Noncommutative Geometry€¦ · TOC Fundamental interactions Noncommutative geometry Spectral...

Noncommutative Geometry

Raimar Wulkenhaar

Mathematisches Institut der Westfalischen Wilhelms-Universitat

Munster, GermanyaRaimar Wulkenhaar (Munster) Noncommutative Geometry

Contents

1 Introduction

2 Operator algebras

3 Tools in noncommutative geometry

4 The standard model in NCG

Raimar Wulkenhaar (Munster) Noncommutative Geometry

TOC Fundamental interactions Noncommutative geometry Spectral geometry Dirac operators QFT on NCG

Part I

Introduction

1 Fundamental interactions

2 Noncommutative geometry

3 Spectral geometry

4 Dirac operators

5 QFT on NCG



Fundamental interactions

Two distinct frameworks for fundamental interations in Nature:1 General Relativity to describe gravity2 Standard Model of particle physics to describe

electro-weak and strong interaction

Both frameworkds are geometrical, albeit of different type:

1 General Relativity is the dynamics of semi-Riemannianmanifolds (M,g) in response to the stress-energy tensor.

2 Standard Model is the geometry of (principal+vector) fibrebundles over a fixed (3+1)-dimensional space-timemanifold.

Their mutual relation is described by Einstein’s equation

Rµν −12gµνR + Λgµν = κTµν



Kaluza-Klein theory

First geometrical unification of General Relativity with (part of)Standard Model was proposed by [Kaluza, 1919] & [Klein, 1926]:

General Relativity in 5 dimensions, but viewed on 4Dsubmanifold, is 4D General Relativity plus Maxwellelectrodynamics.

More generally one studies a geometrical theory on totalmanifold M × Σ restricted to M.

Revived in String Theory (a different unification of gravityand Yang-Mills theory):

The total 10D target space manifold is split as M × Σ into4D space-time manifold M and a ‘tiny’ 6D manifold Σ(typically Calabi-Yau).

[Connes, Douglas & Schwarz, 1998] studied Σ being thenoncommutative torus.



Noncommutative Geometry

NCG is an enormous extension of topology and geometryformulated in the language of operator algebras.

Some influential milestones:

Mathematical foundation of quantum mechanics [vonNeumann, 1932], extended to theory of operator algebras[Murray & von Neumann, 1936–1943].

[Gelfand & Naimark, 1943] duality theorem: commutativeC∗-algebras are 1:1 with locally-compact Hausdorff spaces

Classification of factors in von Neumann algebras[Tomita, 1967], [Takesaki, 1970], [Connes, 1973–1976]

Bivariant K-theory [Kasparov, 1980]

Cyclic (co)homology and noncommutative index theory[Connes, 1979–1985]



Fredholm modules

Key ingredients in Kasparov’s KK-theory are Fredholm modules(A,H,F ) .

A is a ∗-algebra acting on a Hilbert space H

F is a self-adjoint operator on H with F 2 = 1 and [A,F ]compact

Such Fredholm modules describe topology.

[Connes, 1986] emphasised that geometry is encoded inspectral data of Dirac operator D = |D|F .

D contains:1 the spectrum of the Laplace operator −∆ = |D|2

(‘Can one hear the shape of a drum?’)2 the topological information of its phase F .



The Pioneer plaque [NASA, 1972]

Geometry encoded in spectral data:1 Hyperfine transition of H-atom as unit for length (21cm)

and frequency (1420 MHz)2 Spectra of strong pulsars to locate position of the sun.

Change of rotation period with time yields fabrication date.



Euclidean gravity and heat kernel expansion

[Weyl, 1919]-formula: Laplace op. on compact n-dim. manifold Mhas tr(χ[0,Λ](−∆)) ≈ ωn

(2π)n vol(M)Λn2 eigenvalues ≤ Λ

Any function tr(f (D2)) =∫∞

0 dt f (t)tr(e−tD2) of D2 computable

from the the heat operator e−tD2(functional calculus).

Asymptotic expansion tr(e−tD2) ∼

∑∞k=0 t−

n2+kak (D2)

yields Seeley-de Witt coefficients ak (D2).

The ak(D2) are integrals over curvature invariants:a0(D2) = 1

16π2

∫

M dx√

|det g|, a2(D2) = 196π2

∫

M dx√

|det g|R, .

tr(f (D2)), restricted to 0th and 1st order of the asymptoticexpansion, is Einstein-Hilbert action of Euclidean gravity.

Spectral action principle [Chamseddine & Connes,1996]

tr(f (D2)) is the physical action of a geometrical theory (A,H,D)



What is a Dirac operator?

Clear for spin manifolds, but we ask for a purelyoperator-algebraic characterisation of (A,H,D).

1 Dimension d : N th eigenvalue of |D|−1 is ∼ N−1/d

2 First order: [[D,A],A] = 0 (for A commutative)

3 Smoothness of A: δmA and δm[D,A] are all bounded,for δT := [|D|,T ]

4 Orientability: volume form ∈ span(A [D,A] · · · [D,A]︸︷︷︸

d

)

5 Spin structure: smooth part of H is of the form eAn,for e a projection.



Reconstruction & noncommutative generalisation

These requirements can be made so precise that they allow (forA commutative) to reconstruct the differentiable manifold Mfrom (A,H,D) [Connes, 2007].

Obvious generalisation to noncommutative A withTomita-Takesaki operator J that implements Aop = JAJ−1:

[A,Aop] = 0 and [[D,A],Aop] = 0 (first order)

Commutation relations of J with D and volume form definea Clifford dimension KO.

Simplest example is matrix-valued functions for A.‘Reconstruction’ corresponds to noncommutative manifoldof Kaluza-Klein type M × Σ with Σ a discrete space.

The conditions give, for KO=6, almost uniquely theStandard Model as simplest solution.

tr(f (D2)) is Einstein-Hilbert gravity coupled to full bosonicstandard model (strong+electroweak+Higgs).



Quantum field theory on noncommutative geometriesThere are truly noncommutative manifolds (A,H,D) which aresimple enough to do QFT (e.g. NC torus, Moyal space)

Yang-Mills+Higgs fields obtained via ‘inner fluctuations’D 7→ D +A[D,A] + JA[D,A]J−1

Scalar fields obtained as elements of eAn.

Euclidean QFT for (Yang-Mills-Higgs, scalar) fields Φ

defined via partition function Z =∫DΦ exp(−S[Φ])

S[Φ] is spectral action for Yang-Mills-Higgs or naturally builtfor scalar fields

Directly Lorentzian NCGs are a major open problem. Firstattempts by [Paschke & Verch, 2004].

Osterwalder-Schrader only established for Moyal spacewith commutative time [Grosse, Lechner, Ludwig & Verch, 2013]


TOC Topological spaces Banach & C∗-algebras Operator topologies Von Neumann algebras Projective modules Dictionary

Part II

Operator algebras

1 Topological spaces

2 Banach & C∗-algebras

3 Operator topologies

4 Von Neumann algebras

5 Projective modules

6 DictionaryRaimar Wulkenhaar (Munster) Noncommutative Geometry


Topological spaces

topological space X : set of points together with collection T(topology) of open subsets

1 Y ⊂ X closed if X \ Y open2 sufficies to define convergence of sequences3 suffices to define continuity: φ : X → Y continuous if for

every open Z ⊂ Y the pre-image φ−1(Z ) is open in X4 φ : X → Y homeomorphism if bijective and both φ, φ−1

continuous5 X is compact if any open cover has a finite subcover;

X is sequentially compact if any sequence has aconvergent subsequence (equivalent for metric spaces)

X may carry different topologies which leads to different notionsof convergence and continuity!



Hausdorff spaces

Hausdorff space X : topological space in which distinct pointsare separated by open neighbourhoods

1 limit of a convergent sequence is unique2 compact subsets are closed3 all metric spaces are Hausdorff

Hausdorff space X is locally-compact if every point has acompact neighbourhood

a locally-compact Hausdorff space can be embedded in acompact Hausdorff space which has only one extra point atinfinity



C∗-algebras

algebra A: vector space over K + ring + compatibilitywe assume 1 ∈ A and K = C

normed algebra: ‖ ‖ : A → R satisfying norm axioms ofvector spaces, ‖ab‖ ≤ ‖a‖ ‖b‖ and ‖1‖ = 1Banach algebra: completeness, i.e. Cauchy sequences inA have a limit in Ainvolution: (a+λb)∗ = a∗+λb∗ , (ab)∗ = b∗a∗ , (a∗)∗ = aC∗-algebra: Banach ∗-algebra with ‖a∗a‖ = ‖a‖2

The C∗-property is very restrictive:

1 ‖ ‖ unique: ‖a‖2 = sup|λ| : a∗a − λ1 not invertible in A2 φ : A → B isomorphism ⇒ ‖φ(a)‖ = ‖a‖3 any C∗-algebra is ∗-isomorphic to a norm-closed

subalgebra of B(H) (bounded operators on Hilbert space)



Standard example: continuous functions

A = C(X ) continuous functions on compact Hausdorff space Xwith ‖f‖ := supx∈X |f (x)|, (f ⋆)(x) := f (x)

Theorem. C(X ) is a C∗-algebra

1 norm-closed: Cauchy sequence fk in C(X ) definespointwise limit function f ; it follows ‖f − fk‖ → 0ǫ3 -trick proves continuity of f

2 C∗-property: ∃p ∈ X with ‖f‖ = supx∈X |f (x)| = |f (p)|

‖f ∗f‖ = supx∈X |f (x)|2 = |f (p)|2 = ‖f‖2

For X locally-compact: continuous functions vanishing at ∞C0(X ) = f ∈ C(X ) : ∀ǫ > 0 ∃K ⊂ X compact with

|f (x)| < ǫ ∀x ∈ X \ K



Standard example: bounded operators

H complex Hilbert space, B(H) algebra of bounded linearoperators on H with ‖a‖ := supx∈H,‖x‖=1 ‖ax‖

adjoint operator 〈a∗x , y〉 = 〈x ,ay〉 from Riesz representationtheorem, with ‖a∗‖ = ‖a‖

Theorem. B(H) is a C∗-algebra

1 norm-closed: same argument as for C(X )

2 C∗-property: from Cauchy-Schwarz

0≤‖ax‖2=〈ax ,ax〉=〈a∗ax , x〉≤‖a∗ax‖‖x‖≤‖a∗a‖‖x‖2

Example: The compact operators K(H) form an important(sub-) C∗-algebra.

One of many characterisations is norm-closure offinite-rank operators.



Important subclasses of bounded operators

selfadjoint: a = a∗

normal: aa∗ = a∗apositive: a = b∗bidempotent: e2 = eprojection: p = p∗ = p2

unitary: uu∗ = u∗u = 1 ≡ idHisometry: v∗v = 1partial isometry: v∗v = p = p∗ = p2

Further remarks:1 Operators in a subclass have particular spectrum.2 The classification of projections and unitaries is central in

K-theory.3 The comparison of projections is central in von Neumann

algebras.



The spectrum

A - Banach algebra, 1 ∈ A

sp(a) = λ ∈ C : a − λ1 not invertible in Ais the spectrum of a ∈ A

resolvent set R(a) = C \ sp(a), always open;resolvent function z 7→ (a − z1)−1 holomorphic on R(a)

sp(a) nonempty and compact in C

Further remarks:

Selfadjoint, positive, unitary operators and projections haveparticular spectrum.

A functional calculus which defines functions of a ∈ Arelies on the spectrum.

The spectrum can also be defined for closed denselydefined unbounded operators on Hilbert space.



Locally-convex algebras

Norm topology and norm convergence are often too restrictive.

Many useful topologies arise from family pk of seminorms:pk (a+b)≤pk (a)+pk (b), pk (λa)=|λ|pk (a), (pk (a)=0) 6⇔ (a=0)

The subsets a : pkj(b − a) < ǫ for finitely many pkj

playthe role of the open ǫ-balls centered at b.

pk defines a Hausdorff topology iff⋂

k p−1k (0) = 0.

Example:1 Topology of pointwise convergence: Functions f on

manifold M with px (f ) = |f (x)|

2 Smooth topology C∞(M): Functions f on compact manifoldM with pα(f ) = supx∈M |(∂αf )(x)|

3 Schwartz topology S(M): C∞-functions on M withpm,α(f ) = supx∈M(1 + ‖x‖m)|(∂αf )(x)|



Locally-convex operator topologies

On B(H), natural in quantum mechanics:1 Strong operator topology: px(a) = ‖ax‖H, for x ∈ H

2 Weak operator topology: px,y(a) = |〈x ,ay〉|, for x , y ∈ H

These are coarser (or weaker) than norm topology (less open &closed sets, more compact sets, more convergent sequences).

X–Banach space, X ∗=f : X → C linear & bounded dual space

1 Weak topology on X : pf (x) = |f (x)|, for f ∈ X ∗

(coarsest topology in which all f ∈ X ∗ continuous)2 Weak∗ topology on X ∗: px(f ) = |f (x)|, for x ∈ X

(Unit ball in X ∗ is weak∗-compact [Banach-Alaoglu])



The Gelfand-Naimark duality theoremLet X := Spec(A) = χ : χ character on A

A character on commutative Banach-algebra A is anon-zero ∗-algebra homomorphism χ : A → C

X ⊂ A∗1 (the unit ball in dual space A∗)

Equipped with weak-∗-topology, X is a compact topologicalHausdorff space.

Gelfand transform: ρ : A ∋ a 7→ a ∈ C(Spec(A))defined for commutative Banach algebra A by a(χ) := χ(a)

Theorem

Let A be a commutative C∗-algebra and X = Spec(A).Then ρ : A → C(X ) is an isometric isomorphism.

Proof establishes correspondence between λ ∈ sp(a) andcharacter with χ(a) = λ. Surjectivity by Stone-Weierstraß.



Von Neumann algebrasIf A ⊂ B(H), then A′ = b ∈ B(H) : [b,A]=0 is the commutant.

Von Neumann bicommutant theorem

For M ⊂ B(H) a ∗-subalgebra with 1 ∈ M are equivalent:1 M′′ = M

2 M is closed in strong operator topology.3 M is closed in weak operator topology.

Such M is called von Neumann algebra.

S ⊂ B(H) arbitrary subset, then (S ∪ S∗)′′ is von Neumannalgebra.

Von Neumann algebras are generated by its projections.

Von Neumann algebras are (large) C∗-algebras.

There are less von Neumann algebras than C∗-algebras,which made it possible to classify von Neumann algebras.



Factors

A factor is a von Neumann algebra M with M∩M′ = C1.

Any von Neumann algebra is isomorphic to a direct integralof factors.

Classification of factors rests on a comparison of projections:

There is a natural partial order p ≤ q iff pH ⊂ qH.

We define p - q if p = uu∗ and u∗u ≤ q for a partialisometry u ∈ M

p ≈ q (Murray-von Neumann equivalent) if p = uu∗ andq = u∗u

In a factor one has either p q, q p or p ≈ q.



Types of factors

Type I

Factor M with minimal projection.

p 6= 0 is minimal projection if (q ≤ p) ⇒ (q = 0 or q = p).Minimal projections fulfil pMp = Cp.

Standard form: M = u∗(B(H)⊗ 1)u

Type II

Infinite factor M with ultraweakly-continuous trace tr 6= 0.

A factor is infinite if there is p < 1 with p ≈ 1.

tr(ab) = tr(ba), tr(a∗a) ≥ 0; faithful if tr(a∗a) = 0 ⇔ a = 0

Standard form: GNS construction for (faithful) tracial state.

Type III

everything elseRaimar Wulkenhaar (Munster) Noncommutative Geometry


[Haag & Kastler, 1964] axioms in Algebraic QFTNet of abstract C∗-algebras A(O) assigned to bounded, open,causally complete regions O in Minkowski space, satisfying:

1 Isotony: O1 ⊂ O2 ⇒ A(O1) ⊂ A(O2)

2 Covariance: Poincare group is realised as group ofautomorphisms of the net.

3 Locality: For O1,O2 causally independent,[A(O1),A(O2)] = 0

4 Time slice axiom: If O1 ⊂ O2 have a common Cauchyhypersurface, then A(O1) = A(O2).

Further remarks:The A(O) are usually Type-III von Neumann algebras.QFT profits from deep results about operator algebras.Bounded functions of field op’s. provide coordinates on A(O).Applies (in contrast to Wightman) to curved space-time.



Projective modules

A (right) module over an algebra A is a space E with a mapE ×A → E satisfying η(ab) = (ηa)b.

Easiest example is a free module E =⊕

I A.We write this as E = An for finite I.E is projective if there is another A-module F such thatE ⊕ F is free.In this case e : E ⊕ F → E is a idempotent e2 = e.A finite projective module is of the form E = eAn.

Example: Space Γ(E) of sections of vector bundle E

Ep

−→ X vector bundle over topological space Xfibre p−1(x) is finite-dim. vector space ∀x ∈ X .Sections Γ(E ,X ) := s : X → E : p s = idX form amodule over algebra of functions: (sf )(x) = s(x)f (x).



The Serre-Swan theorem

Theorem [Swan, 1962]

Let X be a compact Hausdorff space. A C(X )-module E isisomorphic to a module Γ(E ,X ) for a locally-trivial vector

bundle Ep→ X if and only iff E is finitely projective.

Remarks:

Local triviality: For every x ∈ X there is a neighbourhoodUx and m ∈ N such that p−1(Ux ) ≃ Cm × U (homeomorphic)

Finitely many sx,1, . . . sx,k generate p−1(Ux).

By compactness, finitely many s1, . . . , sn generate Γ(E ,X )

Image of projection e : Γ(Cn × X ,X ) → Γ(E ,X ) has locallyconstant dimension.



The dictionary of NCG

compact Hausdorff space commutative C∗-algebrameasure space (X , µ) commut. von Neumann algebragroup commutative Hopf algebravector bundle over X finitely generated projective mod-

ule over C(X )

vector field derivationK-theory K-theoryde Rham complex Hochschild cohomologyde Rham cohomology cyclic homologydifferentiable manifold M commutative spectral triplediffeomorphism of M automorphism of commutative

spectral triple(real/infinitesimal) variable linear (selfadjoint/compact) oper-

ator on Hilbert spaceintegral trace


TOC K-theory Hochschild & cyclic (co)homology Compact operators Spectral triples

Part III

Tools in noncommutative geometry

1 K-theory

2 Hochschild & cyclic (co)homology

3 Compact operators

4 Spectral triples



Operator K-theoryFor A a C∗-algebra, let [p] be the Murray-von Neumannequivalence class of projection p ∈ A. We assume 1 ∈ A.

K0(A) = [p]− [q] : p,q projections in A

K0(A) is abelian group with [p] + [q] =[( p 0

0 q

)]

Covariance: ∗-homomorphism φ : A → B inducesφ∗ : K0(A) → K0(B)

Homotopic ∗-homomorphisms φ,ψ : A → B induceφ∗ = ψ∗ : K0(A) → K0(B)

Exact sequence 0 → I → A → A/I → 0 inducesK0(I) → K0(A) → (A/I) which is exact at K0(A)

Morita invariance K0(A) = K0(A⊗K(H))

Serre-Swan: K0(C(X )) ≃ K 0(X ) (topological K-theory)

Remark: α−→ A

β−→ exact at A if ker β = imα



Higher K -theory groupsgroup Un(A) := U(Mn(A)) of unitaries in A-valued matricesU1(A) ⊂ U2(A) ⊂ · · · ⊂ U∞(A) by identifying u ≡

( u 00 1

)

K1(A) = π0(U∞(A)) = homotopy classes [u] : u ∈ U∞(A)

K1(A) is abelian group with [u][v ] =[( u 0

0 v

)]

Observation: K1(A) isomorphic to K0(SA) (suspension):For u ∈ Un(A) there is path αt ∈ U2n(A) connectingdiag(u,u−1) ∼ diag(1n,1n).

[u] 7→ [αt diag(1n,0)α−1t ]− [diag(1n,0)] is isomorphism

between K1(A) and K0(SA) := K0(C0(]0,1[,A))

all functorial properties of K0 transfer to K1

Generalisation to Kn+1(A) := Kn(C0(R,A)).Raimar Wulkenhaar (Munster) Noncommutative Geometry


The index map

Given an exact sequence 0 → Ii

−→ Aπ

−→ A/I → 0 andu ∈ Un(A/I).

Since π is surjective, there is v ∈ U2n(A) withπ(v) = diag(u,u−1).

Consider v diag(1n,0)v−1]− diag(1n,0) ∈ M2n(A).

Since v diag(1n,0)v−1 − diag(1n,0)) ∈ kerπ, there is byexactness a p ∈ I with i(p) = v diag(1n,0)v−1 − diag(1n,0)).

By injectivity of i , this p is unique and defines byfunctoriality a map ∂ : K1(A/I) ∋ [u] 7→ [p] ∈ K0(I).

Proposition

The following induced sequence is everywhere exact:. . .

∂−→K1(I)

i∗−→K1(A)π∗−→K1(A/I)

∂−→K0(I)

i∗−→K0(A)π∗−→K0(A/I)



Bott periodicityFor z ∈ S1 and p projection in Mn(A) definefp(z) = zp + (1n − p) ∈ Un(A) with fp(1) = 1n for all p.

Viewing fp ∈ C(]0,1[,Mn(A)) up to homotopy, one getsβ : K0(A) → K1(SA) = K2(A)

Theorem

The Bott map β : K0(A) → K2(A) is an isomorphism.

Corollary: Six-term exact sequence of K -theory

K0(I)i∗ K0(A)

π∗ K0(A/I)

∂β

K1(A/I)

∂

K1(A)π∗

K1(I)i∗

This (and other) exact sequence(s) make K-theory computable!K-theory goups are important to distinguish C∗-algebras.



Hochschild homologyA unital algebra, M bimodule over A and Cn(A,M) = M⊗A⊗n

Hochschild boundary operator b : Cn(A,M) → Cn−1(A,M)

b(m⊗a1 ⊗ . . . an):=ma1 ⊗ a2 ⊗ · · · ⊗ an

+n−1∑

i=1

(−1)im ⊗ a1 ⊗ . . .⊗ ai−1 ⊗ aiai+1 ⊗ ai+2 ⊗ . . .⊗ an

+(−1)nanm ⊗ a1 ⊗ · · · ⊗ an−1

b2 = 0

Definition

The homology of the chain complex (C∗(A,M),b) is calledHochschild homology and denoted HH∗(A,M)

If Zn(A,M) ∋ cn is the subspace of (closed) Hochschildn-cycles bcn = 0, then HHn(A,M) = Zn(A,M)/bCn+1(A,M)



Hochschild and cyclic cohomology

On (dual) space Cn(A) = Hom(A⊗(n+1),C) of (n+1)-linearfunctionals, define Hochschild differential b : Cn(A) → Cn+1(A):

(bϕ)(a0, . . . ,an+1)=n∑

i=0

(−1)iϕ(a0, . . . ,aiai+1, . . . ,an+1)

+(−1)n+1ϕ(an+1a0,a1, . . . ,an)

b2 = 0, cohomology of complex (C∗(A),b) is Hochschildcohomology HH∗(A)

ϕ ∈ Cn(A) is cyclic if ϕ(a0,a1, . . . ,an) = (−1)nϕ(an,a0, . . . ,an−1)

Cnλ(A) space of cyclic cochains, preserved by b

Cyclic cohomology HC∗(A) is cohomology of complex(C∗

λ(A),b)Similarly, there is a dual cyclic homology HC∗(A)

Computable in Connes’ (b,B)-bicomplex.



Commutative cases

Theorem (Hochschild-Kostant-Rosenberg-Connes)

HH∗(C∞(M), C∞(M)) ≃ Ω∗(M) (exterior differential algebra)

Remarkable fact: HH∗(C∞(M), C∞(M)) is local; only the

diagonal in (C∞(M))⊗(n+1) ≃ C∞(M × · · · × M) contributesVolume form in NCG is Hochschild d -cycle.

For M a n-dim. closed smooth manifold, define ϕ ∈ Cn(C(M))by ϕ(f0, . . . , fn) =

∫

M f0df1 ∧ · · · ∧ dfnBy Leibniz rule, ϕ is Hochschild cocycle, bϕ = 0By Stokes’ law, ϕ is cyclic, hence cyclic cocyle

Theorem

HCk(C∞(M)) = Ωk (M)/

(im d ⊕

⊕[k/2]i=0 Hk−2i

dR (M))

Cyclic homology is NCG-analogon of de Rham cohomology.Raimar Wulkenhaar (Munster) Noncommutative Geometry


Pairing with K-theoryidentify ϕn(m0 ⊗ a0, . . . ,mn ⊗ an) := tr(m0 · · ·mn)ϕn(a0, . . . ,an)

Theorem

HC2n(A)× K0(A) ∋ ([ϕ2n], [p]) 7→ ϕ2n( p, . . . ,p︸︷︷︸

2n+1

) ∈ C and

HC2n−1(A)×K1(A)∋([ϕ2n−1], [u]) 7→ ϕ2n−1( u,u−1, . . .,u,u−1︸︷︷︸

n+n

)∈C

provide well defined pairings:

these maps are bilinear,

they only depend on the K-theory classes of p, u,

they only depend on cyclic cohomology classes of ϕk .

∃ periodicity operator S : HCn(A) → HCn+2(A) with〈[ϕ2n], [p]〉 = 〈[Sϕ2n], [p]〉 and 〈[ϕ2n−1], [u]〉 = 〈[Sϕ2n−1], [u]〉

Periodic cyclic cohomology is S-limit HP(A) = lim→

HCk(A).

Chern characters ∈ HP∗(A) yield integer pairings.Raimar Wulkenhaar (Munster) Noncommutative Geometry


K-homology and Chern characterFredholm module (A,H,F ) is p-summable if [A,F ]p trace class.

Chern characterLet 2n ≥ p, projection P = 1

2(1 + F ) andϕ2n−1(a0, . . .,a2n−1) = (−1)n2(n−1

2)· · ·12 tr(F [F,a0] · · · [F,a2n−1]):

ϕ2n−1 ∈ HC2n−1(A) and Sϕ2n−1 = ϕ2n+1

Chern-Connes character Chodd(F ,H) ∈ HP1(A) isperiodic cyclic cohomology class of ϕ2n−1

For u ∈ Uk(A), PuP is Fredholm operator, andindex(PuP) = 2−(2n+1)

(n− 12 )···

12〈Ch2n−1(F ,H), [u]〉 ∈ Z

Even case Cheven(F ,H) requires γ with Fγ=−γF and [γ,A]=0

F is non-local, but Ch(F ,H) has local representative.

Corresponding local index formula [Connes & Moscovici, 1995]related to BPHZ renormalisation [Connes & Kreimer, 1998]NC Chern-Simons action is not naıve [Pfante, 2012]



Compact operators

Let K ⊂ B(H) be the ideal of compact operators on H andT ∈ K.

There is a null sequence of eigenvalues µi > 0 of |T |

The eigenspace Ei = ker(µi1 − |T |) is finite-dimensionalsp(|T |) = µii∈N ∪ 0

sk (T ) = inf‖T∣∣E⊥‖ : dim(E) = k – characteristic value

sk (T ) – k th eigenvalue µk (|T |) if arrangend decreasingly andwith multiplicity

Schatten ideal Lp

Lp :=

T ∈ K : ‖T‖p :=( ∞∑

k=0

(sk (T ))p) 1

p<∞

All (Lp, ‖ ‖p) are Banach spaces.



L1 – trace classTr(T ) =

∑

n〈ψn,Tψn〉 for T ∈ L1 and ψn ONB

L2 – Hilbert-Schmidt class

L∞ = K with ‖T‖∞ = ‖T‖

Inequalities

Tr(T ) ≤ Tr(|T |) = ‖T‖1 for T ∈ L1

Tr(TS) = Tr(ST ) if TS,ST ∈ L1

Holder: ‖TS‖1 ≤ ‖T‖p‖S‖q for T ∈ Lp, S ∈ Lq with1p + 1

q = 1

Lp ⊂ Lr if p ≤ r

It turns out that Tr on L1 is not the right generalisation of theintegral.



Dixmier ideal L1+

partial sums σn(T ) =∑n−1

k=0 sk (T )

σn(T ) =sup‖TPE‖1 : PE projector to n-dim. subspace E ⊂ H

Lemma

σn(T + S) ≤ σn(T ) + σn(S) ≤ σ2n(T + S) for 0 ≤ T ,S ∈ K

Proof: first ≤ from norm, second from inequalities

‖TPE‖1 + ‖SPF‖1 = Tr(PETPE) + Tr(PF SPF )

≤ Tr(PE+F TPE+F ) + Tr(PE+F SPE+F )

= Tr(PE+F (T + S)PE+F )



Definition (Dixmier-Ideal L1+ ⊂ K)

L1+ :=

T ∈ K : ‖T‖1+ := supn≥2

σn(T )

log n<∞

L1 ⊂ L1+ ⊂ Lp for any p > 1

If(σn(T )

log n

)

n≥2 convergent, then T ∈ L1+ is called measurable

Definition (noncommutative integral)∫

−T := limn→∞

σn(T )

log n

∫− additive on positive elements, linear by extensionσn(UTU∗) = σn(T ):∫− is trace on subspace of measurable elements of L1+∫− vanishes on L1

can be (not uniquely) generalised to Dixmier trace on L1+



Example

∆ = −∑p

µ=1∂2

∂x2µ

Laplace operator on Tp

sp(∆) = ‖k‖2 : k ∈ Zp

Replace ∆ by 1 on ker∆ = C ⇒ ∆− q2 compact for q > 0

Determine sn(‖k‖)(∆− q

2 ) = ‖k‖−q asymptotically:n(‖k‖) = #(lattice points in ball of radius ‖k‖ in Rp)

= Vp‖k‖p with Vp = πp2

Γ( p+22 )

sn(∆− q

2 ) =( n

Vp

)− qp

σn(∆− q

2 ) =

∫ n

1du(Vp

u

) qp , ‖∆− q

2 |1+ =

∞ for q < p0 for q > p

∫

−∆− p2 = lim

n→∞

1log n

∫ n

1du

Vp

u= Vp



Spectral triples

Definition [Connes, 1996]

(A,H,D) – commutative spectral triple, i.e. H a Hilbert space, A acommutative involutive unital algebra represented in H,D a selfadjoint operator in H with compact resolvent, p an integer.

1 Dimension: k th characteristic value of resolvent of D is O(k−1p )

2 Order one: [[D, f ], g] = 0 ∀f , g ∈ A

3 Regularity: f and [D, f ] belong to the domain of δm, for any f ∈ Aand m ∈ Z, where δT := [|D|,T ]

4 Orientability: ∃ Hochschild p-cycle c ∈ Zp(A,A) s.t. πD(c) = 1for p odd, πD(c) = γ for p even with γ = γ∗, γ2 = 1, γD = −Dγ

5 Finiteness and absolute continuity: H∞ :=⋂

m dom(Dm) ⊂ H isfinitely generated projective A-module, H∞ = eAn, withe = e∗ = e2 ∈ Mn(A). Hermitian structure(ξ|aη) =

∑ni=1 aξ∗i ηi ∈ A satisfies 〈ξ, η〉 =

∫−(ξ|η)|D|−p



Reconstruction theorem

Theorem [Connes, 2008]

Let (A,D,H) be a spectral triple, A commutative and unital. Letthe conditions (1)–(5) be realised in stonger form:

1 All T ∈ EndA(H∞) are regular2 The Hochschild cycle is antisymmetric,

c =∑

α a0α ⊗

∑

β∈Spǫ(β)aβ(1)

α ⊗ · · · ⊗ aβ(p)α

Then there exists a compact oriented differentialble manifold Xwith A = C∞(X ).Conversely, every compact oriented differentiable manifoldarises in this way.

Second theorem: If in addition the multiplicity of A′′ in H is 2p/2,then A = C∞(X ) for a smooth oriented compact spinc-manifoldX . The stronger condition (1) is automatic.



Definition (smooth compact p-dimensional manifold)

. . . is a compact Hausdorff space X together with a system oflocal charts (Uα, sα) such that

the Uα are open in X and X =⋃

α Uα

sα : Uα → sα(Uα) ⊂ Rp is a homeomorphism. In particular,sα(Uα) is open in Rp and sα is injective

sα s−1β : sβ(Uα ∩ Uβ) → sα(Uα ∩ Uβ) is smooth

Strategy:

norm-completion of A is unital commutative C∗-algebraA = C(X ) for compact Hausdorff space X = Spec(A)

build tentative charts (up to injectivity) from c

prove that there exists restriction to subsets where sα isinjective (very complicated)



Noncommutative spectral triplesGeneralisation to noncommutative algebras requires realstructure J to formulate first-order condition. Examples:

Noncommutative torus: C∗-algebra generated by unitariesU,V with UV = e2πiθVU, smooth algebra by [Rieffel, 1993]

Crossed products [Connes & Moscovici, 1998]

Noncommutative 4-sphere S4θ [Connes & Landi, 2001]

application to instantons [Landi & van Suijlekom]

NC 3-manifolds [Connes & Dubois-Violette, 2002]

Moyal space: non-compact variant of nc torus; difficultiessolved in [Gayral, Gracia-Bondıa, Iochum, Schucker & Varilly, 2004]

SUq(2) quantum group[Dabrowski, Landi, Sitarz, van Suijlekom & Varilly, 2005]

Moyal space with harmonic propagation [Gayral & W., 2011]


TOC Real structure Order-one Fluctuations Spectral action

Part IV

The standard model in NCG

1 Real structure

2 Order-one

3 Fluctuations

4 Spectral action



Some historical notes1 Differential geometry on matrices describes Higgs potential

[Dubois-Violette, Kerner & Madore, 1989], [Connes, 1990]

2 [Connes & Lott, 1991] model of two-sheeted universe

3 Full standard model formulated with bimodules [Connes, 1994]

4 Real structure [Connes, 1995]

5 Spectral action principle [Chamseddine & Connes, 1996]

6 Various articles by [Iochum, Kastler & Schucker (et al), 1990s]

7 Neutrino mixing in KO-dimension 6[Barret, 2006], [Connes, 2006]

8 New scalar field from right neutrinos[Chamseddine & Connes, 2010]

We follow [Chamseddine, Connes & Marcoli, 2006] at level of (7).Raimar Wulkenhaar (Munster) Noncommutative Geometry


Real structure

Definition

A spectral triple (A,H,D), with A not necessarily commutative,satisfies the noncommutative order-one condition if there existsan anti-unitary operator J on H such that [a, JbJ−1] = 0 and[[D,a], JbJ−1] = 0 for all a,b ∈ A.

J defines a real structure of KO-dimension k ∈ Z/8 if

J2 = ǫ , JD = ǫ′DJ , Jγ = ǫ′′γJ for p even

where the signs ǫ, ǫ′, ǫ′′ = ±1 are functions of k mod 8:

k 0 1 2 3 4 5 6 7

ε 1 1 −1 −1 −1 −1 1 1ε′ 1 −1 1 1 1 −1 1 1ε′′ 1 −1 1 −1



Matrix-valued spectral triples in KO-dimension 6We look for matrix solutions of axioms for (A,H,D, J, γ) with

J2 = 1 , JD = DJ , Jγ = −γJ , [a, JbJ] = 0 , [[D,a], JbJ] = 0

γ = γ∗ , γ2 = 1 , Dγ = −γD ,

The conditions and further (more or less natural) requirementsrestrict an initial matrix algebra A eventually to AF .

Requirements

1 H has a separating vector, i.e. there is a ξ ∈ H such thatA ∋ a 7→ aξ ∈ H is injective.

2 (A,H, J) is irreducible, i.e. there is no non-trivial (e 6= 0,1)projector e ∈ B(H) which commutes with A and J.

Proposition

Let AC be the complexification of A. Then Z (AC) = Ce1 ⊕ Ce2

in KO-dimension 6 for minimal projections e1,e2 = J−1e1J.Raimar Wulkenhaar (Munster) Noncommutative Geometry


Requirements from particle physics

1 One block of the real algebra A is quaternionic

A = Mk (H)⊕ M2k (C)2 The quaternionic block has a non-trivial Z2-grading γ1

The minimal solution of these requirements isA = M2(H)⊕ M4(C).

Up to isomorphisms, we have

J(

xy

)

=

(y∗

x∗

)

,

γ = (γ1,0)− J(γ1,0)J−1 ∈ AAop , γ

(xy

)

=

(γ1x−yγ1

)

with γ1 = diag(12,−12) ∈ M2(H).

Result:

The even part Aev of A is Aev = (H⊕H)⊕ M4(C)



Order-one condition

Proposition

Up to isomorphisms of Aev there is a unique involutivesubalgebra AF ⊂ Aev of maximal dimension which under theorder-one condition permits a connecting D, i.e. e1De2 6= 0.

This solution is the matrix algebra of the standard modelAF= H⊕ C⊕ M3(C) ⊂ M2(H)⊕ M4(C)

∋

λ 0 0 00 λ 0 00 0 α β0 0 −β α

⊕

λ 0 0 00 m11 m12 m13

0 m21 m22 m23

0 m31 m32 m33

Non-zero part of e1De2 connects the two λ-places.

The Hilbert space is M4(C)⊕ M4(C) ∈(x

y

). We have

J(x

y

)=(y∗

x∗

), γ

(xy

)=( γ1x−yγ1

)



The Hilbert space is M4(C)⊕ M4(C). Its elements areparametrised by elementary fermions:

H ∋

νR uRr uRb uRg

eR dRr dRb dRg

νL uLr uLb uLg

eL dLr dLb dLg

νcR ec

Ruc

Rr dcRr

ucRb dc

Rbuc

Rg dcRg

νcL ec

Luc

Lr dcLr

ucLb dc

Lbuc

Lg dcLg

We now look for the complete D (not only e1De2) which is evenand satisfies order-one.



Lepton/quark decomposition

We want to represent D as a matrix acting on HF :

HF = H1 ⊕H3 , H1 ≃ C8 ∋

ℓR

ℓL

ℓcRℓc

L

, H3 ≃ C24 ∋

qRqLqc

Rqc

L

with leptons ℓR =(νR

eR

), ℓL =

(νLeL

), ℓc

R =(νc

Rec

R

), ℓc

L =(νc

Lec

L

)and

quarks qR =(uR

dR

), qL =

(uLdL

), qc

R =(uc

Rdc

R

), qc

L =(uc

Ldc

L

).

Write operators on HF as TF =

(

T11 T13

T31 T33

)

with T11 ∈ M8(C)

and T33 ∈ M24(C).

AF , JF , γF diagonal (no 13,31-blocks)D13 = 0 and D31 = 0 from order-one and DFγF = −γF DF



diagonal parts of AF :

(q, λ,m)11 =

qλ 0 0 00 q 0 00 0 λ12 00 0 0 λ12

, qλ =

(

λ 00 λ

)

(q, λ,m)33 =

qλ ⊗ 13 0 0 00 q ⊗ 13 0 00 0 12 ⊗ m 00 0 0 12 ⊗ m

,

with q ∈ H, λ ∈ C, m ∈ M3(C)



diagonal part of JF :

J11 =

0 0 12 00 0 0 12

12 0 0 00 12 0 0

CC

J33 =

0 0 12 ⊗ 13 00 0 0 12 ⊗ 13

12 ⊗ 13 0 0 00 12 ⊗ 13 0 0

CC

where CC means complex conjugation



diagonal part of γF :

γ11 =

12 0 0 00 −12 0 00 0 −12 00 0 0 12

,

γ33 =

12 ⊗ 13 0 0 00 −12 ⊗ 13 0 00 0 −12 ⊗ 13 00 0 0 12 ⊗ 13

.

γ2F = 1, γF = γ∗F , J2

F = 1 and γF JF = −JFγF satisfied



Diagonal part of DF ; from DF , γF = 0, [DF , JF ] = 0 andconnecting part:

D11 =

0 Y1 T 0Y ∗

1 0 0 0T ∗ 0 0 Y1

0 0 Y t1 0

, D33 =

0 Y 3 0 0Y ∗

3 0 0 00 0 0 Y 3

0 0 Y t3 0

,

with Y1 ∈ M2(C), Y 3 ∈ M2(C)⊗ M3(C) and

T = T t =

(

YR 00 0

)

with YR ∈ C

order-one: Y 3 = Y3 ⊗ 13 with Y3 ∈ M2(C)

Physical condition: DF commutes with representation of Cin AF , i.e. with (qλ, λ,0)

leads to Y1 =

(

Yν 00 Ye

)

, Y3 =

(

Yu 00 Yd

)



Cabibbo-Kobayashi-Maskawa matrices

Standard model needs 3 copies of HF .leptons ν,e ∈ C3, quarks u,d ∈ C3 ⊗ C3

Yν,e,u,d ,R ∈ M3(C) in D

We say that DF ,D′F are equivalent if D′

F = UDF U∗ for a unitarymatrix U which commutes with AF , JF , γF .

U11 =

diag(V1,V2) 0 0 00 diag(V3,V3) 0 00 0 diag(V1,V2) 00 0 0 diag(V3,V3)

U33 =

diag(W1,W2)⊗ 13 0 0 00 diag(W3,W3)⊗ 13 0 00 0 diag(W1,W2)⊗ 13 00 0 0 diag(W3,W3)⊗ 13

with Vi ,Wi ∈ U(3)Raimar Wulkenhaar (Munster) Noncommutative Geometry


Product of spectral triples

A = AM ⊗AF , H = HM ⊗HF , D = DM ⊗ 1F + γM ⊗ DF ,

J = JM ⊗ JF , γ = γM ⊗ γF

withAM = C∞(M), HM = L2(M,S), γM = γ5, JM = γ2 CC and

DM = ieµaγ

a∇sµ , ∇S

µ = ∂µ +18ωabµ [γa, γb]

eaµ vierbein, ωab

µ spin connection form:

δabeaµeb

ν = gµν , ∂µeνa − Γνµρeρ

a + ωabµ eν

b = 0

Lichnerowicz formula

D2M = ∆LC+

14

R, ∆LC = −gµν(∂µ∂ν − Γρµν∂ρ), R = gµνRµν



Fluctuationsreplace D by fluctuated operator DA = D + A + JAJ−1,where A = A∗ =

∑

α aα[D,bα]

DA satisfies the same axioms as D

DM ⊗ 1 generates 1-forms G = eµaγ

aGµ withGµ =

∑

α aα∂µ(bα) ∈ A

γM ⊗ DF generates 1-forms γ5Φ = γ5∑α aα[DF ,bα]

Φ11 =

0 Φℓ 0 0Φ∗ℓ 0 0 0

0 0 0 00 0 0 0

, Φℓ =

(

MνCℓφ1 MνCℓφ2

−Meφ2 Meφ1

)

Φ33 =

0 Φq ⊗ 13 0 0Φq ⊗ 13 0 0 0

0 0 0 00 0 0 0

, Φq =

(

MuCqφ1 MuCqφ2

−Mdφ2 Mdφ1

)



lepton part of DA is:

DA,11 =

iγaeµa∇

S,Bµ γ5Φℓ T 0

γ5Φ∗ℓ iγaeµ

a∇S,B,Wµ 0 0

T ∗ 0 iγaeµa∇

S,Bµ γ5Φℓ

0 0 γ5Φtℓ iγaeµ

a∇S,B,Wµ

with

∇S,Bµ =

(

∇Sµ 0

0 ∇Sµ−2iBµ

)

1Y ,

∇S,B,Wµ =

(

∇Sµ + i(−Bµ − W 3

µ) −i(W 1µ − iW 2

µ)

−i(W 1µ + iW 2

µ) ∇Sµ + i(−Bµ + W 3

µ )

)

13

Bµ,W aµ ∈ C∞(X ) real-valued



quark part of DA is

DA,33 =

iγaeµa∇

S,B,Gµ γ5Φq 0 0

γ5Φ∗q iγaeµ

a∇S,W ,Gµ 0 0

0 0 iγaeµa∇

S,B,Gµ γ5Φq

0 0 γ5Φtq iγaeµ

a∇S,W ,Gµ

with

∇S,B,Gµ =

∇S

µ13 + i((−G0µ + Bµ)13 − Gµ) 0

0 ∇Sµ13 + i(−(G0

µ + Bµ)13 − Gµ)

13

∇S,W ,Gµ =

∇S

µ13 + i((−G0µ − W 3

µ )13 + Gµ) −i(W 1µ − iW 2

µ )13

−i(W 1µ + iW 2

µ)13 ∇Sµ13 + i((−G0

µ + W 3µ ) + Gµ)

13

G0µ ∈ C∞(X ) real Gµ ∈ M3(C∞(X )) hermitian and traceless

Unimodularity condition tr(A) = 0 ⇒ G0µ = −1

3Bµ



The action functional

Fermionic action functional

SF = 〈Jψ,DAψ〉

where ψ = γψ ∈ H+ are Grassmann-valued

Spectral action principle [Chanseddine-Connes]

The bosonic action is a functional only of the spectrum of D2A.

functional calculus and Laplace transformation

SA = Tr(χ(D2A)) =

∫ ∞

0dt χ(t)Tr(e−tD2

A) ,

with χ(s) =∫∞

0 dt e−st χ(t)



Proposition (heat kernel expansion)

Let F be a vector bundle over (M,g). A second-orderdifferential operator P = −(gµν∂µ∂ν + Aρ∂ρ + B) (locally),where Aµ,B ∈ End(F), has an asymptotic expansion

Tr(e−tP) ∼∞∑

k=0

tk−p

2

∫

Mdx ak (x ,P) ,

where ak (x ,P) are the Seeley-de Witt coefficients.

inversion of Laplace transformation

SA ∼

∞∑

k=0

χ k−p2

∫

Mdx ak(x ,D

2A) ,

χz=

∫ ∞

0dt tz χ(t) =

1

Γ(−z)

∫∞0 ds s−z−1χ(s) for z /∈ N

(−1)kχ(k)(0) for z = k ∈ N



The Seeley-de Witt coefficients are given in the book of Gilkey,but expressed in terms of P = ∆F − E , where ∆F is theconnection Laplacian for ∇f = dxµ ⊗ (∂µf + ωµf ). One finds

P = ∆F − E ⇔ ωµ =12

gµν(Aν + gρσΓνρσ)

E = B − gµν(ωµων + ∂µων − Γρµνωρ)

The first coefficients are

a0(x ,P) = (4π)−p2 tr(id) ,

a2(x ,P) =16(4π)−

p2 tr(−Rid + 6E) ,

a4(x ,P) =(4π)−

p2

360tr((5R2 − 2RµνRµν + 2RµνρσRµνρσ − 12∆LC(R))id

+ 60∆F (E)− 60RE + 180E2 + 30ΩµνΩµν)

where Ωµν = ∂µων − ∂νωµ + ωµων − ωνωµ



Spectral action

SA =1π2

(48χ−2 − cχ−1 + dχ0

)∫

d4x√

det g

+1

24π2

(96χ−1 − cχ0

)∫

d4x√

det g R

+χ0

10π2

∫

d4x√

det g(11

6R∗R∗ − 3CµνρσCµνρσ

)

+1π2 (−2aχ−1 + eχ0)

∫

d4x√

det g |φ|2

+χ0

2π2

∫

d4x√

det g a(

|Dµφ|2 −

16

R|φ|2)

+2χ0

π2

∫

d4x√

det g(1

2tr3(GµνGµν) +

12

tr2(WµνWµν) +53

BµνBµν)

+χ0

2π2

∫

d4x√

det g|φ|4



with

Gµν = ∂µGν − ∂νGµ − i(GµGν − GνGµ) ∈ M3(C∞(M)) ,

Wµν = ∂µWν − ∂νWµ − i(WµWν − WνWµ) ∈ M2(C∞(M)) ,

Bµν = ∂µBν − ∂νBµ ∈ C∞(M)

|φ|2 := |φ1|2 + |φ2|

2 ,(

Dµφ1

Dµφ2

)

=

(∂µφ1

∂µφ2

)

+ i

(W 3

µ − Bµ W 1µ − iW 2

µ

W 1µ − iW 2

µ − W 3µ − Bµ

)(φ1

φ2

)

and

a = tr(Y ∗ν Yν + Y ∗

e Ye + 3Y ∗u Yu + Y ∗

d Yd) ,

b = tr((Y ∗ν Yν)

2 + (Y ∗e Ye)

2 + 3(Y ∗u Yu)

2 + (Y ∗d Yd )

2) ,

c = tr(Y ∗RYR) , d = 4tr((Y ∗

RYR)2) , e = tr((Y ∗

RYR)(Y∗ν Yν)) .



Summary

The resulting action is gravity (including second ordercorrections) coupled to the full bosonic standard model(including Higgs sector with spontaneous symmetry breaking).

The spectral action predicts:

1 Equal coupling constants for U(1), SU(2) and SU(3). Thespectral action can only hold at GUT scale (smalldisagreement with experiment).

2 Extrapolating renormalisation group flow of a, . . . ,e to GUTscale and the resulting Higgs mass back to electroweakscale predicts mH ≈ 174GeV. Today this is known to large.

There are modified versions which yield a smaller Higgs mass.

I personally think that the discrepancy signals new physicsbetween standard model and GUT scales.


Noncommutative Geometry€¦ · TOC Fundamental interactions Noncommutative geometry Spectral...

Documents

Transcript of Noncommutative Geometry€¦ · TOC Fundamental interactions Noncommutative geometry Spectral...