EXTREMA OF THE EINSTEIN-HILBERT ACTION FOR NONCOMMUTATIVE 4-TORI - PowerPoint PPT Presentation

EXTREMA OF THE EINSTEIN-HILBERT ACTION FOR NONCOMMUTATIVE 4-TORI Farzad Fathizadeh joint with Masoud Khalkhali 1 / 39

The Heat Kernel of a Riemannian Manifold ( M, g ) △ g : C ∞ ( M ) → C ∞ ( M ) , K : R > 0 × M × M → C , � e − t △ g f � � ( x ) = K ( t, x, y ) f ( y ) dvol ( y ) . M � ∞ K ( t, x, y ) ∼ e − dist( x,y ) 2 / 4 t U i ( x, y ) t i � � ( t → 0) , (4 πt ) n/ 2 i =0 U i : N ( Diag ( M × M )) → C (geometric information), U 0 ( x, x ) = 1 ( ⇒ Weyl’s law ) , U 1 ( x, x ) = scalar curvature. 2 / 39

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 3 / 39

Local Geometric Invariants of ( A , H , D ) These invariants such as scalar curvature can be computed by con- sidering small time heat kernel expansions of the form ∞ π ( a ) e − tD 2 � � a n ( a, D ) t ( n − d ) / 2 , � Trace ∼ t → 0 + n =0 where d is the spectral dimension. 4 / 39

Noncommutative 4-Torus T 4 θ C ( T 4 θ ) is the universal C ∗ -algebra generated by 4 unitaries U 1 , U 2 , U 3 , U 4 , satisfying U k U ℓ = e 2 πiθ kℓ U ℓ U k , for a skew symmetric matrix θ = ( θ kℓ ) ∈ M 4 ( R ) . 5 / 39

Action of T 4 = ( R / 2 π Z ) 4 on C ( T 4 θ ) R 4 ∋ s �→ α s ∈ Aut � � C ( T 4 θ ) , α s ( U m ) := e is · m U m , U m := U m 1 U m 2 U m 3 U m 4 , m j ∈ Z . 1 2 3 4 ∂ s =0 α s : C ∞ ( T 4 θ ) → C ∞ ( T 4  δ j = θ ) ,  ∂s j δ j ( U k ) := U k if k = j, := 0 if k � = j. 6 / 39

Complex Structure on T 4 θ ∂ = ¯ ¯ ∂ 1 ⊕ ¯ ∂ = ∂ 1 ⊕ ∂ 2 , ∂ 2 , ∂ 1 = 1 ∂ 2 = 1 2 ( δ 1 − iδ 3 ) , 2 ( δ 2 − iδ 4 ) , ∂ 1 = 1 ∂ 2 = 1 ¯ ¯ 2 ( δ 1 + iδ 3 ) , 2 ( δ 2 + iδ 4 ) . 7 / 39

Volume Form on T 4 θ ϕ 0 : C ( T 4 θ ) → C , ϕ 0 (1) := 1 , ϕ 0 ( U m 1 U m 2 U m 3 U m 4 ) := 0 , ( m 1 , m 2 , m 3 , m 4 ) � = (0 , 0 , 0 , 0) . 1 2 3 4 a, b ∈ C ( T 4 ϕ 0 ( a b ) = ϕ 0 ( b a ) , θ ) . ϕ 0 ( a ∗ a ) > 0 , a � = 0 . 8 / 39

Conformal Perturbation (Connes-Tretkoff) Let h = h ∗ ∈ C ∞ ( T 4 θ ) and replace the trace ϕ 0 by ϕ : C ( T 4 θ ) → C , ϕ ( a ) := ϕ 0 ( a e − 2 h ) , a ∈ C ( T 4 θ ) . ϕ is a KMS state with the modular group σ t ( a ) = e 2 ith a e − 2 ith , a ∈ C ( T 4 θ ) , and the modular automorphism ∆( a ) := σ i ( a ) = e − 2 h a e 2 h , a ∈ C ( T 4 θ ) . a, b ∈ C ( T 4 � � ϕ ( a b ) = ϕ b ∆( a ) , θ ) . 9 / 39

Perturbed Laplacian on T 4 θ d = ∂ ⊕ ¯ ∂ : H ϕ → H (1 , 0) ⊕ H (0 , 1) , ϕ ϕ △ ϕ := d ∗ d. Remark. If h = 0 then ϕ = ϕ 0 and △ ϕ 0 = δ 2 1 + δ 2 2 + δ 2 3 + δ 2 4 = ∂ ∗ ∂ (the underlying manifold is K¨ ahler). 10 / 39

Explicit Formula for △ ϕ Lemma. Up to an anti-unitary equivalence △ ϕ is given by e h ¯ ∂ 1 e − h ∂ 1 e h + e h ∂ 1 e − h ¯ ∂ 1 e h + e h ¯ ∂ 2 e − h ∂ 2 e h + e h ∂ 2 e − h ¯ ∂ 2 e h , where ∂ 1 , ∂ 2 are analogues of the Dolbeault operators. 11 / 39

Connes’ Pseudodifferential Calculus (1980) A smooth map ρ : R 4 → C ∞ ( T 4 θ ) is a symbol of order m ∈ Z , if for any i, j ∈ Z 4 ≥ 0 , there exists a constant c such that || ∂ j δ i � || ≤ c (1 + | ξ | ) m −| j | , � ρ ( ξ ) and if there exists a smooth map k : R 4 \{ 0 } → C ∞ ( T 4 θ ) such that ξ ∈ R 4 \ { 0 } . λ →∞ λ − m ρ ( λξ ) = k ( ξ ) , lim 12 / 39

• Given a symbol ρ : R 4 → C ∞ ( T 4 θ ) , the corresponding ψ DO is: � � e − is.ξ ρ ( ξ ) α s ( a ) ds dξ, P ρ ( a ) = (2 π ) − 4 a ∈ C ∞ ( T 4 θ ) . • Differential operators: � � a ℓ ξ ℓ , a ℓ ∈ C ∞ ( T 4 a ℓ δ ℓ . ρ ( ξ ) = θ ) ⇒ P ρ = • Ψ DO’s on T 4 θ form an algebra: 1 � ℓ ! ∂ ℓ ξ ρ ( ξ ) δ ℓ ( ρ ′ ( ξ )) . σ ( P Q ) ∼ ℓ ∈ Z 4 ≥ 0 13 / 39

• A symbol ρ : R 4 → C ∞ ( T 4 θ ) of order m is elliptic if ρ ( ξ ) is invertible for any ξ � = 0 , and if there exists a constant c such that || ρ ( ξ ) − 1 || ≤ c (1 + | ξ | ) − m , when | ξ | is sufficiently large. • Example of an elliptic operator: △ ϕ = e h ¯ ∂ 1 e − h ∂ 1 e h + e h ∂ 1 e − h ¯ ∂ 1 e h + e h ¯ ∂ 2 e − h ∂ 2 e h + e h ∂ 2 e − h ¯ ∂ 2 e h . 14 / 39

Symbol of △ ϕ Lemma. The symbol of △ ϕ is equal to a 2 ( ξ ) + a 1 ( ξ ) + a 0 ( ξ ) , where 4 4 � � a 2 ( ξ ) = e h ξ 2 δ i ( e h ) ξ i , i , a 1 ( ξ ) = i =1 i =1 4 i ( e h ) − δ i ( e h ) e − h δ i ( e h ) � δ 2 � � a 0 ( ξ ) = . i =1 15 / 39

Mellin Transform and Asymptotic Expansions � ∞ 1 e − t △ ϕ t s dt △ − s = t , ϕ Γ( s ) 0 ∞ Trace ( a e − t △ ϕ ) ∼ t → 0 + t − 2 � B n ( a, △ ϕ ) t n/ 2 . n =0 Approximate e − t △ 2 ϕ by pseudodifferential operators: 1 � e − tλ ( △ ϕ − λ ) − 1 dλ, e − t △ ϕ = 2 πi C B λ ( △ ϕ − λ ) ≈ 1 , σ ( B λ ) = b 0 + b 1 + b 2 + · · · . 16 / 39

Analogue of Weyl’s Law for T 4 θ Theorem. For the eigenvalue counting function N ( λ ) = # { λ j ≤ λ } of the Laplacian △ ϕ on T 4 θ , we have N ( λ ) ∼ π 2 ϕ 0 ( e − 2 h ) λ 2 ( λ → ∞ ) . 2 Corollary. √ 2 πϕ 0 ( e − 2 h ) 1 / 2 j 1 / 2 λ j ∼ ( j → ∞ ) , = π 2 � (1 + △ ϕ ) − 2 � 2 ϕ 0 ( e − 2 h ) . Tr ω 17 / 39

Dixmier Trace Tr ω : L 1 , ∞ ( H ) → C For any T ∈ K ( H ) , let µ 1 ( T ) ≥ µ 2 ( T ) ≥ · · · ≥ 0 1 be the sequence of eigenvalues of | T | = ( T ∗ T ) 2 . N � L 1 , ∞ ( H ) := � � • T ∈ K ( H ); µ n ( T ) = O ( log N ) . n =1 N 1 � � � 0 ≤ T ∈ L 1 , ∞ ( H ) . • Tr ω ( T ) := lim µ n ( T ) , log N ω n =1 18 / 39

Noncommutative Residue (Wodzicki) Let P be a classical ψ DO acting on smooth sections of a vector bundle E over a closed smooth manifold M of dimension n . • Definition: � Res ( P ) = (2 π ) − n tr ( ρ − n ( x, ξ )) dx dξ, S ∗ M where S ∗ M ⊂ T ∗ M is the unit cosphere bundle on M and ρ − n is the component of order − n of the complete symbol of P . • Theorem: Res is the unique trace on Ψ( M, E ) . 19 / 39

A Noncommutative Residue for T 4 θ Classical symbols: ρ : R 4 → C ∞ ( T 4 θ ) ∞ � ρ ( ξ ) ∼ ρ m − i ( ξ ) ( ξ → ∞ ) , i =0 ρ m − i ( t ξ ) = t m − i ρ m − i ( ξ ) , ξ ∈ R 4 . t > 0 , Theorem. The linear functional � � � Res ( P ρ ) := S 3 ϕ 0 ρ − 4 ( ξ ) dξ is the unique trace on classical pseudodifferential operators on T 4 θ . 20 / 39

Analogue of Connes’ Trace Theorem for T 4 θ Theorem. For any classical symbol ρ of order − 4 on T 4 θ , we have P ρ ∈ L 1 , ∞ ( H 0 ) , and Tr ω ( P ρ ) = 1 4 Res ( P ρ ) . Remark. Weyl’s law is a special case of this theorem: let 1 ρ ( ξ ) = (1 + | ξ | 2 ) 2 . 21 / 39

Scalar Curvature for T 4 θ It is the unique element R ∈ C ∞ ( T 4 θ ) such that a ∈ C ∞ ( T 4 Res s =1 ζ a ( s ) = ϕ 0 ( a R ) , θ ) , ζ a ( s ) := Trace ( a △ − s ϕ ) , ℜ ( s ) ≫ 0 . 22 / 39

Connes’ Rearrangement Lemma For any m = ( m 0 , m 1 , . . . , m ℓ ) ∈ Z ℓ +1 > 0 , ρ 1 , . . . , ρ ℓ ∈ C ∞ ( T 4 θ ) : � ∞ ℓ u | m |− 2 ρ j ( e h u + 1) − m j du � ( e h u + 1) m 0 0 1 ℓ = e − ( | m |− 1) h F m (∆ , . . . , ∆) � � � ρ j , 1 where � ∞ j ℓ x | m |− 2 � − m j � � � F m ( u 1 , . . . , u ℓ ) = x u k + 1 dx. ( x + 1) m 0 0 1 1 23 / 39

Examples of F m F (3 , 4) ( u ) = 60 u 3 log( u ) + ( u − 1)( u ( u (3( u − 9) u − 47) + 13) − 2) 6( u − 1) 6 u 3 F (2 , 2 , 1) ( u, v ) = ( v − 1) ( ( u − 1)( uv − 1)( u ( u ( v − 1)+ v ) − 1) − u 2 ( v − 1)(2 uv + u − 3) log( uv ) ) +( u (2 v − 3)+1)( uv − ( u − 1) 3 u 2 ( v − 1) 2 ( uv − 1) 2 24 / 39

Identities Relating δ i ( e h ) and δ i ( h ) e − h δ i ( e h ) = g 1 (∆) � � δ i ( h ) , e − h δ 2 i ( e h ) = g 1 (∆) δ 2 � � � � i ( h ) + 2 g 2 (∆ (1) , ∆ (2) ) δ i ( h ) δ i ( h ) , where g 1 ( u ) = u − 1 log u , g 2 ( u, v ) = u ( v − 1) log( u ) − ( u − 1) log( v ) log( u ) log( v )(log( u ) + log( v )) . 25 / 39

Final Formula for the Scalar Curvature of T 4 θ Theorem. 4 4 R = e − h k ( ∇ ) � � + e − h H ( ∇ , ∇ ) � δ i ( h ) 2 � � δ 2 � i ( h ) , i =1 i =1 where ∇ ( a ) := 1 a ∈ C ( T 4 2 log ∆( a ) = [ − h, a ] , θ ) , k ( s ) = 1 − e − s , 2 s H ( s, t ) = − e − s − t (( − e s − 3) s ( e t − 1) + ( e s − 1) (3 e t + 1) t ) . 4 s t ( s + t ) 26 / 39

Recalling the Scalar Curvature of T 2 θ Theorem. (Connes-Moscovici; Khalkhali-F.) Up to an overall factor ℑ ( τ ) , the scalar curvature of T 2 − π of θ 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 )] . 27 / 39

The One Variable Function for T 4 θ 4 + s 2 12 − s 3 48 + s 4 s 5 k ( s ) = 1 2 − s s 6 � � 240 − 1440 + O . 3.0 2.5 2.0 1.5 1.0 0.5 � 2 2 4 28 / 39