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 September 1, 2009 09GENCO: Noncommutative Geometry and Quantum Physics (Vietri sul Mare) Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 1 / 23
Introduction s sphere S ✷ q = SU q ( ✷ ) /U ( ✶ ) (here ✵ < q < ✶ ). A case study: the standard Podle´ L. D ˛ abrowski – A. Sitarz Dirac operator on the standard Podle´ s quantum sphere Banach Center Publ. 61 (2003), 49–58. K. Schmüdgen – E. Wagner Dirac operator and a twisted cyclic cocycle on the standard Podle´ s quantum sphere J. Reine Angew. M. 574 (2004), 219–235. R. Oeckl Braided Quantum Field Theory CMP 217 (2001) 451–473. Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 2 / 23
Introduction s sphere S ✷ q = SU q ( ✷ ) /U ( ✶ ) (here ✵ < q < ✶ ). A case study: the standard Podle´ L. D ˛ abrowski – A. Sitarz Dirac operator on the standard Podle´ s quantum sphere Banach Center Publ. 61 (2003), 49–58. K. Schmüdgen – E. Wagner Dirac operator and a twisted cyclic cocycle on the standard Podle´ s quantum sphere J. Reine Angew. M. 574 (2004), 219–235. R. Oeckl Braided Quantum Field Theory CMP 217 (2001) 451–473. Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 2 / 23
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) September 1, 2009 2 / 23
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) September 1, 2009 2 / 23
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) September 1, 2009 2 / 23
Outline 1 Preliminary definitions The quantum SU ( ℓ + ✶ ) group The QUEA U q (su( ℓ + ✶ )) S ✷ ℓ + ✶ and CP ℓ q q 2 K-theory and K-homology K-theory K-homology 3 Antiholomorphic forms and real spectral triples The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator 4 From GDAs to spectral triples Reality and the first order condition A family of spectral triples Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 3 / 23
Outline 1 Preliminary definitions The quantum SU ( ℓ + ✶ ) group The QUEA U q (su( ℓ + ✶ )) S ✷ ℓ + ✶ and CP ℓ q q 2 K-theory and K-homology K-theory K-homology 3 Antiholomorphic forms and real spectral triples The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator 4 From GDAs to spectral triples Reality and the first order condition A family of spectral triples Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 3 / 23
Outline 1 Preliminary definitions The quantum SU ( ℓ + ✶ ) group The QUEA U q (su( ℓ + ✶ )) S ✷ ℓ + ✶ and CP ℓ q q 2 K-theory and K-homology K-theory K-homology 3 Antiholomorphic forms and real spectral triples The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator 4 From GDAs to spectral triples Reality and the first order condition A family of spectral triples Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 3 / 23
Outline 1 Preliminary definitions The quantum SU ( ℓ + ✶ ) group The QUEA U q (su( ℓ + ✶ )) S ✷ ℓ + ✶ and CP ℓ q q 2 K-theory and K-homology K-theory K-homology 3 Antiholomorphic forms and real spectral triples The quantum Grassmann algebra The algebra of forms Vector fields and the Dolbeault operator 4 From GDAs to spectral triples Reality and the first order condition A family of spectral triples Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 3 / 23
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 � u i j u k l = u k l u i p ∈ S ℓ + ✶ (− ✶ ) || p || u ✶ p ( ✶ ) u ✷ p ( ✷ ) ✳ ✳ ✳ u ℓ + ✶ (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 � p ( n ✷ ) ✳ ✳ ✳ u k ℓ ( u i p ∈ S ℓ (− q ) || p || u k ✶ p ( n ✶ ) u k ✷ p ( n ℓ ) Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 4 / 23
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 � u i j u k l = u k l u i p ∈ S ℓ + ✶ (− ✶ ) || p || u ✶ p ( ✶ ) u ✷ p ( ✷ ) ✳ ✳ ✳ u ℓ + ✶ (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 � p ( n ✷ ) ✳ ✳ ✳ u k ℓ ( u i p ∈ S ℓ (− q ) || p || u k ✶ p ( n ✶ ) u k ✷ p ( n ℓ ) Geometry of quantum CP ℓ Francesco D’Andrea (UCL) September 1, 2009 4 / 23
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