(Not only) Line bundles over noncommutative spaces

(Not only) Line bundles over noncommutative spaces Giovanni Landi Trieste Gauge Theory and Noncommutative Geometry Radboud University Nijmegen ; April 4 8, 2016

Work done over few years with Francesca Arici Simon Brain Francesco D Andrea Jens Kaad

Abstract: Pimsner algebras of tautological line bundles: Total spaces of principal bundles out of a Fock-space construction Gysin-like sequences in KK-theory Quantum lens spaces as direct sums of line bundles over weighted quantum projective spaces Self-dual connections: on line bundles: monopole connections on higher rank bundles: instanton connections some hint to T-dual noncommutative bundles

grand motivations : Gauge fields on noncommutative spaces T-duality for noncommutative spaces Chern-Simons theory A Gysin sequence for U(1)-bundles relates H-flux (three-forms on the total space E) to line bundles (two-forms on the base space M) also giving an isomorphism between Dixmier-Douady classes on E and line bundles on M

The classical Gysin sequence Long exact sequence in cohomology; for any sphere bundle In particular, for circle bundles: U(1) E π X H k (E) π H k 1 (X) c 1(E) H k+1 (X) π H k+1 (E) H 3 (E) π H 2 (X) c 1(E) H 4 (X) H 3 (E) H π (H) = F = c 1 (E )

E M E E π M π E H 3 (E ) π H 2 (X) c 1(E ) H 4 (X) F F = 0 = F F H 3 (E ) H π (H) = F = c 1 (E) T-dual (E, H) and (E, H ) Bouwknegt, Evslin, Mathai, 2004

difficult to generalize to quantum spaces rather go to K-theory ; a six term exact sequence ( see later )

Projective spaces and lens spaces CP n = S 2n+1 /U(1) and L (n,r) = S 2n+1 /Z r assemble in principal bundles : S 2n+1 L (n,r) π CP n This leads to the Gysin sequence in topological K-theory: 0 K 1 (L (n,r) ) δ K 0 (CP n ) α K 0 (CP n ) π K 0 (L (n,r) ) 0 δ is a connecting homomorphism α is multiplication by the Euler class χ(l r ) := 1 [L r ] From this: K 1 (L (n,r) ) ker(α) and K 0 (L (n,r) ) coker(α) torsion groups

U(1)-principal bundles The Hopf algebra H = O(U(1)) := C[z, z 1 ]/ 1 zz 1 : z n z n z n ; S : z n z n ; ɛ : z n 1 Let A be a right comodule algebra over H with coaction R : A A H B := {x A R (x) = x 1} be the subalgebra of coinvariants Definition 1. The datum ( A, H, B ) is a quantum principal U(1)- bundle when the canonical map is an isomorphism can : A B A A H, x y x R (y).

Z-graded algebras A = n Z A n a Z-graded algebra. A right H-comodule algebra: R : A A H x x z n, for x A n, with the subalgebra of coinvariants given by A 0. Proposition 2. The triple ( A, H, A 0 ) is a quantum principal U(1)- bundle if and only if there exist finite sequences {ξ j } N j=1, {β i} M i=1 in A 1 and {η j } N j=1, {α i} M i=1 in A 1 such that: N j=1 ξ jη j = 1 A = M i=1 α iβ i.

Corollary 3. Same conditions as above. The right-modules A 1 and A 1 are finitely generated and projective over A 0. Proof. For A 1 : define the module homomorphisms Φ 1 : A 1 (A 0 ) N, Φ 1 (ζ) = Ψ 1 : (A 0 ) N A 1, Ψ 1 x 1 x 2. x N η 1 ζ η 2 ζ. η N ζ and = j ξ j x j. Then Ψ 1 Φ 1 = Id A1. Thus E 1 := Φ 1 Ψ 1 is an idempotent in M N (A 0 ).

The above results show that ( A, H, A 0 ) is a quantum principal U(1)-bundle if and only if A is strongly Z-graded, that is A n A (m) = A (n+m) Equivalently, the right-modules A (±1) are finitely generated and projective over A 0 if and only if A is strongly Z-graded C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory K.H. Ulbrich, 1981

More generally: G any group with unit e An algebra A is G-graded if A = g G A g, and A g A h A gh If H := CG the group algebra, then A is G-graded if and only if A is a right H-comodule algebra for the coaction δ : A A H δ(a g ) = a g g, a g A g ; coinvariants given by A coh = A e, the identity components. Proposition 4. The datum ( A, H, A e ) is a noncommutative principal H-bundle for the canonical map can : A Ae A A H, a b g ab g g, if and only if A is strongly graded, that is A g A h = A gh. When G = Z = U(1), then CG = O(U(1)) as before.

More general scheme: Pimsner algebras M.V. Pimsner 97 The right-modules A 1 and A 1 before are line bundles over A 0 The slogan: a line bundle is a self-morita equivalence bimodule E a (right) Hilbert module over B B-valued hermitian structure, on E L(E) adjointable operators; K(E) L(E) compact operators with ξ, η E, denote θ ξ,η K(E) defined by θ ξ,η (ζ) = ξ η, ζ There is an isomorphism φ : B K(E) and E is a B-bimodule

Comparing with before: A 0 B and A 1 E Look for the analogue of A O E Pimsner algebra Examples B = O(CP n q ) E = L r (L 1 ) r quantum (weighted) projective spaces (powers of) tautological line bundle O E = O(L (n,r) q ) quantum lens spaces

Define the B-module E := N Z E φ N, E 0 = B E φ E the inner tensor product: a B-Hilbert module with B- valued hermitian structure ξ 1 η 1, ξ 2 η 2 = η 1, φ( ξ 1, ξ 2 )η 2 E 1 = E the dual module; its elements are written as λ ξ for ξ E : λ ξ (η) = ξ, η

For each ξ E a bounded adjointable operator S ξ : E E generated by S ξ : E φ N E φ (N+1) : S ξ (b) := ξ b, b B, S ξ (ξ 1 ξ N ) := ξ ξ 1 ξ N, N > 0, S ξ (λ ξ1 λ ξ N ) := λ ξ2 φ 1 (θ ξ1,ξ) λ ξ 3 λ ξ N, N < 0. Definition 5. The Pimsner algebra O E of the pair (φ, E) is the smallest subalgebra of L(E ) which contains the operators S ξ : E E for all ξ E. Pimsner: universality of O E

There is a natural inclusion B O E a generalized principal circle bundle roughly: as a vector space O E E and E φ N η ηλ N, λ U(1) Two natural classes in KK-theory: 1. the class [E] KK 0 (B, B) of the even Kasparov module (E, φ, 0) (with trivial grading) the map 1 [E] has the role of the Euler class χ(e) := 1 [E] of the line bundle E over the noncommutative space B

2. the class [ ] KK 1 (O E, B) of the odd Kasparov module (E, φ, F ): F := 2P 1 L(E ) of the projection P : E E with and inclusion φ : O E L(E ). Im(P ) = ( N=0 E φ N ) E The Kasparov product induces group homomorphisms [E] : K (B) K (B), [E] : K (B) K (B) and [ ] : K (O E ) K +1 (B), [ ] : K (B) K +1 (O E ),

Associated six-terms exact sequences Gysin sequences: in K-theory: K 0 (B) [ ] K 1 (O E ) 1 [E] K 0 (B) i i K 1 (B) 1 [E] K 0 (O E ) [ ] ; K 1 (B) the corresponding one in K-homology: K 0 (B) 1 [E] K 0 (B) i K 0 (O E ) [ ] [ ]. K 1 (O E ) In fact in KK-theory i K 1 (B) 1 [E] K 1 (B)

The quantum spheres and the projective spaces The coordinate algebra O(S 2n+1 q ) of quantum sphere S 2n+1 -algebra generated by 2n + 2 elements {z i, zi } i=0,...,n s.t.: q : [z n, z n ] = 0, [z i, z i] = (1 q 2 ) z i z j = q 1 z j z i 0 i < j n, zi z j = qz j zi i j, n j=i+1 z j z j i = 0,..., n 1, and a sphere relation: 1 = z 0 z 0 + z 1z 1 +... + z nz n. L. Vaksman, Ya. Soibelman, 1991 ; M. Welk, 2000

The -subalgebra of O(S 2n+1 q ) generated by p ij := z i z j coordinate algebra O(CP n q ) of the quantum projective space CP n q Invariant elements for the U(1)-action on the algebra O(S 2n+1 q ): (z 0, z 1,..., z n ) (λz 0, λz 1,..., λz n ), λ U(1). the fibration S 2n+1 q CP n q is a quantum U(1)-principal bundle: O(CP n q ) = O(S 2n+1 q ) U(1) O(S 2n+1 q ).

The C -algebras C(S 2n+1 q ) and C(CP n q ) of continuous functions: completions of O(S 2n+1 q ) and O(CP n q ) in the universal C -norms these are graph algebras J.H. Hong, W. Szymański 2002 K 0 (CP n q ) Z n+1 K 0 (C(CP n q )) F. D Andrea, G. L. 2010 Generators of the homology group K 0 (C(CP n q )) given explicitly as (classes of) even Fredholm modules µ k = (O(CP n q ), H (k), π (k), γ (k), F (k) ), for 0 k n.

Generators of the K-theory K 0 (CP n q ) also given explicitly as projections whose entries are polynomial functions: line bundles & projections For N Z, vector-valued functions Ψ N := (ψ N j 0,...,j n ) s.t. Ψ N Ψ N = 1 P N := Ψ N Ψ N is a projection: P N M dn (O(CP n q )), d N := ( N + n), n Entries of P N are U(1)-invariant and so elements of O(CP n q )

Proposition 6. For all N N and for all 0 k n it holds that [µk ], [P N ] := Tr Hk (γ (k) (π (k) (Tr P N )) = ( N k ), [µ 0 ],..., [µ n ] are generators of K 0 (C(CP n q )), and [P 0 ],..., [P n ] are generators of K 0 (CP n q ) The matrix of couplings M M n+1 (Z) is invertible over Z: M ij := [µ i ], [P j ] = ( ) j i, (M 1 ) ij = ( 1) i+j( ) j i. These are bases of Z n+1 as Z-modules; they generate Z n+1 as an Abelian group.

The inclusion O(CP n q ) O(S 2n+1 q ) is a U(1) q.p.b. To a projection P N there corresponds an associated line bundle L N (O(CP n q )) d NP N P N (O(CP n q )) d N L N made of elements of O(S 2n+1 q ) transforming under U(1) as ϕ N ϕ N λ N, λ U(1) Each L N is indeed a bimodule over L 0 = O(CP n q ); the bimodule of equivariant maps for the IRREP of U(1) with weight N. Also, L N O(CP n q ) L M L N+M

Denote [P N ] = [L N ] in the group K 0 (CP n q ). The module L N is a line bundle, in the sense that its rank (as computed by pairing with [µ 0 ]) is equal to 1 Completely characterized by its first Chern number (as computed by pairing with the class [µ 1 ]): Proposition 7. For all N Z it holds that [µ 0 ], [L N ] = 1 and [µ 1 ], [L N ] = N.

The line bundle L 1 emerges as a central character: its only non-vanishing charges are [µ 0 ], [L 1 ] = 1 [µ 1 ], [L 1 ] = 1 L 1 is the tautological line bundle for CP n q, with Euler class u = χ([l 1 ]) := 1 [L 1 ]. Proposition 8. It holds that K 0 (CP n q ) Z[u]/u n+1 Z n+1. [µ k ] and ( u) j are dual bases of K-homology and K-theory

The quantum lens spaces Fix an integer r 2 and define O(L (n,r) q ) := N Z L rn. Proposition 9. O(L (n,r) q ) is a -algebra; all elements of O(Sq 2n+1 ) invariant under the action α r : Z r Aut(O(Sq 2n+1 )) of the cyclic group Z r : (z 0, z 1,..., z n ) (e 2πi/r z 0, e 2πi/r z 1,..., e 2πi/r z n ). The dual L (n,r) q : the quantum lens space of dimension 2n + 1 (and index r) There are algebra inclusions j : O(CP n q ) O(L (n,r) q ) O(S 2n+1 q ).

Pulling back line bundles Proposition 10. The algebra inclusion j : O(CP n q ) O(L (n,r) q ) is a quantum principal bundle with structure group Ũ(1) := U(1)/Z r : O(CP n q ) = O(L (n,r) q )Ũ(1). Then one can pull-back line bundles from CP n q to L(n,r) q. L N j L N O(L (n,r) q) O(CP n q ). j

Definition 11. For each L N an O(CP n q )-bimodule (a line bundle over CP n q ), its pull-back to L(n,r) q is the O(L (n,r) q )-bimodule L N = j (L N ) := O(L (n,r) q ) O(CP n q ) L N. The algebra inclusion j : O(CP n q ) O(L (n,r) q ) induces a map j : K 0 (CP n q ) K 0 (L (n,r) q )

Each L N over CP n q is not free when N 0, this need not be the case for L N over L (n,r) q : the pull-back L r of L r is tautologically free : L r = O(L (n,r) q ) L0 L r O(L (n,r) q ) = L 0. ( L N ) r L rn also has trivial class for any N Z L N define torsion classes; they generate the group K 0 (L (n,r) q )

Multiplying by the Euler class A second crucial ingredient α : K 0 (CP n q ) K 0(CP n q ), α is multiplication by χ(l r ) := 1 [L r ] the Euler class of the line bundle L r

Assembly these into an exact sequence, the Gysin sequence 0 K 1 (L (n,r) q ) K 0 (CP n q ) α K 0 (CP n q ) K 0 (L (n,r) q ) 0 0 K 1 (L (n,r) q ) Ind D K 0 (CP n q )... and... K 0 (L (n,r) q ) Ind D 0 Ind D comes from Kasparov theory

Some practical and important applications, notably, the computation of the K-theory of the quantum lens spaces L (n,r) q. Thus K 1 (L (n,r) q ) ker(α), K 0 (L (n,r) q ) coker(α). Moreover, geometric generators of the groups K 1 (L (n,r) q ) K 0 (L (n,r) q ) for the latter as pulled-back line bundles from CP n q to L (n,r) q Explicit generators as integral combinations of powers of the pull-back to the lens space L (n,r) q of the generator u := 1 [L 1 ]

The K-theory of quantum lens spaces Proposition 12. The (n + 1) (n + 1) matrix α has rank n: K 1 (C(L (n,r) q )) Z. On the other hand, the structure of the cokernel of the matrix A depends on the divisibility properties of the integer r. This leads to K 0 (L (n,r) q ) = Z Z/α 1 Z Z/α n Z. for suitable integers α 1,..., α n.

Example 13. For n = 1 K 0 (C(L (1,r) q )) = Z Z r. From definition [ L r ] = 1, thus L 1 generates the torsion part. Alternatively, from u 2 = 0 it follows that L j = (j 1) + jl 1 ; upon lifting to L q (1,r), for j = r this yields or 1 [ L 1 ] is cyclic of order r. r(1 [ L 1 ]) = 0

Example 14. If r = 2 L (n,2) q = Sq 2n+1 /Z 2 = RPq 2n+1, the quantum real projective space, we get K 0 (C(RPq 2n+1 )) = Z Z 2 n the generator 1 [ L 1 ] is cyclic with the correct order 2 n. Example 15. For n = 2 there are two cases. When r = 2k + 1: When r = 2k: r ũ = 0, r ũ 2 = 0, K 0 (L (2,r) q ) = Z Z r Z r 1 2 r (ũ 2 + 2 ũ) = 0, 2r ũ = 0, K 0 (C(L (2,r) q )) = Z Z r Z 2 2r

T-dual Pimsner algebras: a simple example 0 K 1 (L (1,r) q ) K 0 (CP 1 q ) 1 [L r] K 0 (CP 1 q ) K 0 (L (1,r) q ) 0 ker(1 [L r ]) =< u >=< 1 [L 1 ] > and K 1 (L (1,r) q ) h (h) = h(1 [L 1 ]) 1 [L h ] (1 [L r ])(1 [L h ]) = 0 = (1 [L h ])(1 [L r ])

The exactness of the dual sequence for 0 K 1 (L (1,h) q ) K 0 (CP 1 q ) 1 [L h] K 0 (CP 1 q ) K 0 (L (1,h) q ) 0 implies there exists a r K 1 (L q (1,r) ) such that K 1 (L (1,h) q ) r (r) = r(1 [L 1 ]) 1 [L r ] The couples ( L (1,r) q, h K 1 (L (1,r) ) ) and ( L (1,h) q q, r K 1 (L (1,h) q ) ) are T-dual

More generally : Quantum w. projective lines and lens spaces: B = O(W q (k, l)) = quantum weighted projective line the fixed point algebra for a weighted circle action on O(S 3 q ) z 0 λ k z 0, z 1 λ l z 1, λ U(1) The corresponding universal enveloping C -algebra C(W q (k, l)) does not in fact depend on the label k: isomorphic to the unitalization of l copies of K = compact operators on l 2 (N 0 ) C(W q (k, l)) = l s=0 K Then: K 0 (C(W q (k, l))) = Z l+1, K 1 (C(W q (k, l))) = 0 a partial resolution of singularity, since classically K 0 (C(W (k, l))) = Z 2.

O E = O(L q (lk; k, l)) = quantum lens space Indeed, a vector space decomposition O(L q (lk; k, l)) = N Z L n (k, l), with E = L 1 (k, l) a right finitely projective module L 1 (k, l) := (z 1 )k O(W q (k, l)) + (z 0 )l O(W q (k, l)) Also, O(L q (lk; k, l)) the fixes point algebra of a cyclic action Z/(lk)Z S 3 q S 3 q z 0 exp( 2πi ) z 0, z 1 exp( 2πi l k ) z 1.

K-theory and K-homology of quantum lens space Denote the diagonal inclusion by ι : Z Z l, 1 (1,..., 1) with transpose ι t : Z l Z, ι t (m 1,..., m l ) = m 1 +... + m l. Proposition 16. (Arici, Kaad, L.) With k, l N coprime: K 0 ( Lq (lk; k, l)) ) coker(1 E) Z ( Z l /Im(ι) ) as well as K 1 ( Lq (lk; k, l)) ) ker(1 E) Z l K 0( L q (lk; k, l)) ) ker(1 E t ) Z ( ker(ι t ) ) K 1( L q (lk; k, l)) ) coker(1 E t ) Z l. Again there is no dependence on the label k.

grand motivations / applications : Gauge fields on noncommutative spaces T-duality for noncommutative spaces Chern-Simons theory A Gysin sequence for U(1)-bundles relates H-flux (three-forms on the total space E) to line bundles (two-forms on the base space M) also giving an isomorphism between Dixmier-Douady classes on E and line bundles on M

Summing up: many nice and elegant and useful geometry structures hope you enjoyed it Thank you!!