Modules over the noncommutative torus, elliptic curves and cochain quantization Francesco D Andrea ( joint work with G. Fiore & D. Franco ) ((A B) C) D Φ (12)34 Φ 123 (A B) (C D) (A (B C)) D Φ 12(34) Φ 1(23)4 A (B (C D)) Φ 234 A ((B C) D)) Non-commutative geometry s interactions with mathematics HIM, Bonn, 15-19 september 2014
Summary. Aim: describe modules for the noncommutative torus from a deformation point of view. For 0 < θ < 1, let A θ be the universal C -algebra generated by unitaries U and V with UV = e 2πiθ V U. The dense -subalgebra { } A θ := a m,nu m V n : {a m,n } S(Z 2 ) A θ m,n Z is a strict deformation quantization of T 2 associated to the natural action of R 2 [Rieffel, 1993]. Finitely-generated projective A θ -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 R 2 on line bundles: Rieffel s tecnique cannot be used. Modules as Moyal deformation of complex line bundles on the elliptic curve E τ for non-trivial τ. Modules from a quasi-associative cochain quantization of the Heisenberg manifold H 3 (R)/H 3 (Z). 1 / 16
PART I Elliptic curves & nc tori 2 / 16
Moyal product & the nc torus. On the Schwartz space S(R 2 ) one has an associative product: (f θ g)(z) = 4 θ 2 R 2 R 2 f(z + p)g(z + q)e i θ ω(p,q) d 2 pd 2 q, with p, q, z R 2 and ω(p, q) = 4π(p 1 q 2 p 2 q 1 ). Extends to several function spaces (cf. [Gayral, Gracia-Bondía, Iochum, Schücker & Várilly, CMP 246, 2004] & references therein), e.g. to B(R 2 ), the set of smooth functions that are bounded together with all their derivatives. On generators u(x, y) := e 2πix and v(x, y) := e 2πiy of C (T 2 ) B(R 2 ): u θ u = v θ v = 1, u θ v = e 2πiθ v θ u. By Fourier analysis the map a m,n u m v n a m,n e πimnθ U m V n, {a m,n } S(Z 2 ), m,n Z m,n Z is a -algebra isomorphism (C (T 2 ), θ ) A θ. 3 / 16
Moyal deformation of bimodules. Identify (C (T 2 ), 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) = e 2πix f ( x, y + 1 θ), v 2 θ f(x, y) = e 2πiy f(x 1 θ, y), 2 f θ u(x, y) = e 2πix f ( x, y 1 θ), f 2 θ v(x, y) = e 2πiy f(x + 1 θ, y). 2 It can be extended to C (R 2 ). 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 has image in A θ (f, g) f θ g (it is then a Hermitian structure). Hint: smooth sections of line bundles L E τ are elements of C (R 2 ). 4 / 16
Complex line bundles on a torus. ω 2 ω1 Complex structures on C/Λ T 2 parametrized by τ = ω 2 /ω 1 ( Λ := ω 1 Z + ω 2 Z ). Take ω 1 = 1 and τ = ω 2 H := { z C : Im(z) > 0 }, call E τ the complex manifold. 5 / 16
Factors of automorphy. Let us identify (x, y) R 2 with z = x + iy C. Fix τ H and let E τ = 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 End C (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 E τ. There is a corresponding line bundle L α E τ with total space L α = C C/, where (z + λ, w) (z, α(λ, z)w), z, w C, λ Λ, All line bundles on E τ are of this form (Appell-Humbert thm.). (Cpx. l.b. if α holomorphic.) 6 / 16
Sections of line bundles. Smooth sections of L α subset Γ α C (C) of invariant functions under π(λ) in ( ). if α = 1, Γ α C (E τ ) are Λ-periodic functions (and C (C) is a C (E τ )-module). if α holomorphic holomorphic elements of Γ α are the well-known theta functions: they form a finite-dimensional vector space. if α unitary elements of Γ α are quasi-periodic functions ( f is periodic ). For any α, Γ α is a finitely-generated projective C (E τ )-submodule of C (C). For α(m + nτ, x + iy) = e πi Re(τ) n2 e 2πinx the unitary f.a. α p gives the smooth line bundle with degree p (unique for each p Z). Let M p,τ denote the set of smooth sections. Proposition. M p,τ C (R 2 ) is an right A θ -submodule iff τ i pθ 2 Z + iz. If τ = i(1 + pθ 2 ), a Hermitian structure M p,τ M p,τ A θ is given by (f, g) f θ g. 7 / 16
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 L 2 (R), a, b R, given by { } W(a, b)ψ (t) = e πiab e 2πibt ψ(t a), ψ L 2 (R). Proposition. Let [n] := n mod p (from now on p 1). Every f M p,τ is of the form f(z) = n Z e 2πinx e πi Re(τ)n2 /p f [n] (y + n ωy p ), for a unique f = (f [1],..., f [p] ) S(R) C p. The map f f is called WBZ transform. For simplicitly, fix τ = i(s + pθ 2 ) with s Z, and let E p,s := S(R) C p. Proposition. The WBZ transform M p,τ E p,s, f f, is a right A θ -module isomorphism if the right action on f E p,s is defined by f u = { W( s p + θ, 0) S} f, f v = { W(0, 1) (C ) s} f, with C, S M p (C) are the clock and shift operators. 8 / 16
The Weil-Brezin-Zak transform, II. Remarks. Modulo slighly different notations, these are the modules in [Connes 1980, Rieffel 1983]. A posteriori, Mp,τ E p,s is finitely generated and projective as a right A θ -module. Let f, g M p,τ. While fg is Λ-periodic, so it belongs to C (E τ ), if s = 1 the product f θ g is Z 2 -periodic, hence it belongs to C (T 2 ). For all ψ = (ψ 1,..., ψ p ) and ϕ = (ϕ 1,..., ϕ p ) E p,s, let: Proposition ψ ϕ t p := ψ r (t)ϕ r (t). r=1 Let s = 1 (so τ = i(1 + pθ 2 ) ). For all f, g M p,τ: f θ g = + m,n Z um v n f u m v n g dt, t with f and g the WBZ transform of f and g, respectively. 9 / 16
PART II Non-associative deformations of Heisenberg manifolds 10 / 16
Where were we? Two constructions of line bundles on T 2 : 1 From a representation of a lattice Λ: 2 As associated to a principal U(1)-bundle: L C Λ,α C H 3 (R)/H 3 (Z) U(1) T 2 C/Λ T 2 For people familiar with U(1)-bundles, 2 is more natural (analogous to S 3 U(1) S 2 or 2n+1 U(1) S CP n ). It has the advantage that: C (total space) = n Z Γ (line bundle (we can work with an algebra instead of a module) L ) of deg. n T 2 There is a quasi-associative (formal) deformation of H 3 (R)/H 3 (Z) encoding the module structure of A θ (the pairing of bimodules, H0 (L, ), etc.) 11 / 16
Hopf cohomology Let (H,, ɛ) be a Hopf algebra and G n = { multiplicative group of invertible h H n}. For 0 i n + 1, define i : G n G n+1 as follows: 0 (h) = 1 h, n+1 (h) = h 1, i = id i 1 id n i in all other cases. A map : G n G n+1 is given by h = ( + h)( h 1 ), where + h = 0 (h) 2 (h)... h = 1 (h) 3 (h)... H is commutative (G, ) = cochain complex of abelian groups ( = group homomorphism & 2 = 1). In general: 2 = 1 on G 1 and on H-invariant elements of G 2, but not on every element (there are counterexamples). Also, is in general not a group homomorphism. Lazy chains (dual to invariant cochains) studied by Schauenburg, Bichon, Carnovale, Guillot, Kassel and others lazy homology. Invariant chain trivial deformation of the coproduct (cf. next slide). Not what we want in deformation theory. 12 / 16
Cochain quantization In the definition of quasi-hopf algebra H we relax the coassociativity condition: (id ) (h) = Φ( id) (h)φ 1 h H, where Φ G 3 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 as a 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 map m : A A A, and F G 2 a 2-cochain. Let H F be H with a new coproduct F, and A F be A with a new product m F given by: F (h) := F (h)f 1, m F (a b) := m F 1 (a b). Then H F is a quasi-hopf algebra with coassociator Φ F := F (a 3-cocycle w.r.t. F!), A F is a left H F -module algebra, and m F (m F id) = m F (id m F )Φ F. F = coassociative iff Φ F is H F -invariant, a sufficient condition being F = 1 (cocycle twist). In the latter case, A F is associative. 13 / 16
Heisenberg manifolds Let 1 x t H 3 (R) := 0 1 y : x, y, t R, 0 0 1 H 3 (Z) the subgroup of integer matrices and M 3 = H 3 (R)/H 3 (Z) There is a right circle action of Z(H 3 (R)) R, and M 3 /U(1) T 2. One can think of U(h 3 (R)) as the Hopf algebra of right-invariant differential operators, generated by: Note that X := x + y t, T is central and [X, Y] = T; C (M 3 ) is a left U(h 3 (R))-module algebra; Y := y, T := t. C (M 3 ) = { f C (S 1 R S 1 ) : f(x, y + 1, t + x) = f(x, y, t) } ; C (M 3 ) = L k where L k := { f C (M 3 ) : Tf = 2πikf } ( ) and C (T 2 ) = L 0 ; k Z 14 / 16
Non-associative geometry of quantum tori Let F h = e i h 2π X Y and m h(f g) = f h g := m F 1 h (f g) the corresponding star product. Let A h = ( C (T 2 )[[ h]], h) B h = ( ) C (M 3 )[[ h]], h Using Baker-Campbell-Hausdorff formula one computes the coassociator: Corollaries: F h = e ( h 2π) 2 X T Y. (a h b) h c = a h (b h c) b L 0 = ker(t). A h is associative generated by u(x, y) := e 2πix and v(x, y) := e 2πiy with relations: u h u = v h v = 1, u h v = e 2πi h v h u. B h is not associative; L k is a left and a right A h-module (w.r.t. h) but not a bimodule. (L k, h) Connes-Rieffel modules (modulo a replacement h θ). 15 / 16
Pairing of bimodules Let h C[[ h]]. Then m h (m h id) = m h(id m h )Φ h, h where One has Φ h, h = (1 F h )(id )(F h)( id)(f 1 h )(F 1 h 1) { = exp i ( ) } 2π X h h h h T Y 2πi ( ) a h (b h c) = (a h b) h c a, c L, b L k, h if and only if h = = g h, where g is the fractional linear transformation: 1+k h ( ) 1 0 g = SL(2, Z). k 1 Thus, (L k, h, h) is a A g h-a h bimodule (compare with Rieffel imprimitivity bimodules). With the WBZ transform one proves that θ : L k L n L k+n is the pairing of bimodules of [Polishchuk & Schwarz, CMP 236, 2003]. Associativity of the pairing (diagram 1.4 in PS s paper) can also be derived from ( ). 16 / 16
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