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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

September 1, 2009

09GENCO: Noncommutative Geometry and Quantum Physics (Vietri sul Mare)

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 1 / 23

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.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 2 / 23

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.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 2 / 23

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 September 1, 2009 2 / 23

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 September 1, 2009 2 / 23

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 September 1, 2009 2 / 23

Outline

1 Preliminary definitions

The quantum SU(ℓ + ✶) group The QUEA Uq(su(ℓ + ✶)) S✷ℓ+✶

q

and CPℓ

q

2 K-theory and K-homology

K-theory K-homology

3 Antiholomorphic forms and real spectral triples

The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator

4 From GDAs to spectral triples

Reality and the first order condition A family of spectral triples

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 3 / 23

Outline

1 Preliminary definitions

The quantum SU(ℓ + ✶) group The QUEA Uq(su(ℓ + ✶)) S✷ℓ+✶

q

and CPℓ

q

2 K-theory and K-homology

K-theory K-homology

3 Antiholomorphic forms and real spectral triples

The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator

4 From GDAs to spectral triples

Reality and the first order condition A family of spectral triples

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 3 / 23

Outline

1 Preliminary definitions

The quantum SU(ℓ + ✶) group The QUEA Uq(su(ℓ + ✶)) S✷ℓ+✶

q

and CPℓ

q

2 K-theory and K-homology

K-theory K-homology

3 Antiholomorphic forms and real spectral triples

The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator

4 From GDAs to spectral triples

Reality and the first order condition A family of spectral triples

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 3 / 23

Outline

1 Preliminary definitions

The quantum SU(ℓ + ✶) group The QUEA Uq(su(ℓ + ✶)) S✷ℓ+✶

q

and CPℓ

q

2 K-theory and K-homology

K-theory K-homology

3 Antiholomorphic forms and real spectral triples

The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator

4 From GDAs to spectral triples

Reality and the first order condition A family of spectral triples

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 3 / 23

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 September 1, 2009 4 / 23

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 September 1, 2009 4 / 23

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 relations ( aij = Cartan matrix )

[Ki✱ Kj] = ✵ ✱ KiEjK−✶

i

= qaij/✷Ej ✱ [Ei✱ Fj] = δij

K✷

i−K−✷ i

q−q−✶

EiE✷

j − (q + q−✶)EiEjEi + EjE✷ i = ✵

✐❢ |i − j| = ✶ ✱ [Ei✱ Ej] = ✵ ✐❢ |i − j| > ✶ ✳ 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 ✳ With Ki = qHi, at the ✵-th order in

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

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

✷ ✳ ✳ ✳ Kℓ ℓ. We enlarge the algebra with

✂ K := (K✶K✷

✷ ✳ ✳ ✳ Kℓ ℓ)

✷ ℓ+✶ ✱

and its inverse, and call Uq(u(ℓ)) the algebra generated by Uq(su(ℓ)), ✂ K and ✂ K−✶.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 5 / 23

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 relations ( aij = Cartan matrix )

[Ki✱ Kj] = ✵ ✱ KiEjK−✶

i

= qaij/✷Ej ✱ [Ei✱ Fj] = δij

K✷

i−K−✷ i

q−q−✶

EiE✷

j − (q + q−✶)EiEjEi + EjE✷ i = ✵

✐❢ |i − j| = ✶ ✱ [Ei✱ Ej] = ✵ ✐❢ |i − j| > ✶ ✳ 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 ✳ With Ki = qHi, at the ✵-th order in

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

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

✷ ✳ ✳ ✳ Kℓ ℓ. We enlarge the algebra with

✂ K := (K✶K✷

✷ ✳ ✳ ✳ Kℓ ℓ)

✷ ℓ+✶ ✱

and its inverse, and call Uq(u(ℓ)) the algebra generated by Uq(su(ℓ)), ✂ K and ✂ K−✶.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 5 / 23

A(SUq(ℓ + ✶)) as a function algebra

The set of linear maps Uq(su(ℓ + ✶)) → C is a Hopf ∗-algebra with operations dual to those of Uq(su(ℓ + ✶)). For f✱ g two such linear maps we define the product by (f · g)(x) := f(x(✶))g(x(✷)) ∀ x ∈ Uq(su(ℓ + ✶)) ✱ the unity is the map ✶(x) := ε(x), coproduct, counit, antipode and ∗-involution are ∆(f)(x✱ y) := f(xy) ✱ ε(f) := f(✶) ✱ S(f)(x) := f(S(x)) ✱ f∗(x) := f(S(x)∗) ✳ The Hopf ∗-subalgebra generated by the matrix elements of type ✶ irreps is A(SUq(ℓ + ✶)). The algebra A(SUq(ℓ + ✶)) is a Uq(su(ℓ + ✶))-bimodule ∗-algebra for the actions: (x ⊲ f)(y) := f(yx) ❛♥❞ (f ⊳ x)(y) := f(xy) ✳ If Lxf := f ⊳ S−✶(x) , the two left actions ⊲ and L are unitary w.r.t. the inner product ✱ associated to the Haar state h : A(SUq(ℓ + ✶)) → C: f✱ g := h(f∗g) ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 6 / 23

A(SUq(ℓ + ✶)) as a function algebra

The set of linear maps Uq(su(ℓ + ✶)) → C is a Hopf ∗-algebra with operations dual to those of Uq(su(ℓ + ✶)). For f✱ g two such linear maps we define the product by (f · g)(x) := f(x(✶))g(x(✷)) ∀ x ∈ Uq(su(ℓ + ✶)) ✱ the unity is the map ✶(x) := ε(x), coproduct, counit, antipode and ∗-involution are ∆(f)(x✱ y) := f(xy) ✱ ε(f) := f(✶) ✱ S(f)(x) := f(S(x)) ✱ f∗(x) := f(S(x)∗) ✳ The Hopf ∗-subalgebra generated by the matrix elements of type ✶ irreps is A(SUq(ℓ + ✶)). The algebra A(SUq(ℓ + ✶)) is a Uq(su(ℓ + ✶))-bimodule ∗-algebra for the actions: (x ⊲ f)(y) := f(yx) ❛♥❞ (f ⊳ x)(y) := f(xy) ✳ If Lxf := f ⊳ S−✶(x) , the two left actions ⊲ and L are unitary w.r.t. the inner product ✱ associated to the Haar state h : A(SUq(ℓ + ✶)) → C: f✱ g := h(f∗g) ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 6 / 23

A(SUq(ℓ + ✶)) as a function algebra

The set of linear maps Uq(su(ℓ + ✶)) → C is a Hopf ∗-algebra with operations dual to those of Uq(su(ℓ + ✶)). For f✱ g two such linear maps we define the product by (f · g)(x) := f(x(✶))g(x(✷)) ∀ x ∈ Uq(su(ℓ + ✶)) ✱ the unity is the map ✶(x) := ε(x), coproduct, counit, antipode and ∗-involution are ∆(f)(x✱ y) := f(xy) ✱ ε(f) := f(✶) ✱ S(f)(x) := f(S(x)) ✱ f∗(x) := f(S(x)∗) ✳ The Hopf ∗-subalgebra generated by the matrix elements of type ✶ irreps is A(SUq(ℓ + ✶)). The algebra A(SUq(ℓ + ✶)) is a Uq(su(ℓ + ✶))-bimodule ∗-algebra for the actions: (x ⊲ f)(y) := f(yx) ❛♥❞ (f ⊳ x)(y) := f(xy) ✳ If Lxf := f ⊳ S−✶(x) , the two left actions ⊲ and L are unitary w.r.t. the inner product ✱ associated to the Haar state h : A(SUq(ℓ + ✶)) → C: f✱ g := h(f∗g) ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 6 / 23

A(SUq(ℓ + ✶)) as a function algebra

The set of linear maps Uq(su(ℓ + ✶)) → C is a Hopf ∗-algebra with operations dual to those of Uq(su(ℓ + ✶)). For f✱ g two such linear maps we define the product by (f · g)(x) := f(x(✶))g(x(✷)) ∀ x ∈ Uq(su(ℓ + ✶)) ✱ the unity is the map ✶(x) := ε(x), coproduct, counit, antipode and ∗-involution are ∆(f)(x✱ y) := f(xy) ✱ ε(f) := f(✶) ✱ S(f)(x) := f(S(x)) ✱ f∗(x) := f(S(x)∗) ✳ The Hopf ∗-subalgebra generated by the matrix elements of type ✶ irreps is A(SUq(ℓ + ✶)). The algebra A(SUq(ℓ + ✶)) is a Uq(su(ℓ + ✶))-bimodule ∗-algebra for the actions: (x ⊲ f)(y) := f(yx) ❛♥❞ (f ⊳ x)(y) := f(xy) ✳ If Lxf := f ⊳ S−✶(x) , the two left actions ⊲ and L are unitary w.r.t. the inner product ✱ associated to the Haar state h : A(SUq(ℓ + ✶)) → C: f✱ g := h(f∗g) ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 6 / 23

S✷ℓ+✶

q

and CPℓ

q

The algebra of ‘functions’ on S✷ℓ+✶

q

is the left Uq(su(ℓ + ✶))-module ∗-algebra A(S✷ℓ+✶

q

) := A(SUq(ℓ + ✶))Uq(su(ℓ)) ✳ It is generated by zi := uℓ+✶

ℓ+✶−i , i = ✵✱ ✳ ✳ ✳ ✱ ℓ, with relations

zizj = q−✶zjzi ∀ ✵ i < j ℓ ✱ z∗

izj = qzjz∗ i

∀ i = j ✱ [z∗

i✱ zi] = (✶ − q✷)

ℓ

j=i+✶ zjz∗ j

∀ i = ✵✱ ✳ ✳ ✳ ✱ n − ✶ ✱ [z∗

ℓ✱ zℓ] = ✵ ✱

z✵z∗

✵ + z✶z∗ ✶ + ✳ ✳ ✳ + zℓz∗ ℓ = ✶ ✳

The algebra of ‘functions’ on CPℓ

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

A(CPℓ

q) := A(SUq(ℓ + ✶))Uq(u(ℓ)) ≡ A(S✷ℓ+✶ q

)U(✶) ✳ It is generated by the matrix entries Pij := z∗

izj of a projection.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 7 / 23

S✷ℓ+✶

q

and CPℓ

q

The algebra of ‘functions’ on S✷ℓ+✶

q

is the left Uq(su(ℓ + ✶))-module ∗-algebra A(S✷ℓ+✶

q

) := A(SUq(ℓ + ✶))Uq(su(ℓ)) ✳ It is generated by zi := uℓ+✶

ℓ+✶−i , i = ✵✱ ✳ ✳ ✳ ✱ ℓ, with relations

zizj = q−✶zjzi ∀ ✵ i < j ℓ ✱ z∗

izj = qzjz∗ i

∀ i = j ✱ [z∗

i✱ zi] = (✶ − q✷)

ℓ

j=i+✶ zjz∗ j

∀ i = ✵✱ ✳ ✳ ✳ ✱ n − ✶ ✱ [z∗

ℓ✱ zℓ] = ✵ ✱

z✵z∗

✵ + z✶z∗ ✶ + ✳ ✳ ✳ + zℓz∗ ℓ = ✶ ✳

The algebra of ‘functions’ on CPℓ

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

A(CPℓ

q) := A(SUq(ℓ + ✶))Uq(u(ℓ)) ≡ A(S✷ℓ+✶ q

)U(✶) ✳ It is generated by the matrix entries Pij := z∗

izj of a projection.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 7 / 23

S✷ℓ+✶

q

and CPℓ

q

The algebra of ‘functions’ on S✷ℓ+✶

q

is the left Uq(su(ℓ + ✶))-module ∗-algebra A(S✷ℓ+✶

q

) := A(SUq(ℓ + ✶))Uq(su(ℓ)) ✳ It is generated by zi := uℓ+✶

ℓ+✶−i , i = ✵✱ ✳ ✳ ✳ ✱ ℓ, with relations

zizj = q−✶zjzi ∀ ✵ i < j ℓ ✱ z∗

izj = qzjz∗ i

∀ i = j ✱ [z∗

i✱ zi] = (✶ − q✷)

ℓ

j=i+✶ zjz∗ j

∀ i = ✵✱ ✳ ✳ ✳ ✱ n − ✶ ✱ [z∗

ℓ✱ zℓ] = ✵ ✱

z✵z∗

✵ + z✶z∗ ✶ + ✳ ✳ ✳ + zℓz∗ ℓ = ✶ ✳

The algebra of ‘functions’ on CPℓ

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

A(CPℓ

q) := A(SUq(ℓ + ✶))Uq(u(ℓ)) ≡ A(S✷ℓ+✶ q

)U(✶) ✳ It is generated by the matrix entries Pij := z∗

izj of a projection.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 7 / 23

S✷ℓ+✶

q

and CPℓ

q

The algebra of ‘functions’ on S✷ℓ+✶

q

is the left Uq(su(ℓ + ✶))-module ∗-algebra A(S✷ℓ+✶

q

) := A(SUq(ℓ + ✶))Uq(su(ℓ)) ✳ It is generated by zi := uℓ+✶

ℓ+✶−i , i = ✵✱ ✳ ✳ ✳ ✱ ℓ, with relations

zizj = q−✶zjzi ∀ ✵ i < j ℓ ✱ z∗

izj = qzjz∗ i

∀ i = j ✱ [z∗

i✱ zi] = (✶ − q✷)

ℓ

j=i+✶ zjz∗ j

∀ i = ✵✱ ✳ ✳ ✳ ✱ n − ✶ ✱ [z∗

ℓ✱ zℓ] = ✵ ✱

z✵z∗

✵ + z✶z∗ ✶ + ✳ ✳ ✳ + zℓz∗ ℓ = ✶ ✳

The algebra of ‘functions’ on CPℓ

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

A(CPℓ

q) := A(SUq(ℓ + ✶))Uq(u(ℓ)) ≡ A(S✷ℓ+✶ q

)U(✶) ✳ It is generated by the matrix entries Pij := z∗

izj of a projection.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 7 / 23

Outline

1 Preliminary definitions

The quantum SU(ℓ + ✶) group The QUEA Uq(su(ℓ + ✶)) S✷ℓ+✶

q

and CPℓ

q

2 K-theory and K-homology

K-theory K-homology

3 Antiholomorphic forms and real spectral triples

The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator

4 From GDAs to spectral triples

Reality and the first order condition A family of spectral triples

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 8 / 23

K-theory

Some notations: [✵]q✦ := ✶ ✱ [n]q✦ := [n]q · [n − ✶]q✦ ∀ n ✶ ✱ [j✵✱ ✳ ✳ ✳ ✱ jn]q✦ := [j✵ + ✳ ✳ ✳ + jn]q✦ [j✵]q✦ ✳ ✳ ✳ [jn]q✦ ∀ j✵✱ ✳ ✳ ✳ ✱ jn ✵ ✳ For N ✵ let ΨN = (ψN

j✵✱✳✳✳✱jℓ) be the vector-valued ‘function’ on S✷ℓ+✶ q

with components ψN

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q− ✶ ✷

• r<s jrjs(zjℓ

ℓ ✳ ✳ ✳ zj✵ ✵ )∗ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✱ ψ−N

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q ✶ ✷

• r<s jrjs+ℓ

r=✵ rjrzj✵

✵ ✳ ✳ ✳ zjℓ ℓ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✳

Proposition

For all N ∈ Z, Ψ†

NΨN = ✶. Thus, PN := ΨNΨ† N are projections with entries in A(CPℓ q).

We know that K✵(CPℓ

q) ≃ Zℓ+✶ and K✶(CPℓ q) = ✵. We’ll see that {[P✵]✱ [P−✶]✱ ✳ ✳ ✳ ✱ [P−ℓ]}

are generators of K✵. In fact, {PN}N∈Z are representatives of the equivariant K✵ group.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 9 / 23

K-theory

Some notations: [✵]q✦ := ✶ ✱ [n]q✦ := [n]q · [n − ✶]q✦ ∀ n ✶ ✱ [j✵✱ ✳ ✳ ✳ ✱ jn]q✦ := [j✵ + ✳ ✳ ✳ + jn]q✦ [j✵]q✦ ✳ ✳ ✳ [jn]q✦ ∀ j✵✱ ✳ ✳ ✳ ✱ jn ✵ ✳ For N ✵ let ΨN = (ψN

j✵✱✳✳✳✱jℓ) be the vector-valued ‘function’ on S✷ℓ+✶ q

with components ψN

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q− ✶ ✷

• r<s jrjs(zjℓ

ℓ ✳ ✳ ✳ zj✵ ✵ )∗ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✱ ψ−N

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q ✶ ✷

• r<s jrjs+ℓ

r=✵ rjrzj✵

✵ ✳ ✳ ✳ zjℓ ℓ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✳

Proposition

For all N ∈ Z, Ψ†

NΨN = ✶. Thus, PN := ΨNΨ† N are projections with entries in A(CPℓ q).

We know that K✵(CPℓ

q) ≃ Zℓ+✶ and K✶(CPℓ q) = ✵. We’ll see that {[P✵]✱ [P−✶]✱ ✳ ✳ ✳ ✱ [P−ℓ]}

are generators of K✵. In fact, {PN}N∈Z are representatives of the equivariant K✵ group.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 9 / 23

K-theory

Some notations: [✵]q✦ := ✶ ✱ [n]q✦ := [n]q · [n − ✶]q✦ ∀ n ✶ ✱ [j✵✱ ✳ ✳ ✳ ✱ jn]q✦ := [j✵ + ✳ ✳ ✳ + jn]q✦ [j✵]q✦ ✳ ✳ ✳ [jn]q✦ ∀ j✵✱ ✳ ✳ ✳ ✱ jn ✵ ✳ For N ✵ let ΨN = (ψN

j✵✱✳✳✳✱jℓ) be the vector-valued ‘function’ on S✷ℓ+✶ q

with components ψN

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q− ✶ ✷

• r<s jrjs(zjℓ

ℓ ✳ ✳ ✳ zj✵ ✵ )∗ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✱ ψ−N

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q ✶ ✷

• r<s jrjs+ℓ

r=✵ rjrzj✵

✵ ✳ ✳ ✳ zjℓ ℓ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✳

Proposition

For all N ∈ Z, Ψ†

NΨN = ✶. Thus, PN := ΨNΨ† N are projections with entries in A(CPℓ q).

We know that K✵(CPℓ

q) ≃ Zℓ+✶ and K✶(CPℓ q) = ✵. We’ll see that {[P✵]✱ [P−✶]✱ ✳ ✳ ✳ ✱ [P−ℓ]}

are generators of K✵. In fact, {PN}N∈Z are representatives of the equivariant K✵ group.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 9 / 23

K-theory

Some notations: [✵]q✦ := ✶ ✱ [n]q✦ := [n]q · [n − ✶]q✦ ∀ n ✶ ✱ [j✵✱ ✳ ✳ ✳ ✱ jn]q✦ := [j✵ + ✳ ✳ ✳ + jn]q✦ [j✵]q✦ ✳ ✳ ✳ [jn]q✦ ∀ j✵✱ ✳ ✳ ✳ ✱ jn ✵ ✳ For N ✵ let ΨN = (ψN

j✵✱✳✳✳✱jℓ) be the vector-valued ‘function’ on S✷ℓ+✶ q

with components ψN

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q− ✶ ✷

• r<s jrjs(zjℓ

ℓ ✳ ✳ ✳ zj✵ ✵ )∗ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✱ ψ−N

j✵✱✳✳✳✱jℓ := [j✵✱ ✳ ✳ ✳ ✱ jℓ]q✦

✶ ✷ q ✶ ✷

• r<s jrjs+ℓ

r=✵ rjrzj✵

✵ ✳ ✳ ✳ zjℓ ℓ ✱

∀ j✵ + ✳ ✳ ✳ + jℓ = N ✳

Proposition

For all N ∈ Z, Ψ†

NΨN = ✶. Thus, PN := ΨNΨ† N are projections with entries in A(CPℓ q).

We know that K✵(CPℓ

q) ≃ Zℓ+✶ and K✶(CPℓ q) = ✵. We’ll see that {[P✵]✱ [P−✶]✱ ✳ ✳ ✳ ✱ [P−ℓ]}

are generators of K✵. In fact, {PN}N∈Z are representatives of the equivariant K✵ group.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 9 / 23

K-homology I: representations

Here ✶ n ℓ. Let m = (m✶✱ ✳ ✳ ✳ ✱ mn) ∈ Nn and, for ✵ i < k n, let εk

i := ( i t✐♠❡s

• ✵✱ ✵✱ ✳ ✳ ✳ ✱ ✵ ✱

k−i t✐♠❡s

• ✶✱ ✶✱ ✳ ✳ ✳ ✱ ✶ ✱

n−k t✐♠❡s

• ✵✱ ✵✱ ✳ ✳ ✳ ✱ ✵) ✳

Let Hn := ℓ✷(Nn), with orth. basis |m. For any ✵ k n, a ∗-rep. π(n)

k

: A(S✷n+✶

q

) → B(Hn) is defined as follows: π(n)

k (zi) = ✵

for all i > k ✶, while for the remaining generators (with notation m✵ := ✵) π(n)

k (zi) |m = qmi

✶ − q✷(mi+✶−mi+✶) m + εk

i

• ✱

∀ ✵ i k − ✶ ✱ π(n)

k (zk) |m = qmk |m ✱

• n the subspace linear span of basis vectors |m satisfying the restrictions

✵ m✶ m✷ ✳ ✳ ✳ mk ✱ mk+✶ > mk+✷ > ✳ ✳ ✳ > mn ✵ ✱ and they are zero on the orthogonal subspace.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 10 / 23

K-homology I: representations

Here ✶ n ℓ. Let m = (m✶✱ ✳ ✳ ✳ ✱ mn) ∈ Nn and, for ✵ i < k n, let εk

i := ( i t✐♠❡s

• ✵✱ ✵✱ ✳ ✳ ✳ ✱ ✵ ✱

k−i t✐♠❡s

• ✶✱ ✶✱ ✳ ✳ ✳ ✱ ✶ ✱

n−k t✐♠❡s

• ✵✱ ✵✱ ✳ ✳ ✳ ✱ ✵) ✳

Let Hn := ℓ✷(Nn), with orth. basis |m. For any ✵ k n, a ∗-rep. π(n)

k

: A(S✷n+✶

q

) → B(Hn) is defined as follows: π(n)

k (zi) = ✵

for all i > k ✶, while for the remaining generators (with notation m✵ := ✵) π(n)

k (zi) |m = qmi

✶ − q✷(mi+✶−mi+✶) m + εk

i

• ✱

∀ ✵ i k − ✶ ✱ π(n)

k (zk) |m = qmk |m ✱

• n the subspace linear span of basis vectors |m satisfying the restrictions

✵ m✶ m✷ ✳ ✳ ✳ mk ✱ mk+✶ > mk+✷ > ✳ ✳ ✳ > mn ✵ ✱ and they are zero on the orthogonal subspace.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 10 / 23

K-homology II: Fredholm modules

Since K✵(CPℓ

q) ≃ Zℓ+✶ (and K✶(CPℓ q) = ✵) we look for ℓ + ✶ even Fredholm modules.

1st = pull-back of the unique non-trivial even Fred. mod. of C, via the only char. of A(CPℓ

q).

Lemma For |j − k| > ✶, and for all a✱ b ∈ A(S✷n+✶

q

), we have π(n)

j

(a) π(n)

k (b) = ✵.

As a corollary, we have two ∗-reps π(n)

±

: A(S✷n+✶

q

) → B(Hn) π(n)

+ (a) :=

• ✵kn

k ❡✈❡♥

π(n)

k (a) ✱

π(n)

− (a) :=

• ✵kn

k ♦❞❞

π(n)

k (a) ✳

Proposition

π(n)

+ (a) − π(n) − (a) ∈ L✶(Hn) for all a ∈ A(CPℓ q).

For n = ✶✱ ✳ ✳ ✳ ✱ ℓ a ✶-summable Fredholm module is given on Hn ⊕ Hn by the (restriction

• f) the rep. π+ ⊕ π−, grading γn = ✶ ⊕ −✶, and Fn(v ⊕ w) := w ⊕ v.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 11 / 23

K-homology II: Fredholm modules

Since K✵(CPℓ

q) ≃ Zℓ+✶ (and K✶(CPℓ q) = ✵) we look for ℓ + ✶ even Fredholm modules.

1st = pull-back of the unique non-trivial even Fred. mod. of C, via the only char. of A(CPℓ

q).

Lemma For |j − k| > ✶, and for all a✱ b ∈ A(S✷n+✶

q

), we have π(n)

j

(a) π(n)

k (b) = ✵.

As a corollary, we have two ∗-reps π(n)

±

: A(S✷n+✶

q

) → B(Hn) π(n)

+ (a) :=

• ✵kn

k ❡✈❡♥

π(n)

k (a) ✱

π(n)

− (a) :=

• ✵kn

k ♦❞❞

π(n)

k (a) ✳

Proposition

π(n)

+ (a) − π(n) − (a) ∈ L✶(Hn) for all a ∈ A(CPℓ q).

For n = ✶✱ ✳ ✳ ✳ ✱ ℓ a ✶-summable Fredholm module is given on Hn ⊕ Hn by the (restriction

• f) the rep. π+ ⊕ π−, grading γn = ✶ ⊕ −✶, and Fn(v ⊕ w) := w ⊕ v.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 11 / 23

K-homology II: Fredholm modules

Since K✵(CPℓ

q) ≃ Zℓ+✶ (and K✶(CPℓ q) = ✵) we look for ℓ + ✶ even Fredholm modules.

1st = pull-back of the unique non-trivial even Fred. mod. of C, via the only char. of A(CPℓ

q).

Lemma For |j − k| > ✶, and for all a✱ b ∈ A(S✷n+✶

q

), we have π(n)

j

(a) π(n)

k (b) = ✵.

As a corollary, we have two ∗-reps π(n)

±

: A(S✷n+✶

q

) → B(Hn) π(n)

+ (a) :=

• ✵kn

k ❡✈❡♥

π(n)

k (a) ✱

π(n)

− (a) :=

• ✵kn

k ♦❞❞

π(n)

k (a) ✳

Proposition

π(n)

+ (a) − π(n) − (a) ∈ L✶(Hn) for all a ∈ A(CPℓ q).

For n = ✶✱ ✳ ✳ ✳ ✱ ℓ a ✶-summable Fredholm module is given on Hn ⊕ Hn by the (restriction

• f) the rep. π+ ⊕ π−, grading γn = ✶ ⊕ −✶, and Fn(v ⊕ w) := w ⊕ v.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 11 / 23

K-homology II: Fredholm modules

Since K✵(CPℓ

q) ≃ Zℓ+✶ (and K✶(CPℓ q) = ✵) we look for ℓ + ✶ even Fredholm modules.

1st = pull-back of the unique non-trivial even Fred. mod. of C, via the only char. of A(CPℓ

q).

Lemma For |j − k| > ✶, and for all a✱ b ∈ A(S✷n+✶

q

), we have π(n)

j

(a) π(n)

k (b) = ✵.

As a corollary, we have two ∗-reps π(n)

±

: A(S✷n+✶

q

) → B(Hn) π(n)

+ (a) :=

• ✵kn

k ❡✈❡♥

π(n)

k (a) ✱

π(n)

− (a) :=

• ✵kn

k ♦❞❞

π(n)

k (a) ✳

Proposition

π(n)

+ (a) − π(n) − (a) ∈ L✶(Hn) for all a ∈ A(CPℓ q).

For n = ✶✱ ✳ ✳ ✳ ✱ ℓ a ✶-summable Fredholm module is given on Hn ⊕ Hn by the (restriction

• f) the rep. π+ ⊕ π−, grading γn = ✶ ⊕ −✶, and Fn(v ⊕ w) := w ⊕ v.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 11 / 23

K-homology II: Fredholm modules

Since K✵(CPℓ

q) ≃ Zℓ+✶ (and K✶(CPℓ q) = ✵) we look for ℓ + ✶ even Fredholm modules.

1st = pull-back of the unique non-trivial even Fred. mod. of C, via the only char. of A(CPℓ

q).

Lemma For |j − k| > ✶, and for all a✱ b ∈ A(S✷n+✶

q

), we have π(n)

j

(a) π(n)

k (b) = ✵.

As a corollary, we have two ∗-reps π(n)

±

: A(S✷n+✶

q

) → B(Hn) π(n)

+ (a) :=

• ✵kn

k ❡✈❡♥

π(n)

k (a) ✱

π(n)

− (a) :=

• ✵kn

k ♦❞❞

π(n)

k (a) ✳

Proposition

π(n)

+ (a) − π(n) − (a) ∈ L✶(Hn) for all a ∈ A(CPℓ q).

For n = ✶✱ ✳ ✳ ✳ ✱ ℓ a ✶-summable Fredholm module is given on Hn ⊕ Hn by the (restriction

• f) the rep. π+ ⊕ π−, grading γn = ✶ ⊕ −✶, and Fn(v ⊕ w) := w ⊕ v.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 11 / 23

K-homology II: Fredholm modules

Since K✵(CPℓ

q) ≃ Zℓ+✶ (and K✶(CPℓ q) = ✵) we look for ℓ + ✶ even Fredholm modules.

1st = pull-back of the unique non-trivial even Fred. mod. of C, via the only char. of A(CPℓ

q).

Lemma For |j − k| > ✶, and for all a✱ b ∈ A(S✷n+✶

q

), we have π(n)

j

(a) π(n)

k (b) = ✵.

As a corollary, we have two ∗-reps π(n)

±

: A(S✷n+✶

q

) → B(Hn) π(n)

+ (a) :=

• ✵kn

k ❡✈❡♥

π(n)

k (a) ✱

π(n)

− (a) :=

• ✵kn

k ♦❞❞

π(n)

k (a) ✳

Proposition

π(n)

+ (a) − π(n) − (a) ∈ L✶(Hn) for all a ∈ A(CPℓ q).

For n = ✶✱ ✳ ✳ ✳ ✱ ℓ a ✶-summable Fredholm module is given on Hn ⊕ Hn by the (restriction

• f) the rep. π+ ⊕ π−, grading γn = ✶ ⊕ −✶, and Fn(v ⊕ w) := w ⊕ v.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 11 / 23

Pairing between K-theory and K-homology

Proposition

If µk, ✵ n ℓ, denotes the n-th Fredholm module previously introduced, we have [µn]✱ [P−N] := ❚rHn(π(n)

+

− π(n)

− )(❚r P−N) =

N

n

• ✱

for all N ∈ N, where N

n

• := ✵ when n > N.

Proof.

In ✸ steps: 1st) the pairing [µn]✱ [P−N] as a function of q is continuous (it is given by a series that is absolutely convergent for ✵ q < ✶); 2nd) the pairing is integer being the index of a Fredholm operator, and a continuous function [✵✱ ✶) → Z is constant; 3rd) compute it in the limit q → ✵+.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 12 / 23

Pairing between K-theory and K-homology

Proposition

If µk, ✵ n ℓ, denotes the n-th Fredholm module previously introduced, we have [µn]✱ [P−N] := ❚rHn(π(n)

+

− π(n)

− )(❚r P−N) =

N

n

• ✱

for all N ∈ N, where N

n

• := ✵ when n > N.

Proof.

In ✸ steps: 1st) the pairing [µn]✱ [P−N] as a function of q is continuous (it is given by a series that is absolutely convergent for ✵ q < ✶); 2nd) the pairing is integer being the index of a Fredholm operator, and a continuous function [✵✱ ✶) → Z is constant; 3rd) compute it in the limit q → ✵+.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 12 / 23

Pairing between K-theory and K-homology

Proposition

The elements [µ✵]✱ ✳ ✳ ✳ ✱ [µℓ] are generators of K✵(A(CPℓ

q)), and the elements

[P✵]✱ ✳ ✳ ✳ ✱ [P−ℓ] are generators of K✵(A(CPℓ

q)).

Proof.

The matrix M with entries Mij := [µi]✱ [P−j] is in GLℓ+✶(Z), the inverse being (M−✶)ij = (−✶)i+jj

i

• ✳

Thus, the elements above are a basis of Zn+✶ as a Z-module, i.e. they generate Zn+✶ as abelian group. Remark: the Fredholm module µℓ can be realized as ‘conformal class’ of a regular spectral triple (A(CPℓ

q)✱ Hn ⊕ Hn✱ D) — i.e. Fn := D|D|−✶ — of any summability d ∈ R+.

Next: a ‘geometric’ Dirac operator (✵+-summable and real) is discussed in the following.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 12 / 23

Pairing between K-theory and K-homology

Proposition

The elements [µ✵]✱ ✳ ✳ ✳ ✱ [µℓ] are generators of K✵(A(CPℓ

q)), and the elements

[P✵]✱ ✳ ✳ ✳ ✱ [P−ℓ] are generators of K✵(A(CPℓ

q)).

Proof.

The matrix M with entries Mij := [µi]✱ [P−j] is in GLℓ+✶(Z), the inverse being (M−✶)ij = (−✶)i+jj

i

• ✳

Thus, the elements above are a basis of Zn+✶ as a Z-module, i.e. they generate Zn+✶ as abelian group. Remark: the Fredholm module µℓ can be realized as ‘conformal class’ of a regular spectral triple (A(CPℓ

q)✱ Hn ⊕ Hn✱ D) — i.e. Fn := D|D|−✶ — of any summability d ∈ R+.

Next: a ‘geometric’ Dirac operator (✵+-summable and real) is discussed in the following.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 12 / 23

Pairing between K-theory and K-homology

Proposition

The elements [µ✵]✱ ✳ ✳ ✳ ✱ [µℓ] are generators of K✵(A(CPℓ

q)), and the elements

[P✵]✱ ✳ ✳ ✳ ✱ [P−ℓ] are generators of K✵(A(CPℓ

q)).

Proof.

The matrix M with entries Mij := [µi]✱ [P−j] is in GLℓ+✶(Z), the inverse being (M−✶)ij = (−✶)i+jj

i

• ✳

Thus, the elements above are a basis of Zn+✶ as a Z-module, i.e. they generate Zn+✶ as abelian group. Remark: the Fredholm module µℓ can be realized as ‘conformal class’ of a regular spectral triple (A(CPℓ

q)✱ Hn ⊕ Hn✱ D) — i.e. Fn := D|D|−✶ — of any summability d ∈ R+.

Next: a ‘geometric’ Dirac operator (✵+-summable and real) is discussed in the following.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 12 / 23

Outline

1 Preliminary definitions

The quantum SU(ℓ + ✶) group The QUEA Uq(su(ℓ + ✶)) S✷ℓ+✶

q

and CPℓ

q

2 K-theory and K-homology

K-theory K-homology

3 Antiholomorphic forms and real spectral triples

The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator

4 From GDAs to spectral triples

Reality and the first order condition A family of spectral triples

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 13 / 23

Homogeneous vector bundles over CPℓ

q

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

• v ∈ A(SUq(ℓ + ✶))n

{Lx(✶) ⊗ σ(x(✷))}v = 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)

Which σ gives the bimodule of antiholomorphic forms? Uq(u(ℓ)) is a central extension of Uq(su(ℓ)) by ✂ K and ✂ K−✶. Any irrep of Uq(u(ℓ)) is

• btained from an irrep σ of Uq(su(ℓ)) by defining σ(✂

K) as a (non-zero) multiple of the identity transformation.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 14 / 23

Homogeneous vector bundles over CPℓ

q

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

• v ∈ A(SUq(ℓ + ✶))n

{Lx(✶) ⊗ σ(x(✷))}v = 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)

Which σ gives the bimodule of antiholomorphic forms? Uq(u(ℓ)) is a central extension of Uq(su(ℓ)) by ✂ K and ✂ K−✶. Any irrep of Uq(u(ℓ)) is

• btained from an irrep σ of Uq(su(ℓ)) by defining σ(✂

K) as a (non-zero) multiple of the identity transformation.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 14 / 23

Homogeneous vector bundles over CPℓ

q

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

• v ∈ A(SUq(ℓ + ✶))n

{Lx(✶) ⊗ σ(x(✷))}v = 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)

Which σ gives the bimodule of antiholomorphic forms? Uq(u(ℓ)) is a central extension of Uq(su(ℓ)) by ✂ K and ✂ K−✶. Any irrep of Uq(u(ℓ)) is

• btained from an irrep σ of Uq(su(ℓ)) by defining σ(✂

K) as a (non-zero) multiple of the identity transformation.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 14 / 23

Homogeneous vector bundles over CPℓ

q

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

• v ∈ A(SUq(ℓ + ✶))n

{Lx(✶) ⊗ σ(x(✷))}v = 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)

Which σ gives the bimodule of antiholomorphic forms? Uq(u(ℓ)) is a central extension of Uq(su(ℓ)) by ✂ K and ✂ K−✶. Any irrep of Uq(u(ℓ)) is

• btained from an irrep σ of Uq(su(ℓ)) by defining σ(✂

K) as a (non-zero) multiple of the identity transformation.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 14 / 23

Fundamental representations of Uq(su(ℓ))

Let k = ✶✱ ✳ ✳ ✳ ✱ ℓ − ✶, Λk :=

• i = (i✶✱ i✷✱ ✳ ✳ ✳ ✱ ik) ∈ Zk

✶ i✶ < i✷ < ✳ ✳ ✳ < ik ℓ

• ✱

and Wk ≃ C(ℓ

k) be the vector space with basis vectors labelled by elements in Λk,

represented pictorially by Young tableaux (YT). Generators of Uq(su(ℓ)) are represented by insertion/deletion operators:

◮ Kj(YT) = q

✶ ✷ NjYT where Nj = number of rows of length j in the Young

tableau minus the number of rows of length j + ✶ (either ✵ or ±✶);

◮ Fj add a slot to the row of length j, if any, otherwise it gives ✵; Ej is the adjoint of Fj.

Example: ℓ = ✸ and k = ✷. F✶

• = ✵

F✶

• =
• F✶
• = ✵

F✷

• =
• F✷
• = ✵

F✷

• = ✵

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 15 / 23

Fundamental representations of Uq(su(ℓ))

Let k = ✶✱ ✳ ✳ ✳ ✱ ℓ − ✶, Λk :=

• i = (i✶✱ i✷✱ ✳ ✳ ✳ ✱ ik) ∈ Zk

✶ i✶ < i✷ < ✳ ✳ ✳ < ik ℓ

• ✱

and Wk ≃ C(ℓ

k) be the vector space with basis vectors labelled by elements in Λk,

represented pictorially by Young tableaux (YT). Generators of Uq(su(ℓ)) are represented by insertion/deletion operators:

◮ Kj(YT) = q

✶ ✷ NjYT where Nj = number of rows of length j in the Young

tableau minus the number of rows of length j + ✶ (either ✵ or ±✶);

◮ Fj add a slot to the row of length j, if any, otherwise it gives ✵; Ej is the adjoint of Fj.

Example: ℓ = ✸ and k = ✷. F✶

• = ✵

F✶

• =
• F✶
• = ✵

F✷

• =
• F✷
• = ✵

F✷

• = ✵

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 15 / 23

Fundamental representations of Uq(su(ℓ))

Let k = ✶✱ ✳ ✳ ✳ ✱ ℓ − ✶, Λk :=

• i = (i✶✱ i✷✱ ✳ ✳ ✳ ✱ ik) ∈ Zk

✶ i✶ < i✷ < ✳ ✳ ✳ < ik ℓ

• ✱

and Wk ≃ C(ℓ

k) be the vector space with basis vectors labelled by elements in Λk,

represented pictorially by Young tableaux (YT). Generators of Uq(su(ℓ)) are represented by insertion/deletion operators:

◮ Kj(YT) = q

✶ ✷ NjYT where Nj = number of rows of length j in the Young

tableau minus the number of rows of length j + ✶ (either ✵ or ±✶);

◮ Fj add a slot to the row of length j, if any, otherwise it gives ✵; Ej is the adjoint of Fj.

Example: ℓ = ✸ and k = ✷. F✶

• = ✵

F✶

• =
• F✶
• = ✵

F✷

• =
• F✷
• = ✵

F✷

• = ✵

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 15 / 23

Fundamental representations of Uq(su(ℓ))

Let k = ✶✱ ✳ ✳ ✳ ✱ ℓ − ✶, Λk :=

• i = (i✶✱ i✷✱ ✳ ✳ ✳ ✱ ik) ∈ Zk

✶ i✶ < i✷ < ✳ ✳ ✳ < ik ℓ

• ✱

and Wk ≃ C(ℓ

k) be the vector space with basis vectors labelled by elements in Λk,

represented pictorially by Young tableaux (YT). Generators of Uq(su(ℓ)) are represented by insertion/deletion operators:

◮ Kj(YT) = q

✶ ✷ NjYT where Nj = number of rows of length j in the Young

tableau minus the number of rows of length j + ✶ (either ✵ or ±✶);

◮ Fj add a slot to the row of length j, if any, otherwise it gives ✵; Ej is the adjoint of Fj.

Facts:

◮ Wk is the irrep with highest weight δk = (

k−✶ t✐♠❡s

✵✱ ✳ ✳ ✳ ✱ ✵✱ ✶✱

ℓ−k−✶ t✐♠❡s

✵✱ ✳ ✳ ✳ ✱ ✵ ) ;

◮ for q = ✶, Wk ≃ ∧kW✶ ∀ ✵ k ℓ where W✵ = Wℓ = C is the rep. given by ε.

Next point: to define an intertwiner ∧q : Wj ⊗ Wk → Wj+k deformation of the wedge product, for all j + k ℓ.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 15 / 23

The quantum Grassmann algebra, I

Elements of Sk✱n−k := (Sk × Sn−k)\Sn are called (k✱ n − k)-shuffles. The map p → p−✶ sends Sk✱n−k into S(k)

n

:= Sn/(Sk × Sn−k), explicitly given by S(k)

n

= {p ∈ Sn | p(✶) < p(✷) < ✳ ✳ ✳ < p(k) and p(k + ✶) < p(k + ✷) < ✳ ✳ ✳ < p(n)} ✳ Any p ∈ Sn can be uniquely factorized as p = p′p′′ with p′ ∈ S(k)

n , p′′ ∈ Sk × Sn−k, and

||p|| = ||p′|| + ||p′′||.

Def./Prop.

A map ∧q : Wh ⊗ Wk → Wh+k is given by the formula (v ∧q w)i =

• p∈S(h)

h+k

(−q−✶)||p|| vip(✶)✱✳✳✳✱ip(h) wip(h+✶)✱✳✳✳✱ip(h+k) for all v = (vi′) ∈ Wh, w = (wi′′) ∈ Wk, h + k ℓ. We set v ∧q w := ✵ if h + k > ℓ. The product ∧q is surjective, associative, and a left Uq(su(ℓ))-module map. Crucial to prove associativity is the factorization property above.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 16 / 23

The quantum Grassmann algebra, I

Elements of Sk✱n−k := (Sk × Sn−k)\Sn are called (k✱ n − k)-shuffles. The map p → p−✶ sends Sk✱n−k into S(k)

n

:= Sn/(Sk × Sn−k), explicitly given by S(k)

n

= {p ∈ Sn | p(✶) < p(✷) < ✳ ✳ ✳ < p(k) and p(k + ✶) < p(k + ✷) < ✳ ✳ ✳ < p(n)} ✳ Any p ∈ Sn can be uniquely factorized as p = p′p′′ with p′ ∈ S(k)

n , p′′ ∈ Sk × Sn−k, and

||p|| = ||p′|| + ||p′′||.

Def./Prop.

A map ∧q : Wh ⊗ Wk → Wh+k is given by the formula (v ∧q w)i =

• p∈S(h)

h+k

(−q−✶)||p|| vip(✶)✱✳✳✳✱ip(h) wip(h+✶)✱✳✳✳✱ip(h+k) for all v = (vi′) ∈ Wh, w = (wi′′) ∈ Wk, h + k ℓ. We set v ∧q w := ✵ if h + k > ℓ. The product ∧q is surjective, associative, and a left Uq(su(ℓ))-module map. Crucial to prove associativity is the factorization property above.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 16 / 23

The quantum Grassmann algebra, I

Elements of Sk✱n−k := (Sk × Sn−k)\Sn are called (k✱ n − k)-shuffles. The map p → p−✶ sends Sk✱n−k into S(k)

n

:= Sn/(Sk × Sn−k), explicitly given by S(k)

n

= {p ∈ Sn | p(✶) < p(✷) < ✳ ✳ ✳ < p(k) and p(k + ✶) < p(k + ✷) < ✳ ✳ ✳ < p(n)} ✳ Any p ∈ Sn can be uniquely factorized as p = p′p′′ with p′ ∈ S(k)

n , p′′ ∈ Sk × Sn−k, and

||p|| = ||p′|| + ||p′′||.

Def./Prop.

A map ∧q : Wh ⊗ Wk → Wh+k is given by the formula (v ∧q w)i =

• p∈S(h)

h+k

(−q−✶)||p|| vip(✶)✱✳✳✳✱ip(h) wip(h+✶)✱✳✳✳✱ip(h+k) for all v = (vi′) ∈ Wh, w = (wi′′) ∈ Wk, h + k ℓ. We set v ∧q w := ✵ if h + k > ℓ. The product ∧q is surjective, associative, and a left Uq(su(ℓ))-module map. Crucial to prove associativity is the factorization property above.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 16 / 23

The quantum Grassmann algebra, I

Elements of Sk✱n−k := (Sk × Sn−k)\Sn are called (k✱ n − k)-shuffles. The map p → p−✶ sends Sk✱n−k into S(k)

n

:= Sn/(Sk × Sn−k), explicitly given by S(k)

n

= {p ∈ Sn | p(✶) < p(✷) < ✳ ✳ ✳ < p(k) and p(k + ✶) < p(k + ✷) < ✳ ✳ ✳ < p(n)} ✳ Any p ∈ Sn can be uniquely factorized as p = p′p′′ with p′ ∈ S(k)

n , p′′ ∈ Sk × Sn−k, and

||p|| = ||p′|| + ||p′′||.

Def./Prop.

A map ∧q : Wh ⊗ Wk → Wh+k is given by the formula (v ∧q w)i =

• p∈S(h)

h+k

(−q−✶)||p|| vip(✶)✱✳✳✳✱ip(h) wip(h+✶)✱✳✳✳✱ip(h+k) for all v = (vi′) ∈ Wh, w = (wi′′) ∈ Wk, h + k ℓ. We set v ∧q w := ✵ if h + k > ℓ. The product ∧q is surjective, associative, and a left Uq(su(ℓ))-module map. Crucial to prove associativity is the factorization property above.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 16 / 23

The quantum Grassmann algebra, I

Elements of Sk✱n−k := (Sk × Sn−k)\Sn are called (k✱ n − k)-shuffles. The map p → p−✶ sends Sk✱n−k into S(k)

n

:= Sn/(Sk × Sn−k), explicitly given by S(k)

n

= {p ∈ Sn | p(✶) < p(✷) < ✳ ✳ ✳ < p(k) and p(k + ✶) < p(k + ✷) < ✳ ✳ ✳ < p(n)} ✳ Any p ∈ Sn can be uniquely factorized as p = p′p′′ with p′ ∈ S(k)

n , p′′ ∈ Sk × Sn−k, and

||p|| = ||p′|| + ||p′′||.

Def./Prop.

A map ∧q : Wh ⊗ Wk → Wh+k is given by the formula (v ∧q w)i =

• p∈S(h)

h+k

(−q−✶)||p|| vip(✶)✱✳✳✳✱ip(h) wip(h+✶)✱✳✳✳✱ip(h+k) for all v = (vi′) ∈ Wh, w = (wi′′) ∈ Wk, h + k ℓ. We set v ∧q w := ✵ if h + k > ℓ. The product ∧q is surjective, associative, and a left Uq(su(ℓ))-module map. Crucial to prove associativity is the factorization property above.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 16 / 23

The quantum Grassmann algebra, I

Elements of Sk✱n−k := (Sk × Sn−k)\Sn are called (k✱ n − k)-shuffles. The map p → p−✶ sends Sk✱n−k into S(k)

n

:= Sn/(Sk × Sn−k), explicitly given by S(k)

n

= {p ∈ Sn | p(✶) < p(✷) < ✳ ✳ ✳ < p(k) and p(k + ✶) < p(k + ✷) < ✳ ✳ ✳ < p(n)} ✳ Any p ∈ Sn can be uniquely factorized as p = p′p′′ with p′ ∈ S(k)

n , p′′ ∈ Sk × Sn−k, and

||p|| = ||p′|| + ||p′′||.

Def./Prop.

A map ∧q : Wh ⊗ Wk → Wh+k is given by the formula (v ∧q w)i =

• p∈S(h)

h+k

(−q−✶)||p|| vip(✶)✱✳✳✳✱ip(h) wip(h+✶)✱✳✳✳✱ip(h+k) for all v = (vi′) ∈ Wh, w = (wi′′) ∈ Wk, h + k ℓ. We set v ∧q w := ✵ if h + k > ℓ.

Let W• := ⊕ℓ

k=✵Wk. ●rℓ q := (W•✱ ∧q) is a graded associative algebra – generated

by W✶ – and a left Uq(su(ℓ))-module algebra. ❞✐♠C ●rℓ

q = ✷ℓ.

Crucial to prove associativity is the factorization property above.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 16 / 23

The quantum Grassmann algebra, II

For x ∈ W✶, we define left/right ‘exterior product’ eL✱R

x

: Wk → Wk+✶ as eL

x w = x ∧q w ✱

eR

x w = (−q)k w ∧q x ✱

and left/right ‘contraction’ as the adjoint iL✱R

x

• f eL✱R

x

w.r.t. the canonical inner product on W• . An antilinear map J : Wk → Wℓ−k is given by (Jw)i = (−q−✶)|i|q

✶ ✹ ℓ(ℓ+✶) w ic ✱

where |i| := i✶ + ✳ ✳ ✳ + iℓ−k, ic = (✶✱ ✳ ✳ ✳ ✱ ℓ) i, and ✠ z is the complex conjugate of z ∈ C.

Proposition

The map J is equivariant (i.e. x∗J = J S(x) for all x ∈ Uq(su(ℓ))), and has square J✷ = (−✶)⌊ ℓ+✶

✷

⌋ ✳

Conjugating with J transforms the left exterior product into the right contraction: JeL

xJ−✶ = −qiR x ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 17 / 23

The quantum Grassmann algebra, II

For x ∈ W✶, we define left/right ‘exterior product’ eL✱R

x

: Wk → Wk+✶ as eL

x w = x ∧q w ✱

eR

x w = (−q)k w ∧q x ✱

and left/right ‘contraction’ as the adjoint iL✱R

x

• f eL✱R

x

w.r.t. the canonical inner product on W• . An antilinear map J : Wk → Wℓ−k is given by (Jw)i = (−q−✶)|i|q

✶ ✹ ℓ(ℓ+✶) w ic ✱

where |i| := i✶ + ✳ ✳ ✳ + iℓ−k, ic = (✶✱ ✳ ✳ ✳ ✱ ℓ) i, and ✠ z is the complex conjugate of z ∈ C.

Proposition

The map J is equivariant (i.e. x∗J = J S(x) for all x ∈ Uq(su(ℓ))), and has square J✷ = (−✶)⌊ ℓ+✶

✷

⌋ ✳

Conjugating with J transforms the left exterior product into the right contraction: JeL

xJ−✶ = −qiR x ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 17 / 23

The quantum Grassmann algebra, II

For x ∈ W✶, we define left/right ‘exterior product’ eL✱R

x

: Wk → Wk+✶ as eL

x w = x ∧q w ✱

eR

x w = (−q)k w ∧q x ✱

and left/right ‘contraction’ as the adjoint iL✱R

x

• f eL✱R

x

w.r.t. the canonical inner product on W• . An antilinear map J : Wk → Wℓ−k is given by (Jw)i = (−q−✶)|i|q

✶ ✹ ℓ(ℓ+✶) w ic ✱

where |i| := i✶ + ✳ ✳ ✳ + iℓ−k, ic = (✶✱ ✳ ✳ ✳ ✱ ℓ) i, and ✠ z is the complex conjugate of z ∈ C.

Proposition

The map J is equivariant (i.e. x∗J = J S(x) for all x ∈ Uq(su(ℓ))), and has square J✷ = (−✶)⌊ ℓ+✶

✷

⌋ ✳

Conjugating with J transforms the left exterior product into the right contraction: JeL

xJ−✶ = −qiR x ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 17 / 23

The quantum Grassmann algebra, II

For x ∈ W✶, we define left/right ‘exterior product’ eL✱R

x

: Wk → Wk+✶ as eL

x w = x ∧q w ✱

eR

x w = (−q)k w ∧q x ✱

and left/right ‘contraction’ as the adjoint iL✱R

x

• f eL✱R

x

w.r.t. the canonical inner product on W• . An antilinear map J : Wk → Wℓ−k is given by (Jw)i = (−q−✶)|i|q

✶ ✹ ℓ(ℓ+✶) w ic ✱

where |i| := i✶ + ✳ ✳ ✳ + iℓ−k, ic = (✶✱ ✳ ✳ ✳ ✱ ℓ) i, and ✠ z is the complex conjugate of z ∈ C.

Proposition

The map J is equivariant (i.e. x∗J = J S(x) for all x ∈ Uq(su(ℓ))), and has square J✷ = (−✶)⌊ ℓ+✶

✷

⌋ ✳

Conjugating with J transforms the left exterior product into the right contraction: JeL

xJ−✶ = −qiR x ✳

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 17 / 23

Antiholomorphic forms on CPℓ

q

For N ∈ Z, the irrep. of Uq(su(ℓ)) with h.w. δk is lifted to a σN

k : Uq(u(ℓ)) → ❊♥❞(Wk) by

σN

k (✂

K) = qk−

ℓ ℓ+✶ N · idWk ✳

Since σN

k ≃ σ✵ k ⊗ σN ✵ ,

Ωk

N := M(σN k ) ≃ Ωk ✵ ⊗A(CPℓ

q) Ω✵

N ✳

Ωk

✵ = antiholomorphic k-forms. Ω✵ N = ‘sections of line bundles’.

If ℓ is odd: Ω✵

✶ ✷ (ℓ+✶) = square root of the canonical bundle, Ωk ✶ ✷ (ℓ+✶) = chiral spinors.

An associative product ∧q : Ωk

N × Ωk′ N′ → Ωk+k′ N+N′ is given by

ω ∧q ω′ := a · a′(v ∧q v′) ✱ where ω = av ∈ Ωk

N and ω′ = a′v′ ∈ Ωk′ N′, a✱ a′ ∈ A(SUq(ℓ + ✶)), v ∈ Wk, v′ ∈ Wk′.

Ω•

✵ := ℓ k=✵ Ωk ✵ is a graded associative algebra.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 18 / 23

Antiholomorphic forms on CPℓ

q

For N ∈ Z, the irrep. of Uq(su(ℓ)) with h.w. δk is lifted to a σN

k : Uq(u(ℓ)) → ❊♥❞(Wk) by

σN

k (✂

K) = qk−

ℓ ℓ+✶ N · idWk ✳

Since σN

k ≃ σ✵ k ⊗ σN ✵ ,

Ωk

N := M(σN k ) ≃ Ωk ✵ ⊗A(CPℓ

q) Ω✵

N ✳

Ωk

✵ = antiholomorphic k-forms. Ω✵ N = ‘sections of line bundles’.

If ℓ is odd: Ω✵

✶ ✷ (ℓ+✶) = square root of the canonical bundle, Ωk ✶ ✷ (ℓ+✶) = chiral spinors.

An associative product ∧q : Ωk

N × Ωk′ N′ → Ωk+k′ N+N′ is given by

ω ∧q ω′ := a · a′(v ∧q v′) ✱ where ω = av ∈ Ωk

N and ω′ = a′v′ ∈ Ωk′ N′, a✱ a′ ∈ A(SUq(ℓ + ✶)), v ∈ Wk, v′ ∈ Wk′.

Ω•

✵ := ℓ k=✵ Ωk ✵ is a graded associative algebra.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 18 / 23

Antiholomorphic forms on CPℓ

q

For N ∈ Z, the irrep. of Uq(su(ℓ)) with h.w. δk is lifted to a σN

k : Uq(u(ℓ)) → ❊♥❞(Wk) by

σN

k (✂

K) = qk−

ℓ ℓ+✶ N · idWk ✳

Since σN

k ≃ σ✵ k ⊗ σN ✵ ,

Ωk

N := M(σN k ) ≃ Ωk ✵ ⊗A(CPℓ

q) Ω✵

N ✳

Ωk

✵ = antiholomorphic k-forms. Ω✵ N = ‘sections of line bundles’.

If ℓ is odd: Ω✵

✶ ✷ (ℓ+✶) = square root of the canonical bundle, Ωk ✶ ✷ (ℓ+✶) = chiral spinors.

An associative product ∧q : Ωk

N × Ωk′ N′ → Ωk+k′ N+N′ is given by

ω ∧q ω′ := a · a′(v ∧q v′) ✱ where ω = av ∈ Ωk

N and ω′ = a′v′ ∈ Ωk′ N′, a✱ a′ ∈ A(SUq(ℓ + ✶)), v ∈ Wk, v′ ∈ Wk′.

Ω•

✵ := ℓ k=✵ Ωk ✵ is a graded associative algebra.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 18 / 23

Antiholomorphic forms on CPℓ

q

For N ∈ Z, the irrep. of Uq(su(ℓ)) with h.w. δk is lifted to a σN

k : Uq(u(ℓ)) → ❊♥❞(Wk) by

σN

k (✂

K) = qk−

ℓ ℓ+✶ N · idWk ✳

Since σN

k ≃ σ✵ k ⊗ σN ✵ ,

Ωk

N := M(σN k ) ≃ Ωk ✵ ⊗A(CPℓ

q) Ω✵

N ✳

Ωk

✵ = antiholomorphic k-forms. Ω✵ N = ‘sections of line bundles’.

If ℓ is odd: Ω✵

✶ ✷ (ℓ+✶) = square root of the canonical bundle, Ωk ✶ ✷ (ℓ+✶) = chiral spinors.

An associative product ∧q : Ωk

N × Ωk′ N′ → Ωk+k′ N+N′ is given by

ω ∧q ω′ := a · a′(v ∧q v′) ✱ where ω = av ∈ Ωk

N and ω′ = a′v′ ∈ Ωk′ N′, a✱ a′ ∈ A(SUq(ℓ + ✶)), v ∈ Wk, v′ ∈ Wk′.

Ω•

✵ := ℓ k=✵ Ωk ✵ is a graded associative algebra.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 18 / 23

Antiholomorphic forms on CPℓ

q

For N ∈ Z, the irrep. of Uq(su(ℓ)) with h.w. δk is lifted to a σN

k : Uq(u(ℓ)) → ❊♥❞(Wk) by

σN

k (✂

K) = qk−

ℓ ℓ+✶ N · idWk ✳

Since σN

k ≃ σ✵ k ⊗ σN ✵ ,

Ωk

N := M(σN k ) ≃ Ωk ✵ ⊗A(CPℓ

q) Ω✵

N ✳

Ωk

✵ = antiholomorphic k-forms. Ω✵ N = ‘sections of line bundles’.

If ℓ is odd: Ω✵

✶ ✷ (ℓ+✶) = square root of the canonical bundle, Ωk ✶ ✷ (ℓ+✶) = chiral spinors.

An associative product ∧q : Ωk

N × Ωk′ N′ → Ωk+k′ N+N′ is given by

ω ∧q ω′ := a · a′(v ∧q v′) ✱ where ω = av ∈ Ωk

N and ω′ = a′v′ ∈ Ωk′ N′, a✱ a′ ∈ A(SUq(ℓ + ✶)), v ∈ Wk, v′ ∈ Wk′.

Ω•

✵ := ℓ k=✵ Ωk ✵ is a graded associative algebra.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 18 / 23

Vector fields and the Dolbeault operator

Let {ei}i=✶✱✳✳✳✱ℓ be the canonical basis of W✶ ≃ Cℓ and Xi := KiKi+✶ ✳ ✳ ✳ Kℓ✂ K−✶ [Ei✱ [Ei+✶✱ ✳ ✳ ✳ [Eℓ−✶✱ Eℓ]q]q]q ✱ i = ✶✱ ✳ ✳ ✳ ✱ ℓ ✱ where [a✱ b]q := ab − q−✶ba. Then X =

ieiXi ∈ Uq(su(ℓ + ✶)) ⊠σ W✶

(on the first leg we use the right adjoint action composed with S−✶.) By invariance of X, ✠ ∂ := ℓ

i=✶ L✂ KXi ⊗ eL ei

maps Ωk

N in Ωk+✶ N

. The Xi’s satisfy some useful commutation rules, for example (X ∧q X)i✶✱i✷ = Xi✶Xi✷ − q−✶Xi✷Xi✶ = ✵ ✱ for all ✶ i✶ < i✷ ℓ ✱ together with associativity of ∧q implies ✠ ∂✷ =

i✶<i✷ Lq✂ K✷(X∧qX)i✶✱i✷ ⊗ eL ei✶ eL ei✷ = ✵ ✳

Facts:

◮ (Ω•

N✱ ✠

∂) is a left covariant cohomology complex;

◮ (Ω•

✵✱ ✠

∂) is a left covariant differential calculus;

◮ for all a ∈ A(CPℓ

q) and ω ∈ Ωk N: ✠

∂(aω) = a(✠ ∂ω) + (✠ ∂a) ∧q ω .

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 19 / 23

Vector fields and the Dolbeault operator

Let {ei}i=✶✱✳✳✳✱ℓ be the canonical basis of W✶ ≃ Cℓ and Xi := KiKi+✶ ✳ ✳ ✳ Kℓ✂ K−✶ [Ei✱ [Ei+✶✱ ✳ ✳ ✳ [Eℓ−✶✱ Eℓ]q]q]q ✱ i = ✶✱ ✳ ✳ ✳ ✱ ℓ ✱ where [a✱ b]q := ab − q−✶ba. Then X =

ieiXi ∈ Uq(su(ℓ + ✶)) ⊠σ W✶

(on the first leg we use the right adjoint action composed with S−✶.) By invariance of X, ✠ ∂ := ℓ

i=✶ L✂ KXi ⊗ eL ei

maps Ωk

N in Ωk+✶ N

. The Xi’s satisfy some useful commutation rules, for example (X ∧q X)i✶✱i✷ = Xi✶Xi✷ − q−✶Xi✷Xi✶ = ✵ ✱ for all ✶ i✶ < i✷ ℓ ✱ together with associativity of ∧q implies ✠ ∂✷ =

i✶<i✷ Lq✂ K✷(X∧qX)i✶✱i✷ ⊗ eL ei✶ eL ei✷ = ✵ ✳

Facts:

◮ (Ω•

N✱ ✠

∂) is a left covariant cohomology complex;

◮ (Ω•

✵✱ ✠

∂) is a left covariant differential calculus;

◮ for all a ∈ A(CPℓ

q) and ω ∈ Ωk N: ✠

∂(aω) = a(✠ ∂ω) + (✠ ∂a) ∧q ω .

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 19 / 23

Vector fields and the Dolbeault operator

Let {ei}i=✶✱✳✳✳✱ℓ be the canonical basis of W✶ ≃ Cℓ and Xi := KiKi+✶ ✳ ✳ ✳ Kℓ✂ K−✶ [Ei✱ [Ei+✶✱ ✳ ✳ ✳ [Eℓ−✶✱ Eℓ]q]q]q ✱ i = ✶✱ ✳ ✳ ✳ ✱ ℓ ✱ where [a✱ b]q := ab − q−✶ba. Then X =

ieiXi ∈ Uq(su(ℓ + ✶)) ⊠σ W✶

(on the first leg we use the right adjoint action composed with S−✶.) By invariance of X, ✠ ∂ := ℓ

i=✶ L✂ KXi ⊗ eL ei

maps Ωk

N in Ωk+✶ N

. The Xi’s satisfy some useful commutation rules, for example (X ∧q X)i✶✱i✷ = Xi✶Xi✷ − q−✶Xi✷Xi✶ = ✵ ✱ for all ✶ i✶ < i✷ ℓ ✱ together with associativity of ∧q implies ✠ ∂✷ =

i✶<i✷ Lq✂ K✷(X∧qX)i✶✱i✷ ⊗ eL ei✶ eL ei✷ = ✵ ✳

Facts:

◮ (Ω•

N✱ ✠

∂) is a left covariant cohomology complex;

◮ (Ω•

✵✱ ✠

∂) is a left covariant differential calculus;

◮ for all a ∈ A(CPℓ

q) and ω ∈ Ωk N: ✠

∂(aω) = a(✠ ∂ω) + (✠ ∂a) ∧q ω .

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 19 / 23

Vector fields and the Dolbeault operator

Let {ei}i=✶✱✳✳✳✱ℓ be the canonical basis of W✶ ≃ Cℓ and Xi := KiKi+✶ ✳ ✳ ✳ Kℓ✂ K−✶ [Ei✱ [Ei+✶✱ ✳ ✳ ✳ [Eℓ−✶✱ Eℓ]q]q]q ✱ i = ✶✱ ✳ ✳ ✳ ✱ ℓ ✱ where [a✱ b]q := ab − q−✶ba. Then X =

ieiXi ∈ Uq(su(ℓ + ✶)) ⊠σ W✶

(on the first leg we use the right adjoint action composed with S−✶.) By invariance of X, ✠ ∂ := ℓ

i=✶ L✂ KXi ⊗ eL ei

maps Ωk

N in Ωk+✶ N

. The Xi’s satisfy some useful commutation rules, for example (X ∧q X)i✶✱i✷ = Xi✶Xi✷ − q−✶Xi✷Xi✶ = ✵ ✱ for all ✶ i✶ < i✷ ℓ ✱ together with associativity of ∧q implies ✠ ∂✷ =

i✶<i✷ Lq✂ K✷(X∧qX)i✶✱i✷ ⊗ eL ei✶ eL ei✷ = ✵ ✳

Facts:

◮ (Ω•

N✱ ✠

∂) is a left covariant cohomology complex;

◮ (Ω•

✵✱ ✠

∂) is a left covariant differential calculus;

◮ for all a ∈ A(CPℓ

q) and ω ∈ Ωk N: ✠

∂(aω) = a(✠ ∂ω) + (✠ ∂a) ∧q ω .

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 19 / 23

Vector fields and the Dolbeault operator

Let {ei}i=✶✱✳✳✳✱ℓ be the canonical basis of W✶ ≃ Cℓ and Xi := KiKi+✶ ✳ ✳ ✳ Kℓ✂ K−✶ [Ei✱ [Ei+✶✱ ✳ ✳ ✳ [Eℓ−✶✱ Eℓ]q]q]q ✱ i = ✶✱ ✳ ✳ ✳ ✱ ℓ ✱ where [a✱ b]q := ab − q−✶ba. Then X =

ieiXi ∈ Uq(su(ℓ + ✶)) ⊠σ W✶

(on the first leg we use the right adjoint action composed with S−✶.) By invariance of X, ✠ ∂ := ℓ

i=✶ L✂ KXi ⊗ eL ei

maps Ωk

N in Ωk+✶ N

. The Xi’s satisfy some useful commutation rules, for example (X ∧q X)i✶✱i✷ = Xi✶Xi✷ − q−✶Xi✷Xi✶ = ✵ ✱ for all ✶ i✶ < i✷ ℓ ✱ together with associativity of ∧q implies ✠ ∂✷ =

i✶<i✷ Lq✂ K✷(X∧qX)i✶✱i✷ ⊗ eL ei✶ eL ei✷ = ✵ ✳

Facts:

◮ (Ω•

N✱ ✠

∂) is a left covariant cohomology complex;

◮ (Ω•

✵✱ ✠

∂) is a left covariant differential calculus;

◮ for all a ∈ A(CPℓ

q) and ω ∈ Ωk N: ✠

∂(aω) = a(✠ ∂ω) + (✠ ∂a) ∧q ω .

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 19 / 23

Vector fields and the Dolbeault operator

Let {ei}i=✶✱✳✳✳✱ℓ be the canonical basis of W✶ ≃ Cℓ and Xi := KiKi+✶ ✳ ✳ ✳ Kℓ✂ K−✶ [Ei✱ [Ei+✶✱ ✳ ✳ ✳ [Eℓ−✶✱ Eℓ]q]q]q ✱ i = ✶✱ ✳ ✳ ✳ ✱ ℓ ✱ where [a✱ b]q := ab − q−✶ba. Then X =

ieiXi ∈ Uq(su(ℓ + ✶)) ⊠σ W✶

(on the first leg we use the right adjoint action composed with S−✶.) By invariance of X, ✠ ∂ := ℓ

i=✶ L✂ KXi ⊗ eL ei

maps Ωk

N in Ωk+✶ N

. The Xi’s satisfy some useful commutation rules, for example (X ∧q X)i✶✱i✷ = Xi✶Xi✷ − q−✶Xi✷Xi✶ = ✵ ✱ for all ✶ i✶ < i✷ ℓ ✱ together with associativity of ∧q implies ✠ ∂✷ =

i✶<i✷ Lq✂ K✷(X∧qX)i✶✱i✷ ⊗ eL ei✶ eL ei✷ = ✵ ✳

Facts:

◮ (Ω•

N✱ ✠

∂) is a left covariant cohomology complex;

◮ (Ω•

✵✱ ✠

∂) is a left covariant differential calculus;

◮ for all a ∈ A(CPℓ

q) and ω ∈ Ωk N: ✠

∂(aω) = a(✠ ∂ω) + (✠ ∂a) ∧q ω .

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 19 / 23

Outline

1 Preliminary definitions

The quantum SU(ℓ + ✶) group The QUEA Uq(su(ℓ + ✶)) S✷ℓ+✶

q

and CPℓ

q

2 K-theory and K-homology

K-theory K-homology

3 Antiholomorphic forms and real spectral triples

The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator

4 From GDAs to spectral triples

Reality and the first order condition A family of spectral triples

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 20 / 23

Reality and the first order condition

The adjoint of ✠ ∂ is ✠ ∂† = ℓ

i=✶ LX∗

i ✂

K ⊗ iL ei

≡ −qN−k+✶ℓ

i=✶ q−✷iLS(X∗

i ✂

K) ⊗ iR ei ✳

An equivariant antilinear map J✵ : Ωk

N → Ωℓ−k ℓ+✶−N is given by

J✵ := (∗ ⊗ J)(LK−✶

✷ρ ⊗ id) ✱

where K✷ρ = (Kℓ

✶K✷(ℓ−✶) ✷

✳ ✳ ✳ Kj(ℓ−j+✶)

j

✳ ✳ ✳ Kℓ

ℓ)✷.

Proposition

The antiunitary part J of J✵ satisfies (i) J✷ = (−✶)⌊ ℓ+✶

✷

⌋;

(ii) Ja∗J−✶ω = ω · (K

✶ ✷

✷ρ ⊲ a) for all ω ∈ Ωk N and a ∈ A(CPℓ q);

(iii) J✠ ∂J−✶|Ωk

N = qk−N✠

∂†.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 21 / 23

Reality and the first order condition

The adjoint of ✠ ∂ is ✠ ∂† = ℓ

i=✶ LX∗

i ✂

K ⊗ iL ei

≡ −qN−k+✶ℓ

i=✶ q−✷iLS(X∗

i ✂

K) ⊗ iR ei ✳

An equivariant antilinear map J✵ : Ωk

N → Ωℓ−k ℓ+✶−N is given by

J✵ := (∗ ⊗ J)(LK−✶

✷ρ ⊗ id) ✱

where K✷ρ = (Kℓ

✶K✷(ℓ−✶) ✷

✳ ✳ ✳ Kj(ℓ−j+✶)

j

✳ ✳ ✳ Kℓ

ℓ)✷.

Proposition

The antiunitary part J of J✵ satisfies (i) J✷ = (−✶)⌊ ℓ+✶

✷

⌋;

(ii) Ja∗J−✶ω = ω · (K

✶ ✷

✷ρ ⊲ a) for all ω ∈ Ωk N and a ∈ A(CPℓ q);

(iii) J✠ ∂J−✶|Ωk

N = qk−N✠

∂†.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 21 / 23

Reality and the first order condition

The adjoint of ✠ ∂ is ✠ ∂† = ℓ

i=✶ LX∗

i ✂

K ⊗ iL ei

≡ −qN−k+✶ℓ

i=✶ q−✷iLS(X∗

i ✂

K) ⊗ iR ei ✳

An equivariant antilinear map J✵ : Ωk

N → Ωℓ−k ℓ+✶−N is given by

J✵ := (∗ ⊗ J)(LK−✶

✷ρ ⊗ id) ✱

where K✷ρ = (Kℓ

✶K✷(ℓ−✶) ✷

✳ ✳ ✳ Kj(ℓ−j+✶)

j

✳ ✳ ✳ Kℓ

ℓ)✷.

Proposition

The antiunitary part J of J✵ satisfies (i) J✷ = (−✶)⌊ ℓ+✶

✷

⌋;

(ii) Ja∗J−✶ω = ω · (K

✶ ✷

✷ρ ⊲ a) for all ω ∈ Ωk N and a ∈ A(CPℓ q);

(iii) J✠ ∂J−✶|Ωk

N = qk−N✠

∂†.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 21 / 23

Reality and the first order condition

The adjoint of ✠ ∂ is ✠ ∂† = ℓ

i=✶ LX∗

i ✂

K ⊗ iL ei

≡ −qN−k+✶ℓ

i=✶ q−✷iLS(X∗

i ✂

K) ⊗ iR ei ✳

An equivariant antilinear map J✵ : Ωk

N → Ωℓ−k ℓ+✶−N is given by

J✵ := (∗ ⊗ J)(LK−✶

✷ρ ⊗ id) ✱

where K✷ρ = (Kℓ

✶K✷(ℓ−✶) ✷

✳ ✳ ✳ Kj(ℓ−j+✶)

j

✳ ✳ ✳ Kℓ

ℓ)✷.

Proposition

The antiunitary part J of J✵ satisfies (i) J✷ = (−✶)⌊ ℓ+✶

✷

⌋;

(ii) Ja∗J−✶ω = ω · (K

✶ ✷

✷ρ ⊲ a) for all ω ∈ Ωk N and a ∈ A(CPℓ q);

(iii) J✠ ∂J−✶|Ωk

N = qk−N✠

∂†.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 21 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

A family of spectral triples

Data: HN := Hilbert space completion of Ω•

N;

γN := +✶ on even forms and −✶ on odd forms; A(CPℓ

q) acts by left multiplication on Ω• N (completed to bounded operators);

J extends to an (antilinear) isometry HN → Hℓ+✶−N; an unbounded densely defined symmetric operator DN on HN is DN := q

✶ ✷ (k−N)✠

∂ + q

✶ ✷ (k−N−✶)✠

∂† ✱ and satisfies JDN = Dℓ+✶−NJ on Ω•

• N. For q = ✶:

◮ D✵ = Dolbeault-Dirac operator, ◮ DN = twist of D✵ with the Grassmannian connection Ω✵

N,

◮ D ✶

✷ (ℓ+✶) = Dirac operator of the Fubini-Study metric (for odd ℓ).

Theorem

(A(CPℓ

q)✱ HN✱ DN✱ γN) is a ✵+-dimensional equivariant even spectral triple. If ℓ is

• dd and N = ✶

✷(ℓ + ✶), the spectral triple is real with KO-dimension ✷ℓ ♠♦❞ ✽.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 22 / 23

References

• F. D’A. and L. D ˛

abrowski, Dirac operators on quantum projective spaces, preprint [arXiv:0901.4735].

• F. D’A. and G. Landi,

Bounded and unbounded Fredholm modules for quantum projective spaces, preprint [arXiv:0903.3553], to appear on Journal of K-theory.

Francesco D’Andrea (UCL) Geometry of quantum CPℓ September 1, 2009 23 / 23