Spectral Zeta Functions and Gauss-Bonnet Theorems in Noncommutative Geometry

Transcription

1 Spectral Zeta Functions and Gauss-Bonnet Theorems in Noncommutative Geometry Masoud Khalkhali (joint work with Farzad Fathizadeh) Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 1 / 44

2 Spectral Geometry and Classical-Quantum Correspondence One of the backbones of Alain Connes program of NCG, specially its metric and spectral aspects, is Spectral Geometry and the Correspondence Principle which relates QM to CM. The correspondence principle has its roots in Planck s derivation of his celebrated Radiation Law and in Bohr-Sommerfeld Quantization Rules. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 2 / 44

3 Figure: Black body spectrum Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 3 / 44

4 Planck s Radiation Law From 1859 (Kirchhoff) till 1900 (Planck) a great effort went into finding the right formula for spectral energy density function of a radiating black body (T = temperature, ν = frequency, h= Planck s constant, k= Boltzmann s constant, c = speed of light): ρ(ν, T ) = 8πhν3 c 3 1 e hν/kt 1 Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 4 / 44

5 Planck s Radiation Law From 1859 (Kirchhoff) till 1900 (Planck) a great effort went into finding the right formula for spectral energy density function of a radiating black body (T = temperature, ν = frequency, h= Planck s constant, k= Boltzmann s constant, c = speed of light): ρ(ν, T ) = 8πhν3 c 3 1 e hν/kt 1 Kirchhoff predicted: ρ will be independent of the shape of the cavity and should only depend on its volume. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 4 / 44

6 Limits of Planck s Law Quantum Limit (high-frequency or low temperature regime; hν/kt 1) ρ(ν, T ) Aν 3 e Bν/T (T 0) Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 5 / 44

7 Limits of Planck s Law Quantum Limit (high-frequency or low temperature regime; hν/kt 1) ρ(ν, T ) Aν 3 e Bν/T (T 0) Semiclassical Limit (low frequency or high temperature; hν/kt 1) ρ(ν, T ) = 8πν2 (kt )(1 + O(h)) (T ) c3 RHS is the Rayleigh-Jeans-Einstein radiation formula. It can be established, assuming the Weyl s Law: For high frequencies there are approximately V (8πν 3 dν/c 3 ) modes of oscillations in the frequency interval v, ν + dν. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 5 / 44

8 Moral: To relate classical and quantum worlds, Weyl s law is needed: One can hear the volume of a cavity. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 6 / 44

9 Moral: To relate classical and quantum worlds, Weyl s law is needed: One can hear the volume of a cavity. The conjecture of Lorentz (1910; proved by Weyl in 1911): It is here that there arises the mathematical problem to prove that the number of sufficiently high overtones which lie between ν and ν + dν is independent of the shape of the enclosure and is simply proportional to its volume....there is no doubt that it holds in general even for multiply connected spaces. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 6 / 44

10 Moral: To relate classical and quantum worlds, Weyl s law is needed: One can hear the volume of a cavity. The conjecture of Lorentz (1910; proved by Weyl in 1911): It is here that there arises the mathematical problem to prove that the number of sufficiently high overtones which lie between ν and ν + dν is independent of the shape of the enclosure and is simply proportional to its volume....there is no doubt that it holds in general even for multiply connected spaces. But the ultimate question is What else can one hear about the shape of a cavity? Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 6 / 44

11 Bohr-Sommerfeld Quantization Stationary Schrodinger equation: WKB approximation ansatz ( 2 + V )ϕ(x) = λϕ(x) 2m ϕ(x) = Ae i B(x) S = pdq, the total action; the Bohr-Sommerfeld quantization rule: Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 7 / 44

12 exp { i S} = 1 or pdq = 2πn, n = 1, 2,... It relates classical periodic orbits to energy levels in the corresponding quantum system. Can one hear the periodic orbits of a classical system? Yes, for Riemann surfaces, In general, trace formula (Selberg, Connes) relates lengths of periodic orbits in chaotic systems to the spectrum. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 8 / 44

13 How to Quantize? Weyl s law imposes some constraints. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 9 / 44

14 How to Quantize? Weyl s law imposes some constraints. Consider a Classical System (X, h); X = symplectic manifold, h : X R, Hamiltonian. Assume {x X ; h(x) λ} are compact for all λ (confined system). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 9 / 44

15 How to Quantize? Weyl s law imposes some constraints. Consider a Classical System (X, h); X = symplectic manifold, h : X R, Hamiltonian. Assume {x X ; h(x) λ} are compact for all λ (confined system). Typical example: X = T M, (M, g) = compact Riemannian manifold, h = T + V. T = kinetic energy, V = potential energy. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 9 / 44

16 How to quantize this? (X, h) (H, H), where H = Hilbert space, H = self-adjoint operator on H. No one knows! No functor!, but...dirac rules, geometric quantization, deformation quantization,...and the correspondence principle: Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 10 / 44

17 Looking for a pair (H, H), H = Hilbert space, H = self-adjoint operator on H, Hamiltonian, with discrete spectrum s.t. λ 1 λ 2 λ 3 N(λ) c Volume (h λ) λ N(λ) = #{λ i λ} Eigenvalue Counting Function Thus:quantized energy levels are approximated by phase space volumes. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 11 / 44

18 Apply this to X = T M, (M, g) = compact Riemannian manifold, h(q, p) = p 2 ; set obtain Weyl s Law: H = L 2 (M), H = Laplacian N(λ) c Vol (M)λ m/2 (λ ) Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 12 / 44

19 A Heuristic, Physical Proof of Weyl s Law Classical partition function from Gibbs equilibrium state at inverse temperature β = 1/kT Z = e h/β dvol = e x/β dµ(x) X µ[0, λ] = Vol (h λ) Quantum partition function Z q = Trace (e H/β ) = Eigenvalue counting measure 0 µ q [0, λ] = #{λ i λ} 0 e x/β dµ q (x) Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 13 / 44

20 Experimental fact: classical statistical mechanics gives good results at high temperatures; in particular specific heat obtained from Z should converge to its quantum value from Z q. C = E T = 1 2 lnz kt 2 2 β In particular, the measures µ[0, λ] and µ q [0, λ] are asymptotically proportional: µ[0, λ] µ q [0, λ] (2π )N (dim (X ) = 2N) Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 14 / 44

21 Weyl s Law (M, g) = compact Riemannian manifold = d d : L 2 (M) L 2 (M), Laplacian Is a s. a. positive operator. In local coordinates: = g µν µ ν + A µ µ + B Spectrum of (counting multiplicities): 0 λ 0 λ 1 λ 2 Eigenvalue counting function: N(λ) := #{λ i λ} Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 15 / 44

22 Weyl s Law: N(λ) = Vol (M) (4π) m/2 Γ(1 + m/2) λm/2 + O(λ m/2 ) One can hear the Volume and Dimension of a Riemannian manifold. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 16 / 44

23 Weyl s Law: N(λ) = Vol (M) (4π) m/2 Γ(1 + m/2) λm/2 + O(λ m/2 ) One can hear the Volume and Dimension of a Riemannian manifold. Asymptotic expansion of the trace of the heat kernel: Trace (e t ) a n = M 0 a n (x, )dvol x a n t n m 2 (t 0) local invariants Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 16 / 44

24 Seeley-DeWitt coefficients a n (x, ), n 0 a 0 (x, ) = (4π) m/2 a 0 = a 0 (x, )dvol = (4π) m/2 Vol(M) Tauberian theorems Weyl s law. M Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 17 / 44

25 Spectral Triples (A, H, D), A= involutive unital algebra, acting by bounded operators on a Hilbert space H, D = a s.a. operator on H with compact resolvent such that all commutators [D, a] are bounded. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 18 / 44

26 Spectral Triples (A, H, D), A= involutive unital algebra, acting by bounded operators on a Hilbert space H, D = a s.a. operator on H with compact resolvent such that all commutators [D, a] are bounded. Assume: an asymptotic expansion of the form holds. Trace (e td2 ) a α t α (t 0) Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 18 / 44

27 Spectral Triples (A, H, D), A= involutive unital algebra, acting by bounded operators on a Hilbert space H, D = a s.a. operator on H with compact resolvent such that all commutators [D, a] are bounded. Assume: an asymptotic expansion of the form holds. Trace (e td2 ) a α t α (t 0) Let = D 2. Spectral zeta function ζ D (s) = Tr ( D s ) = Tr ( s/2 ), Re(s) 0. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 18 / 44

28 Spectral Triples Using the Mellin transform and the asymptotic expansion, easy to show that: ζ D has a meromorphic extension to all of C and non-zero terms a α, α < 0, give a pole of ζ D at 2α with Res s= 2α ζ D (s) = Also, ζ D (s) is holomorphic at s = 0 and 2a α Γ( α). ζ D (0) + dim ker D = a 0 Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 19 / 44

29 Gauss-Bonnet for Noncommutative Torus Fix θ R. A θ = C -algebra generated by unitaries U and V satisfying VU = e 2πiθ UV. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 20 / 44

30 Gauss-Bonnet for Noncommutative Torus Fix θ R. A θ = C -algebra generated by unitaries U and V satisfying VU = e 2πiθ UV. Dense subalgebra of smooth functions : A θ A θ, a A θ iff a = a mn U m V n where (a mn ) S(Z 2 ) is rapidly decreasing: for all k N. sup (1 + m 2 + n 2 ) k a mn < m,n Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 20 / 44

31 A θ has a normalized, faithful, and positive trace (unique if θ is irrational): τ 0 : A θ C τ 0 ( a mn U m V n ) = a 00. Derivations δ 1, δ 2 : A θ A θ ; uniquely defined by: δ 1 (U) = U, δ 1 (V ) = 0 δ 2 (U) = 0, δ 2 (V ) = V. We have δ 1 δ 2 = δ 2 δ 1, δ i (a ) = δ i (a), Invariance property: τ 0 (δ i (a)) = 0. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 21 / 44

32 The Hilbert space H 0 = L 2 (A θ, τ 0 ), completion of A θ w.r.t. inner product a, b = τ 0 (b a). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 22 / 44

33 The Hilbert space H 0 = L 2 (A θ, τ 0 ), completion of A θ w.r.t. inner product a, b = τ 0 (b a). The derivations δ 1, δ 2 : H 0 H 0 are formally selfadjoint unbounded operators (analogues of 1 i d dx, 1 d i dy ). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 22 / 44

34 Complex structure Metrics on A θ will be defined through their conformal class. Fix τ = τ 1 + iτ 2, τ 2 > 0, and define = δ 1 + τδ 2, = δ 1 + τδ 2. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 23 / 44

35 Complex structure Metrics on A θ will be defined through their conformal class. Fix τ = τ 1 + iτ 2, τ 2 > 0, and define = δ 1 + τδ 2, = δ 1 + τδ 2. Define the Hilbert space (analogue of (1, 0)-forms) H (1,0) H 0 as the completion of the subspace spanned by finite sums a b, a, b A θ. Connes and Tretkoff consider τ = i. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 23 / 44

36 Laplacian View = δ 1 + τδ 2 : H 0 H (1,0) as an unbounded operator with the adjoint given by = δ 1 + τδ 2. Define the Laplacian := = δ τ 1 δ 1 δ 2 + τ 2 δ2. 2 Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 24 / 44

37 Conformal perturbation of the metric To investigate the Gauss-Bonnet theorem for general metrics, vary the metric by a Weyl factor e h, h = h A θ : Define a positive linear functional ϕ : A θ C by ϕ(a) = τ 0 (ae h ), a A θ. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 25 / 44

38 Conformal perturbation of the metric To investigate the Gauss-Bonnet theorem for general metrics, vary the metric by a Weyl factor e h, h = h A θ : Define a positive linear functional ϕ : A θ C by It is a twisted trace ϕ(a) = τ 0 (ae h ), a A θ. ϕ(ba) = ϕ(aσ i (b)) which is the KMS condition at β = 1 for the automorphisms σ t : A θ A θ, t R, In fact with the modular operator σ t (x) = e ith xe ith. σ t = it (x) = e h xe h. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 25 / 44

39 The perturbed Laplacian Let H ϕ = completion of A θ w.r.t., ϕ, where a, b ϕ = ϕ(b a), a, b A θ. Let ϕ = = δ 1 + τδ 2 : H ϕ H (1,0). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 26 / 44

40 The perturbed Laplacian Let H ϕ = completion of A θ w.r.t., ϕ, where a, b ϕ = ϕ(b a), a, b A θ. Let ϕ = = δ 1 + τδ 2 : H ϕ H (1,0). It has a formal adjoint ϕ given by ϕ = R(e h ) where R(e h ) is the right multiplication operator by e h (R(e h )(x) = e h x). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 26 / 44

41 Define the new Laplacian: = ϕ ϕ : H ϕ H ϕ. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 27 / 44

42 Define the new Laplacian: = ϕ ϕ : H ϕ H ϕ. Lemma (Connes-Tretkoff; continues to hold for general τ) is anti-unitarily equivalent to the positive unbounded operator k k acting on H 0, where k = e h/2. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 27 / 44

43 Spectral Zeta Function ζ(s) = λ s i = Trace ( s ), Re(s) > 1. Mellin transform gives us where λ s = 1 Γ(s) ζ(s) = 1 Γ(s) 0 0 e tλ t s 1 dt Trace + (e t )t s 1 dt, Trace + (e t ) = Trace (e t ) Dim Ker( ). ζ has a moromorphic extension to C \ 1 with a simple pole at s = 1. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 28 / 44

44 The Gauss-Bonnet theorem Theorem (Gauss-Bonnet for classical Riemann surfaces) Let Σ = compact connected oriented Riemann surface with metric g. Then ζ(0) + 1 = 1 R = 1 12π 6 χ(σ), where ζ is the zeta function associated to the Laplacian g = d d, and R is the (scalar) curvature. In particular ζ(0) is a topological invariant; e.g. is invariant under conformal perturbations of the metric g e f g. Σ Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 29 / 44

45 Theorem (Gauss-Bonnet for NC torus) Let k A θ be an invertible positive element. Then the value ζ(0) of the zeta function ζ of the operator k k is independent of k. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 30 / 44

46 Pseudodifferential calculus Recall: Connes (1980; C -algebras and Noncommutative Differential Geometry) Differential operators of order n: P : A θ A θ, P = j a j δ j 1 1 δ j 2 2 with a j A θ, j = (j 1, j 2 ), j n. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 31 / 44

47 Pseudodifferential calculus Recall: Connes (1980; C -algebras and Noncommutative Differential Geometry) Differential operators of order n: P : A θ A θ, P = j a j δ j 1 1 δ j 2 2 with a j A θ, j = (j 1, j 2 ), j n. Operator valued symbols of order n Z: smooth maps ρ : R 2 A θ Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 31 / 44

48 s.t. δ i 1 1 δ i 2 2 ( j 1 1 j 2 2 ρ(ξ)) c(1 + ξ ) n j, where i = ξ i, and ρ is homogeneous of order n at infinity: lim λ n ρ(λξ 1, λξ 2 ), λ exists and is smooth. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 32 / 44

49 Given a symbol ρ, define a pseudodifferential operator P ρ : A θ A θ by where P ρ (a) = (2π) 2 R 2 R 2 e is.ξ ρ(ξ)α s (a)dsdξ, α s (U n V m ) = e is.(n,m) U n V m. For pseudodifferential operators P, Q, with symbols σ(p) = ρ, σ(q) = ρ : σ(pq) 1 l 1!l 2! l 1 1 l 2 2 (ρ(ξ))δl 1 1 δl 2 2 (ρ (ξ)). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 33 / 44

50 Elliptic Symbols: A symbol ρ(ξ) of order n is called elliptic if ρ(ξ) is invertible for ξ 0, and, for ξ large enough, ρ(ξ) 1 c(1 + ξ ) n. Example: = δ τ 1 δ 1 δ 2 + τ 2 δ2 2 is an elliptic operator with an elliptic symbol σ( ) = ξ τ 1 ξ 1 ξ 2 + τ 2 ξ2. 2 Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 34 / 44

51 Computing ζ(0) Recall: ζ(s) = 1 Γ(s) 1 = Dim Ker( ). Cauchy integral formula: 0 e t = 1 2πi (Trace(e t )t s 1 1)dt, C e tλ ( λ1) 1 dλ gives the asymptotic expansion as t 0 + : Trace(e t ) t 1 B 2n ( )t n. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 35 / 44 0

52 It follows that: ζ(0) = B 2 ( ), where Can assume λ = 1: B 2 ( ) = 1 e λ τ 0 (b 2 (ξ, λ))dλdξ 2πi R 2 C (b 0 (ξ, λ) + b 1 (ξ, λ) + b 2 (ξ, λ) + )σ( λ) 1, b j (ξ, λ) is a symbol of order 2 j. ζ(0) = τ 0 (b 2 (ξ, 1))dξ. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 36 / 44

53 where σ( + 1) = σ(k k + 1) = (a 2 + 1) + a 1 + a 0 a 2 = k 2 ξ τ 1 k 2 ξ 1 ξ 2 + τ 2 k 2 ξ 2 2 a 1 = (2kδ 1 (k) + 2τ 1 kδ 2 (k))ξ 1 + (2τ 1 kδ 1 (k) + 2 τ 2 kδ 2 (k))ξ 2 a 0 = kδ 2 1(k) + 2τ 1 kδ 1 δ 2 (k) + τ 2 kδ 2 2(k). Using the calculus for symbols: b 0 = (a 2 + 1) 1 b 1 = (b 0 a 1 b 0 + i (b 0 )δ i (a 2 )b 0 ) b 2 = (b 0 a 0 b 0 + b 1 a 1 b 0 + i (b 0 )δ i (a 1 )b 0 + i (b 1 )δ i (a 2 )b 0 + (1/2) i j (b 0 )δ i δ j (a 2 )b 0 ). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 37 / 44

54 Integrating b 2 (ξ, 1) over the plane Pass to these coordinates: ξ 1 = r cos θ r τ 1 τ 2 sin θ ξ 2 = r τ 2 sin θ where θ ranges from 0 to 2π and r ranges from 0 to. After integrating 2π 0 dθ we have terms such as 4τ 1 r 3 b 3 0k 2 δ 2 (k)δ 1 (k), 2r 3 b 2 0k 2 δ 1 (k)b 0 δ 1 (k), 2r 5 b 2 0k 2 δ 1 (k)b 2 0k 2 δ 1 (k), where b 0 = (1 + r 2 k 2 ) 1. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 38 / 44

55 Lemma (Connes-Tretkoff) For ρ A θ and every non-negative integer m: 0 k 2m+2 u m (k 2 u + 1) m+1 ρ 1 (k 2 u + 1) du = D m(ρ) where D m = L m ( ), = the modular automorphism, L m (u) = ( 1) m (u 1) (m+1)( log u 0 x m (x + 1) m+1 1 (xu + 1) dx = m j+1 (u 1)j ) ( 1) j j=1 (modified logarithm). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 39 / 44

56 Lemma Let k be an invertible positive element of A θ. Then the value ζ(0) of the zeta function ζ of the operator k k is given by ζ(0) + 1 = 2π ϕ(f ( )(δ 1 (k))δ 1 (k)) + 2π τ 2 ϕ(f ( )(δ 2 (k))δ 2 (k))+ τ 2 2πτ 1 ϕ(f ( )(δ 1 (k))δ 2 (k)) + 2πτ 1 ϕ(f ( )(δ 2 (k))δ 1 (k)), τ 2 where ϕ(x) = τ 0 (xk 2 ), τ 0 is the unique trace on A θ, is the modular automorphism, and f (u) = 1 6 u 1/ L 1(u) 2(1 + u 1/2 )L 2 (u) + (1 + u 1/2 ) 2 L 3 (u). (L m is the modified logarithm.) τ 2 τ 2 Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 40 / 44

57 The following theorem was proved by Alain Connes and Paula Tretkoff for conformal parameter τ = i, and then for all conformal parameters by Farzad Fathizadeh and M.K. Theorem (Gauss-Bonnet for NC torus) Let k A θ be an invertible positive element. Then the value ζ(0) of the zeta function ζ of the operator k k is independent of k. Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 41 / 44

58 Proof. ϕ(f ( )(δ j (k))δ j (k)) = 0 for j = 1, 2, ϕ(f ( )(δ 1 (k))δ 2 (k)) = ϕ(f ( )(δ 2 (k))δ 1 (k)). Therefore ζ(0) + 1 = 2π ϕ(f ( )(δ 1 (k))δ 1 (k)) + 2π τ 2 ϕ(f ( )(δ 2 (k))δ 2 (k))+ τ 2 2πτ 1 ϕ(f ( )(δ 1 (k))δ 2 (k)) + 2πτ 1 ϕ(f ( )(δ 2 (k))δ 1 (k)) τ 2 = 0. τ 2 τ 2 Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 42 / 44

59 An argument of Moscovici: variational method. A technique of Branson-Orsted in the commutative case can be extended to the NC case, when there is a good pseudodifferential calculus and good resolvent approximation. Write, for P= a NC polynomial in D and elements of A, Tr (Pe td2 sh ) j=0 a j (P, s)t j n p 2 (t 0) Term by term differentiate w.r.t. s and observe that d ds a p(s) = 0. This brings you back to h = 0 (still you have to evaluate a zeta value using the spectrum of on A θ ). Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 43 / 44

60 Back to the Scalar Curvature But: we are really interested in computing the scalar curvature as a variable function on T 2 θ. Gauss-Bonnet computes its total integral. Let (A, H, D) be a finitely summable regular spectral triple. Consider the zeta function ζ D (P, z) = Tr (P D z ), P Ψ(A, H, D) For the NC torus, the scalar curvature can be defined as the functional on the NC torus: a ζ (a, 0) Ongoing work: compute the scalar curvature! Masoud Khalkhali (joint work with Farzad Fathizadeh) Spectral Zeta () Functions and Gauss-Bonnet Theorems in Noncommutative Geometry 44 / 44