Quantum Complex Projective Spaces Fredholm modules, K-theory, - PowerPoint PPT Presentation

Quantum Complex Projective Spaces Fredholm modules, K-theory, spectral triples Francesco D’Andrea Université Catholique de Louvain Chemin du Cyclotron 2, Louvain-La-Neuve, Belgium MFO, 8 September 2009 Joint work with G. Landi and L. D ˛ abrowski Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 1 / 17

Introduction s sphere S ✷ q = SU q ( ✷ ) /U ( ✶ ) (here ✵ < q < ✶ ). A case study: the standard Podle´ ◮ L. D ˛ abrowski – A. Sitarz Prescribed Hilbert space + Dirac operator on the standard Podle´ s SU q ( ✷ ) equivariance = unique quantum sphere real spectral triple (modulo Banach Center Publ. 61 (2003), 49–58. equivalences). � � K. Schmüdgen – E. Wagner Spectrum ( D ) = ± [ n ] q n � ✶ Dirac operator and a twisted cyclic cocycle with [ n ] q := q n − q − n q − q − ✶ . on the standard Podle´ s quantum sphere The spectrum of D diverges J. Reine Angew. M. 574 (2004), 219–235. exponentially � the resolvent ( D ✷ + m ✷ ) − ✶ of the Laplacian is R. Oeckl of trace class. Braided Quantum Field Theory CMP 217 (2001) 451–473. Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 2 / 17

Introduction s sphere S ✷ q = SU q ( ✷ ) /U ( ✶ ) (here ✵ < q < ✶ ). A case study: the standard Podle´ L. D ˛ abrowski – A. Sitarz The representation in DS Dirac operator on the standard Podle´ s spectral triple is the direct sum quantum sphere of two copies of the left regular Banach Center Publ. 61 (2003), 49–58. representation. ◮ K. Schmüdgen – E. Wagner Generators of U q (su( ✷ )) are Dirac operator and a twisted cyclic cocycle (external) derivations on S ✷ q . on the standard Podle´ s quantum sphere With these one constructs D . J. Reine Angew. M. 574 (2004), 219–235. D ✷ is proportional to the Casimir of U q (su( ✷ )) : this explains why R. Oeckl eigenv. diverge exponentially. Braided Quantum Field Theory CMP 217 (2001) 451–473. Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 2 / 17

Introduction s sphere S ✷ q = SU q ( ✷ ) /U ( ✶ ) (here ✵ < q < ✶ ). A case study: the standard Podle´ L. D ˛ abrowski – A. Sitarz On S ✷ q the tadpole diagram Dirac operator on the standard Podle´ s – the only basic divergence of quantum sphere φ ✹ theory in 2D – becomes Banach Center Publ. 61 (2003), 49–58. finite at q � = ✶ . K. Schmüdgen – E. Wagner � Reason: the propagator ( D ✷ + m ✷ ) − ✶ is of trace class. Dirac operator and a twisted cyclic cocycle on the standard Podle´ s quantum sphere Regularization of QFT with J. Reine Angew. M. 574 (2004), 219–235. quantum groups symmetries: what about higher dimensional ◮ R. Oeckl spaces? Braided Quantum Field Theory CMP 217 (2001) 451–473. Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 2 / 17

Geometry of quantum projective spaces Some results: ◮ generators of K -homology and K -theory groups, computation of the pairing [DL09b]: q ) ։ A ( CP ℓ − ✶ ◮ by induction, using A ( CP ℓ ) ; q ◮ Fredholm modules are ‘conformal classes’ of spectral triples (regular, in general they are not real/equivariant, arbitrary summability n ∈ R + ); ◮ family of U q (su( ℓ + ✶ )) -equivariant spectral triples [DD09]: ◮ for generic ℓ , ✵ + -dimensional equivariant even spectral triples labelled by N ∈ Z ; ◮ if ℓ is odd and N = ✶ ✷ ( ℓ + ✶ ) , the spectral triple is real with KO-dimension ✷ ℓ ♠♦❞ ✽ . ◮ there is a map τ : K U ✵ ( A ) → HC U n ( A ) . With the differential calculus associated to an equivariant spectral triple one can construct twisted Hochschild cocycle that can be ✵ ( A ) ⊃ Z ∞ if q is transcendental. paired with τ ([ p ]) . As a byproduct, we prove that K U Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 3 / 17

The quantum SU ( ℓ + ✶ ) group Let ℓ > ✶ . For G := SU ( ℓ + ✶ ) , the functions u i u i j ( g ) := g i j : G → C , , generate j a Hopf ∗ -algebra A ( G ) . As abstract ∗ -algebra it is defined by the relations � p ∈ S ℓ + ✶ (− ✶ ) || p || u ✶ p ( ✶ ) u ✷ p ( ✷ ) ✳ ✳ ✳ u ℓ + ✶ u i j u k l = u k l u i (1) j ✱ p ( ℓ + ✶ ) = ✶ ✱ where || p || = length of the permutation p ∈ S ℓ + ✶ , and with ∗ -structure j ) ∗ = (− ✶ ) j − i � ( u i p ∈ S ℓ (− ✶ ) || p || u k ✶ p ( n ✶ ) u k ✷ p ( n ✷ ) ✳ ✳ ✳ u k ℓ (2) p ( n ℓ ) where { k ✶ ✱ ✳ ✳ ✳ ✱ k ℓ } = { ✶✱ ✳ ✳ ✳ ✱ ℓ + ✶ } � { i } and { n ✶ ✱ ✳ ✳ ✳ ✱ n ℓ } = { ✶✱ ✳ ✳ ✳ ✱ ℓ + ✶ } � { j } (as ordered sets). Coproduct, counit and antipode are of ‘matrix type’ � i ) ∗ ✳ ∆ ( u i k u i k ⊗ u k ε ( u i j ) = δ i S ( u i j ) = ( u j j ) = j ✱ j ✱ Similarly coproduct, counit and antipode of A ( G q ) , ✵ < q < ✶ , are given by the same formulas above, while (1) and (2) becomes: � R ij n = u j kl ( q ) u k m u l l u i k R kl p ∈ S ℓ + ✶ (− q ) || p || u ✶ p ( ✶ ) u ✷ p ( ✷ ) ✳ ✳ ✳ u ℓ + ✶ mn ( q ) ✱ p ( ℓ + ✶ ) = ✶ ✱ j ) ∗ = (− q ) j − i � ( u i p ∈ S ℓ (− q ) || p || u k ✶ p ( n ✶ ) u k ✷ p ( n ✷ ) ✳ ✳ ✳ u k ℓ p ( n ℓ ) Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 4 / 17

The QUEA U q (su( ℓ + ✶ )) Symmetries are described by the Hopf ∗ -algebra U q ( su ( ℓ + ✶ )) , generated by { K i ✱ K − ✶ i ✱ E i ✱ F i } i = ✶✱✳✳✳✱ ℓ , with K i = K ∗ i ✱ F i = E ∗ i , and with some relations. . . � with the rescaling K i = q H i , at the ✵ -th order in � h := ❧♦❣ q one gets Serre’s � presentation of U (su( ℓ + ✶ )) . Coproduct, counit and antipode are given by (with i = ✶✱ ✳ ✳ ✳ ✱ ℓ ) ∆ ( E i ) = E i ⊗ K i + K − ✶ ∆ ( K i ) = K i ⊗ K i ✱ ⊗ E i ✱ i S ( K i ) = K − ✶ ε ( K i ) = ✶ ✱ ε ( E i ) = ✵ ✱ ✱ S ( E i ) = − qE i ✳ i The Hopf ∗ -subalgebra with generators { K i ✱ E i ✱ F i } i = ✶✱✷✱✳✳✳✱ ℓ − ✶ is U q (su( ℓ )) ; its central extension by K ✶ K ✷ ✷ ✳ ✳ ✳ K ℓ ℓ and its inverse gives U q (u( ℓ )) . There is a non-degenerate dual pairing A ( SU q ( ℓ + ✶ )) × U q (su( ℓ + ✶ )) → C . The algebra A ( SU q ( ℓ + ✶ )) is a U q (su( ℓ + ✶ )) -bimodule ∗ -algebra for the left ⊲ and right ⊳ canonical actions. Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 5 / 17

CP ℓ q and homogeneous vector bundles The algebra of ‘functions’ on CP ℓ q is the left U q (su( ℓ + ✶ )) -module ∗ -algebra q ) := A ( SU q ( ℓ + ✶ )) U q (u( ℓ )) ✳ A ( CP ℓ Let σ : U q (u( ℓ )) → ❊♥❞ ( C n ) be a ∗ -representation. The set M ( σ ) = A ( SU q ( ℓ + ✶ )) ⊠ σ C n � � � σ ( x ( ✷ ) ) v ⊳ S − ✶ ( x ( ✶ ) ) = ǫ ( x ) v ∀ x ∈ U q (u( ℓ )) v ∈ A ( SU q ( ℓ + ✶ )) n � := ✱ is an A ( CP ℓ q ) -bimodule and a left A ( CP ℓ q ) ⋊ U q (su( ℓ + ✶ )) -module. It is the analogue of (sections of) an homogeneous vector bundle of rank n over CP ℓ q . A non-degenerate inner product is induced by the canonical one on A ( CP ℓ q ) n : � n i = ✶ h ( v ∗ � v ✱ w � = i w i ) M ( σ ) is projective as one-sided module. Given a differential calculus ( Ω • ✱ ❞ ) , a canonical connection (the Grassmannian connection ) can be obtained by projecting the flat one. Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 6 / 17

The case ℓ = ✷ From now on, let ℓ = ✷ . For short: A := A ( CP ✷ O := A ( SU q ( ✸ )) ✱ q ) ✱ U := U q (su( ✸ )) ✱ K := U q (u( ✷ )) ✳ Recall: O is an U -bimodule ∗ -algebra, A = O K , K ⊂ U generated by K ✶ ✱ K ✷ ✱ E ✶ ✱ F ✶ . In general for a ✱ b ∈ O and x ∈ U , ( ab ) ⊳ x = ( a ⊳ x ( ✶ ) )( b ⊳ x ( ✷ ) ) where ∆x = x ( ✶ ) ⊗ x ( ✷ ) . Thus for example ∆ ( E ✷ ) = E ✷ ⊗ K ✷ + K − ✶ ⊗ E ✷ ✱ ✷ and the map ◦ ⊳ E ✷ : O → O is not a derivation. But if a ∈ A , then a ⊳ K ✷ = a and ( ab ) ⊳ E ✷ = ( a ⊳ E ✷ ) b + a ( b ⊳ E ✷ ) ✱ ∀ a ✱ b ∈ A ✳ Therefore ◦ ⊳ E ✷ is a derivation on A (an ‘exterior’ derivation, since A ⊳ E ✷ / ∈ A ). This and similar considerations allow to construct a differential calculus on A . Geometry of quantum CP ℓ Francesco D’Andrea (UCL) MFO, 8 September 2009 7 / 17