“Spectral Triples”

Spectral Triples

Giovanni Landi

Trieste

Noncommutative Topics

Noncommutative geometry

has roots in Heisenberg’ (and Born and Jordan and ...) matrix formulation of quantum mechanics

which resulted in the replacement of the phase space by a

‘noncommutative space’

or, more precisely

the replacement of classical observable ( usual functions )

Commutativity of algebra of functions on space X

is

localization of points of X

Quantum mechanics phase space noncommutativity:

[p, x] = 1 ⇒ ∆p∆x ≥ 1

To incorporate limitations on spatial resolution one makes posi-tions noncommuting as well

yx − xy = iθ

as a consequence

∆y ∆x ≥ θ

The idea that spacetime geometry may be noncommutative

(as a way to regularize infinities)

goes back to Schr¨odinger and Heisenberg

Heisenberg mentioned this possibility in a letter to Rudolph Peierls in the 30s.

Peierls mentions Heisenberg’s ideas to Wolfgang Pauli

Pauli explains it to Hartland Snyder

A basic duality

(commutative) Gel’fand-Naimark thm:

equivalence of categories C₀(X) ↔ X,

commutative C∗-algebras ↔ locally compact Haus. spaces;

points x ∈ X are characters of C₀(X)

x(f) := f(x)

or irreducible representations;

A (complex, unital) Banach algebra is a complex algebra equipped with a complete normed vector space structure ; the norm and the multiplicative structures are related by the identity

kabk ≤ kakkbk

An involutive Banach algebra satisfying the C∗-identity

ka∗ak = kak2

becomes a C∗-algebra

Pathological spaces: not a good point-set theoretical description

Equivalence relation R on X;

the quotient Y = X/R can be bad even for good X

Classically: function on the quotient

A(Y ) := {f ∈ A(X) ; f is R − invariant}

NCG approach:

the noncommutative algebra

A(Y ) := A(Γ_R)

of functions on the graph Γ_R ⊂ X ×X of the equivalence relation (of compact support, of rapid decay,...)

convolution product:

(f₁ ∗ f₂)(x, y) = X

x∼u∼y

f₁(x, u)f₂(u, y)

involution:

the quotient Y = X/R is a noncommutative space

with a noncommutative algebra of functions

A(Y ) := A(Γ_R)

as good as X to do geometry:

exterior forms, metric, integration,

vector bundles, connections, curvature, ...

The celebrated irrational rotation algebra a.k.a. the noncommutative torus

A_θ = C∞(_T2_θ) ' C∞(S1/θ_Z) θ ∈ _R − _Q

a = X

(m,n)∈_Z2

a_mn U₁m U₂n {amn} ∈ S(Z2)

U₂ U₁ = e2πiθ U₁ U₂

Yang-Mills theory on _T2

A dictionary :

Classical Noncommutative

locally compact space C∗-algebra

compact space unital C∗-algebra

vector bundle finite projective module

smooth manifold C∗-algebra with ‘smooth’ subalgebra

partial derivative unbounded derivation

integral tracial state

spin structure spectral triple

Requirements on noncommutative geometry dictated by physics

The noncommutative geometry of the standard model

Present knowledge of spacetime is described by two theories :

• General Relativity

• The Standard Model of particle physics

Gravity minimally coupled to matter :

the gravitational potential g_µν

ds2 = g_µνdxµdxν S_EH[g_µν] = 1

G

Z

M

R√g d4x

the standard model S_SM based on the gauge theory of the group

G = U(1) × SU(2) × SU(3)

The symmetries of S = S_EH + S_SM

G = Map(M, G) _o Diff (M)

Is there a space so that G = Diff (X) ?

it can be a noncommutative space !!

Almost commutative geometry: X = M × F

M Riemannian spin 4-manifold

F finite (i.e. metric 0-dimensional) geometry of KO-dimension ( inner signature ) = 6

A(X) = C∞(M) ⊗ A_F, A_F = _C ⊕ _H ⊕ _M3(_C)

Full standard model minimally coupled to gravity obtained from

S(geometry) = tr(χ(D_A/Λ)) + hJ ξ, D_Aξi

D_A = D + A + ε0J AJ−1

J extra structure dictating the KO-dimension (the signature) ;

Λ a mass scale, χ is a cut-off function.

The simplest example

A = C∞(M; M_n(_C)) = C∞(M) ⊗ M_n(_C)

Algebra of n × n matrices of smooth functions on manifold M

The group Inn(A) of inner automorphisms locally isomorphic to the group G of smooth maps from M to the gauge group SU(n)

1 → Inn(A) → Aut(A) → Out(A) → 1

becomes identical to

(Connes-Chamseddine)

The pure gravity on this space yields Einstein gravity on M mini-mally coupled with Yang-Mills theory for the gauge group SU(n)

The gauge potential appears as the inner part of the metric,

What is a metric in spectral geometry

d(A; B) = Inf

Z

γ

q

g_µν dxµdxν

Dirac’s square root of the Laplacian

(A,H, D) ds = D−1

Kinematic relations

• The algebra A is commutative.

• The commutator [[D, a], b] = 0 for a, b ∈ A

• It holds that P

α aα0 h

[D, aα₁][D, aα₂], . . . ,[D, aα_n]i = 1

for some aα_j ∈ A [T₁, T₂, . . . T_n] = P

σ (σ)Tσ(1)Tσ(2)· · ·Tσ(n)

Even case: 1 → γ₅ (grading)

this means that the determinant of the metric gµν does not vanish

and its square root multiplied by the volume form

P

Spectral conditions

• The k-th characteristic value of the resolvent of D goes like

λ_k ∼ k−1/n

this condition gives the metric dimension n

• Regularity for the geodesic flow expit|D|

The reconstruction (Connes)

Let (A,H, D) be a spectral geometry fullfing the kinematic rela-tions and the spectral condirela-tions

Assume that the multiplicity of the action of A in H is 2n/2

Then there exists a smooth oriented compact (spinc) manifold

M such that A = C∞(M).

To get spin manifolds one needs additional data, a real structure i.e. an antilinear unitary operator J acting in H

It plays the same role and has the same algebraic properties as the charge conjugation operator in physics .

There are additional relations D, J and γ

J2 = ε(n)1, J D = ε0(n)DJ, J γ = ε00(n)γJ ε(n) = (1, 1, −1, −1, −1, −1, 1, 1)

ε0(n) = (1, −1, 1, 1, 1, −1, 1, 1)

ε00(n) = (1, , −1, , 1, , −1, )

n = 0,1, . . . ,7.

In the classical case of spin manifolds

there is thus a relation between the metric (or spectral) dimen-sion given by the rate of growth of the spectrum of D

and

the integer modulo 8 which appears in the above table

The dimension modulo 8 is called the KO-dimension because of its origin in K-theory ; it is a signature

With an ordinary spin geometry M of dimension n and taking the product space M × F, has metric dimension which is still n

but KO-dimension the sum of n with the KO-dimension of F

For the Standard Model with neutrino mixing favors

Noncommutative spin geometries

A spectral triple (A,H, D) is:

• a unital ∗-algebra A with a faithful ∗-rep. π : A → B(H),

H a (separable) Hilbert space,

• a self-adjoint operator D on H (“the Dirac operator”)

(i) (D + i)−1 is compact

(ii) [D, π(a)] is bounded for all a ∈ A

The spectral triple is graded (or even) if there exists a _Z2-grading operator γ on H, γ = γ∗, γ2 = 1, s.t.

With 0 < µ < ∞, the spectral triple is µ+-summable (of metric dimension µ) if (D2 + 1)−1/2 is in the Dixmier ideal Lµ+(H);

roughly µ_k(D) ∼ k1/µ as k → ∞

A tracial state (an integral)

Z

− T = lim_ω 1 logN

N−1 X

k=0

• M a compact oriented manifold (no boundary)

the spectral triple on M: (C∞(M),H, D)

H := L2(M,V∗

(M)) ; square integrable differential forms;

D = d + d∗ the signature operator M;

• M a compact Riemannian spin manifold (no boundary)

the spectral triple on M: (C∞(M),H, D/ , J)

H := L2(M,S); square integrable spinors;

D/ the Dirac operator of the metric of M;

(D2 + 1)−1/2 ∈ Lm+(H) is Weyl’s formula for the eigen.s of |D|:

µ_k(|D|) ∼ 2π m

Ω_mvol(M)

!1/m

k1/m as k → ∞

The geodesic distance between any two points on M:

d(p, q) = sup

f∈C∞(M)

{|f(p) − f(q)| ; k[D/ , f]k ≤ 1}

The Riemannian measure on M: R

M dµg f ∼

R

(f|D/|−m)

The right hand sides in both furmulæ above make sense for any spectral triple ; in particular p, q are states of the algebra A

The (Eucledian) Einstein-Hilbert action obtained as:

Z

− |D/|2−m = −(m − 2)2

[m/2]_Ω

m

12m(2π)m

Z

M R dµg

An evolution is the Spectral action :

S(geometry) = tr(χ(|D/|/Λ))

with Λ a mass scale, χ is a cut-off function.

The action counts the eigenvalues of D smaller than Λ.

An expansion in Λ :

tr(χ(D/ /Λ)) ∼ Λ0vol(M) + Λ2 Z

M

A real structure J of KO-dimension n ∈ _Z/8_Z; an antilinear isometry J on H ,

J2 = ε(n)1, J D = ε0(n)DJ, J γ = ε00(n)γJ ε(n) = (1, 1, −1, −1, −1, −1, 1, 1)

ε0(n) = (1, −1, 1, 1, 1, −1, 1, 1)

ε00(n) = (1, , −1, , 1, , −1, )

n = 0,1, . . . ,7 .

J maps to the commutant

[π(a), J π(b)J−1] = 0, a, b ∈ A,

the first order condition on D

Recal the noncommutative torus _T2_θ.

Let M be a compact Riemannian spin manifold (no boundary) whose isometry group has rank r ≥ 2. Then M admits natural isospectral deformations to noncommutative geometries M_θ.

θ = (θ_jk = −θ_kj) θ_jk ∈ _R j, k = 1, . . . , r

idea: deform the standard spectral triple describing the Rie-mannian geometry of M along a torus embedded in the isometry group to get an isospectral triple

gauge theory on M_θ

E a noncommutative vector bundle over M_θ:

C∞(M_θ)N = E + E0

a connection on E: a linear map satisfying a Leibniz rule

∇ : E 7→ E ⊗_C∞_(M

θ) Ω

1₍

M_θ), ∇(ηa) = (∇η)a + η ⊗_A da

F = ∇2 is its curvature

on E a natural C∞(M_θ)-valued Hermitian metric a connection ∇ is metric compatible if

the Yang-Mills functional

YM(∇) = Z

− ∗_θtr(F ∗_θ F)

tr is the C∞(M_θ)-valued trace on End_C∞_(M

θ)(E) from the

Hermi-tian metric on E; the Hodge operator ∗_θ via the Dirac operator

the equations for critical points

[∇,∗_θF] = 0

(antiself )self-dual configurations solutions from Bianchi identity

∗_θF = ±F

a family of gauge-nonequivalent SU(2) instantons on a S_θ4

noncommutative parameters

The noncommutative geometry of quantum groups and quantum homogeneous spaces

The guiding principle:

equivariance with respect to a ‘quantum symmetry’ this will build all the geometry from scratch

Symmetries

implemented by the action of a Hopf ∗-algebra U_q(g), a quantum universal enveloping algebra;

main example :

Symmetries

U = (U,∆, S, ε) a Hopf ∗-algebra

A a left U-module ∗-algebra: there is a left action . of U on A,

h . xy = (h₍₁₎ . x)(h₍₂₎ . y),

h . 1 = ε(h)1, (h . x)∗ = S(h)∗ . x∗,

notation ∆(h) = h₍₁₎ ⊗ h₍₂₎.

A U-equivariant ∗-representation π of A on V (a dense subspace of H) is a ∗-representation of the left crossed product ∗-algebra

defined as the ∗-algebra generated by the two ∗-subalgebras A

and U with crossed commutation relations

hx = (h₍₁₎ . x)h₍₂₎, h ∈ U, x ∈ A

that is, there is also a ∗-representation λ of U on V s. t.

λ(h) π(x) ξ = π(h₍₁₎ . x) λ(h₍₂₎) ξ,

for all h ∈ U, x ∈ A and ξ ∈ V.

A linear operator D on V is equivariant if it commutes with λ(h), for all h ∈ U and ξ ∈ V

The nc geometry of the quantum group A(SU_q(2))

The algebra:

A = A(SU_q(2)) is ∗-algebra generated by a and b, with relations:

ba = qab, b∗a = qab∗, bb∗ = b∗b,

a∗a + q2b∗b = 1, aa∗ + bb∗ = 1

these state that the defining matrix is unitary

U = a b

−qb∗ a∗

!

A = A(SU_q(2)) is a Hopf ∗-algebra (a quantum group) with

• coproduct:

∆ : A → A ⊗ A; ∆ a b

−qb∗ a∗

!

:= a b

−qb∗ a∗

!

.

⊗ a b

−qb∗ a∗

!

∆(a) = a ⊗ a − qb ⊗ b∗, ....

• antipode:

S : A → A; S a b −qb∗ a∗

!

= a

∗ _−qb

b∗ a

!

• counit:

ε : A → _C; ε a b −qb∗ a∗

!

:= 1 0 0 1

These dualize classical operations

A₀ = A(SU(2), polynomial functions on SU(2)

∆ : A₀ → A₀ ⊗ A₀; (∆f)(x ⊗ y) = f(xy)

S : A₀ → A₀; (Sf)(x) = f(x−1)

An isospectral nc geometry on the ‘manifold underlying’ SU_q(2)

The equivariance is with respect to an action of the quantum universal envelopping algebra U = U_q(su(2))

The spectral triple (A(SU_q(2)),H, D) is regular and 3+-summable.

It has simple dimension spectrum given by Σ = {1,2,3}.

Its KO-dimension is 3.