Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli MAT1845HS Winter 2020, University of Toronto M 5-6 and T 10-12 BA6180 Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Some References Christensen, Erik; Ivan, Cristina; Lapidus, Michel L., Dirac operators and spectral triples for some fractal sets built on curves. Adv. Math. 217 (2008), no. 1, 42{78. Chamseddine, Ali H.; Connes, Alain, The spectral action principle. Comm. Math. Phys. 186 (1997), no. 3, 731{750. A. Ball, M. Marcolli, Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology, Classical and Quantum Gravity, 33 (2016), no. 11, 115018, 39 pp. Farzad Fathizadeh, Yeorgia Kafkoulis, Matilde Marcolli, Bell polynomials and Brownian bridge in Spectral Gravity models on multifractal Robertson-Walker cosmologies, arXiv:1811.02972 Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Von Neumann Algebras • Hilbert space H (infinite dimensional, separable, over C) algebra of bounded operators B(H) with operator norm • Commutant of M ⊂ B(H): M0 := fT 2 B(H): TS = ST ; 8S 2 Mg: • von Neumann algebra: M = M00 (double commutant) , weakly closed • Center: Z(M) = L1(X ; µ). • If Z(M) = C: factor Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli C ∗-algebras • involutive (∗ anti-isomorphism) Banach algebra (complete in norm, kabk ≤ kak · kbk, ka∗ak = kak2) • Gel'fand{Naimark correspondence: locally compact Hausdorff topological space , commutative C ∗-algebra: X , C0(X ) • representation of a C ∗-algebra A π : A!B(H) C ∗-algebra homomorphism • state: continuous linear functional ' : A! C with positivity '(a∗a) ≥ 0 for all a 2 A and '(1) = 1 Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli GNS representation • cyclic vector ξ for a representation π : A!B(H) of a C ∗-algebra if set fπ(a)ξ : a 2 Ag dense in H • state from unit norm cyclic vector '(a) = hπ(a)ξ; ξi • given a state ' : A! C construct a representation (GNS) where state is as above • define ha; bi = '(a∗b) for a; b 2 A •N = fa 2 A : '(a∗a) = 0g linear subspace but for C ∗-algebras also a left ideal in A •H = A=N with ha; bi = '(a∗b) Hilbert space • the representation π(a)b + N = ab + N • cyclic vector ξ = 1 + N unit of A Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Spectral triple( A; H; D) A = C∗-algebra H Hilbert space: ρ : A!B(H) D unbounded self{adjoint operator on H −1 (D − λ) compact operator, 8λ2 = R [D; a] bounded operator, 8a 2 A0 ⊂ A, dense involutive subalgebra of A. Riemannian spin manifold X : A = C(X ), H = L2-spinors, 1 D = Dirac operator, A0 = C (X ) Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Zeta functions • spectral triple (A; H; D) ) family of zeta functions: for a 2 A0 [ [D; A0] −z X −z ζa;D (z) := Tr(ajDj ) = Tr(a Π(λ, jDj))λ λ Dimension of a spectral triple (A; H; D) Simpler definition: dimension n (n{summable) if jDj−n −n −1 infinitesimal of order one: λk (jDj ) = O(k ) Refined definition: dimension spectrum Σ ⊂ C: set of poles of the zeta functions ζa;D (z). (all zetas extend holomorphically to C n Σ) in sufficiently nice cases (almost commutative geometries) poles of ζD (s) = ζ1;D (s) suffice Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Example: Fractal string Ω bounded open in R (e.g. complement of Cantor set Λ in [0; 1]) L = f`k gk≥1 lengths of connected components of Ω with `1 ≥ `2 ≥ `3 ≥ · · · ≥ `k ··· > 0: Geometric zeta function (Lapidus and van Frankenhuysen) X s ζL(s) := `k k Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Cantor set: spectral triple 3−s Λ = middle-third Cantor set: ζL(s) = 1−2·3−s algebra commutative C∗-algebra C(Λ). Hilbert space: E = fxk;±g endpoints of intervals Jk ⊂ Ω = [0; 1] r Λ, with xk;+ > xk;− H := `2(E) action C(Λ) acts on H f · ξ(x) = f (x)ξ(x); 8f 2 C(Λ); 8ξ 2 H; 8x 2 E: sign operator subspace Hk of coordinates ξ(xk;+) and ξ(xk;−), 0 1 F j = : Hk 1 0 Dirac operator ξ(xk;+) −1 ξ(xk;−) DjHk = `k · ξ(xk;−) ξ(xk;+) Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli • verify [D; a] bounded for a 2 A0: (f (xk;+) − f (xk;−)) 0 1 [D; f ]jHk = : `k −1 0 for f Lipschitz: k[D; f ]k ≤ C(f ) take dense A0 ⊂ C(Λ) to be locally constant or more generally Lipschitz functions • same for any self-similar set in R (Cantor-like) Zeta function X 2 · 3−s Tr(jDj−s ) = 2ζ (s) = 2k 3−sk = L 1 − 2 · 3−s k≥1 Dimension spectrum log 2 2πin Σ = + log 3 log 3 n2Z Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Example: AF algebras (noncommutative Cantor sets) C ∗-algebras approximated by finite dimensional algebras (direct limits of a direct system of finite dimensional algebras) determined by Bratteli diagram: Fk = fin dim algebras φk;k+1 embeddings with specified multiplicities C(Λ) = commutative AF algebra corresponding to the diagram Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Example: Fibonacci spectral triple Fibonacci AF algebra Fn = MFn+1 ⊕ MFn , embeddings from partial embedding 1 1 1 0 Fibonacci Cantor set from the interval I = [0; 4] remove Fn+1 open n intervals Jn;j of lengths `n = 1=2 , according to the rule: Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Hilbert space E of endpoints xn;j;± of the intervals Jn;j : H = `2(E), completion of 2 3 5 C ⊕ C ⊕ C ⊕ C ⊕ C ··· 2 3 ⊕ C ⊕ C ⊕ C ⊕ C ··· Action of MFn+1 ⊕ MFn ) of AF algebra Sign on subspace Hn;j spanned by ξ(xn;j;±) 0 1 F j = : Hn;j 1 0 Dirac operator ξ(xn;j;+) −1 ξ(xn;j;−) DjHn;j = `n (xn;j;−) ξ(xn;j;+) Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli ) spectral triple with zeta function 2 Tr(jDj−s ) = 2ζ (s) = F 1 − 2−s − 4−s P −ns geometric zeta function ζF (s) = n Fn+12 • bounded commutators condition: [D; a] bounded for a 2 A0: ξ(xn;j;+) −1 (An;+ − An;−)ξ(xn;j;−) [D; U]jHn;j = `n : ξ(xn;j;−) (An;− − An;+)ξ(xn;j;+) ) for U 2 [k Fk (dense subalgebra) p 1+ 5 Dimension spectrum with φ = 2 log φ 2πin log φ 2πi(n + 1=2) + [ − + log 2 log 2 log 2 log 2 n2Z n2Z Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli The spectral action functional Ali Chamseddine, Alain Connes, The spectral action principle, Comm. Math. Phys. 186 (1997), no. 3, 731{750. A good action functional for noncommutative geometries Tr(f (D=Λ)) D Dirac, Λ mass scale, f > 0 even smooth function (cutoff approx) Simple dimension spectrum ) expansion for Λ ! 1 Z X k −k Tr(f (D=Λ)) ∼ fk Λ − jDj + f (0) ζD (0) + o(1); k R 1 k−1 with fk = 0 f (v) v dv momenta of f −s where DimSp(A; H; D) = poles of ζb;D (s) = Tr(bjDj ) Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Asymptotic expansion of the spectral action −t∆ X α Tr(e ) ∼ aα t (t ! 0) and the ζ function −s=2 ζD (s) = Tr(∆ ) Non-zero term aα with α < 0 ) pole of ζD at −2α with 2 a Res ζ (s) = α s=−2α D Γ(−α) No log t terms ) regularity at 0 for ζD with ζD (0) = a0 Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Get first statement from Z 1 −s −s=2 1 −t∆ s=2−1 jDj = ∆ = s e t dt Γ 2 0 R 1 α+s=2−1 −1 with 0 t dt = (α + s=2) . Second statement from 1 s s ∼ as s ! 0 Γ 2 2 contrib to ζD (0) from pole part at s = 0 of Z 1 Tr(e−t∆) ts=2−1 dt 0 R 1 s=2−1 2 given by a0 0 t dt = a0 s Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Zeta function and heat kernel (manifolds) • Mellin transform Z 1 −s 1 −tD2 s −1 jDj = e t 2 dt Γ(s=2) 0 • heat kernel expansion −tD2 X α Tr(e ) = t cα for t ! 0 α • zeta function expansion X cα ζ (s) = Tr(jDj−s ) = + holomorphic D Γ(s=2)(α + s=2) α • taking residues 2c Res ζ (s) = α s=−2α D Γ(−α) Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Spectral Action as a model of Euclidean (Modified) Gravity D X λ S = Tr(f ( )) = f ( ) Λ Λ Λ λ2Spec(D) D Dirac operator ∗ Λ 2 R+ energy scale f (x) test function (smooth approximation to cutoff function) Why a model of (Euclidean) Gravity? • M compact Riemannian 4-manifold 4 2 Tr(f (D=Λ)) ∼ 2Λ f4a0 + 2Λ f2a2 + f0a4 48f Λ4 Z p 96 f Λ2 Z p = 4 g d 4x + 2 R g d 4x π2 24π2 f Z 11 p + 0 ( R∗R∗ − 3C C µνρσ) g d 4x 10π2 6 µνρσ coefficients a0, a2 and a4 cosmological, Einstein{Hilbert, and Weyl curvature C µνρσ and Gauss{Bonnet R∗R∗ gravity terms Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Example spectral action of the round 3-sphere S3 X 1 S 3 (Λ) = Tr(f (D 3 =Λ)) = n(n + 1)f ((n + )=Λ) S S 2 n2Z • zeta function 3 1 3 ζD (s) = 2ζ(s − 2; ) − ζ(s; ) S3 2 2 2 ζ(s; q) = Hurwitz zeta function • by asymptotic expansion 3 1 S 3 (Λ) ∼ Λ f − Λf S 3 4 1 • can also compute using Poisson summation formula (Chamseddine{Connes): estimate error term O(Λ−∞) Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli 3 Example: round 3-sphere Sa radius a s 3 1 3 ζD 3 (s) = a (2ζ(s − 2; ) − ζ(s; )) Sa 2 2 2 3 1 S 3 (Λ) ∼ (Λa) f3 − (Λa)f1 Sa 4 3 ´ Example: spherical space form Y = Sa =Γ(Ca´ci´c,Marcolli, Teh) 1 SY (Λ) ∼ S 3 (Λ) #Γ Sa Fractals and Spectral Triples Introduction to Fractal Geometry and Chaos Matilde Marcolli Spectral Triples on Fractals obtained by gluing smooth spaces Sierpinski gasket (treat