreover constructed the holographic dual theory and compared it to the 7D description,generalizing the 6D setup to the non conformal and interacting case, in analogy to the 4D model.

The structure of this thesis is composed by two parts. The first part is devoted to noncommutativeintegrable field theories and to my results on noncommutative integrable sineGordon. The secondpart is dedicated to braneworld holographic cosmology and 7D RS results.

An introduction to integrable systems is given in the first chapter. In chapter 2, I review theWeylMoyal formalism for noncommutative geometry, its application to noncommutative quantumfield theories and the relation to string theory. The third and fourth chapters contain the two gener-alizations to noncommutative geometry of sineGordon model that I proposed with my collaborators.The first theory exhibits acausality and non factorization of the Smatrix, which is calculated at tree

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level, as shown in chapter 3. As a result, the connection to NC 4dimensional selfdual YangMillsand to NC Thirring model are also illustrated. The second theory, examined in chapter 4 displays in-

tegrability properties of the Smatrix. It is shown how this model comes from dimensional reductionfrom NC (2 + 2)dimensional selfdual YangMills, via the intermediate (2 + 1)dimensional modifiedsigma model. I give the procedure allowing to construct the noncommutative multisoliton solutionsand calculate tree level amplitudes.

Chapter 5 is a review of AdS/CFT correspondence, particularly focusing on holographic renor-malization and RS dual. It is followed by a summary of conventional cosmology issues and by anintroduction on braneworlds in chapter 6. Cosmological evolution in the 5D RS braneworld and thecomparison to the holographic dual scenario is also reviewed in chapter 6. The new results on 7D RSbraneworld cosmology and holography are illustrated in the two following chapters. In particular,chapter 7 is devoted to the critical point analysis and brane cosmological evolution on the 7D gravityside. I construct the 6D dual theory in chapter 8, deriving the Friedmannlike equations and the

matching with the 7D description. A summary on results concludes the thesis.

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Part I

Noncommutative integrable theories

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Chapter 1

Integrable systems and thesine-Gordon model

It is well known that integrable theories can be related to statistical models in their continuous limit.Statistical systems are of high interest in physics literature, for the study of correlation functions,critical exponents, and other physical measurable quantities. Integrable models are of interest ontheir own since they are by definition endowed with a number of conserved currents equal to thenumber of degrees of freedom. In the case of integrable field theories this number is infinite. Dueto this property, integrable models are exactly solvable and in many cases the exact mass spectrumand Smatrix are calculable. The presence of an infinite number of local conserved currents is aconsequence of the equations of motions of the theory and do not need to be generated by a specific

action for the fields. Nevertheless, in some cases of particular interest the action leading to theequations of motion is known and classical and quantum characteristics of the theory may be derived.One of these models is the sineGordon model that I will briefly review in section 1.2. The first partof my thesis is based on the sineGordon generalization to noncommutative geometry. Before facingthe sineGordon quantum theory, I will clarify its relation to the statistical XY model in subsection1.1.1 and to the fermionic Thirring theory in subsection 1.1.2. I will then sketch some properties ofthe soliton solutions and their construction in the last section of this chapter.

I will focus on the properties possessed by the Smatrix for a two dimensional integrable the-ory. This issue has been studied in the noncommutative generalizations of the sineGordon modelsconstructed in my first two publications [63, 46]. Also noncommutative solitons solutions have beensystematically produced, being another important aspect of integrable systems.

1.1 SineGordon and relations to other models

I will sketch in this section a couple of interesting links between the sineGordon theory on one handand apparently different models on the other hand. The first topic shows how sineGordon field candescribe a 2D Coulomb gas via XY model in the continuous limit. The second correspondence isbosonizations, which relates sineGordon to massive Thirring fermionic theory.

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1. Integrable systems and the sine-Gordon model

1.1.1 The XY model as the sineGordon theory

The XY model describes two dimensional spin variables S on a N site lattice of dimension L and

step a. The components of S are (Sx, Sy) = (cos , sin ), since S is normalized to be |S|2 = 1. Thesystem partition function is given by

Z =

i

di2

eKP

cos(ij) (1.1.1)

where K = JkBT (J is the spin coupling, kB is the Boltzman constant) and < i,j > indicates firstneighbors.

This two dimensional spin variable model gives a nice description of a 2D classical Coulomb gas(but also has applications for thin films and fluctuating surfaces). It is thus interesting to note that,due to the equivalence (in the continuous limit) to the sineGordon theory, renormalization group

(RG) equation of the quantum field theory describes the dynamics of such statistical systems.The spin lattice displays two different behaviors and hence a phase transition in the high and

low temperature regimes. At high temperatures, the correlation function for two spins located at twodifferent sites of the lattice is exponentially decreasing with the distance between the two sites, whileat low temperatures one gets a power dependence. In terms of the Coulomb gas this is interpretedas a transition between a plasma phase at high temperatures and a neutral gas with coupled chargesat low temperatures, where the effect of vortices can be neglected. The critical temperature canbe evaluated as the temperature for which vortices are no more negligible. This gives 2kBTc = J(renormalization group analysis have also been performed and gives as a result the RG flow of thesineGordon model, that I will shortly sketch in section 1.2).

The equivalence with the sineGordon theory is derived by rewriting the partition function (1.1.1)

on the dual lattice in the continuous limit a 0 (integrating over the angular spin variables)ZV =

n

f2n

(2n)!

D exp

d2x

1

2(x)(x)

d2x

e2i

K(x) + e2i

K(x)2n

(1.1.2)

Here (x) is the continuous limit of the dual lattice variables and f eKlog ar0 represents thefugacity of vortices in the Coulomb gas language (r0 regulates the UV). Identifying the sineGordoncoupling constants by

= 2K , = 2f = 2eKlog ar

0 (1.1.3)

one obtains the exact (euclidean) sineGordon partition function once summed over n in (1.1.2)

ZSG =

D exp

d2x

1

2(x)(x) cos (x)

(1.1.4)

The critical point for the temperature phase transition is thus translated for the sineGordonparameters to 2 = 8. In the short overview of the renormalization results of section 1.2 it will beclear indeed why this phase transition occurs in the quantum field theory.

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1.1. SineGordon and relations to other models

1.1.2 Massive Thirring model and bosonization

A second interesting duality relates the massive Thirring model to sineGordon via bosonization.

Bosonization acts by means of an integration over the fermionic fields in the Thirring partitionfunction, leaving as a result a path integral over a scalar field, which becomes the sineGordon field[30, 78, 79]. Explicitly, the partition function for the massive Thirring lagrangian

LM T = (i m) gMT2

(1.1.5)

cab be put in the following form (up to an overall normalization coefficient and gauge fixing)

Z =

DDAD [A]exp

LMT + iA + 1

2F

(1.1.6)

The Lagrange multiplier has been introduced imposing a null gauge for the U(1) field strength F

associated to the gauge field A coupled to the fermions. The FaddeevPopov determinant is givenby [A] =

x,t [

A(x, t)]. Fixing the residue gauge to be A = 0, (1.1.6) becomes the partitionfunction for the Thirring model.

Roughly1 performing in the first place the integration over the fermions, then over the gaugefields, one obtains disregarding for the moment the mass term that the partition function(1.1.6) corresponds the the bosonic theory governed by the lagrangian

L = 12

(1.1.7)

( has been rescaled to /

+ g2M T).

The mass term contribution to the bosonic theory is computed exploiting the properties of chiral

symmetry breaking. In order to cancel the chiral anomaly when no mass is present one has to imposethat the scalar field transforms under infinitesimal chiral transformations L eiL, R eiR, parametrized by as + . Obviously, the mass term m = m

RL + LR

m

M[] + M[] breaks chiral symmetry explicitly. Its transformation rules give a direct evaluationof the term appearing in the bosonic lagrangian M[] and M[] once integrated over fermions. Infact, one gets the following equations

M

+

= e2iM [] , M

+

= e2iM [] (1.1.8)

yielding M[] e2i and M[] e2i. The mass term thus gives a cosine potential for therescaled field .

Putting all together, the bosonic theory is a sineGordon

LSG = 12

cos (1.1.9)

where = Cm (C is a constant) and = 2+g2MT

. From the expression for it is clear that

bosonization is a strong/weak coupling duality.

1The complete calculation needs the introduction of an additional vector field h that appears quadratically in theaction coupled to the U(1) current, whose integration immediately gives (1.1.6). To be more precise the Thirringlagrangian without mass term is rewritten as LT = i 12hh + igMTh.

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1.2 SineGordon at classical and quantum level

I will briefly mention some important characteristics of the sineGordon model. In particular, inthe next subsection I will derive the classical action from the equations of motion that one obtainsusing the bicomplex formalism. Following [59], this ensures the integrability of the theory. It will bethe starting point for the noncommutative generalizations constructed and studied in [63, 46]. Thereduction from selfdual YangMills is also outlined in subsection 1.2.1, while quantum propertiesare described in subsection 1.2.2. Finally, subsection 1.2.3 contains some general remarks on theSmatrix.

1.2.1 Gauging the bicomplex

The bicomplex technique guarantees to supply the theory, whose equations of motion can be derivedfrom a matrix valued equation, with an infinite number of local conserved currents. Hence it offers

a systematic method to build integrable field theories. Since this procedure acts at the level of theequations of motion it is not assured that an action can be found.

In two euclidean dimensions the bicomplex technique is illustrated as follows. Space is spannedby complex coordinates

z =1

2(x0 + ix1) , z =

12

(x0 ix1) (1.2.1)

The bicomplex is a triple (M, d, ) where M = r0Mr is an N0graded associative (not necessarilycommutative) algebra, M0 is the algebra of functions on R2 and d, : Mr Mr+1 are two linearmaps satisfying the conditions d2 = 2 = {d, } = 0. Mr is therefore a space of rforms. The linearequation characterizing the bicomplex is

= d (1.2.2)where is a real parameter and Ms for a given s. If a non trivial solution exists, we wish toexpand it in powers of the parameter as

=

i=0

i(i) (1.2.3)

The components (i) Ms are then related by an infinite set of linear equations

(0) = 0

(i) = d(i1) , i

1 (1.2.4)

which give us the desired chain of closed and exact forms

(i+1) d(i) = (i+1) , i 0 (1.2.5)

We remark that for the chain not to be trivial (0) must not be exact. Now, the equations of motionof the theory should come from the conditions d2 = 2 = {d, } = 0. When the two differential mapsd and are defined in terms of ordinary derivatives in R2, these conditions are trivial and the chainof conserved currents (1.2.4) is not associated to any second order differential equation.

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1.2. SineGordon at classical and quantum level

To have non trivial equations the bicomplex must be gauged. The procedure gets modified byintroducing a connection such that

Dd = d + A , D = + B (1.2.6)

The flatness conditions now amount to D2d = D2 = {Dd, D} = 0 and are non trivial. In fact, the

gauged bicomplex provides the differential equations

F(A) dA + A2 = 0F(B) B + B2 = 0G(A, B) dB + A + {A, B} = 0 (1.2.7)

In analogy to the trivial setup, the theory is equipped with an infinite number of conserved currentsoriginating from the solution to the linear differential equation

D (D Dd) = 0 (1.2.8)

The nonlinear equations (1.2.7) play the role of the compatibility conditions for (1.2.8)

0 = D2 = F(B) + 2F(A) G(A, B) (1.2.9)A solution Ms to (1.2.8) can be expanded as = i=0 i(i), giving as a consequence the possiblyinfinite chain of relations

D(0) = 0

D(i) = Dd

(i1) , i

1 (1.2.10)

In analogy to (1.2.5), the Dclosed and Dexact forms (i) can be constructed when (0) is not

Dexact.For suitable connections A and B we obtain an infinite number of local2

SineGordon from gauged bicomplex

We define the elements of the bicomplex to be M = M0 , where M0 is the space of 22 matriceswith entries in the algebra of smooth functions on ordinary R2 and = 2i=0i is a two dimensionalgraded vector space. We call the 1 basis (, ) and impose 2 = 2 = {, } = 0. Finally, we definethe non gauged differential maps

= R , d = Sf + (1.2.11)

in terms of the commuting constant matrices R and S. The flatness conditions d2 = 2 = {d, } = 0are trivially satisfied in this case. But when gauging the bicomplex dressing the d map

D G1d(G) (1.2.12)2The (i) currents may in general not be local functions of the coordinates. However it is possible to define local

conserved currents in terms of the (i) which will have physical meaning.

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by means of a generic invertible matrix G M0, the condition D2 = 0 is trivially satisfied, while{, D} = 0 yields the nontrivial second order differential equation

G1G = R, G1SG (1.2.13)In order to specify to the sineGordon equation, we choose R, S to be

R = S =

0 0

0 1

(1.2.14)

and G SU(2) asG = e

22 =

cos 2 sin

2

sin 2 cos 2

(1.2.15)

The sine-Gordon equation then follows from the offdiagonal part of the matrix equation (1.2.13)

= sin (1.2.16)

while the diagonal part gives a trivially satisfied identity.

The bicomplex approach is straightforward generalizable to noncommutative geometry, inducingnoncommutative equations of motion for the theory under exam sineGordon for our purposes.We note that deriving the action is a non trivial calculation. In [63] we constructed such an actionstarting from the deformed equations of motion obtained by generalizing the bicomplex with theintroduction of the noncommutative product Dd = d + A and D = + B. This will be explainedin chapter 3. Moreover, noncommutativity implies an extension of the SU(2) symmetry group, which

is no longer closed. It will be necessary to consider U(2), rather that SU(2), leading to an extra U(1)factor. The extension of the symmetry group must be carried carefully, as we have shown in [46] (seechapter 4).

From selfdual YangMills to sineGordon

Selfdual YangMills was conjectured by Ward to give origin to all integrable equations in two di-mensions via dimensional reduction. The conjecture has been tested over the last years for the mostimportant known integrable systems. Indeed, sineGordon can be obtained both from euclidean (R4)and kleinian (R(2,2)) signature for the four dimensional YangMills equations. In the first case thedimensional reduction leads to euclidean sineGordon, while in the second one gets minkowskiansignature for the two dimensional metric.

Selfdual YangMills in short Let me summarize the relation between the YangMills selfduality equations and the associated matrix valued equation. Four dimensional YangMills theorywith signature (+ + ) has a stringy origin since it describes N = 2 strings, which indeed livein a real (2 + 2)dimensional target space [9]. Self-duality of YangMills models in R4 or R(2,2) isexpressed by the following equation [70]

1

2F

= F (1.2.17)

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where F is the field strength of the A gauge field F = AA + [A, A]. Equation (1.2.17)is integrable. In fact, taking for instance the gauge group to be SU(N), the selfduality equation can

be rewritten in terms of complex coordinates y, y,z, z in four dimensions performing an analyticalcontinuation on AFyz = Fyz = 0 , Fyy Fzz = 0 (1.2.18)

The sign in the second equation depends on the signature of the metric, being + is the euclidean caseand in the kleinian case. The zero value for the mixed yz and yz components of the field strengthmakes the fields Ay, Az (for fixed y and z) and Ay , Az (for fixed y and z) pure gauges. Gauge fieldsmay thus be expressed in terms of two N N complex matrices B and B

Ay = B1yB , Az = B1zB

Ay = B1yB , Az = B1zB (1.2.19)

Finally, the Yang formulation of YangMills theory in lightcone gauge is obtained defining a complexN N matrix J = BB1. In terms of J the selfduality equations read

y(J1yJ) z(J1zJ) = 0 (1.2.20)

The action whose variation leads to such equations of motion is

S =

d2yd2z tr(yJyJ

1)

d2yd2z

10

d tr

J1J[J1y J , J1yJ]

+

d2yd2z tr(zJzJ

1)

d2yd2z

10

d tr

J1J[J1zJ , J1zJ]

(1.2.21)

where J(y, y,z, z, ) is a homotopy path satisfying J( = 0) = 1 and J( = 1) = J.

A Leznov formulation [71] of the same equations (1.2.18) has also been derived. It correspondsto a different choice of lightcone gauge and is ruled by a cubic action in terms of an algebra valuedfield. I will tell more about this formulation in its noncommutative version in the reduction procedurethrough the three dimensional modified non linear sigma model in chapter 4. It is interesting to notethat YangMills in the Leznov gauge completely describes N = 2 strings at tree level, while Yangformulation is related to the zero instanton sector of the same theory [48].

Through dimensional reduction The dimensional reduction from four dimensional selfdualYangMills in order to get two dimensional integrable theories must satisfy a requirement about group

invariance. More precisely, the theory must be invariant under any arbitrary subgroup of the groupof conformal transformations in four dimensional spacetime. The dependence on the disregardedcoordinates is eliminated through an algebraic constraint on the arbitrary matrices involved in thereduction.

As I anticipated, sineGordon equations of motion, both in their euclidean and lorentzian version,can arise from the selfduality equations of YangMills theory in four dimensions. It is not trivialthat the action of the bidimensional models in general can be obtained from the YangMills actionvia the dimensional reduction operated at the level of the equations of motion (as an example we willdiscuss two different cases in the noncommutative generalizations, in chapters 3 and 4).

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Euclidean sineGordon comes from the euclidean version of Yang equation (1.2.20) when the Band B matrices are chosen to be

B = ez21ei

2 3 , B = e

z21 (1.2.22)

where = (y, y). In fact, it immediately turns out that the field satisfies (1.2.16) with 4 = 1.Kleinian selfdual Yang-Mills equations instead lead to lorentzian sineGordon through a two

step reduction procedure. Yang equation (1.2.20) is required to have no dependence on one of thereal coordinates xi, i = 1, 2, 3, 4, lets say x4. This first step brings to the (2 + 1)dimensional sigmamodel equations

( + V)(J

1J) = 0 (1.2.23)

with V defined to be a constant vector in spacetime. Non zero V implies the breaking of Lorentz

invariance but guarantees integrability if it is chosen to be a spacelike vector with unit length(nonlinear sigma models in (2 + 1) dimensions can be either Lorentz invariant or integrable butcannot share both these properties [69]). Once we fix the value for V as V = (0, 1, 0), the secondstep consists in performing a reduction on the matrix J [18] factorizing the dependence on the thirdcoordinate x x3

J =

cos 2 e

i2x sin 2e i2x sin 2 cos 2

SU(2) (1.2.24)

Here is a function of two coordinates only = (t, y), t x1, y x2 (not to be confused with thecomplex coordinates y, y) with different signature. The field satisfies the sineGordon equations ofmotion in (1 + 1) dimensions.

1.2.2 Quantum properties of ordinary sineGordon

We already expect from the discussion about the XY model and its relation to sineGordon to geta phase transition for the critical value of the coupling constant 2 = 8. However, this estimateis naive, since it doesnt take account of the running of the coupling. The sineGordon theoryundergoes a change of regime from superrenormalizability, for < 8, to nonrenormalizability, for 8. In the nonrenormalizable regime, the theory may still be finite pertubatively in 2 8,but the renormalization of (or equivalently) is needed in addition to renormalization (thelagrangian of the theory is precisely given by (1.1.9)).

Superrenormalizability regime: renormalizing Renormalization in the 2 < 8 regimehas to cure divergences which only come from tadpoles with multiple legs (multitadpoles). The valueof such Feynmann diagrams is the same for any arbitrary number of external legs (more precisely,the coefficient depending on the number M of external propagators factors out for every M). Theimplication of this feature is that all correlation functions can be renormalized at the same time.Moreover, the series over N of (IR and UV) regulated multitadpoles with N internal propagators

(N tadpoles) sums to an exponential (up to overall factors) exp

2

8 logm2

2

, where m and are

respectively the IR and UV cutoff. Renormalization of alone is thus needed and amounts to define

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the renormalized coupling constant at a scale according to

R = 22

228 (1.2.25)

at all orders in .

The renormalization group flow driven from the betafunction = 2R

2

8 1

displays UV

fixed points for 2 < 8 and would have IR fixed points in the 2 > 8 regime (this will hold in the2 8 analysis). Trajectories in the R/

2 phase space are straight lines parallel to the R axe,since doesnt get renormalized in this regime.

Nonrenormalizability regime: renormalizing New divergences appear in the nonrenor-

malizability regime. They emerge in the two vertices correlation functions, i.e. in the secondorder contribution to the effective action. Individual Feynmann diagrams are convergent, but theirsum over all orders in the (or field) expansion diverges. The divergence has a different naturedistinguishing the two super or nonrenormalizable phases: it is IR for 2 < 8 and turns toUV for larger values of 2, 2 > 8 (the critical value 2 = 8 gives logarithmical UV divergences).Renormalization of the coupling constant is needed. We consider to be in the proximity of the naivecritical point 2 8 0. The renormalized coupling constants (and renormalized field since renormalization also implies field renormalization not to get the cosine potential renormalized)read

R = Z1

2 Z = 22 2

8

R = Z1/2 , R = Z

1/2 Z = 1

222 log

2

2(1.2.26)

The renormalization group flow equations are obtained from the betafunctions = 2 and = 8

22. Noting that 2 2 2 is a RG invariant, we can divide the RG phase space 2R/Raccording to the sing of . For > 0 ( < 0) one gets IR (UV) fixed points where the theorybecomes asymptotically free, but in opposition to the superrenormalizability analysis trajectoriesare hyperboloids with axes 2 = 0. For imaginary , 2 < 0, the hyperbolic trajectories intersect the axe and flow to large negative in the IR.

The transition from dipole to plasma phase happens at = (intersected with the equationgiving in terms of in the XY model of section 1.1.1). For < (dielectric) the trajectories gotowards the region of validity of our small approximation, while for > (conductor) they rapidlyflow away.

1.2.3 The Smatrix and its features for 2D integrable theories

Here I make some comments on the Smatrix of integrable two dimensional theories and in particularof the sineGordon model.

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The theorem: integrability vs. factorization It is important to note that the property ofintegrability for a system in two dimensions is equivalent to the property of factorization of the

Smatrix. More in detail, if a theory possesses an infinite number of local conserved currents and hence an infinite number of conserved charges that must be components of Lorentz tensors ofincreasing rank it follows that the Smatrix is constrained to be elastic and factorized in twoparticle scattering [11, 10]. The number of particles involved in a process with given mass andmomentum is thus always conserved and all processes can be described only by some number of twoparticle scattering, i.e. no production or annihilation occurs. Moreover the two particle Smatrix isassociated to a cubic equation. the solution to this cubic equation can give in most cases the exactform of the Smatrix. For example this is the case for the sineGordon theory.

It is worthwhile to note some unavoidable restrictions that must be applied to the integrable theoryin order to prove the integrability vs. factorization theorem just stated. This restrictions play animportant role in the generalization to noncommutative geometry, since they fail in noncommutative

theories. Precisely, we must have locality and unitarity in the theory. Both these two properties aretypically absent in noncommutative generalizations of quantum field theories, as I will point out insubsection 2.3.4. So, we dont expect the theorem on integrable two dimensional models to be validin general. Indeed, I will show two noncommutative examples: the first [63] gives a non factorized Smatrix at tree level, non vanishing production processes and acausality while the second [46] displaysnice properties such as factorization, absence of production, causality.

1.3 Generalities on solitons

I now move to illustrate a very peculiar and useful characteristic of integrable models: the solitonicsolutions. Solitons are widely studied in literature (see [11] for a review). They are defined as

localized solutions of the non linear equations of motion carrying a finite amount of energy. Theywere originally thought of as a kind of solitonic wave that doesnt change shape and velocity in timeor after scattering processes with other such waves. Since at infinity they have to approach a constantvalue labeled by an integer sol

x 2n, they come along with an associated (integer) topologicalcharge defined by

Qsol =1

2

+

dxsol

x= n+ n (1.3.1)

Topological charges Qsol of 1 are associated the the simplest solutions: one(anti)soliton.For the ordinary euclidean sineGordon system such a classical solutions of the equations of motion

is known to be

sol(x0, x1) = 4 arctan e

2

x1x10ivx

01v2 = antisol(x0, x1) (1.3.2)

where v is the velocity parameter of the soliton and is the sineGordon coupling constant.

Dressing the solitons Multisoliton solutions can be constructed using a recursive procedure thatI will refer to as the dressing method [72], inspired by the original work by Belavin and Zakharov [13].Babelon and Bernard worked out multisoliton solutions recursively from the onesoliton [12]. In our

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1.3. Generalities on solitons

paper [46] sineGordon noncommutative solitons are obtained by reduction from selfdual YangMills and (2 + 1)dimensional sigma model. Since most of the known integrable theories descend

from selfdual YangMills, as I already pointed out, I will briefly mention the procedure for derivingits multisoliton solutions.The key observation is that selfduality equations (1.2.17) can be interpreted as the integrability

conditions of a linear system containing the spectral parameter CP1 (in complex coordinates)D1 D2

= 0 ,

D2 D1

= 0 , = 0 (1.3.3)

Covariant derivatives are defined as Di = i + Ai (and analogously for Di) where Ai are the complexgauge fields A1 =

12 (Ax1 iAx2), A2 = 12 (Ax3 iAx4) and their complex conjugate A1, A2, follows

from the definition. Equations (1.3.3) must be solved for the arbitrary field . Gauge fields are thendeduced by reverting (1.3.3) as

A1

A2 = 1 2

1 , A2

A1 = 2 11 (1.3.4)

and imposing the reality condition on

1(x, ) =

x, 1 (1.3.5)This comes from noticing that

x, 11 also solves equations (1.3.3). The field isassumed to be meromorphic in the spectral parameter , so that it solves (1.3.4) only if the residuesvanish. In fact, the l.h.s. of (1.3.4) is linear in since the gauge fields are independent. Hence nopoles exist. The requirement of vanishing residues allows to calculate simple soliton solutions for Aiand Ai [13] such as the BPST oneinstanton [14].

The dressing procedure generates new solutions to (1.3.3) from a known solution , multiplyingit by a dressing factor on the left (x, ) = (x, )(x, ). The factor is a function of the complexcoordinates. Being meromorphic in , it can be expanded as 3

= R1 + R0 +r

i=1

Rii+ i

(1.3.6)

where Rn are some independent complex matrices. Equations (1.3.4) written in terms of thedressing factor read

1D D2

= A1 A2 ,

2D D1

= A2 A1 (1.3.7)

where covariant derivatives are though in terms of the old Ai,

iA gauge fields. Again, since Ai, Ai areindependent, the solutions to (1.3.7) are found imposing zero value for all the residues associatedto the poles appearing in (1.3.6). This yields a set of differential equations for the independentcoefficients R1, R0 and Ri for i = 1, . . . , r. In turn, using (1.3.7) and substituting the derivedexpression for , we get a set of differential equations for the new solutions Ai and Ai.

Specific multisoliton solutions for selfdual YangMills were constructed in [13], while solutionsby dressing method for noncommutative selfdual YangMills are described in (1 + 1)dimensionalNC sineGordon multisolitons from (2 + 1)dimensional NC sigma model solutions, which is itselfderived by dimensional reduction from (2 + 2)dimensional selfdual YangMills.

3Here I use the notations of [15].

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Chapter 2

Basics and origins of noncommutativefield theories

The aim of this chapter is to motivate and introduce the study of noncommutative geometry andnoncommutative field theories. The interest in noncommutativity has grown over the years thanksto the important improvements in understanding string theories and the consequent emergence ofnoncommutative backgrounds in its context. Some examples of spacetime coordinate noncommu-tativity originating from string theory will be mentioned in subsection 2.2.2. Nonetheless, thereexists a famous example of quantum mechanics which already introduces noncommutativity relationsamong coordinates: the quantum Hall effect, which I will sketch in subsection 2.2.1. Even earlier,motivations to noncommutative geometry applied to quantum field theories were adopted, such as

the novelty of an intrinsic UV cutoff furnished to the theory, due to the noncommutation relationsamong space coordinates. However, the intrinsic cutoff doesnt seem to give better renormalizationresults in comparison to the usual regularization schemes. In addition it shows a typical feature innoncommutative theories mixing UV with IR divergences, in such a way that the two high and lowenergy limits dont commute (more about this will be discussed together with the main commonproblems and properties of NC field theories in subsection 2.3.4).

Noncommutative geometry formalism has hence been developed for more than twenty years [19, 2]and has been recently understood in terms of strings and branes [47]. Extensive studies have sincethen been performed on noncommutative generalization of quantum field theories. The fundamentalrelation characterizing noncommutative geometry is the non vanishing commutator

[x, x] = i

which is determined by the noncommutativity (antisymmetric) parameter (the constant valueof will be justified in the next section). How an algebra of functions (fields) can b e defined innoncommutative geometry is the subject of the next section, while concrete construction of NC FTare discussed in section 2.3 and, in particular, integrable NC deformations are described in the lastsection.

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2. Basics and origins of noncommutative field theories

2.1 Weyl formalism and Moyal product

Non vanishing commutation relation among coordinates remind of the quantum phase space forparticles, which is described by a non trivial commutator between momenta and positions

xi, pj

= iijxi, xj

= [pi, pj] = 0 (2.1.1)

In the same way as the quantum mechanics commutators imply the wellknown uncertainty principlexipj 2 ij, the noncommutative geometry coordinate algebra

[x, x] = i (2.1.2)

generally time/space noncommutativity 0i = 0 can be considered and related problems will beillustrated in section 2.3.4 gives rise to the spacetime uncertainty relations

xx 12|| (2.1.3)

Hence, at distances lower that the order of the noncommutativity parameter||, ordinary geom-

etry can no longer be used to describe spacetime. In fact there are reasons to believe that at veryshort distances (i.e. shorter than the Planck length) known geometry should be replaced by somenew physics, since quantum effects of gravity could arise. When the NC parameter vanishes, ordinarygeometry is recovered.

The parallel between quantum mechanics phase space and noncommutative geometry can bepushed further. Analogously to the correspondence between functions of the phase space variablesxi, pj and the associated operators expressed in terms of the quantum momentum and position oper-

ators xi

, pj, one can construct a map going from the commutative algebra of functions overRd

(wherea noncommutative product is implemented) to the noncommutative algebra of operators generatedby the coordinate operators obeying to (2.1.2). This is formally achieved by the Weyl transform.

2.1.1 Moyal product arising from Weyl transform

The case of my interest is two dimensional space with variables x1, x2 associated to noncommutingoperators x1, x2 that satisfy

x1, x2

= i12 with constant 12 . The Weyl transform associates an

operator Wf(x1, x2) to a function f(x1, x2) of the coordinates provided with the usual pointwiseproduct. The Weyl operator Wf(x1, x2) is defined via the Fourier transform of the function f

Wf(x

1

, x

2

) = d2x

(2)2

f(k1, k2) e

ikixi

= d2xf(x1, x2) (x1, x2) (2.1.4)where the Fourier transform is as usual

f(k1, k2) =

d2x eikix

if(x1, x2) (2.1.5)

The map is hermitian and equal to

(x1, x2) =

d2k

(2)2eikix

ieikix

i(2.1.6)

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2.1. Weyl formalism and Moyal product

It is interpreted as a mixed basis for operators and fields on the two dimensional space. Moreover, itreduces to 0(x) =

2(x x) when commutativity is restored 0. The trace of the Weyl operatorgives an integral of the associated function over the space

trWf(x1, x2)

=

d2x f(x1, x2) (2.1.7)

if we normalize tr (x) = 1. From the trace normalization

tr

(x1, x2)(y1, y2)

= (2)(x y) (2.1.8)

and the product of two maps (the BakerCampbellHausdorff formula should be used) we candeduce that the map between functions f(x1, x2) and operators Wf(x1, x2) via (x1, x2) is invert-ible and represents a onetoone correspondence between Weyl operators and Wigner distributionfunctions. Indeed, these functions are obtained by means of the following inverted relation

f(x1, x2) = trWf(x1, x2)(x1, x2) (2.1.9)

A noncommutative product among functions belonging to the commutative algebra is introducedas the image via the inverse map of the product of Weyl operators. In fact, using the expressionfor the product of two operators

(x1, x2)(y1, y2) =1

2| det |

d2z(z1, z2) e2i(1)

ij(xz)i(yz)j

(2.1.10)

it follows that

tr Wf(x)Wg(x)(x) =1

2

|det

| d2yd2z f(y)g(z)e

2i(1)ij

(xy)i(yz)j(2.1.11)

The product between two Weyl operators is thus mapped to a noncommutative product betweenfunctions

Wf(x1, x2)Wg(x1, x2) = Wfg(x1, x2) (2.1.12)where the noncommutative product is defined by

(f g) (x) = ei2

ijij f(x)g(x)

x=x= f(x)e

i2

i

ijj g(x) (2.1.13)

Obviously, the usual product is recovered in (2.1.13) when 0. For n functions the productformula is straightforward generalized to

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2. Basics and origins of noncommutative field theories

Properties product There are three main properties of product, which are fundamental forperturbative calculations in noncommutative field theories.

(i) Associativity remains a property of this noncommutative product. In fact, the product de-fined in (2.1.13) is a special example of the associative products that arise in the deformationquantization [20]. The deformation of an algebra is defined by a formal power series expan-sion in the deformation parameter , such that the 0 order restores the algebra itself. Themultiplication rule between elements of the algebra f, g is defined as

f g = f g +

n=1

nCn(f, g) (2.1.16)

Equality (2.1.13) hence defines a unique deformation of the algebra of function to a noncom-mutative associative algebra (up to local redefinitions of the elements of the algebra), since itcan be rewritten in the following form

f g = f g +

n=1

i

2

n 1n!

i1j1 injni1 inf(x) j1 jng(x) (2.1.17)

I will further discuss algebra deformations in the next subsection.

(ii) The product is closed under complex conjugation. For complex valued functions we get(f g) = g f.

(iii) Ciclic invariance under integration is a very important property of product, which directlycomes from the ciclicity of trace of Weyl operators

d2xf1 fn = tr Wf1 Wfn (2.1.18)In particular, the product of two functions is equivalent to the usual product when integrated.

2.1.2 Moyal product defined by translation covariance and associativity

As I mentioned above, the product appearing in the Weyl formalism can be interpreted as comingfrom a special algebra deformation where the product (2.1.16) is defined by the Poisson bracket offunctions. I will now formulate in a more rigorous way how the product we chose can be uniquelyderived by imposing the properties of associativity and translation invariance to a Poisson structureproduct over a generic manifold M.

The manifold

Mis endowed with a Poisson structure P if for any two functions f, g

Ain the

algebra A (in general a Calgebra) defined over M we specify the Poisson bracketP(f, g) = {f, g}P = Pf(x)g(x)

x=x

= fPg (2.1.19)

A generic product on M is then defined through the derivatives defined on the manifold itself(with vanishing torsion and curvature) by

f g =

n=0

rann!

Pn(f, g) (2.1.20)

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2.1. Weyl formalism and Moyal product

where the coefficients a0 and a1 must be chosen to be a0 = a1 = 1 and

Pn(f, g) = P11

Pnn

1 . . .

nf

1 . . .

ng (2.1.21)

Comparing (2.1.20) to (2.1.16) we can immediately relate the deformation coefficients Cn(f, g) to thePoisson structure P(f, g) by Cn(f, g) =

ann! P

n(f, g). A necessary hypothesis on P in order to haveassociativity for the Moyal product (2.1.20) is P = 0, i.e. the Poisson structure must be constant.However this is not the only constraint that associativity implies. The property

(f g) h = f (g h) (2.1.22)

gives order by order equations for the coefficients an. As a result, we get that all the an must beequal to an = 1. It is essential for the Poisson structure to be constant and for the derivatives tobe curvature and torsion free. If any of these assumptions on P and is dropped, the Moyalproduct defined by (

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Liuba Mazzanti- Topics in Noncommutative Integrable Theories and Holographic Brane-World Cosmology