Vector bundles on the noncommutative torus from deformation quantization Francesco D’Andrea ( joint work with G. Fiore & D. Franco ) 15/07/2014 Frontiers of Fundamental Physics 14 Marseille, 15-18 July 2014
Line bundles: an example. M¨ obius strip (Non-trivial) real line bundle on S 1 . A copy of R attached to any point of S 1 . • Section = function associating to any point P of S 1 a point on the line through P . • • In quantum mechanics: charged particle interacting with a magnetic monopole. ◮ wave function = section of a non-trivial vector bundle ◮ magnetic field = curvature of a connection on the bundle 1 / 20
Complex line bundles on a torus. ⇒ ω 2 ω 1 • Torus algebra C ( T 2 ) : universal C ∗ -algebra generated by two commuting unitaries. 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. 2 / 20
Summary. For 0 < θ < 1 , let A θ be the universal C ∗ -algebra generated by unitaries U and V with UV = e 2 πiθ V U . echet pre C ∗ -algebra with suitable seminorms. . . ) The dense ∗ -subalgebra (Fr´ � � � m , n ∈ Z a m , n U m V n : { a m , n } ∈ S ( Z 2 ) A ∞ θ := ⊂ A θ is a strict deformation quantization of C ( T 2 ) associated to the 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 on the torus? No action of R 2 on line bundles: Rieffel’s tecnique cannot be used. ◮ they are deformations of line bundles on the elliptic curve E τ = C /Λ τ , provided τ − pθ 2 i ∈ Z + i Z , where p ∈ Z is the 1st Chern number of the bundle. 3 / 20
Credits & motivations. • Works on nc-tori and elliptic curves (quantum theta functions, etc.): Y.I. Manin, M. Marcolli, F.P . Boca, J. Plazas, A. Polishchuk, A. Schwarz, M. Vlasenko, I. Nikolaev, S. Mahanta, W.D. van Suijlekom, . . . Why? Hilbert’s 12th problem: description of the maximal abelian extension K ab of a • number field K in terms of special values of suitable meromorphic functions. ◮ K = Q : K ab generated by roots of unity, i.e. special values of e iz corresponding to torsion points of C ∗ (Kronecker-Weber theorem). ◮ K = Q ( τ ) imaginary quadratic field: K ab generated by the j -invariant j ( E τ ) and by the values of the Weierstrass’s elliptic function ℘ ( z ; τ ) at torsion points of E τ . ◮ Manin’s idea: for a real quadratic field K = Q ( θ ) , A θ plays the role of E τ . (Very ample) line bundles � realize E τ (and A θ ?) as complex projective variety. • √ • I. Nikolaev (arxiv:1404.4999) found generators of the Hilbert class field of K = Q ( D ) , i.e. the analogue of the j -invariant for nc-tori. 4 / 20
— P ART I — Nc-torus and Heisenberg modules 5 / 20
The deformation point of view: twisted group algebras. Let x , y ∈ R 2 . The C ∗ -algebra C ( T 2 ) is generated by the two functions: u ( x , y ) := e 2 πix , v ( x , y ) := e 2 πiy . By standard Fourier analysis, � � � a m , n u m v n : { a m , n } ∈ S ( Z 2 ) C ∞ ( T 2 ) ≡ f = . m , n ∈ Z An associative product ∗ θ on C ∞ ( T 2 ) is given on monomials by u j + m v k + n , ( u j v k ) ∗ θ ( u m v n ) = σ � � ( j , k ) , ( m , n ) ∀ j , k , m , n ∈ Z , where σ : Z 2 × Z 2 → C ∗ is a 2 -cocycle in the group cohomology complex of Z 2 , given by: := e iπθ ( jn − km ) . � � σ ( j , k ) , ( m , n ) ( C ∞ ( T 2 ) , ∗ θ ) is a ∗ -algebra (with undeformed involution), cocycle quantization of S ( Z 2 ) with convolution product. The C ∗ -completion is the twisted group C ∗ -algebra C ∗ ( Z 2 , σ ) . 6 / 20
Quantization map & Moyal product. θ is given on f = � m , n ∈ Z a m , n u m v n by: An isomorphism T θ : ( C ∞ ( T 2 ) , ∗ θ ) → A ∞ � a m , n e − πimnθ U m V n . T θ ( f ) := m , n ∈ Z The phase factors are chosen to have T θ ( f ) ∗ = T θ ( f ∗ ) for all f . On the Schwartz space S ( R 2 ) one has the Moyal product (see e.g. Gayral et al., CMP 246, 2004 & references therein): � ( f ∗ θ g )( z ) = 4 4 πi θ Im ( ξ η ) dξdη , f ( z + ξ ) g ( z + η ) e θ 2 C × C where z = x + iy and dz = dx dy . Extended by duality to tempered distributions, allows to define an associative product (by restriction) on several interesting function spaces. E.g. B( R 2 ) , the set of smooth functions echet pre C ∗ -algebra. that are bounded together with all their derivatives, with ∗ θ is a Fr´ On C ∞ ( T 2 ) ⊂ B( R 2 ) one recovers the star product of the nc-torus. 7 / 20
Heisenberg modules. [Connes 1980, Rieffel 1983] θ -module structure on E p , s := S ( R ) ⊗ C p is given by Let p , s ∈ Z and p � 1 . A right A ∞ � p + θ , 0 ) ⊗ ( S ∗ ) s � � � W ( s W ( 0, 1 ) ⊗ C ψ ⊳ U = ψ , ψ ⊳ V = ψ , where C , S ∈ M p ( C ) are the clock and shift operators, and W ( a , b ) unitaries on L 2 ( R ) : � � ( t ) = e − πiab e 2 πibt ψ ( t − a ) , ψ ∈ L 2 ( R ) , a , b ∈ R . W ( a , b ) ψ θ -valued Hermitian structure on E p , s is given by An A ∞ � ∞ � U m V n ( ψ ⊳ U m V n | ϕ ) t d t . � ψ , ϕ � = − ∞ m , n ∈ Z where for all ψ = ( ψ 1 , . . . , ψ p ) and ϕ = ( ϕ 1 , . . . , ϕ p ) ∈ E p , s : p � ( ψ | ϕ ) t := ψ r ( t ) ϕ r ( t ) . r = 1 θ ) p or If θ ∈ R � Q , any finitely-gen. projective right A θ -module is isomorphic either to ( A ∞ to a module E p , s , with p and s coprime ( p > 0 and s � = 0 ) or p = 1 and s = 0 . 8 / 20
— P ART II — Elliptic curves and the WBZ transform 9 / 20
Basics on elliptic curves. Let us identify ( x , y ) ∈ R 2 with z = x + iy ∈ C . Fix τ ∈ H and let E τ = C /Λ , Λ := Z + τ Z , 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 ∀ z ∈ C , λ , λ ′ ∈ Λ . α ( λ + λ ′ , z ) = α ( λ , z + λ ′ ) α ( λ ′ , z ) , ( ‡ ) 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 theorem). 10 / 20
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 called theta functions: they form a finite-dimensional vector space. ◮ if α unitary � elements of Γ α are quasi-periodic functions ( | f | is periodic ). Note that in this case Γ α ⊂ B( R 2 ) is in the domain of Moyal product. For any α , Γ α is a finitely-generated projective C ∞ ( E τ ) -submodule of C ∞ ( C ) . Let τ = ω x + iω y with ω x , ω y ∈ R and ω y > 0 . The smooth line bundle with degree p (unique for each p ) can be obtained from the unitary factor of automorphy β p , where: β ( m + nτ , x + iy ) = e − πiω x n 2 e − 2 πinx 11 / 20
WBZ transform. For fixed τ , p , let F τ , p be the module with factor of automorphy β p . Let [ n ] := n mod p . Proposition Every f ∈ F τ , p is of the form � e 2 πinx e πin 2 ωx p f [ n ] ( y + n ω y f ( z ) = p ) , n ∈ Z for some (unique) Schwartz functions f [ 1 ] , . . . , f [ p ] ∈ S ( R ) , We denote by ϕ τ , p : F τ , p → S ( R ) ⊗ C p the bijection f �→ f = ( f [ 1 ] , . . . , f [ p ] ) t . ϕ τ , p is very similar to the Weil-Brezin-Zak transform of solid state physics (Folland, 1989). F τ , p is a pre Hilbert C ∞ ( E τ ) -module with canonical Hermitian structure ( f , g ) �→ f ∗ g . 12 / 20
— P ART III — Vector bundles over the nc-torus 13 / 20
Moyal deformation of bimodules. C ∞ ( R 2 ) is a A ∞ θ -bimodule, with module structure: ( U ⊲ f )( x , y ) = e 2 πix f � x , y + 1 � ( f ⊳ U )( x , y ) = e 2 πix f � x , y − 1 � 2 θ , 2 θ , ( V ⊲ f )( x , y ) = e 2 πiy f ( x − 1 ( f ⊳ V )( x , y ) = e 2 πiy f ( x + 1 2 θ , y ) , 2 θ , y ) . 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. The relation with Moyal is as follows. The space B( R 2 ) is an A ∞ θ sub-bimodule of C ∞ ( R 2 ) . For a ∈ A ∞ θ and f ∈ B( R 2 ) , one can check that a ⊲ f = σ θ ( a ) ∗ θ f and f ⊳ a = f ∗ θ σ θ ( a ) , where ∗ θ is Moyal product, T θ : C ∞ ( T 2 ) → A ∞ θ the quantization map introduced before and σ θ := T − 1 the symbol map. θ We will focus on right modules. . . 14 / 20
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