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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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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.
8 / 16
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.
9 / 16
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
);
14 / 16
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 (‡).
16 / 16