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Noncommutative Geometry Models for Particle Physics and Cosmology, Lecture I Noncommutative Geometry models for Particle Physics and Cosmology, Lecture I Matilde Marcolli Villa de Leyva school, July 2011 Matilde Marcolli NCG models for particles and cosmology, I Plan of lectures 1 Noncommutative geometry approach to elementary particle physics; Noncommutative Riemannian geometry; finite noncommutative geometries; moduli spaces; the finite geometry of the Standard Model 2 The product geometry; the spectral action functional and its asymptotic expansion; bosons and fermions; the Standard Model Lagrangian; renormalization group flow, geometric constraints and low energy limits 3 Parameters: relations at unification and running; running of the gravitational terms; the RGE flow with right handed neutrinos; cosmological timeline and the inflation epoch; effective gravitational and cosmological constants and models of inflation 4 The spectral action and the problem of cosmic topology; cosmic topology and the CMB; slow-roll inflation; Poisson summation formula and the nonperturbative spectral action; spherical and flat space forms Matilde Marcolli NCG models for particles and cosmology, I Geometrization of physics Kaluza-Klein theory: electromagnetism described by circle bundle over spacetime manifold, connection = EM potential; Yang{Mills gauge theories: bundle geometry over spacetime, connections = gauge potentials, sections = fermions; String theory: 6 extra dimensions (Calabi-Yau) over spacetime, strings vibrations = types of particles NCG models: extra dimensions are NC spaces, pure gravity on product space becomes gravity + matter on spacetime Matilde Marcolli NCG models for particles and cosmology, I Why a geometrization of physics? The standard model of elementary particles Matilde Marcolli NCG models for particles and cosmology, I Standard model Lagrangian: can compute from simpler data? 1 a a abc a b c 1 2 abc ade b c d e + SM = 2 ∂ν gµ∂ν gµ gsf ∂µgν gµgν 4 gs f f gµgνgµgν ∂ν Wµ ∂νWµ− L 2 + − 1 0 − 0 1 2 0 −0 1 − 0 + − M W W − ∂ν Z ∂νZ 2 M Z Z ∂µAν∂µAν igcw(∂νZ (W W − µ µ 2 µ µ 2cw µ µ 2 µ µ ν + − 0 + − +− 0 + − + − Wν Wµ−) Zν (Wµ ∂ν Wµ− Wµ−∂ν Wµ )+ Zµ(Wν ∂νWµ− Wν−∂ν Wµ )) −+ + − + +− + − igsw(∂νAµ(Wµ Wν− Wν Wµ−) Aν (Wµ ∂ν Wµ− Wµ−∂ν Wµ )+ Aµ(Wν ∂ν Wµ− + 1 2 −+ + − 1 2 + − + 2 2 0 + 0 − Wν−∂ν Wµ )) 2 g Wµ Wµ−Wν Wν− + 2 g Wµ Wν−Wµ Wν− + g cw(ZµWµ Zν Wν− 0 0 + − 2 2 + + 2 0 + − ZµZµWν Wν−)+ g sw(AµWµ AνWν− AµAµWν Wν−)+ g swcw(AµZν (Wµ Wν− + 0 + 1 − 2 2 + 1 0 0− W W −) 2A Z W W −) ∂ H∂ H 2M α H ∂ φ ∂ φ− ∂ φ ∂ φ ν µ − µ µ ν ν − 2 µ µ − h − µ µ − 2 µ µ − 2M 2 2M 1 2 0 0 + 2M 4 β + H + (H + φ φ +2φ φ−) + α h g2 g 2 g2 h − 3 0 0 + gαhM (H + Hφ φ +2Hφ φ−) 1 2 4 0 4 + 2 0 2 + 2−+ 0 2 2 8 g αh (H +(φ ) + 4(φ φ−) + 4(φ ) φ φ− +4H φ φ− + 2(φ ) H ) + 1 M 0 0 − gMW W −H g 2 Z Z H µ µ 2 cw µ µ 1 + 0 0 − 0 +− + 0 2 ig Wµ (φ ∂µφ− φ−∂µφ ) Wµ−(φ ∂µφ φ ∂µφ ) + 1 + − + − + 1− 1 0 0 0 g W (H∂ φ− φ−∂ H)+ W −(H∂ φ φ ∂ H) + g (Z (H∂ φ φ ∂ H)+ 2 µ µ µ µ µ µ 2 cw µ µ µ − 2 − − 1 0 0 + + sw 0 + + + M ( Z ∂µφ +W ∂µφ− +W −∂µφ ) ig MZ (W φ− W −φ )+igswMAµ(W φ− cw µ µ µ cw µ µ µ µ 2 − − − + 1 2cw 0 + + + + W −φ ) ig − Z (φ ∂ φ− φ−∂ φ )+ igs A (φ ∂ φ− φ−∂ φ ) µ 2cw µ µ µ w µ µ µ 1 2 + −2 0 2 + − 1 2 1 0 0 2 0 2 − 2 2 +− g W W − (H +(φ ) +2φ φ−) g 2 Z Z (H +(φ ) + 2(2s 1) φ φ−) 4 µ µ 8 cw µ µ w 2 −2 − − 1 2 sw 0 0 + + 1 2 sw 0 + + 1 2 0 + g Z φ (W φ− + W −φ ) ig Z H(W φ− W −φ )+ g s A φ (W φ− + 2 cw µ µ µ 2 cw µ µ µ 2 w µ µ + 1 2 − + + −2 sw 2 0 + W −φ )+ ig s A H(W φ− W −φ ) g (2c 1)Z A φ φ− µ 2 w µ µ µ cw w µ µ 2 2 + 1 a σ µ σ a− λ − λ λ λ− λ λ −λ g s A A φ φ− + ig λ (¯q γ q )g e¯ (γ∂ + m )e ν¯ (γ∂ + m )ν u¯ (γ∂ + w µ µ 2 s ij i j µ − e − ν − j mλ)uλ d¯λ(γ∂ + mλ)dλ + igs A (¯eλγµeλ)+ 2 (¯uλγµuλ) 1 (d¯λγµdλ) + u j − j d j w µ − 3 j j − 3 j j ig Z0 (¯νλγµ(1 + γ5)νλ)+(¯eλγµ(4s2 1 γ5)eλ)+(d¯λγµ( 4 s2 1 γ5)dλ)+ 4cw µ{ w− − j 3 w − − j (¯uλγµ(1 8 s2 + γ5)uλ) + ig W + (¯νλγµ(1 + γ5)U lep eκ)+(¯uλγµ(1 + γ5)C dκ) + j − 3 w j } 2√2 µ λκ j λκ j ig κ lep µ 5 λ κ µ 5 λ W (¯e U † γ (1 + γ )ν )+(d¯ C† γ (1 + γ )u ) + 2√2 µ− κλ j κλ j ig φ+ mκ(¯νλU lep (1 γ5)eκ)+ mλ(¯νλU lep (1 + γ5)eκ + 2M√2 − e λκ − ν λκ ig g mλ φ mλ(¯eλU lep† (1 + γ5)νκ) mκ(¯eλU lep† (1 γ5)νκ ν H(¯νλνλ) 2M√2 − e λκ ν λκ 2 M λ λ − λ − − − g me H(¯eλeλ)+ ig mν φ0(¯νλγ5νλ) ig me φ0(¯eλγ5eλ) 1 ν¯ M R (1 γ )ˆν 2 M 2 M − 2 M − 4 λ λκ − 5 κ − 1 ν¯ M R (1 γ )ˆν + ig φ+ mκ(¯uλC (1 γ5)dκ)+ mλ(¯uλC (1 + γ5)dκ + 4 λ λκ 5 κ 2M√2 d j λκ j u j λκ j − − − λ ig λ ¯λ 5 κ κ ¯λ 5 κ g mu λ λ φ− m (d C† (1 + γ )u ) m (d C† (1 γ )u H(¯u u ) 2M√2 d j λκ j − u j λκ − j − 2 M j j − mλ λ mλ g d ¯λ λ ig mu 0 λ 5 λ ig d 0 ¯λ 5 λ ¯a 2 a abc ¯a b c 2 M H(dj dj )+ 2 M φ (¯uj γ uj ) 2 M φ (dj γ dj )+ G ∂ G + gsf ∂µG G gµ + − 2 ¯ + 2 2 + ¯ 2 2 ¯ 0 2 M 0 ¯ 2 + ¯ 0 X (∂ M )X + X−(∂ M )X−+X (∂ 2 )X + Y ∂ Y + igcwWµ (∂µX X− − − − cw − ¯ + 0 + ¯ ¯ + ¯ 0 ∂µX X )+igswWµ (∂µYX− ∂µX Y )+ igcwWµ−(∂µX−X ¯ 0 + ¯ − ¯ + 0 ¯ + + − ∂µX X )+igswWµ−(∂µX−Y ∂µYX )+ igcwZµ(∂µX X − + + − ∂µX¯ −X−)+igswAµ(∂µX¯ X − 2 ¯ 1 ¯ + + ¯ 1 ¯ 0 0 1 2cw ¯ + 0 + ¯ 0 ∂µX−X−) gM X X H + X−X−H + 2 X X H + − igM X X φ X−X φ− + − 2 cw 2cw − 1 0 + 0 + 0 + 0 + igMX¯ X−φ X¯ X φ− +igMs X¯X−φ X¯ X φ− + 2cw w 1− + + 0 0 − igM X¯ X φ X¯ −X−φ . 2 − 1 Matilde Marcolli NCG models for particles and cosmology, I Standard model parameters: are there relations? why these values? Matilde Marcolli NCG models for particles and cosmology, I Building mathematical models: essential requirements Conceptualize: complicated things (SM Lagrangian) should follow from simple things (geometry) Enrich: extend existing models with new features Predict: Higgs mass? new particles? new parameter relations? new phenomena? The elephant in the room: Gravity! Coupling gravity to matter: the good: better conceptual structure, links particle physics to cosmology; the bad: worse chances of passing from classical to quantum theory In NCG models: \all forces become gravity" (on an NC space) Matilde Marcolli NCG models for particles and cosmology, I What the NCG model provides: Einstein{Hilbert action: 1 S (g ) = R pg d4x EH µν 16πG ZM Gravity minimally coupled to matter: S = SEH + SSM with SSM = particle physics Standard Model Lagrangian ) Right handed neutrinos with Majorana masses µν Modified gravity model f (R; R ; Cλµνκ) with conformal gravity: Weyl curvature tensor 1 C = R (g R g R g R + g R ) λµνκ λµνκ − 2 λν µκ − λκ µν − µν λκ µκ λν 1 + Rα(g g g g ) 6 α λν µκ − λκ µν Non-minimal coupling of Higgs to gravity 1 2 2 2 2 4 4 D H µ H ξ0 R H + λ0 H pg d x 2j µ j − 0j j − j j j j ZM Matilde Marcolli NCG models for particles and cosmology, I What is Noncommutative Geometry? X compact Hausdorff topological space C(X ) abelian , C ∗-algebra of continuous functions (Gelfand{Naimark) Noncommutative C ∗-algebra : think of as \continuous functions on an NC space" A Describe geometry in terms of algebra C(X ) and dense subalgebras like C 1(X ) if X manifold Differential forms, bundles, connections, cohomology: all continue to make sense without commutativity (NC geometry) Can describe \bad quotients" as good spaces: functions on the graph of the equivalence relation with convolution product To do physics: need the analog of Riemannian geometry Matilde Marcolli NCG models for particles and cosmology, I Spectral triples and NC Riemannian manifolds( ; ; D) A H involutive algebra A representation π : ( ) A!L H self adjoint operator D on , dense domain H compact resolvent (1 + D2) 1=2 − 2 K [a; D] bounded a 8 2 A even if Z=2- grading γ on H [γ; a] = 0; a ; Dγ = γD 8 2 A − 2 Main example( C 1(M); L (M; S); =@M ) with chirality γ5 in 4-dim Alain Connes, Geometry from the spectral point of view, Lett.
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