Spectral geometry of Podle s spheres Francesco DAndrea - PowerPoint PPT Presentation

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Spectral geometry of Podle´ s spheres Francesco D’Andrea International School for Advanced Studies Via Beirut 2-4, I-34014, Trieste, Italy Thematic Program on NC-Geometry and q -Groups Trieste, 22th June 2006

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Preliminary definitions The data ( A , H , D ) is called spectral triple iff: • A ⊂ B ( H ) is a ∗ -algebra with 1 , H a (separable) Hilbert space; • D is a selfadjoint operator on (a dense subspace of) H , ( D + i ) − 1 ∈ K ( H ) and [ D , a ] ∈ B ( H ) ∀ a ∈ A ; ⇒ D is p + - summable iff ∃ p ∈ R + s.t. ( D 2 + 1 ) − 1 / 2 ∈ L p + ( H ) ; ⇒ the triple is even if ∃ γ = γ ∗ , such that γ 2 = 1 , γ D + D γ = 0 and a γ = γ a ∀ a ∈ A . Examples: • The prototype: ( C ∞ ( M ) , L 2 ( M , S ) , D / ) . • Baby example: ( C ∞ ( S 1 ) , L 2 ( S 1 ) , − i ∂ θ ) . • A simple NC-example: ( A , ℓ 2 ( N ) , N ) with | n � can. ortho. basis of ℓ 2 ( N ) , S | n � := | n + 1 � the unilateral shift, A the algebra of polynomials in { S , S ∗ } and N | n � := n | n � the ‘number’ operator.

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Regular spectral triples ( A , H , D ) is called regular if j ∈ N dom δ j , � A ∪ [ D , A ] ⊂ with δ ( . ) := [ | D | , . ] unbounded on B ( H ) . / ) for any A ⊂ C ∞ ( M ) . To select • The prototype: ( A , L 2 ( M , S ) , D C ∞ ( M ) ⇒ ask stability under holomorphic functional calculus. • NC-example. Let T ∞ ⊂ B ( ℓ 2 ( N )) be the set with elements: n ∈ N ( f n S n + f − n − 1 ( S ∗ ) n + 1 ) + j , k ∈ N f jk S j ( 1 − SS ∗ )( S ∗ ) k , � � f = with { f n } ∈ S ( Z ) and { f jk } ∈ S ( N 2 ) . T ∞ is a ∗ -algebra stable under h.f.c. (whose C ∗ -completion is the Toeplitz algebra T ); ( T ∞ , ℓ 2 ( N ) , N ) is a regular spectral triple.

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs ζ -functions and residues Let Ψ 0 be the algebra generated by � j ∈ N δ j ( A ∪ [ D , A ]) . Assume ( A , H , D ) regular and D invertible. To each a ∈ Ψ 0 we associate: ζ a ( z ) := Trace H ( a | D | − z ) , holomorphic for z ∈ C with Re z sufficiently large. Definition A regular spectral triple has dimension spectrum Σ iff Σ ⊂ C is a countable set and all ζ a ( z ) , a ∈ Ψ 0 , extend to meromorphic functions on C with poles in Σ as unique singularities. If Σ is made of simple poles only, the Wodzicki-type residue � − T := Res z = 0 Trace ( T | D | − z ) is tracial on the ∗ -algebra generated by Ψ 0 and | D | .

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Smoothing operators and local computations Let OP 0 := � j ∈ N dom δ j . The class OP −∞ := T ∈ OP 0 : | D | n T ∈ OP 0 ∀ n ∈ N � � is a two-sided ∗ -ideal in the ∗ -algebra OP 0 . Since ζ -functions associated to T ∈ OP −∞ are holomorphic on all C , OP −∞ do not contribute to Σ and one has to look at the image of Ψ 0 in OP 0 / OP −∞ only. A linear map ϕ : Ψ 0 → C is called local if it is insensitive to smoothing perturbations, ϕ | OP −∞ = 0 . Residues of zeta-type functions are local. Locality makes complicated expressions computable, by neglecting irrelevant details.

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Index Theory • D / elliptic over M is Fredholm. If D / V := lift to a v.b. V , ∃ an homo.: Index : K 0 ( M ) → Z , [ V ] �→ Index ( D / V ) . • To generalize it, def. of Fredholm module [Atiyah]: it is a triple ( A , H , F ) , A ⊂ B ( H ) , F = F ∗ , F 2 = 1 and [ F , A ] ⊂ K ( H ) . E.g.: ( A , H , D ) reg. spectral triple ⇒ ( A , H , sign D ) Fredholm module. • Let γ = grading on ( A , H , D ) , a j ∈ A . The class of ch F 2 n ! Γ( n 1 n ( a 0 , . . . , a n ) = 2 + 1 ) Trace ( γ F [ F , a 0 ] . . . [ F , a n ]) in PHC ev ( A ) is indep. of n , ∀ n even and sufficiently large. • Pairing between φ = ( φ 0 , φ 2 , . . . ) ∈ PHC ev ( A ) and K 0 ( A ) : k ∈ N ( − 1 ) k ( 2 k )! k ! φ 2 k ( p − 1 � φ, [ p ] � = φ 0 ( p ) + P 2 , p , . . . , p ) p = p ∗ = p 2 is a projector. The pairing with ch F gives: ˙ ch F , [ p ] ¸ K 0 ( A ) → Z , [ p ] → = Index ( pFp )

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Local index formula A general theorem relates the index of the twisted Dirac operator to residues of zeta-type functions. Theorem (Connes-Moscovici) Let ( A , H , D ) be even, d + -summable with d ∈ 2 N , regular and Σ = { simple poles } . Then [ ϕ ] = ch F in PHC ev , with: ϕ 0 ( a 0 ) = Res z = 0 z − 1 Trace ( γ a 0 | D | − 2 z ) ( − 1 ) k � − γ a 0 [ D , a 1 ] ( k 1 ) . . . [ D , a n ] ( k n ) | D | − ( 2 | k | + n ) � ϕ n ( a 0 , ..., a n ) = k 1 ! ... k n ! α k k ∈ N n where n ≤ d is even, α − 1 = ( k 1 + 1 )( k 1 + k 2 + 2 ) . . . ( k 1 + ... + k n + n ) , k a j ∈ A , T ( 0 ) = T and T ( j + 1 ) = [ D 2 , T ( j ) ] ∀ j ∈ N .

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Real Structure A real structure is an antilinear isometry J on H s.t. ∀ a , b ∈ A : J 2 = ± 1 , JD = ± DJ , [ a , JbJ − 1 ] = 0 , [[ D , a ] , JbJ − 1 ] = 0 . If the spectral triple is real, we require also J γ = ± γ J . The signs ‘ ± ’ depend on the dimension. ( − , + , − if d = 2 ) Motivated by spin manifolds. Let A = C ∞ ( M ) , then: JbJ − 1 = b ∗ and 3rd condition is trivial, [ D , a ] ∈ B ( H ) means that D is a 1st order PDO, [[ D , a ] , b ∗ ] = 0 that means D is a 1st order differential operator. E.g. ( C ∞ ( M ) , L 2 ( M ) , ∆ 1 / 2 ) is a spectral triple, but not real: [[∆ 1 / 2 , a ] , b ∗ ] � = 0 is an order ≤ 0 PDO. If A is a von Neumann algebra, by Tomita-Takesaki theorem ∃ J satisfying all the conditions, except the framed one that is not always possible to satisfy. Typical examples are quantum groups (and q -spaces), where the framed condition is zero modulo OP −∞ .

Dirac for S 2 LIF for S 2 Spectral Triples ζ -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Generalities about Podle´ s spheres qs ) is the ∗ -algebra generated by A = A ∗ , B and B ∗ with relations: A ( S 2 AB = q 2 BA , BB ∗ + ( A + s 2 )( A − 1 ) = 0 , B ∗ B + ( q 2 A + s 2 )( q 2 A − 1 ) = 0 . ] 0 , 1 ] ∋ q = deformation parameter, s ∈ [ 0 , 1 ] an additional parameter. q = 1 ⇒ 2 -sphere with center and radius depending on s . q � = 1 ⇒ NC-algebra, we call Podle´ s sphere the underlying ‘virtual space’. For fixed s , the associated universal C ∗ -algebras C ( S 2 qs ) are a ‘strict deformation quantization’ (Rieffel) of C ∞ ( S 2 ) . Symmetries: q -homo. spaces ( = comodule ∗ -algebra) for SU q ( 2 ) . S 2 SU ( 2 ) − − − − − → (Hopf fibering) ? x (dual) Drinfeld-Jimbo ? ? ? q → 1 deformation y S 2 SU q ( 2 ) − − − − − → (principal coalgebra bundle) qs (quantum group) (Podle´ s spheres)