[PPT] - Quantum Complex Projective Spaces Fredholm modules, K-theory, PowerPoint Presentation

SLIDE 1

Quantum Complex Projective Spaces

Fredholm modules, K-theory, spectral triples Francesco D’Andrea

Université Catholique de Louvain Chemin du Cyclotron 2, Louvain-La-Neuve, Belgium

MFO, 8 September 2009

Joint work with G. Landi and L. D ˛ abrowski

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 1 / 17

SLIDE 2

Introduction

A case study: the standard Podle´ s sphere S✷

q = SUq(✷)/U(✶) (here ✵ < q < ✶).

◮ L. D ˛

abrowski – A. Sitarz Dirac operator on the standard Podle´ s quantum sphere Banach Center Publ. 61 (2003), 49–58.

K. Schmüdgen – E. Wagner

Dirac operator and a twisted cyclic cocycle

n the standard Podle´

s quantum sphere

J. Reine Angew. M. 574 (2004), 219–235.
R. Oeckl

Braided Quantum Field Theory CMP 217 (2001) 451–473. Prescribed Hilbert space + SUq(✷) equivariance = unique real spectral triple (modulo equivalences). Spectrum(D) =

±[n]q
n✶

with [n]q := qn−q−n

q−q−✶ .

The spectrum of D diverges exponentially the resolvent (D✷ + m✷)−✶ of the Laplacian is

f trace class.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 2 / 17

SLIDE 3

Introduction

A case study: the standard Podle´ s sphere S✷

q = SUq(✷)/U(✶) (here ✵ < q < ✶).

L. D ˛

abrowski – A. Sitarz Dirac operator on the standard Podle´ s quantum sphere Banach Center Publ. 61 (2003), 49–58.

◮ K. Schmüdgen – E. Wagner

Dirac operator and a twisted cyclic cocycle

n the standard Podle´

s quantum sphere

J. Reine Angew. M. 574 (2004), 219–235.
R. Oeckl

Braided Quantum Field Theory CMP 217 (2001) 451–473. The representation in DS spectral triple is the direct sum

f two copies of the left regular

representation. Generators of Uq(su(✷)) are (external) derivations on S✷

q.

With these one constructs D. D✷ is proportional to the Casimir

f Uq(su(✷)): this explains why
eigenv. diverge exponentially.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 2 / 17

SLIDE 4

Introduction

A case study: the standard Podle´ s sphere S✷

q = SUq(✷)/U(✶) (here ✵ < q < ✶).

L. D ˛

abrowski – A. Sitarz Dirac operator on the standard Podle´ s quantum sphere Banach Center Publ. 61 (2003), 49–58.

K. Schmüdgen – E. Wagner

Dirac operator and a twisted cyclic cocycle

n the standard Podle´

s quantum sphere

J. Reine Angew. M. 574 (2004), 219–235.

◮ R. Oeckl

Braided Quantum Field Theory CMP 217 (2001) 451–473. On S✷

q the tadpole diagram

– the only basic divergence of φ✹ theory in 2D – becomes finite at q = ✶.

Reason: the propagator

(D✷ + m✷)−✶ is of trace class. Regularization of QFT with quantum groups symmetries: what about higher dimensional spaces?

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 2 / 17

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Geometry of quantum projective spaces

Some results:

◮ generators of K-homology and K-theory groups, computation of the pairing [DL09b]: ◮ by induction, using A(CPℓ

q) ։ A(CPℓ−✶ q

);

◮ Fredholm modules are ‘conformal classes’ of spectral triples (regular, in general

they are not real/equivariant, arbitrary summability n ∈ R+);

◮ family of Uq(su(ℓ + ✶))-equivariant spectral triples [DD09]: ◮ for generic ℓ, ✵+-dimensional equivariant even spectral triples labelled by N ∈ Z; ◮ if ℓ is odd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ

♠♦❞ ✽.

◮ there is a map τ : KU

✵ (A) → HCU n(A). With the differential calculus associated to an

equivariant spectral triple one can construct twisted Hochschild cocycle that can be paired with τ([p]). As a byproduct, we prove that KU

✵ (A) ⊃ Z∞ if q is transcendental.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 3 / 17

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The quantum SU(ℓ + ✶) group

Let ℓ > ✶ . For G := SU(ℓ + ✶) , the functions ui

j : G → C ,

ui

j(g) := gi j

, generate a Hopf ∗-algebra A(G). As abstract ∗-algebra it is defined by the relations (1) ui

juk l = uk l ui j ✱

p∈Sℓ+✶(−✶)||p||u✶

p(✶)u✷ p(✷) ✳ ✳ ✳ uℓ+✶ p(ℓ+✶) = ✶ ✱

where ||p|| = length of the permutation p ∈ Sℓ+✶, and with ∗-structure (2) (ui

j)∗ = (−✶)j−i p∈Sℓ(−✶)||p||uk✶ p(n✶)uk✷ p(n✷) ✳ ✳ ✳ ukℓ p(nℓ)

where {k✶✱ ✳ ✳ ✳ ✱ kℓ} = {✶✱ ✳ ✳ ✳ ✱ ℓ + ✶} {i} and {n✶✱ ✳ ✳ ✳ ✱ nℓ} = {✶✱ ✳ ✳ ✳ ✱ ℓ + ✶} {j} (as

rdered sets). Coproduct, counit and antipode are of ‘matrix type’

∆(ui

j) =

k ui

k ⊗ uk j ✱

ε(ui

j) = δi j ✱

S(ui

j) = (uj i)∗ ✳

Similarly coproduct, counit and antipode of A(Gq), ✵ < q < ✶ , are given by the same formulas above, while (1) and (2) becomes: Rij

kl(q)uk mul n = uj lui kRkl mn(q) ✱

p∈Sℓ+✶(−q)||p||u✶

p(✶)u✷ p(✷) ✳ ✳ ✳ uℓ+✶ p(ℓ+✶) = ✶ ✱

(ui

j)∗ = (−q)j−i p∈Sℓ(−q)||p||uk✶ p(n✶)uk✷ p(n✷) ✳ ✳ ✳ ukℓ p(nℓ)

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 4 / 17

SLIDE 7

The QUEA Uq(su(ℓ + ✶))

Symmetries are described by the Hopf ∗-algebra Uq(su(ℓ + ✶)) , generated by {Ki✱ K−✶

i ✱ Ei✱ Fi}i=✶✱✳✳✳✱ℓ, with Ki = K∗ i✱ Fi = E∗ i, and with some relations. . .

with the rescaling Ki = qHi, at the ✵-th order in

h := ❧♦❣ q one gets Serre’s presentation of U(su(ℓ+✶)).

Coproduct, counit and antipode are given by (with i = ✶✱ ✳ ✳ ✳ ✱ ℓ)

∆(Ki) = Ki ⊗ Ki ✱ ∆(Ei) = Ei ⊗ Ki + K−✶

i

⊗ Ei ✱ ε(Ki) = ✶ ✱ ε(Ei) = ✵ ✱ S(Ki) = K−✶

i

✱ S(Ei) = −qEi ✳ The Hopf ∗-subalgebra with generators {Ki✱ Ei✱ Fi}i=✶✱✷✱✳✳✳✱ℓ−✶ is Uq(su(ℓ)); its central extension by K✶K✷

✷ ✳ ✳ ✳ Kℓ ℓ and its inverse gives Uq(u(ℓ)).

There is a non-degenerate dual pairing A(SUq(ℓ + ✶)) × Uq(su(ℓ + ✶)) → C. The algebra A(SUq(ℓ + ✶)) is a Uq(su(ℓ + ✶))-bimodule ∗-algebra for the left ⊲ and right ⊳ canonical actions.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 5 / 17

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CPℓ

q and homogeneous vector bundles

The algebra of ‘functions’ on CPℓ

q is the left Uq(su(ℓ + ✶))-module ∗-algebra

A(CPℓ

q) := A(SUq(ℓ + ✶))Uq(u(ℓ)) ✳

Let σ : Uq(u(ℓ)) → ❊♥❞(Cn) be a ∗-representation. The set M(σ) = A(SUq(ℓ + ✶)) ⊠σ Cn :=

v ∈ A(SUq(ℓ + ✶))n

σ(x(✷))v ⊳ S−✶(x(✶)) = ǫ(x)v ∀ x ∈ Uq(u(ℓ))

✱

is an A(CPℓ

q)-bimodule and a left A(CPℓ q) ⋊ Uq(su(ℓ + ✶))-module.

It is the analogue of (sections of) an homogeneous vector bundle of rank n over CPℓ

q.

A non-degenerate inner product is induced by the canonical one on A(CPℓ

q)n:

v✱ w = n

i=✶ h(v∗ iwi)

M(σ) is projective as one-sided module. Given a differential calculus (Ω•✱ ❞), a canonical connection (the Grassmannian connection) can be obtained by projecting the flat one.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 6 / 17

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The case ℓ = ✷

From now on, let ℓ = ✷ . For short: O := A(SUq(✸)) ✱ A := A(CP✷

q) ✱

U := Uq(su(✸)) ✱ K := Uq(u(✷)) ✳ Recall: O is an U-bimodule ∗-algebra, A = OK, K ⊂ U generated by K✶✱ K✷✱ E✶✱ F✶ . In general for a✱ b ∈ O and x ∈ U, (ab) ⊳ x = (a ⊳ x(✶))(b ⊳ x(✷)) where ∆x = x(✶) ⊗ x(✷). Thus for example ∆(E✷) = E✷ ⊗ K✷ + K−✶

✷

⊗ E✷ ✱ and the map ◦ ⊳ E✷ : O → O is not a derivation. But if a ∈ A, then a ⊳ K✷ = a and (ab) ⊳ E✷ = (a ⊳ E✷)b + a(b ⊳ E✷) ✱ ∀ a✱ b ∈ A ✳ Therefore ◦ ⊳ E✷ is a derivation on A (an ‘exterior’ derivation, since A ⊳ E✷ / ∈ A). This and similar considerations allow to construct a differential calculus on A.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 7 / 17

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The algebra of antiholomorphic forms

Forms Ω(✵✱•) =

k Ω(✵✱k) are constructed as equiv. maps. Let σℓ✱N be the spin ℓ and

charge N irrep of Uq(u(✷)), and Mℓ✱N the corr. A-bimodule and left A ⋊ U-module: Mℓ✱N := O ⊠σℓ✱N C✷ℓ+✶ ✳ The map Mℓ✱N × Mℓ✱N → A, (v✱ v′) → v†v′, gives the Hermitian structure, and v✱ v′ = h(v†v′) the inner product (h = Haar state). By analogy with the q = ✶ case: Ω(✵✱✵) := M✵✱✵ = A ✱ Ω(✵✱✶) := M ✶

✷ ✱ ✸ ✷ ✱

Ω(✵✱✷) := M✵✱✸ ✳ A generic form is ω = (a✱ v✱ b) , with a ∈ Ω(✵✱✵), v = (v+✱ v−) ∈ Ω(✵✱✶) and b ∈ Ω(✵✱✷). We want a product on Ω(✵✱•) giving a graded ass. algebra. Thus ω✶ · ω✷ = ( a✶a✷ ✱ a✶v✷ + v✶a✷ ✱ a✶b✷ + b✶a✷ + v✶ ∧q v✷ ) ✳ where product by ✵-forms is given by the bimodule structure, and it only remains to define ∧q . This is unique modulo a rescaling: v ∧q v′ :=

✷ [✷]q(q

✶ ✷ v+v′

− − q− ✶

✷ v−v′

+) ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 8 / 17

SLIDE 11

The Dolbeault complex

We want to construct a cochain complex Ω(✵✱✵)

✠ ∂

→ Ω(✵✱✶)

✠ ∂

→ Ω(✵✱✷) → ✵ ✳

Def./Prop.

The maps ✠ ∂ : Ω(✵✱✵) → Ω(✵✱✶) ✱ ✠ ∂a := −(a ⊳ F✷F✶✱ a ⊳ F✷)t ✱ ✠ ∂ : Ω(✵✱✶) → Ω(✵✱✷) ✱ ✠ ∂v := v+ ⊳ F✷ + v− ⊳ F✷F✶ ✱ with v = (v+✱ v−) , are well defined and their composition is ✠ ∂✷ = ✵ . Remark: Ω(✵✱•) is a left U-module algebra, and ✠ ∂ is left U-invariant.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 9 / 17

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A covariant differential calculus

Proposition

The map ✠ ∂ is a graded-derivation: ✠ ∂(ω✶ω✷) = (✠ ∂ω✶)ω✷ + (−✶)❞❣(ω✶) ω✶(✠ ∂ω✷) ✳ If call H+ (resp. H−) the Hilbert space completion of Ω(✵✱✵) ⊕ Ω(✵✱✷) (resp. Ω(✵✱✶)), and H := H+ ⊕ H−, then the commutator [✠ ∂✱ f] is bounded on H for all f ∈ A . In fact: [✠ ∂✱ f]ω = (✠ ∂f) · ω ✱ ∀ ω = (a✱ v✱ b) ✳ Interlude: the full differential calculus. The construction can be extended to a full diff. calc.:

◮ (Ω(•✱•)✱ ∂✱ ✠

∂) is a double cochain complex (∂✠ ∂ + ✠ ∂∂ = ✵);

◮ ∂ and ✠

∂ are graded derivations;

◮ (Ω(•✱•)✱ ❞), with ❞ := ∂ + ✠

∂, is a real left U-cov. diff. calc.;

◮ ∃ a closed inv. integral:

∂ω =

✠ ∂ω = ✵ , a Hodge star s.t.

ω∗ · ω′ = ∗ ω✱ ω′ , ❞† = − ∗ ❞ ∗ , and more. . .

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 10 / 17

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The Dolbeault-Dirac operator

Classically the Dolbeault-Dirac operator is defined as ✠ ∂ + ✠ ∂†, with ✠ ∂† the Hermitian conjugate of ✠ ∂. In our case Dω := (✠ ∂†v✱ ✠ ∂a + s ✠ ∂†b✱ s ✠ ∂v) ✱ ∀ ω = (a✱ v✱ b) ∈ Ω(✵✱•) ✳ where s =

[✷]q/✷ .

Proposition

The datum (A(CP✷

q)✱ H✱ D) is a U-equivariant even spectral triple.

Proof.

By the above considerations, we have [D✱ f] ∈ B(H) for all f ∈ A . A self-adjoint extension of D can be easily defined once it is diagonalized. Also, (D + i)−✶ ∈ K will follow from the asymptotic behaviour of ❙♣(D). How to compute ❙♣(D) ?

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 11 / 17

SLIDE 14

The spectrum of the Dirac operator

In ✸ steps: 1st) D✷ ω = [✷]−✶

q ω ⊳ (Cq − ✷) for all ω ∈ Ω(✵✱•);

2nd) by a previous lemma, ω ⊳ Cq = Cq ⊲ ω for all ω ∈ Ω(✵✱•), being Cq ∈ Z(U); 3rd) Ω(✵✱k) are left U-modules, and decompose into irreps as Ω(✵✱✵) ≃

n∈N(n✱ n) ✱

Ω(✵✱✷) ≃

n∈N(n✱ n + ✸) ✱

Ω(✵✱✶) ≃

n✶(n✱ n) ⊕
n✵(n✱ n + ✸) ✳

If (λn✱ µn) are the eigenvalues of D✷ with corresponding multiplicities, being D odd its eigenvalues +λn and −λn have the same multiplicity µn/✷. We get:

Prop. ❦❡r D = C are the constant ✵-forms, while non-zero eigenvalues are (n ✶)

±

✷

[✷]q[n]q[n + ✷]q

with multiplicity (n + ✶)✸ ✱ ±

[n + ✶]q[n + ✷]q

with multiplicity

✶ ✷n(n + ✸)(✷n + ✸) ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 12 / 17

SLIDE 15

Spectrum of the gauged Laplacians

M✵✱N are finitely generated projective right A-modules. Let Ωn =

j+k=n Ω(j✱k), and

∇N : M✵✱N ⊗A Ω• → M✵✱N ⊗A Ω•+✶ be the Grassmannian connection.

Proposition

The Laplacian ∆N = ∇†

N∇N : M✵✱N → M✵✱N is related to the Casimir operator by

∆N = q− ✸

✷ q ✸ ✷ + q− ✸ ✷

q

N ✸ + q− N ✸

Cq − [ ✶

✸N]✷ q − [ ✶ ✸N + ✶]✷ q − [ ✷ ✸N + ✶]✷ q

+ [✷]q[N]q ✳
Corollary. The spectrum of ∆N is given by (n ∈ N)

λn✱N = (✶ + q−✸)[n]q[n + N + ✷]q + [✷]q[N]q ✐❢ N ✵ ✱ λn✱N = (✶ + q−✸)[n + ✷]q[n − N]q + [✷]q[N]q ✐❢ N < ✵ ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 13 / 17

SLIDE 16

Equivariant cyclic homology

For x ∈ U and ai ∈ A, let x ◮ (a✵ ⊗ a✶ ⊗ ✳ ✳ ✳ ⊗ an) := x(✶) ⊲ a✵ ⊗ x(✷) ⊲ a✶ ⊗ ✳ ✳ ✳ ⊗ x(n+✶) ⊲ an ✱ and denote CU

n(A) the elements ω ∈ ❍♦♠C(U✱ An+✶) which are ‘equivariant’, in the

following sense: (x(✶) ◮ ω)(yx(✷)) = ω(xy) ✳ The face operators and the cyclic operator are bn✱i(a✵ ⊗ a✶ ⊗ ✳ ✳ ✳ ⊗ an)(x) := (a✵ ⊗ ✳ ✳ ✳ ⊗ aiai+✶ ⊗ ✳ ✳ ✳ ⊗ an)(x) ✱ if i = n ✱ bn✱n(a✵ ⊗ a✶ ⊗ ✳ ✳ ✳ ⊗ an)(x) :=

(x(✶) ⊲ an)a✵ ⊗ a✶ ⊗ ✳ ✳ ✳ ⊗ an−✶
(x(✷)) ✱

λn(a✵ ⊗ a✶ ⊗ ✳ ✳ ✳ ⊗ an)(x) := (−✶)n (x(✶) ⊲ an) ⊗ a✵ ⊗ a✶ ⊗ ✳ ✳ ✳ ⊗ an−✶

(x(✷)) ✳

We denote HCU

n(A) the homology of

CU

n(A)/■♠(✶ − λn)✱ bn

, with bn := n

i=✵(−✶)ibn✱i.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 14 / 17

SLIDE 17

“Quantum characteristic classes”

A class in KU

✵ (A) is represented by a pair (e✱ σ), where e is an N × N idempotent matrix

with entries in A, σ : U → ▼❛tN(C) is a representation, and

x(✶) ⊲ e
σ(x(✷))t = σ(x)te ✱

∀ x ∈ U ✳ For ▼❛tσ

N(A) :=

T ∈ ▼❛tN(A)
x(✶) ⊲ T
σ(x(✷))t = σ(x)t T ✱ ∀ x ∈ U
replacing the generalized trace map with the map ❚rσ : ▼❛tσ

N(A)n+✶ → CU n(A),

❚rσ(T✵ ⊗ T✶ ⊗ ✳ ✳ ✳ ⊗ Tn)(x) := ❚r

T✵ ✡

⊗ T✶ ✡ ⊗ ✳ ✳ ✳ ✡ ⊗ Tn σ(x)t ✱

ne proves

Theorem

A map ❝❤n : KU

✵ (A) → HCU n(A) is defined by

❝❤n(e✱ σ) := ❚rσ(e⊗n+✶) ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 15 / 17

SLIDE 18

Pairing with KU

✵

Fix a group-like element K ∈ U and call η the corresponding automorphism of A, η(a) := K ⊲ a ∀ a ∈ A ✳ A pairing Hn(A✱ ηA) × HCU

n(A) → C is given by

[τ]✱ [ω] := τ

ω(K)
✳

On CP✷

q, we take K = (K✶K✷)−✹ so that η is the modular automorphism.

⇒ ✵-cocycles are given by (the restriction of) the Haar state h and the counit ǫ; ⇒ a ✹-cocycle coming from the differential calculus is

τ✹(a✵✱ ✳ ✳ ✳ ✱ a✹) :=

a✵❞a✶ ∧q ✳ ✳ ✳ ∧q ❞a✹ ❀

⇒ a similar ✷-cocycle τ✷ is defined via a pull-back from CP✶

q.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 16 / 17

SLIDE 19

Monopole charge, instanton number, . . .

Equivariant projections PN are associated to the “line bundles” M✵✱N, N ∈ Z, and ∇N denotes the corresponding Grassmannian connection. The curvature is ∇✷

N = qN−✶[N]q ∇✷ ✶

and it is antiselfdual ( “monopole” connection). With some algebraic manipulation,

[τ✹]✱ [❝❤✹(PN✱ σN)]
= q−✷N
∇✷

N ∧q ∇✷ N

is the analogue of the “instanton number”. In particular

[τ✹]✱ [❝❤✹(PN✱ σN)]
∝ [N]✷

q .

Similarly

[τ✷]✱ [❝❤✷(PN✱ σN)]
∝ q−N−✶[N]q ✱
[h]✱ [❝❤✵(PN✱ σN)]
= q−✷N ✱
[ǫ]✱ [❝❤✵(PN✱ σN)]
= q✷N ✳

Last result implies that KU

✵ (A) ⊃ Z∞ if q is transcendental.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ MFO, 8 September 2009 17 / 17