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

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Dirac for S 2 LIF for S 2 Spectral Triples -functions Index theory LIF Real structure Quantum spheres D in PHC qs qs Spectral geometry of Podle s spheres Francesco DAndrea International School for Advanced Studies Via Beirut

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Spectral Triples ζ-functions Index theory LIF Real structure Quantum spheres Dirac for S2

D in PHC LIF for S2

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

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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. (D2 + 1)−1/2 ∈ Lp+(H); ⇒ the triple is even if ∃ γ = γ∗, such that γ2 = 1, γD + Dγ = 0 and aγ = γa ∀ a ∈ A. Examples:

The prototype: (C∞(M), L2(M, S), D

/ ).

Baby example: (C∞(S1), L2(S1), −i∂θ).
A simple NC-example: (A, ℓ2(N), N) with |n can. ortho. basis of

ℓ2(N), S |n := |n + 1 the unilateral shift, A the algebra of polynomials in {S, S∗} and N |n := n |n the ‘number’ operator.

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Regular spectral triples

(A, H, D) is called regular if A ∪ [D, A] ⊂

j∈N dom δ j ,

with δ( . ) := [|D|, . ] unbounded on B(H).

The prototype: (A, L2(M, S), D

/ ) for any A ⊂ C∞(M). To select C∞(M) ⇒ ask stability under holomorphic functional calculus.

NC-example. Let T ∞ ⊂ B(ℓ2(N)) be the set with elements:

f =

n∈N( fnSn + f−n−1(S∗)n+1) +
j,k∈N fjkS j(1 − SS∗)(S∗)k ,

with {fn} ∈ S(Z) and {fjk} ∈ S(N2). T ∞ is a ∗-algebra stable under h.f.c. (whose C∗-completion is the Toeplitz algebra T ); (T ∞, ℓ2(N), N) is a regular spectral triple.

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ζ-functions and residues

Let Ψ0 be the algebra generated by

j∈N δj(A ∪ [D, A]). Assume

(A, H, D) regular and D invertible. To each a ∈ Ψ0 we associate: ζa(z) := TraceH(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 ζa(z), a ∈ Ψ0, extend to meromorphic functions

n C with poles in Σ as unique singularities.

If Σ is made of simple poles only, the Wodzicki-type residue

− T := Resz=0Trace(T|D|−z)

is tracial on the ∗-algebra generated by Ψ0 and |D|.

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Smoothing operators and local computations

Let OP0 :=

j∈N dom δ j. The class

OP−∞ :=

T ∈ OP0 : |D|nT ∈ OP0 ∀ n ∈ N
is a two-sided ∗-ideal in the ∗-algebra OP0.

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 Ψ0 in OP0/OP−∞ only. A linear map ϕ : Ψ0 → 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.

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Index Theory

/ elliptic over M is Fredholm. If D / V := lift to a v.b. V, ∃ an homo.: Index : K0(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∗, F2 = 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), aj ∈ A. The class of

chF

n (a0, . . . , an) = 1 2n! Γ( n 2 + 1) Trace(γF[F, a0] . . . [F, an])

in PHCev(A) is indep. of n, ∀ n even and sufficiently large.

Pairing between φ = (φ0, φ2, . . .) ∈ PHCev(A) and K0(A):

φ, [p] = φ0(p) + P

k∈N(−1)k (2k)! k! φ2k(p − 1 2, p, . . . , p)

p = p∗ = p2 is a projector. The pairing with chF gives: K0(A) → Z , [p] → ˙ chF, [p] ¸ = Index(pFp)

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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 ∈ 2N, regular and Σ = {simple poles}. Then [ϕ] = chF in PHCev, with: ϕ0(a0) =Resz=0z−1Trace(γa0|D|−2z) ϕn(a0, ..., an) =

k∈Nn

(−1)k k1!...kn!αk

− γa0[D, a1](k1) . . . [D, an](kn)|D|−(2|k|+n)

where n ≤ d is even, α−1

= (k1 + 1)(k1 + k2 + 2) . . . (k1 + ... + kn + n) , aj ∈ A , T(0) = T and T(j+1) = [D2, T(j)] ∀ j ∈ N .

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Real Structure

A real structure is an antilinear isometry J on H s.t. ∀ a, b ∈ A: J2 = ±1 , JD = ± DJ , [a, JbJ −1] = 0 , [[D, a], JbJ −1] = 0 . If the spectral triple is real, we require also Jγ = ± γJ. The signs ‘±’ depend

n 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), L2(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−∞.

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Generalities about Podle´ s spheres

A(S2

qs) is the ∗-algebra generated by A = A∗, B and B∗ with relations:

AB = q2BA , BB∗ + (A + s2)(A − 1) = 0 , B∗B + (q2A + s2)(q2A − 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(S2

qs) are a ‘strict

deformation quantization’ (Rieffel) of C∞(S2). Symmetries: q-homo. spaces (= comodule ∗-algebra) for SUq(2). SU(2) − − − − − → S2 (Hopf fibering)

(dual) Drinfeld-Jimbo deformation

? ? y x ? ?q→1 SUq(2)

(quantum group)

− − − − − → S2

qs (Podle´ s spheres)

(principal coalgebra bundle)

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‘Topology’

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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D+ D− S1 S2

Remove D± and replace with Toeplitz operators T .

1 s ∈ [0, 1] S1

From now on, q = 1. Let D± ≃ closed disks. If s = 0: C(S2) ≃ ˘ (f+, f−) ∈ C(D−) ⊕ C(D+) ˛ ˛ f+|S1 = f−|S1 ¯ , C(S2

qs) ≃

˘ (x+, x−) ∈ T ⊕ T ˛ ˛ σ(x+) = σ(x−) ¯ , where σ : T → C(S1) is the symbol map, and replace the “evaluation on the boundary S1”. S1 is a commutative subset: 0 → K ⊕ K → C(S2

qs) → C(S1) → 0 .

If s = 0, D+ collapse to a single point {∗}, C(D+) is replaced by C({∗}) ≃ C,

ne copy of T by C and the exact sequence becomes:

0 → K → C(S2

q0) → C → 0 .

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‘Smooth structure’

Using the fibered product structure, one can define C∞(S2

qs).

Recall the definition of the pre-C∗-algebra T ∞. We define: C∞(S2

qs) =

(x+, x−) ∈ T ∞ ⊕ T ∞

σ(x+) = σ(x−)

if s = 0 ,

C∞(S2

q0) =

(x, c) ∈ T ∞ ⊕ C
σ(x+) = c
.

Then for all s, C∞(S2

qs) is a pre-C∗-algebra and contains the polinomial

algebra A(S2

qs).

C∞(S2

q0) is naturally represented on ℓ2(N) and defines a regular

spectral triple, ‘singular’ for q → 0, with positive D = N. If s = 0, C∞(S2

qs) is naturally represented on ℓ2(N) ⊗ C2 and defines a

regular spectral triple, ‘singular’ for q → 0, with D = N ⊗ 0 1

1 0

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Chiral representations

Deformation of the Bott projector: pqs := 1 1 + s2

s2 + A

B B∗ 1 − q2A

Classically (q = 1) it describes the tautological bundle over CP1 ≃ S2. We define the Hilbert spaces H+ := L2(S2

qs)2pqs ,

H− := L2(S2

qs)2(1 − pqs) .

(q = 1 ⇒ H± = L2-Weyl spinors). The representation of A(S2

qs) on H± can be explicitly computed in a

basis of harmonic spinors1 |l, m±, l ∈ N + 1

2 and m = −l, −l + 1, . . . , l.

1K. Schmüdgen and E. Wagner, math.QA/0305309

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Just to give an idea:

A |l, m+ = − qm−1 s [l + 1

2][l + 3 2]

[2l + 1][2l + 3] p [l + m + 1][l − m + 1] [2l + 2] × × q (ql+ 1

2 + q−l− 1 2 s2)(ql+ 3 2 s2 + q−l− 3 2 ) |l + 1, m+

− qm−1 s [l − 1

2][l + 1 2]

[2l − 1][2l + 1] p [l + m][l − m] [2l] × × q (ql+ 1

2 + q−l− 1 2 s2)(ql− 1 2 s2 + q−l+ 1 2 ) |l − 1, m+

+ (1 − s2)(1 − q2)[l − 1

2][l + 3 2] + 1

1 + q2 − 1 ! × × [l + m + 1][l − m] − q−2[l + m][l − m + 1] [2l][2l + 2] |l, m+ + 1 − s2 1 + q2 |l, m+ ,

plus similar formulas for A |l, m− and B |l, m±. Here [x] := (qx − q−x)/(q − q−1) is the q-analogue of x ∈ C.

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The isospectral Dirac operator

We2 constructed a family of SUq(2)-equivariant spectral triples on S2

qs.

Peculiar example: H := H+ ⊕ H− discussed before, with natural grading, Dirac operator and ‘quasi’ real structure defined by: D |l, m± := (l + 1

2) |l, m∓ ,

J |l, m± := (−1)m+1/2 |l, −m∓ .

Proposition

(A(S2

qs), H, D) is a 2+-summable regular even spectral triple.

Computations are simplified by neglecting smoothing operators. Next step: to quotient the representation by OP−∞.

2Joint work with L. Dabrowski, G. Landi and E. Wagner.

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An ‘approximate’ representation

Let ˆ H be an Hilbert space with ortho. basis ||l, m ±, l ∈ 1

2Z and l + m ∈ N.

Define the inclusion Q : H → ˆ H and the adjoint projection P : ˆ H → H by: Q |l, m± := ||l, m ± , P ||l, m ± :=  |l, m± if l ∈ N + 1

2 and m ≤ l ,

therwise .

The following operators α, β ∈ B( ˆ H) satisfy the comm. relations of A(SUq(2)): α ||l, m ± = p 1 − q2(l+m+1) ||l + 1

2, m + 1 2

± , β ||l, m ± = ql+m ||l + 1

2, m − 1 2

± . The embedding A(S2

qs) ֒

→ A(SUq(2)) leads to a ∗-rep. π : A(S2

qs) → B( ˆ

H) . The sandwich ˜ π(a) := Pπ(a)Q defines a ∗-linear map ˜ π : A(S2

qs) → B(H) .

Lemma

a − ˜ π(a) ∈ OP−∞ for all a ∈ A(S2

qs).

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Let C ⊂ B( ˆ H) be the ∗-algebra generated by α, β, α∗, β∗ and F. Then, Ψ0 ⊂ P CQ + OP−∞ . We determine Σ by looking at singularities of ζ-functions associated to P CQ. Only monomials T := P(ββ∗)kQ contribute to Σ. For such monomials ζT(z) =  4ζ(z − 1) if k = 0 ,

2 1−q2k ζ(z) + holomorphic function

if k = 0 , with ζ(z) the Riemann zeta-function.

Proposition

The dimension spectrum is Σ = {1, 2} and the top residue is given by: Z − a|D|−2 := Resz=2ζa(z) = 2 π Z

S1 σ(a)dθ

∀ a ∈ A(S2

qs) ,

with σ : A(S2

qs) ֒

→ C(S1) the homo. defined by: σ(A) = 0 and σ(B) = seiθ.

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About the PHC-class of D

The Fredholm module associated to D is (A(S2

qs), H, F) with:

F |l, m± = |l, m∓ . The pairing: ˙ chF, [ p] ¸ ≡ chF

0 (Trace p) ,

[ p] ∈ K0(A) , gives a Z-valued invariant analogous to the 1st Chern number. If p = pqs: ˙ chF, [ pqs] ¸ = 1 2 1 − q2 1 + s2 Trace(γF[F, A]) = q−2(1 − q2) X

l,m

[l − m + 1][l + m] [2l][2l + 2] .

Proposition

The cohomology class of F is not trivial: ˙ chF, [ pqs] ¸ = 1 . Proved in 3 steps: 1st) prove that the charge is a continuous function of q; 2nd) a continuous function [0, 1) → Z is constant; 3rd) compute it for q = 0.

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Local index formulas

In general: evaluating

chF, [ p]
by computing kernels and cokernels

(or traces) is very difficult. One needs local formulas. Connes-Moscovici theorem, applied to our case states that chF is cohomologous to the cocycle ϕ with components: ϕ0(a0) = Resz=0 z−1Trace(γa0|D|−2z) , ϕ2(a0, a1, a2) ∝

− γa0[D, a1][D, a2]|D|−2 .

In principle: ϕ2 is local, ϕ0 is not local. With a little analysis one can prove: ϕ2 = 0 and ϕ0 = chF

0 .