Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Spectral Triples on the Sierpinski Gasket

Fabio Cipriani

Dipartimento di MatematicaPolitecnico di Milano - Italy(

Joint works with D. Guido, T. Isola, J.-L. Sauvageot)

AMS Meeting "Analysis, Probability and Mathematical Physics on Fractals"Cornell University Ithaca NY, September, 10th to 13th, 2011

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Sierpinski gasket K ⊂ C and its Dirichlet form (E ,F)

Duomo di Amalfi: Chiostro, sec. XIII

The aim of the work is to construct Spectral Triples (A,D,H) able to describe thegeometric features of K such as:

Dimension, Topology, Volume, Distance and Energy.

Our constructions rely on two facts:

the Dirichlet space F ⊂ C(K), domain of the Dirichlet form (E ,F) of thestandard Laplacian on K, is a semisimple Banach ∗-algebra so that its K-theorycoincides with the K theory of C(K) and with the K-theory of K [F.C. Pacific J.(2006)]

the traces of finite energy functions a ∈ F on K to boundary of lacunas `σ ⊂ Kbelong to fractional Sobolev spaces Hα(`σ) [A. Jonsson Math. Z. (2005)]

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Novelty with respect to [Christensen-Ivan-Schrohe arXiv: 1002.3081 (2010)] is that

the Dirichlet form can be recovered

E [a] = const.Traceω (|[D, a]|2|D|−δD ) a ∈ F

a new energy dimension δD := 2− ln 5/3ln 2 appears as the abscissa of convergence

of the energy functionals

C 3 s 7→ Trace, (|[D, a]|2|D|−s) a ∈ F

The difference w.r.t. the Hausdorff dimension

δD 6= dH =ln 3ln 2

is a consequence of the equivalent facts

Volume and Energy are distributed singularly on K [M. Hino Prob Th. Rel.Fields (2005)]

there are no nontrivial algebras in the domain of the Laplacian

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Spectral Triples (Connes 1985)

The notion of Spectral Triple abstracts some basic properties of a Dirac operator on acomplete Riemannian manifold and lies at the foundation of a metric geometry ingeneralized settings, where the algebra of coordinates of the space maybecommutative or not.

A Spectral Triple (A,H,D) over a C∗-algebra A ⊆ B(H), consists of a smoothsub-∗algebra A ⊂ A and a self-adjoint operator (D, dom (D)) onH subject to:

[D, a] is a bounded operator for all a ∈ Aa(I + D2)−1/2 is a compact operator onH for all a ∈ A

In the framework of the Sierpinski gasket K

A := C(K) is the commutative C∗-algebra of continuous functions

A := F is the domain of the Dirichlet form (E ,F) of the standard Laplacian.

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Parity, Summability and Dimensional Spectrum of Spectral Triples

A Spectral Triple is called even if there exists γ ∈ B(H) such that

γ2 = I , γ = γ∗ , γD + Dγ = 0 , γa = aγ , ∀ a ∈ A

a Spectral Triple is called finitely p-summable, for some p ≥ 1, if

a(I + D2)−1/2 ∈ Lp(H)

a Spectral Triple is called (d,∞)-summable if

a(I + D2)−d/2 ∈ L(1,∞)(H)

a p-summable Spectral Triple has dicrete spectrum if there exists a discrete setF ⊂ C such that all zeta functions associated to elements a ∈ A

ζaD : {z ∈ C : Re z ≥ p} → C ζa

D(s) := Tr (a|D|−s)

extends to meromorphic functions with poles contained in F. The dimensionalspectrum of (A,H,D) is defined to be the smallest of such sets.

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Example: Canonical Spectral Triple of a Riemannian manifold M

H := L2(Λ(M)) be the Hilbert space of square integrable sections of theexterior bundle of differential forms considered as a module over A := C0(M)

D := d + d∗ is the first order differential (Dirac) operator whose square is theHodge-de Rham Laplacian ∆HdR = dd∗ + d∗d of M

a(I + D2)−1/2 is a compact by Sobolev embeddings for all a ∈ A := C∞c (M)

norm of commutators coincides with Lipschiz semi-norms ‖[D, a]‖ = ‖da‖∞the Riemannian distance may be recovered from commutators as

d(x, y) := sup{a(x)− a(y) : a ∈ A , ‖[D, a]‖ ≤ 1}

the Minakshisundaram-Pleijel asymptotic of the heat semigroup e−t∆HdR ast→ 0 allows to identify the dimensional spectrum with

{0, 1, · · · , dim M} ⊂ C

so that the spectral triple is (dim M,∞)-summable.

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Fredholm modules (Athiyah (1969), Kasparov (1975))

Fredholm modules on C∗-algebras generalize elliptic operators on manifolds:

A Fredholm modules (F,H) over a C∗-algebra A ⊆ B(H), consists of a

symmetry F ∈ B(H): F∗ = F, F2 = I such that

[F, a] is compact for all a ∈ A

FM lie at the core of Noncommutative Geometry of A. Connes where thecompact operator da := i[F, a] is the operator theoretic substitute for thedifferential of a ∈ A

FM has been constructed on p.c.f. fractals by [C.-Sauvageot CMP (2009)] andon more general fractals by [Ionescu-Rogers-Teplyaev (2011)]

Proposition. (Connes (1986), Baaj-Julg (1983))

The sign F := sign(D) of the Dirac operator of a Spectral Triple (A,H,D) gives riseto a Fredholm module (F,H).

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Pairing with K-theory and an Index Theorem on SGModifying the proof of the Connes Index Theorem for FM [Connes (1986)], to takeinto account that Dirac operators on SG have

dim Ker (D) = +∞ ,

it is possible to construct topological invariants.

Theorem. (CGIS 2010)

The Spectral Triples on SG determine a group homomorphism on the K-theory

chD : K1(F)→ C

which is integer valued chD(K1(F)) ⊂ Z.These values are the index of suitable Fredholm operators:

for all invertible u ∈ GL∞(F), setting P := (I + F)/2, the operator Gu := PuPis Fredholm and

chD[u] = Index (Gu) =1

22m+1 Trace (u[F, u−1]2m+1) ∀m ≥ 1 .

These values can be interpreted as winding numbers around lacunas in SG.

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Quasi-circles

We will need to consider on the 1-torus T = {z ∈ C : |z| = 1} structures ofquasi-circle associated to the following Dirichlet forms and their associated SpectralTriples for any α ∈ (0, 1).

Lemma. Fractional Dirichlet forms on a circle (CGIS 2010)

Consider the Dirichlet form on L2(T) defined on the fractional Sobolev space

Eα[a] :=

∫T

∫Tϕα(z−w)|a(z)−a(w)|2 dzdw Fα := {a ∈ L2(T) : Eα[a] < +∞}

where ϕα is the Clausen cosine function for 0 < α ≤ 1.ThenHα := L2(T× T) is a symmetric Hilbert C(K)-bimodule w.r.t. actions andinvolutions given by

(aξ)(z,w) := a(z)ξ(z,w) , (ξa)(z,w) := ξ(z,w)a(w) , (J ξ)(z,w) := ξ(w, z) .

The derivation ∂α : Fα → Hα associated to Eα is given by

∂α(a)(z,w) := ϕα(z− w)(a(z)− a(w)) .

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Proposition. Spectral Triples on a circle (CGIS 2010)

Consider on the Hilbert space Kα := L2(T× T)⊕

L2(T), the left C(T)-modulestructure resulting from the sum of those of L2(T× T) and L2(T) and the operator

Dα :=

(0 ∂α∂∗α 0

).

Then Aα := {a ∈ C(T) : supz∈T∫

T|a(z)−a(w)|2

|z−w|2α+1 < +∞} is a uniformly densesubalgebra of C(T) and (Aα,Dα,Kα) is a densely defined Spectral Triple on C(T).The dimensional spectrum is { 1

α}.

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Dirac operators on K.

Identifying isometrically the main lacuna `∅ of the gasket with the circle T, considerthe Dirac operator (C(K),D∅,K∅) where

K∅ := L2(`∅ × `∅)⊕ L2(`∅)

D∅ := Dα

the action of C(K) is given by restriction π∅(a)b := a|`∅ .

Fix γ > 0 and for σ ∈∑

consider the Dirac operators (C(K), πσ,Dσ,Kσ) where

Kσ := K∅Dσ := 2γ|σ|Dα

the action of C(K) is given by contraction/restriction πσ(a)b := (a ◦ Fσ)|`∅ b.

Finally, consider the Dirac operator (C(K), π,D,K) where

K := ⊕σ∈∑Kσπ := ⊕σ∈∑πσD := ⊕σ∈∑Dσ

Notice that dim Ker D = +∞ and that D−1 will be defined to be zero on Ker D.

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Volume functionals and their Spectral dimensions

Theorem. (CGIS 2010)

The zeta function ZD of the Dirac operator (C(K),D,K), i.e. the meromorphicextension of the function C 3 s 7→ Trace(|D|−s) is given by

ZD(s) =4

1− 3c−s z(αs)

where z denotes the Riemann zeta function. The dimensional spectrum is given by

Sdim = { 1α} ∪ { ln 3

ln 2γ−1(

1 +2πiln 3

k)

: k ∈ Z}

and the abscissa of convergence is dD = max(α−1, ln 3ln 2γ

−1). When 0 < γ < ln 3ln 2α

there is a simple pole in dD = ln 3ln 2γ

−1 and the residue of the meromorphic extensionof C 3 s 7→ Trace(f |D|−s) gives the Hausdorff measure of dimension dH = ln 3

ln 2

TraceDix(f |D|−s) = Ress=dD Trace(f |D|−s) =4dln 3

z(d)

(2π)d

∫K

f dµ .

Notice the complex dimensions and the independence of the residue Hausdorffmeasure upon γ > 0.

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Spectral Triples and Connes metrics on the Sierpinski gasket

Theorem. (CGIS 2010)

(C(K),D,K) is a Spectral Triple for any 0 < γ ≤ 1. In particular the seminorm

a 7→ ‖[D, a]‖

is a Lip-seminorm in the sense of Rieffel so that the distance it determines inducesthe original topology on K.If α < γ this distance is bi-Lipschitz w.r.t. the power (ρgeo)

γ .

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

Energy functionals and their Spectral dimensions

By the Spectral Triple it is possible to recover, in addition to dimension, volumemeasure and metric, also the energy form of K

Theorem. (CGIS 2010)

Consider the Spectral Triple (C(K),D,K) for α ≤ α0 := ln 5ln 4 −

12

( ln 3ln 2 − 1

)∼ 0, 87

and assume a ∈ F . Then the abscissa of convergence of

C 3 s 7→ Trace(|[D, a]|2|D|−s)

is δD := max(α−1, 2− γ−1 ln 5/3ln 2 ).

If δD > α−1 then s = δD is a simple pole and the residue is proportional to theDirichlet form

Ress=δD Trace(|[D, a]|2|D|−s) = const. E [a] a ∈ F ;

Spectral Triples, Fredholm modules Dirac operator and Spectral triple

The (γ, α)-plane

Above the line α = γ/dS, the volume measure is a multiple of the Hausdorffmeasure HdS where dS = ln 3

ln 2 and the spectral dimension is dD = dSγ

in the square 0 < α, γ ≤ 1, (A,D,H) is a Spectral Triple whose Fredholmmodule has nontrivial pairing with all generators of the K-theory K1(K) and"generates" the K-homology K0(K)

below the line α = γ, the distance ρD is bi-Lipschitz with (ρgeo)γ

in the strip α < α0 < γ < 1 and above the hyperbola αδD(γ) = 1 the energyfunctional is proportional to the Dirichlet form, whereδD(γ) = 2− ln(5/3)

ln 2 ln γ−1is the energy dimension.