Modules over the noncommutative torus, elliptic curves...

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Modules over the noncommutative torus, elliptic curves and cochain quantization Francesco D’Andrea ( joint work with G. Fiore & D. Franco ) ((A B) C) D (A B) (C D) A (B (C D)) A ((B C) D)) (A (B C)) D Φ (12)34 Φ 12(34) Φ234 Φ 1(23)4 Φ123 Non-commutative geometry’s interactions with mathematics HIM, Bonn, 15-19 september 2014

Transcript of Modules over the noncommutative torus, elliptic curves...

Modules over the noncommutative torus, elliptic

curves and cochain quantization

Francesco D’Andrea

( joint work with G. Fiore & D. Franco )

((A⊗B) ⊗ C) ⊗D

(A⊗B) ⊗ (C ⊗D)

A⊗ (B⊗ (C ⊗D)) A⊗ ((B⊗C) ⊗D))

(A⊗ (B ⊗ C)) ⊗D

Φ(12)34

Φ12(34)

Φ234

Φ1(23)4

Φ123

Non-commutative geometry’s interactions with mathematics

HIM, Bonn, 15-19 september 2014

Summary.I Aim: describe modules for the noncommutative torus from a deformation point of view.

For 0 < θ < 1, let Aθ be the universal C∗-algebra generated by unitaries U and V with

UV = e2πiθVU .

The dense ∗-subalgebra

A∞θ :={ ∑

m,n∈Zam,nU

mVn : {am,n} ∈ S(Z2)}⊂ Aθ

is a strict deformation quantization of T2 associated to the natural action of R2 [Rieffel, 1993].

Finitely-generated projective Aθ-modules classified by Connes and Rieffel in the ‘80s.Can they be obtained as deformations of vector bundles (in fact, line bundles) on the torus?

No action of R2 on linebundles: Rieffel’s tec-nique cannot be used. Modules from a quasi-associative cochain quantiza-

tion of the Heisenberg manifold H3(R)/H3(Z).

Modules as Moyal deformation of complex line bun-dles on the elliptic curve Eτ for non-trivial τ.

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— PART I —

Elliptic curves & nc tori

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Moyal product & the nc torus.

On the Schwartz space S(R2) one has an associative product:

(f ∗θ g)(z) =4

θ2

∫R2×R2

f(z+ p)g(z+ q)eiθω(p,q)d2pd2q ,

with p,q, z ∈ R2 and ω(p,q) = −4π(p1q2 − p2q1).

Extends to several function spaces (cf. [Gayral, Gracia-Bondıa, Iochum, Schucker & Varilly,CMP 246, 2004] & references therein), e.g. to B(R2), the set of smooth functions that arebounded together with all their derivatives.

On generators u(x,y) := e2πix and v(x,y) := e2πiy of C∞(T2) ⊂ B(R2):

u ∗θ u∗ = v ∗θ v∗ = 1 , u ∗θ v = e2πiθv ∗θ u .

By Fourier analysis the map∑m,n∈Z

am,numvn 7→

∑m,n∈Z

am,ne−πimnθUmVn , ∀ {am,n} ∈ S(Z2) ,

is a ∗-algebra isomorphism (C∞(T2), ∗θ)→ A∞θ .

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Moyal deformation of bimodules.

Identify (C∞(T2), ∗θ) with A∞θ and note that, being a ∗-subalgebra of B∞θ := (B(R2), ∗θ),the latter is an A∞θ -bimodule. On generators:

u ∗θ f(x,y) = e2πixf(x,y+ 1

2θ), v ∗θ f(x,y) = e2πiyf(x− 1

2θ,y) ,

f ∗θ u(x,y) = e2πixf(x,y− 1

2θ), f ∗θ v(x,y) = e2πiyf(x+ 1

2θ,y) .

It can be extended to C∞(R2). Let J be the antilinear involutive map:

Jf(x,y) = f(−x,−y) .

J( . )J sends A∞θ into its commutant, and transforms the left action into the right one.We will focus on right modules. . .

We want to find finitely generated (and projective) submodules, possibly such that the map

(f,g) 7→ f ∗θ g

has image in A∞θ (it is then a Hermitian structure).

I Hint: smooth sections of line bundles L→ Eτ are elements of C∞(R2).4 / 16

Complex line bundles on a torus.

ω2

ω1

• Complex structures on C/Λ ' T2 parametrized by τ = ω2/ω1 (Λ := ω1Z+ω2Z ).

• Take ω1 = 1 and τ = ω2 ∈ H :={z ∈ C : Im(z) > 0

}, call Eτ the complex manifold.

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Factors of automorphy.

Let us identify (x,y) ∈ R2 with z = x+ iy ∈ C. Fix τ ∈ H and let

Eτ = C/Λ , Λ := Z+ τZ ,

be the corresponding elliptic curve with modular parameter τ.

Let α : Λ× C→ C∗ be a smooth function and π : Λ→ EndC∞(C) given by

(?) π(λ)f(z) := α−1(λ, z)f(z+ λ) , ∀ λ ∈ Λ, z ∈ C .

Then π is a representation of the abelian group Λ if and only if

(†) α(λ+ λ ′, z) = α(λ, z+ λ ′)α(λ ′, z) , ∀ z ∈ C, λ, λ ′ ∈ Λ.

An α satisfying (†) is called a factor of automorphy for Eτ.

There is a corresponding line bundle Lα → Eτ with total space Lα = C× C/∼ , where

(z+ λ,w) ∼ (z,α(λ, z)w) , ∀ z,w ∈ C, λ ∈ Λ,

All line bundles on Eτ are of this form (Appell-Humbert thm.). (Cpx. l.b. if α holomorphic.)6 / 16

Sections of line bundles.Smooth sections of Lα ≡ subset Γα ⊂ C∞(C) of invariant functions under π(λ) in (?).

I if α = 1, Γα ≡ C∞(Eτ) are Λ-periodic functions (and C∞(C) is a C∞(Eτ)-module).

I if α holomorphic holomorphic elements of Γα are the well-known theta functions:they form a finite-dimensional vector space.

I if α unitary elements of Γα are quasi-periodic functions ( |f| is periodic ).

For any α, Γα is a finitely-generated projective C∞(Eτ)-submodule of C∞(C). For

α(m+ nτ, x+ iy) = e−πiRe(τ)n2e−2πinx

the unitary f.a. αp gives the smooth line bundle with degree p (unique for each p ∈ Z).Let Mp,τ denote the set of smooth sections.

Proposition. Mp,τ ⊂ C∞(R2) is an right A∞θ -submodule iff

τ− ipθ2∈ Z+ iZ .

If τ = i(1+ pθ2) , a Hermitian structure Mp,τ ×Mp,τ → A∞θ is given by (f,g) 7→ f ∗θ g .

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The Weil-Brezin-Zak transform, I.Both the module and Hermitian structure can be described in terms of Weyl operators.These are the unitary operators W(a,b) on L2(R), a,b ∈ R, given by{

W(a,b)ψ}(t) = e−πiabe2πibtψ(t− a) , ψ ∈ L2(R).

Proposition. Let [n] := n mod p (from now on p > 1). Every f ∈Mp,τ is of the form

f(z) =∑n∈Z

e2πinxeπiRe(τ)n2/pf[n](y+ n

ωyp) ,

for a unique f = (f[1], . . . , f[p]) ∈ S(R)⊗ Cp. The map f 7→ f is called WBZ transform.

For simplicitly, fix τ = i(s+ pθ2) with s ∈ Z, and let Ep,s := S(R)⊗ Cp .

Proposition. The WBZ transform Mp,τ → Ep,s , f 7→ f , is a right A∞θ -module isomorphismif the right action on f ∈ Ep,s is defined by

fJu ={W( s

p+ θ, 0)⊗ S

}f , fJ v =

{W(0, 1)⊗ (C∗)s

}f ,

with C,S ∈Mp(C) are the clock and shift operators.

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The Weil-Brezin-Zak transform, II.

Remarks.I Modulo slighly different notations, these are the modules in [Connes 1980, Rieffel 1983].

I A posteriori, Mp,τ ' Ep,s is finitely generated and projective as a right A∞θ -module.

Let f,g ∈Mp,τ. While fg is Λ-periodic, so it belongs to C∞(Eτ), if s = 1 the product f ∗θ gis Z2-periodic, hence it belongs to C∞(T2).

For all ψ = (ψ1, . . . ,ψp) and ϕ = (ϕ1, . . . ,ϕp) ∈ Ep,s, let:

⟨ψ∣∣ϕ⟩

t:=

p∑r=1

ψr(t)ϕr(t) .

Proposition

Let s = 1 (so τ = i(1+ pθ2) ). For all f,g ∈Mp,τ:

f ∗θ g =∑

m,n∈Zumvn

∫+∞−∞

⟨fJumvn

∣∣g⟩tdt ,

with f and g the WBZ transform of f and g, respectively.

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— PART II —

Non-associative deformations of

Heisenberg manifolds

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Where were we?

Two constructions of line bundles on T2:

1 From a representation of a lattice Λ:

L

T2

C×Λ,α C

C/Λ

'

2 As associated to a principal U(1)-bundle:

H3(R)/H3(Z)

T2

U(1)

I For people familiar with U(1)-bundles, 2 is more natural (analogous to S3 U(1)→ S2 or

S2n+1 U(1)→ CPn). It has the advantage that:

C∞(total space) =⊕n∈Z

Γ∞(

line bundleL↓T2

of deg. n)

(we can work with an algebra instead of a module)

I There is a quasi-associative (formal) deformation of H3(R)/H3(Z) encoding themodule structure of A∞θ (the pairing of bimodules, H0(L,∇), etc.)

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Hopf cohomology

Let (H,∆, ε) be a Hopf algebra and Gn ={

multiplicative group of invertible h ∈ H⊗n}

.

For 0 6 i 6 n+ 1, define ∆i : Gn → Gn+1 as follows:

∆0(h) = 1⊗ h , ∆n+1(h) = h⊗ 1 , ∆i = idi−1 ⊗ ∆⊗ idn−i in all other cases.

A map ∂ : Gn → Gn+1 is given by ∂h = (∂+h)(∂−h−1) , where

∂+h = ∆0(h)∆2(h) . . . ∂−h = ∆1(h)∆3(h) . . .

I H is commutative ⇒ (G•,∂) = cochain complex of abelian groups (∂ = grouphomomorphism & ∂2 = 1).

I In general: ∂2 = 1 on G1 and on H-invariant elements of G2, but not on every element(there are counterexamples). Also, ∂ is in general not a group homomorphism.

I Lazy chains (dual to invariant cochains) studied by Schauenburg, Bichon, Carnovale,Guillot, Kassel and others lazy homology.

I Invariant chain trivial deformation of the coproduct (cf. next slide). Not what wewant in deformation theory.

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Cochain quantizationIn the definition of quasi-Hopf algebra H we relax the coassociativity condition:

(id⊗ ∆)∆(h) = Φ(∆⊗ id)∆(h)Φ−1 ∀ h ∈ H,

where Φ ∈ G3 is the coassociator and is a 3-cocycle: ∂Φ = 1

The latter is just the pentagon identity (Mac Lane’s coherence condition) for Φ, seen asa module map (A⊗ B)⊗ C→ A⊗ (B⊗ C) for every three H-modules A,B,C.

Let H be a Hopf algebra, A a (associative) left H-module algebra with multiplication mapm : A⊗A→ A, and F ∈ G2 a 2-cochain.Let HF be H with a new coproduct ∆F, and AF be A with a new product mF given by:

∆F(h) := F∆(h)F−1 , mF(a⊗ b) := m ◦ F−1(a⊗ b) .

Then HF is a quasi-Hopf algebra with coassociator ΦF := ∂F (a 3-cocycle w.r.t. ∆F!),AF is a left HF-module algebra, and

mF(mF ⊗ id) = mF(id⊗mF)ΦF .

∆F = coassociative iffΦF is HF-invariant, a sufficient condition being ∂F = 1 (cocycle twist).In the latter case, AF is associative.

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Heisenberg manifoldsLet

H3(R) :=

1 x t

0 1 y

0 0 1

: x,y, t ∈ R

,

H3(Z) the subgroup of integer matrices and

M3 = H3(R)/H3(Z)

There is a right circle action of Z(H3(R)) ' R, and M3/U(1) ' T2 . One can think ofU(h3(R)) as the Hopf algebra of right-invariant differential operators, generated by:

X :=∂

∂x+ y

∂t, Y :=

∂y, T :=

∂t.

Note that

I T is central and [X, Y] = −T ;

I C∞(M3) is a left U(h3(R))-module algebra;

I C∞(M3) ={f ∈ C∞(S1 × R× S1) : f(x,y+ 1, t+ x) = f(x,y, t)

};

I C∞(M3) =⊕k∈Z

Lk where Lk :={f ∈ C∞(M3) : Tf = 2πikf

} (and C∞(T2) = L0

);

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Non-associative geometry of quantum toriLet

F h = ei h2πX⊗Y

and m h(f⊗ g) = f ∗ h g := m ◦ F−1 h (f⊗ g) the corresponding star product. Let

A h =(C∞(T2)[[ h]], ∗ h

)⊂ B h =

(C∞(M3)[[ h]], ∗ h

)Using Baker-Campbell-Hausdorff formula one computes the coassociator:

∂F h = e( h2π)

2X⊗T⊗Y .

Corollaries:

I (a ∗ h b) ∗ h c = a ∗ h (b ∗ h c) ∀ b ∈ L0 = ker(T).

I A h is associative generated by u(x,y) := e2πix and v(x,y) := e2πiy with relations:

u ∗ h u∗ = v ∗ h v∗ = 1 , u ∗ h v = e2πi h v ∗ h u .

I B h is not associative; Lk is a left and a right A h-module (w.r.t. ∗ h) but not a bimodule.(Lk, ∗ h) ' Connes-Rieffel modules (modulo a replacement h→ θ).

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Pairing of bimodulesLet h′ ∈ C[[ h]] . Then

m h′(m h ⊗ id) = m h(id⊗m h′)Φ h, h′

whereΦ h, h′ = (1⊗ F h′)(id⊗ ∆)(F h)(∆⊗ id)(F−1

h′ )(F−1 h ⊗ 1)

= exp

{−i

2πX⊗

( h− h′ − h h′

T

2πi

)⊗ Y}

One has

(‡) a ∗ h (b ∗ h′ c) = (a ∗ h b) ∗ h′ c ∀ a, c ∈ L•, b ∈ Lk,

if and only if h′ = h

1+k h= g h , where g is the fractional linear transformation:

g =

(1 0

k 1

)∈ SL(2,Z).

Thus, (Lk, ∗ h′ , ∗ h) is a Ag h-A h bimodule (compare with Rieffel imprimitivity bimodules).

With the WBZ transform one proves that ∗θ : Lk ⊗ Ln → Lk+n is the pairing of bimodulesof [Polishchuk & Schwarz, CMP 236, 2003]. Associativity of the pairing (diagram 1.4 inPS’s paper) can also be derived from (‡).

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Thank you for your attention.