Scalar Curvature and Gauss-Bonnet Theorem for Noncommutative Tori - PowerPoint PPT Presentation

Scalar Curvature and Gauss-Bonnet Theorem for Noncommutative Tori Farzad Fathizadeh joint with Masoud Khalkhali COSy 2014 1 / 43

Spectral Triples Noncommutative geometric spaces are described by spectral triples: ( A , H , D ) , π : A → L ( H ) ( ∗ -representation ) , D = D ∗ : Dom ( D ) ⊂ H → H , D π ( a ) − π ( a ) D ∈ L ( H ) . Examples. C ∞ ( M ) , L 2 ( M, S ) , D = Dirac operator � � . C ∞ ( S 1 ) , L 2 ( S 1 ) , 1 ∂ � � . i ∂x 2 / 43

Noncommutative Local Invariants The local geometric invariants such as scalar curvature of ( A, H , D ) are detected by the high frequency behavior of the spectrum of D and the action of A via heat kernel asymptotic expansions of the form ∞ a e − tD 2 � � a j ( a, D 2 ) t ( − n + j ) / 2 , � Trace ∼ t ց 0 a ∈ A. j =0 3 / 43

Noncommutative 2-Torus A θ = C ( T 2 θ ) It is the universal C ∗ -algebra generated by U and V s.t. U ∗ = U − 1 , V ∗ = V − 1 , V U = e 2 πiθ UV, where θ ∈ R is fixed. The geometry of the Kronecker foliation dy = θdx on the ordinary torus R 2 / Z 2 is closely related to the structure of this algebra. A representation of A θ : Uξ ( x ) = e 2 πix ξ ( x ) , ξ ∈ L 2 ( R ) . V ξ ( x ) = ξ ( x + θ ) , 4 / 43

Action of T 2 = ( R 2 π Z ) 2 on A θ and Smooth Elements • s ∈ R 2 , α s : A θ → A θ , α s ( U m V n ) = e is. ( m,n ) U m V n , m, n ∈ Z . • s �→ α s ( a ) is smooth from R 2 to A θ } A ∞ θ := { a ∈ A θ ; a m,n U m V n ∈ A θ ; � � ( a m,n ) ∈ S ( Z 2 ) � = . m,n ∈ Z • ∂  s =0 α s : A ∞ θ → A ∞ δ j = θ .  ∂s j 5 / 43

The Derivations δ 1 , δ 2 and the Volume Form • δ 1 , δ 2 : A ∞ θ → A ∞ θ are defined by: δ 1 ( U ) = U, δ 1 ( V ) = 0 , δ 2 ( U ) = 0 , δ 2 ( V ) = V, a, b ∈ A ∞ δ i ( a b ) = δ i ( a ) b + a δ i ( b ) , θ . • Tracial state ϕ 0 : A θ → C (analog of integration): ϕ 0 ( U m V n ) = 0 ϕ 0 (1) = 1 , if ( m, n ) � = (0 , 0) . 6 / 43

Conformal Structure on A θ (Connes) The Dolbeault operators associated with τ ∈ C , ℑ ( τ ) > 0 are τδ 2 : H 0 → H (1 , 0) , ∂ = δ 1 + ¯ ¯ ∂ = δ 1 + τδ 2 : H 0 → H (0 , 1) . The conformal structure represented by τ is encoded in a ∂ ( b ) ¯ a, b, c ∈ A ∞ � � ψ ( a, b, c ) = − ϕ 0 ∂ ( c ) , θ , which is a positive Hochschild cocycle. 7 / 43

Conformal Perturbation (Connes-Tretkoff) Let h = h ∗ ∈ A ∞ θ and replace the trace ϕ 0 by ϕ : A θ → C , ϕ ( a ) := ϕ 0 ( a e − h ) , a ∈ A θ . ϕ is a KMS state with the modular group σ t ( a ) = e ith a e − ith , a ∈ A θ , and the modular automorphism ∆( a ) := σ i ( a ) = e − h a e h , a ∈ A θ . ϕ ( a b ) = ϕ � b ∆( a ) � , a, b ∈ A θ . 8 / 43

A Spectral Triple ( A ∞ θ , H , D ) H := H ϕ ⊕ H (1 , 0) , � � a 0 a �→ : H → H , 0 a � ∂ ∗ � 0 ϕ D := : H → H , ∂ ϕ 0 τδ 2 : H ϕ → H (1 , 0) . ∂ ϕ := ∂ = δ 1 + ¯ 9 / 43

Anti-Unitary Equivalence of the Laplacians � ∂ ∗ � ϕ ∂ ϕ 0 D 2 = : H ϕ ⊕ H (1 , 0) → H ϕ ⊕ H (1 , 0) . ∂ ϕ ∂ ∗ 0 ϕ Lemma: Let k = e h/ 2 . We have k ¯ ∂ ∗ ϕ ∂ ϕ : H ϕ → H ϕ ∼ ∂∂k : H 0 → H 0 , ϕ : H (1 , 0) → H (1 , 0) ∂k 2 ∂ : H (1 , 0) → H (1 , 0) . ¯ ∂ ϕ ∂ ∗ ∼ 10 / 43

Derivation of the Asymptotic Expansion Approximate e − tD 2 by pseudodifferential operators: e − tD 2 = 1 � e − tλ ( D 2 − λ ) − 1 dλ, 2 πi C B λ ( D 2 − λ ) ≈ 1 , σ ( B λ ) = b 0 + b 1 + b 2 + · · · . 11 / 43

Connes’ pseudodifferential calculus (1980) • Symbols ρ : R 2 → A ∞ θ ⇒ P ρ : A ∞ θ → A ∞ θ � � R 2 e − is.ξ ρ ( ξ ) α s ( a ) ds dξ, P ρ ( a ) = (2 π ) − 2 a ∈ A ∞ θ . R 2 • Differential operators: � 1 ξ j � 1 δ j a ij ξ i a ij ∈ A ∞ a ij δ i ρ ( ξ 1 , ξ 2 ) = 2 , ⇒ P ρ = 2 . θ • Ψ DO’s on A ∞ θ form an algebra: 1 � ℓ 1 ! ℓ 2 ! ∂ ℓ 1 1 ∂ ℓ 2 2 ( ρ ( ξ )) δ ℓ 1 1 δ ℓ 2 2 ( ρ ′ ( ξ )) . σ ( P Q ) ∼ ℓ 1 ,ℓ 2 ≥ 0 12 / 43

Symbol of the first Laplacian σ ( k ¯ ∂∂k ) = a 2 ( ξ ) + a 1 ( ξ ) + a 0 ( ξ ) , where 1 k 2 + | τ | 2 ξ 2 2 k 2 + 2 ℜ ( τ ) ξ 1 ξ 2 k 2 , a 2 ( ξ ) = ξ 2 a 1 ( ξ ) = 2 ξ 1 kδ 1 ( k )+2 | τ | 2 ξ 2 kδ 2 ( k )+2 ℜ ( τ ) ξ 1 kδ 2 ( k )+2 ℜ ( τ ) ξ 2 kδ 1 ( k ) , a 0 ( ξ ) = kδ 2 1 ( k ) + | τ | 2 kδ 2 2 ( k ) + 2 ℜ ( τ ) kδ 1 δ 2 ( k ) . 1 � ℓ 1 ! ℓ 2 ! ∂ ℓ 1 1 ∂ ℓ 2 2 ( b j ) δ ℓ 1 1 δ ℓ 2 b n = − 2 ( a k ) b 0 , n > 0 . 2+ j + ℓ 1 + ℓ 2 − k = n, 0 ≤ j<n, 0 ≤ k ≤ 2 1 k 2 + | τ | 2 ξ 2 2 k 2 + 2 ℜ ( τ ) ξ 1 ξ 2 k 2 − λ ) − 1 . b 0 = a ′− 1 = ( ξ 2 2 13 / 43

Weyl’s law for T 2 θ Theorem. (Khalkhali-F.) Let N ( λ ) = # { λ j ≤ λ } be the eigenvalue counting function of D 2 . We have π ℑ ( τ ) ϕ 0 ( e − h ) λ N ( λ ) ∼ ( λ → ∞ ) . Equivalently: λ j ∼ ℑ ( τ ) π ϕ (1) j ( j → ∞ ) . 14 / 43

Connes’ trace theorem for T 2 θ Classical symbols: ρ : R 2 → A ∞ θ ∞ � ρ ( ξ ) ∼ ρ m − i ( ξ ) ( ξ → ∞ ) , i = − 0 ρ m − i ( t ξ ) = t m − i ρ m − i ( ξ ) , ξ ∈ R 2 . t > 0 , Theorem. (Khalkhali-F.) For any classical symbol ρ of order − 2 on A θ , we have P ρ ∈ L 1 , ∞ ( H 0 ) , and Tr ω ( P ρ ) = 1 � � � S 1 ϕ 0 ρ − 2 ( ξ ) dξ. 2 15 / 43

b 1 = − ( b 0 a 1 b 0 + ∂ 1 ( b 0 ) δ 1 ( a 2 ) b 0 + ∂ 2 ( b 0 ) δ 2 ( a 2 ) b 0 ) , b 2 = − ( b 0 a 0 b 0 + b 1 a 1 b 0 + ∂ 1 ( b 0 ) δ 1 ( a 1 ) b 0 + ∂ 2 ( b 0 ) δ 2 ( a 1 ) b 0 + ∂ 1 ( b 1 ) δ 1 ( a 2 ) b 0 + ∂ 2 ( b 1 ) δ 2 ( a 2 ) b 0 + (1 / 2) ∂ 11 ( b 0 ) δ 2 1 ( a 2 ) b 0 + (1 / 2) ∂ 22 ( b 0 ) δ 2 2 ( a 2 ) b 0 + ∂ 12 ( b 0 ) δ 12 ( a 2 ) b 0 ) . 16 / 43

Connes’ Rearrangement Lemma For any m = ( m 0 , m 1 , . . . , m ℓ ) ∈ Z ℓ +1 > 0 and ρ 1 , . . . , ρ ℓ ∈ A ∞ θ � ∞ ℓ � ρ j ( e h u + 1) − m j du u | m |− 2 ( e h u + 1) − m 0 0 1 ℓ = e − ( | m |− 1) h F m (∆ , . . . , ∆) � � � ρ j , 1 where � ∞ ℓ j x | m |− 2 � − m j � � � F m ( u 1 , . . . , u ℓ ) = x u k + 1 dx. ( x + 1) m 0 0 1 1 17 / 43

Conformal Geometry of T 2 θ with τ = i (Cohen-Connes) Let be the eigenvalues of ∂ ∗ λ 1 ≤ λ 2 ≤ λ 3 ≤ · · · ϕ ∂ ϕ , and � λ − s ζ ( s ) = j , ℜ ( s ) > 1 . Then ζ (0) + 1 = f (∆)( δ 1 ( e h/ 2 )) δ 1 ( e h/ 2 ) f (∆)( δ 2 ( e h/ 2 )) δ 2 ( e h/ 2 ) � � � � ϕ + ϕ , where f ( u ) = 1 6 u − 1 / 2 − 1 3 + L 1 ( u ) − 2(1+ u 1 / 2 ) L 2 ( u )+(1+ u 1 / 2 ) 2 L 3 ( u ) , m ( − 1) j +1 ( u − 1) j L m ( u ) = ( − 1) m ( u − 1) − ( m +1) � � � log u − . j j =1 18 / 43

The Gauss-Bonnet theorem for T 2 θ Theorem. (Connes-Tretkoff; Khalkhali-F.) For any θ ∈ R , complex parameter τ ∈ C \ R and Weyl conformal factor e h , h = h ∗ ∈ A ∞ θ , we have ζ (0) + 1 = 0 . 19 / 43

Final Part of the Proof ζ (0) + 1 = + 2 π | τ | 2 2 π K ( ∇ )( δ 1 ( h 2 )) δ 1 ( h K ( ∇ )( δ 2 ( h 2 )) δ 2 ( h � � � � ℑ ( τ ) ϕ 0 2 ) ℑ ( τ ) ϕ 0 2 ) +2 π ℜ ( τ ) K ( ∇ )( δ 1 ( h 2 )) δ 2 ( h +2 π ℜ ( τ ) K ( ∇ )( δ 2 ( h 2 )) δ 1 ( h � � � � ℑ ( τ ) ϕ 0 2 ) ℑ ( τ ) ϕ 0 2 ) , where � x � 3 x csch 2 � x � � �� � 3 x − 3 sinh − 3 sinh( x ) + sinh 2 2 2 K ( x ) = − 3 x 2 is an odd entire function, and ∇ = log ∆ . 20 / 43

x 3 23 x 5 K ( x ) = − x � x 6 � 20 + 2240 − 806400 + O . 1.0 0.5 � 10 � 5 5 10 � 0.5 � 1.0 21 / 43

Scalar Curvature for ( A ∞ θ , H , D ) It is the unique element R ∈ A ∞ θ such that a ∈ A ∞ ζ a (0) + ϕ 0 ( a ) = ϕ 0 ( a R ) , θ , where ζ a ( s ) := Trace ( a | D | − 2 s ) , Re ( s ) ≫ 0 . Equivalently, consider small-time heat kernel expansions: Trace ( a e − tD 2 ) ∼ � n − 2 B n ( a, D 2 ) t 2 , a ∈ A ∞ θ . n ≥ 0 22 / 43

Final Formula for the Scalar Curvature of T 2 θ Theorem. (Connes-Moscovici; Khalkhali-F.) Up to an overall factor − π of ℑ ( τ ) , R is equal to 1 ( h 2 ) + 2 τ 1 δ 1 δ 2 ( h 2 ( h 2 ) + | τ | 2 δ 2 δ 2 � � R 1 ( ∇ ) 2 ) δ 1 ( h 2 ) 2 + | τ | 2 δ 2 ( h δ 1 ( h 2 ) , δ 2 ( h � 2 ) 2 + ℜ ( τ ) �� � + R 2 ( ∇ , ∇ ) 2 ) ℑ ( τ ) [ δ 1 ( h 2 ) , δ 2 ( h � � + i W ( ∇ , ∇ ) 2 )] . 23 / 43

1 2 − sinh( x/ 2) x R 1 ( x ) = sinh 2 ( x/ 4) . � 0.05 � 0.10 � 0.15 � 0.20 � 0.25 � 100 � 50 50 100 24 / 43

R 2 ( s, t ) = − (1+cosh(( s + t ) / 2))( − t ( s + t ) cosh s + s ( s + t ) cosh t − ( s − t )( s + t +sinh s +sinh t − sinh( s + t ))) st ( s + t ) sinh( s/ 2) sinh( t/ 2) sinh 2 (( s + t ) / 2) 25 / 43

W ( s, t ) = ( − s − t + t cosh s + s cosh t + sinh s + sinh t − sinh( s + t )) . st sinh( s/ 2) sinh( t/ 2) sinh(( s + t ) / 2) 26 / 43

Symbol of the second Laplacian σ ( ∂ ∗ k 2 ∂ ) = c 2 ( ξ ) + c 1 ( ξ ) , where 1 k 2 + 2 τ 1 ξ 1 ξ 2 k 2 + | τ | 2 ξ 2 c 2 ( ξ ) = ξ 2 2 k 2 , c 1 ( ξ ) = ( δ 1 ( k 2 ) + τδ 2 ( k 2 )) ξ 1 + (¯ τδ 1 ( k 2 ) + | τ | 2 δ 2 ( k 2 )) ξ 2 . 27 / 43

K 1 ( x ) = 2 e x/ 2 ( e x ( x − 2) + x + 2) ( e x − 1) 2 x 0.30 0.25 0.20 0.15 0.10 0.05 � 15 � 10 � 5 5 10 15 28 / 43