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Twisted Deformation Quantization of Algebraic Varieties Amnon Yekutieli Department of Mathematics Ben Gurion University Notes available at http://www.math.bgu.ac.il/ amyekut/lectures written 17 June 2009; corrected 11 Sep 2010 Amnon

  1. 1. Some background on Deformation Quantization Let K [[ � ]] be the ring of formal power series in the variable � . Let C [[ � ]] be the set of formal power series with coefficients in C , which we view only as a K [[ � ]] -module. A star product on C [[ � ]] is a function ⋆ : C [[ � ]] × C [[ � ]] → C [[ � ]] which makes C [[ � ]] into an associative K [[ � ]] -algebra, with unit 1 ∈ C , and such that f ⋆ g ≡ fg mod � for any g , f ∈ C . The pair � � is called an associative deformation of C . C [[ � ]] , ⋆ Amnon Yekutieli (BGU) Deformation Quantization 4 / 33

  2. 1. Some background on Deformation Quantization Suppose � � is an associative deformation of C . C [[ � ]] , ⋆ Example 1.1. Given f , g ∈ C , we know that f ⋆ g − g ⋆ f ≡ 0 mod � . Hence there is a unique element { f , g } ⋆ ∈ C such that 1 � � ≡ { f , g } ⋆ mod � . f ⋆ g − g ⋆ f � It is quite easy to show that {− , −} ⋆ is a Poisson bracket on C . We call it the first order bracket of ⋆ . Amnon Yekutieli (BGU) Deformation Quantization 5 / 33

  3. 1. Some background on Deformation Quantization Suppose � � is an associative deformation of C . C [[ � ]] , ⋆ Example 1.1. Given f , g ∈ C , we know that f ⋆ g − g ⋆ f ≡ 0 mod � . Hence there is a unique element { f , g } ⋆ ∈ C such that 1 � � ≡ { f , g } ⋆ mod � . f ⋆ g − g ⋆ f � It is quite easy to show that {− , −} ⋆ is a Poisson bracket on C . We call it the first order bracket of ⋆ . Amnon Yekutieli (BGU) Deformation Quantization 5 / 33

  4. 1. Some background on Deformation Quantization Suppose � � is an associative deformation of C . C [[ � ]] , ⋆ Example 1.1. Given f , g ∈ C , we know that f ⋆ g − g ⋆ f ≡ 0 mod � . Hence there is a unique element { f , g } ⋆ ∈ C such that 1 � � ≡ { f , g } ⋆ mod � . f ⋆ g − g ⋆ f � It is quite easy to show that {− , −} ⋆ is a Poisson bracket on C . We call it the first order bracket of ⋆ . Amnon Yekutieli (BGU) Deformation Quantization 5 / 33

  5. 1. Some background on Deformation Quantization Suppose � � is an associative deformation of C . C [[ � ]] , ⋆ Example 1.1. Given f , g ∈ C , we know that f ⋆ g − g ⋆ f ≡ 0 mod � . Hence there is a unique element { f , g } ⋆ ∈ C such that 1 � � ≡ { f , g } ⋆ mod � . f ⋆ g − g ⋆ f � It is quite easy to show that {− , −} ⋆ is a Poisson bracket on C . We call it the first order bracket of ⋆ . Amnon Yekutieli (BGU) Deformation Quantization 5 / 33

  6. 1. Some background on Deformation Quantization Deformation quantization seeks to reverse Example 1.1. Definition 1.2. Given a Poisson bracket {− , −} on the algebra C , a deformation quantization of {− , −} is an associative deformation � � C [[ � ]] , ⋆ of C whose first order bracket is {− , −} . Amnon Yekutieli (BGU) Deformation Quantization 6 / 33

  7. 1. Some background on Deformation Quantization Deformation quantization seeks to reverse Example 1.1. Definition 1.2. Given a Poisson bracket {− , −} on the algebra C , a deformation quantization of {− , −} is an associative deformation � � C [[ � ]] , ⋆ of C whose first order bracket is {− , −} . Amnon Yekutieli (BGU) Deformation Quantization 6 / 33

  8. 1. Some background on Deformation Quantization In physics � is the Planck constant. For a quantum phenomenon depending on � , the limit as � → 0 is thought of the as the classical limit of this phenomenon. The original idea by the physicists Flato et. al. ([BFFLS], 1978) was that deformation quantization should model the transition from classical Hamiltonian mechanics to quantum mechanics. Special cases (like the Moyal product) were known. The problem arose: does any Poisson bracket admit a deformation quantization? For a symplectic manifold X and C = C ∞ ( X ) the problem was solved by De Wilde and Lecomte ([DL], 1983). A more geometric solution was discovered by Fedosov ([Fe], 1994). The general case, i.e. C = C ∞ ( X ) for a Poisson manifold X , was solved by Kontsevich ([Ko1], 1997). See surveys in the book [CKTB]. Amnon Yekutieli (BGU) Deformation Quantization 7 / 33

  9. 1. Some background on Deformation Quantization In physics � is the Planck constant. For a quantum phenomenon depending on � , the limit as � → 0 is thought of the as the classical limit of this phenomenon. The original idea by the physicists Flato et. al. ([BFFLS], 1978) was that deformation quantization should model the transition from classical Hamiltonian mechanics to quantum mechanics. Special cases (like the Moyal product) were known. The problem arose: does any Poisson bracket admit a deformation quantization? For a symplectic manifold X and C = C ∞ ( X ) the problem was solved by De Wilde and Lecomte ([DL], 1983). A more geometric solution was discovered by Fedosov ([Fe], 1994). The general case, i.e. C = C ∞ ( X ) for a Poisson manifold X , was solved by Kontsevich ([Ko1], 1997). See surveys in the book [CKTB]. Amnon Yekutieli (BGU) Deformation Quantization 7 / 33

  10. 1. Some background on Deformation Quantization In physics � is the Planck constant. For a quantum phenomenon depending on � , the limit as � → 0 is thought of the as the classical limit of this phenomenon. The original idea by the physicists Flato et. al. ([BFFLS], 1978) was that deformation quantization should model the transition from classical Hamiltonian mechanics to quantum mechanics. Special cases (like the Moyal product) were known. The problem arose: does any Poisson bracket admit a deformation quantization? For a symplectic manifold X and C = C ∞ ( X ) the problem was solved by De Wilde and Lecomte ([DL], 1983). A more geometric solution was discovered by Fedosov ([Fe], 1994). The general case, i.e. C = C ∞ ( X ) for a Poisson manifold X , was solved by Kontsevich ([Ko1], 1997). See surveys in the book [CKTB]. Amnon Yekutieli (BGU) Deformation Quantization 7 / 33

  11. 1. Some background on Deformation Quantization In physics � is the Planck constant. For a quantum phenomenon depending on � , the limit as � → 0 is thought of the as the classical limit of this phenomenon. The original idea by the physicists Flato et. al. ([BFFLS], 1978) was that deformation quantization should model the transition from classical Hamiltonian mechanics to quantum mechanics. Special cases (like the Moyal product) were known. The problem arose: does any Poisson bracket admit a deformation quantization? For a symplectic manifold X and C = C ∞ ( X ) the problem was solved by De Wilde and Lecomte ([DL], 1983). A more geometric solution was discovered by Fedosov ([Fe], 1994). The general case, i.e. C = C ∞ ( X ) for a Poisson manifold X , was solved by Kontsevich ([Ko1], 1997). See surveys in the book [CKTB]. Amnon Yekutieli (BGU) Deformation Quantization 7 / 33

  12. 1. Some background on Deformation Quantization In physics � is the Planck constant. For a quantum phenomenon depending on � , the limit as � → 0 is thought of the as the classical limit of this phenomenon. The original idea by the physicists Flato et. al. ([BFFLS], 1978) was that deformation quantization should model the transition from classical Hamiltonian mechanics to quantum mechanics. Special cases (like the Moyal product) were known. The problem arose: does any Poisson bracket admit a deformation quantization? For a symplectic manifold X and C = C ∞ ( X ) the problem was solved by De Wilde and Lecomte ([DL], 1983). A more geometric solution was discovered by Fedosov ([Fe], 1994). The general case, i.e. C = C ∞ ( X ) for a Poisson manifold X , was solved by Kontsevich ([Ko1], 1997). See surveys in the book [CKTB]. Amnon Yekutieli (BGU) Deformation Quantization 7 / 33

  13. 2. Poisson Deformations of Algebraic Varieties 2. Poisson Deformations of Algebraic Varieties In algebraic geometry we have to consider deformations as sheaves. Let X be a smooth algebraic variety over K , with structure sheaf O X . We view O X as a Poisson K -algebra with zero bracket. Definition 2.1. A Poisson deformation of O X is a sheaf A of flat, � -adically complete, commutative Poisson K [[ � ]] -algebras on X , with an isomorphism of Poisson algebras ψ : A / ( � ) ≃ → O X , called an augmentation. Amnon Yekutieli (BGU) Deformation Quantization 8 / 33

  14. 2. Poisson Deformations of Algebraic Varieties 2. Poisson Deformations of Algebraic Varieties In algebraic geometry we have to consider deformations as sheaves. Let X be a smooth algebraic variety over K , with structure sheaf O X . We view O X as a Poisson K -algebra with zero bracket. Definition 2.1. A Poisson deformation of O X is a sheaf A of flat, � -adically complete, commutative Poisson K [[ � ]] -algebras on X , with an isomorphism of Poisson algebras ψ : A / ( � ) ≃ → O X , called an augmentation. Amnon Yekutieli (BGU) Deformation Quantization 8 / 33

  15. 2. Poisson Deformations of Algebraic Varieties 2. Poisson Deformations of Algebraic Varieties In algebraic geometry we have to consider deformations as sheaves. Let X be a smooth algebraic variety over K , with structure sheaf O X . We view O X as a Poisson K -algebra with zero bracket. Definition 2.1. A Poisson deformation of O X is a sheaf A of flat, � -adically complete, commutative Poisson K [[ � ]] -algebras on X , with an isomorphism of Poisson algebras ψ : A / ( � ) ≃ → O X , called an augmentation. Amnon Yekutieli (BGU) Deformation Quantization 8 / 33

  16. 2. Poisson Deformations of Algebraic Varieties 2. Poisson Deformations of Algebraic Varieties In algebraic geometry we have to consider deformations as sheaves. Let X be a smooth algebraic variety over K , with structure sheaf O X . We view O X as a Poisson K -algebra with zero bracket. Definition 2.1. A Poisson deformation of O X is a sheaf A of flat, � -adically complete, commutative Poisson K [[ � ]] -algebras on X , with an isomorphism of Poisson algebras ψ : A / ( � ) ≃ → O X , called an augmentation. Amnon Yekutieli (BGU) Deformation Quantization 8 / 33

  17. 2. Poisson Deformations of Algebraic Varieties 2. Poisson Deformations of Algebraic Varieties In algebraic geometry we have to consider deformations as sheaves. Let X be a smooth algebraic variety over K , with structure sheaf O X . We view O X as a Poisson K -algebra with zero bracket. Definition 2.1. A Poisson deformation of O X is a sheaf A of flat, � -adically complete, commutative Poisson K [[ � ]] -algebras on X , with an isomorphism of Poisson algebras ψ : A / ( � ) ≃ → O X , called an augmentation. Amnon Yekutieli (BGU) Deformation Quantization 8 / 33

  18. 2. Poisson Deformations of Algebraic Varieties A gauge equivalence A → A ′ between Poisson deformations is a K [[ � ]] -linear isomorphism of sheaves of Poisson algebras, that commutes with the augmentations to O X . Given a Poisson deformation A of O X , we may define the first order bracket {− , −} A : O X × O X → O X . This is a Poisson bracket whose formula is � 1 � { ˜ � { f , g } A := ψ f , ˜ g } , where f , g ∈ O X are local sections, and ˜ g ∈ A are arbitrary local lifts. f , ˜ The first order bracket is invariant under gauge equivalence. Amnon Yekutieli (BGU) Deformation Quantization 9 / 33

  19. 2. Poisson Deformations of Algebraic Varieties A gauge equivalence A → A ′ between Poisson deformations is a K [[ � ]] -linear isomorphism of sheaves of Poisson algebras, that commutes with the augmentations to O X . Given a Poisson deformation A of O X , we may define the first order bracket {− , −} A : O X × O X → O X . This is a Poisson bracket whose formula is � 1 � { ˜ � { f , g } A := ψ f , ˜ g } , where f , g ∈ O X are local sections, and ˜ g ∈ A are arbitrary local lifts. f , ˜ The first order bracket is invariant under gauge equivalence. Amnon Yekutieli (BGU) Deformation Quantization 9 / 33

  20. 2. Poisson Deformations of Algebraic Varieties A gauge equivalence A → A ′ between Poisson deformations is a K [[ � ]] -linear isomorphism of sheaves of Poisson algebras, that commutes with the augmentations to O X . Given a Poisson deformation A of O X , we may define the first order bracket {− , −} A : O X × O X → O X . This is a Poisson bracket whose formula is � 1 � { ˜ � { f , g } A := ψ f , ˜ g } , where f , g ∈ O X are local sections, and ˜ g ∈ A are arbitrary local lifts. f , ˜ The first order bracket is invariant under gauge equivalence. Amnon Yekutieli (BGU) Deformation Quantization 9 / 33

  21. 2. Poisson Deformations of Algebraic Varieties A gauge equivalence A → A ′ between Poisson deformations is a K [[ � ]] -linear isomorphism of sheaves of Poisson algebras, that commutes with the augmentations to O X . Given a Poisson deformation A of O X , we may define the first order bracket {− , −} A : O X × O X → O X . This is a Poisson bracket whose formula is � 1 � { ˜ � { f , g } A := ψ f , ˜ g } , where f , g ∈ O X are local sections, and ˜ g ∈ A are arbitrary local lifts. f , ˜ The first order bracket is invariant under gauge equivalence. Amnon Yekutieli (BGU) Deformation Quantization 9 / 33

  22. 2. Poisson Deformations of Algebraic Varieties Let {− , −} 1 be some Poisson bracket on O X . Example 2.2. Define A := O X [[ � ]] . This is a sheaf of K [[ � ]] -algebras, with the usual commutative multiplication, and the obvious augmentation A / ( � ) ∼ = O X . Put on A the K [[ � ]] -bilinear Poisson bracket {− , −} such that { f , g } = � { f , g } 1 for f , g ∈ O X . Then A is a Poisson deformation of O X . The first order bracket in this case is just {− , −} A = {− , −} 1 . Amnon Yekutieli (BGU) Deformation Quantization 10 / 33

  23. 2. Poisson Deformations of Algebraic Varieties Let {− , −} 1 be some Poisson bracket on O X . Example 2.2. Define A := O X [[ � ]] . This is a sheaf of K [[ � ]] -algebras, with the usual commutative multiplication, and the obvious augmentation A / ( � ) ∼ = O X . Put on A the K [[ � ]] -bilinear Poisson bracket {− , −} such that { f , g } = � { f , g } 1 for f , g ∈ O X . Then A is a Poisson deformation of O X . The first order bracket in this case is just {− , −} A = {− , −} 1 . Amnon Yekutieli (BGU) Deformation Quantization 10 / 33

  24. 2. Poisson Deformations of Algebraic Varieties Let {− , −} 1 be some Poisson bracket on O X . Example 2.2. Define A := O X [[ � ]] . This is a sheaf of K [[ � ]] -algebras, with the usual commutative multiplication, and the obvious augmentation A / ( � ) ∼ = O X . Put on A the K [[ � ]] -bilinear Poisson bracket {− , −} such that { f , g } = � { f , g } 1 for f , g ∈ O X . Then A is a Poisson deformation of O X . The first order bracket in this case is just {− , −} A = {− , −} 1 . Amnon Yekutieli (BGU) Deformation Quantization 10 / 33

  25. 2. Poisson Deformations of Algebraic Varieties Let {− , −} 1 be some Poisson bracket on O X . Example 2.2. Define A := O X [[ � ]] . This is a sheaf of K [[ � ]] -algebras, with the usual commutative multiplication, and the obvious augmentation A / ( � ) ∼ = O X . Put on A the K [[ � ]] -bilinear Poisson bracket {− , −} such that { f , g } = � { f , g } 1 for f , g ∈ O X . Then A is a Poisson deformation of O X . The first order bracket in this case is just {− , −} A = {− , −} 1 . Amnon Yekutieli (BGU) Deformation Quantization 10 / 33

  26. 2. Poisson Deformations of Algebraic Varieties Poisson deformations are controlled by a sheaf of DG (differential graded) Lie algebras T poly , X , called the poly derivations. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 11 / 33

  27. 2. Poisson Deformations of Algebraic Varieties Poisson deformations are controlled by a sheaf of DG (differential graded) Lie algebras T poly , X , called the poly derivations. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 11 / 33

  28. 3. Associative Deformations of Algebraic Varieties 3. Associative Deformations of Algebraic Varieties Definition 3.1. An associative deformation of O X is a sheaf A of flat, � -adically complete, associative, unital K [[ � ]] -algebras on X , with an isomorphism of algebras ψ : A / ( � ) ≃ → O X , called an augmentation. There is a suitable notion of gauge equivalence between associative deformations. Amnon Yekutieli (BGU) Deformation Quantization 12 / 33

  29. 3. Associative Deformations of Algebraic Varieties 3. Associative Deformations of Algebraic Varieties Definition 3.1. An associative deformation of O X is a sheaf A of flat, � -adically complete, associative, unital K [[ � ]] -algebras on X , with an isomorphism of algebras ψ : A / ( � ) ≃ → O X , called an augmentation. There is a suitable notion of gauge equivalence between associative deformations. Amnon Yekutieli (BGU) Deformation Quantization 12 / 33

  30. 3. Associative Deformations of Algebraic Varieties 3. Associative Deformations of Algebraic Varieties Definition 3.1. An associative deformation of O X is a sheaf A of flat, � -adically complete, associative, unital K [[ � ]] -algebras on X , with an isomorphism of algebras ψ : A / ( � ) ≃ → O X , called an augmentation. There is a suitable notion of gauge equivalence between associative deformations. Amnon Yekutieli (BGU) Deformation Quantization 12 / 33

  31. 3. Associative Deformations of Algebraic Varieties Given an associative deformation A we may define the first order bracket {− , −} A : O X × O X → O X . The formula is � 1 � (˜ g ⋆ ˜ � { f , g } A := ψ f ⋆ ˜ g − ˜ f ) . The first order bracket is invariant under gauge equivalence. Note that both kinds of deformations – Poisson and associative – include as special cases the classical commutative deformations of O X . Associative deformations are controlled by a quasi-coherent sheaf of DG Lie algebras D poly , X , called the poly differential operators. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 13 / 33

  32. 3. Associative Deformations of Algebraic Varieties Given an associative deformation A we may define the first order bracket {− , −} A : O X × O X → O X . The formula is � 1 � (˜ g ⋆ ˜ � { f , g } A := ψ f ⋆ ˜ g − ˜ f ) . The first order bracket is invariant under gauge equivalence. Note that both kinds of deformations – Poisson and associative – include as special cases the classical commutative deformations of O X . Associative deformations are controlled by a quasi-coherent sheaf of DG Lie algebras D poly , X , called the poly differential operators. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 13 / 33

  33. 3. Associative Deformations of Algebraic Varieties Given an associative deformation A we may define the first order bracket {− , −} A : O X × O X → O X . The formula is � 1 � (˜ g ⋆ ˜ � { f , g } A := ψ f ⋆ ˜ g − ˜ f ) . The first order bracket is invariant under gauge equivalence. Note that both kinds of deformations – Poisson and associative – include as special cases the classical commutative deformations of O X . Associative deformations are controlled by a quasi-coherent sheaf of DG Lie algebras D poly , X , called the poly differential operators. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 13 / 33

  34. 3. Associative Deformations of Algebraic Varieties Given an associative deformation A we may define the first order bracket {− , −} A : O X × O X → O X . The formula is � 1 � (˜ g ⋆ ˜ � { f , g } A := ψ f ⋆ ˜ g − ˜ f ) . The first order bracket is invariant under gauge equivalence. Note that both kinds of deformations – Poisson and associative – include as special cases the classical commutative deformations of O X . Associative deformations are controlled by a quasi-coherent sheaf of DG Lie algebras D poly , X , called the poly differential operators. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 13 / 33

  35. 3. Associative Deformations of Algebraic Varieties Given an associative deformation A we may define the first order bracket {− , −} A : O X × O X → O X . The formula is � 1 � (˜ g ⋆ ˜ � { f , g } A := ψ f ⋆ ˜ g − ˜ f ) . The first order bracket is invariant under gauge equivalence. Note that both kinds of deformations – Poisson and associative – include as special cases the classical commutative deformations of O X . Associative deformations are controlled by a quasi-coherent sheaf of DG Lie algebras D poly , X , called the poly differential operators. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 13 / 33

  36. 3. Associative Deformations of Algebraic Varieties Given an associative deformation A we may define the first order bracket {− , −} A : O X × O X → O X . The formula is � 1 � (˜ g ⋆ ˜ � { f , g } A := ψ f ⋆ ˜ g − ˜ f ) . The first order bracket is invariant under gauge equivalence. Note that both kinds of deformations – Poisson and associative – include as special cases the classical commutative deformations of O X . Associative deformations are controlled by a quasi-coherent sheaf of DG Lie algebras D poly , X , called the poly differential operators. This is explained in Appendix A. Amnon Yekutieli (BGU) Deformation Quantization 13 / 33

  37. 4. Deformation Quantization 4. Deformation Quantization Kontsevich [Ko1] proved that any Poisson deformation of a real C ∞ manifold X can be canonically quantized. In this section we present an algebraic version of this result. But first a definition. Definition 4.1. Let A be a Poisson deformation of O X . A quantization of A is an associative deformation B , such that the first order brackets satisfy {− , −} B = {− , −} A . Recalling Example 2.2, we see that this definition captures the essence of deformation quantization, namely quantizing a Poisson bracket on O X . Amnon Yekutieli (BGU) Deformation Quantization 14 / 33

  38. 4. Deformation Quantization 4. Deformation Quantization Kontsevich [Ko1] proved that any Poisson deformation of a real C ∞ manifold X can be canonically quantized. In this section we present an algebraic version of this result. But first a definition. Definition 4.1. Let A be a Poisson deformation of O X . A quantization of A is an associative deformation B , such that the first order brackets satisfy {− , −} B = {− , −} A . Recalling Example 2.2, we see that this definition captures the essence of deformation quantization, namely quantizing a Poisson bracket on O X . Amnon Yekutieli (BGU) Deformation Quantization 14 / 33

  39. 4. Deformation Quantization 4. Deformation Quantization Kontsevich [Ko1] proved that any Poisson deformation of a real C ∞ manifold X can be canonically quantized. In this section we present an algebraic version of this result. But first a definition. Definition 4.1. Let A be a Poisson deformation of O X . A quantization of A is an associative deformation B , such that the first order brackets satisfy {− , −} B = {− , −} A . Recalling Example 2.2, we see that this definition captures the essence of deformation quantization, namely quantizing a Poisson bracket on O X . Amnon Yekutieli (BGU) Deformation Quantization 14 / 33

  40. 4. Deformation Quantization 4. Deformation Quantization Kontsevich [Ko1] proved that any Poisson deformation of a real C ∞ manifold X can be canonically quantized. In this section we present an algebraic version of this result. But first a definition. Definition 4.1. Let A be a Poisson deformation of O X . A quantization of A is an associative deformation B , such that the first order brackets satisfy {− , −} B = {− , −} A . Recalling Example 2.2, we see that this definition captures the essence of deformation quantization, namely quantizing a Poisson bracket on O X . Amnon Yekutieli (BGU) Deformation Quantization 14 / 33

  41. 4. Deformation Quantization 4. Deformation Quantization Kontsevich [Ko1] proved that any Poisson deformation of a real C ∞ manifold X can be canonically quantized. In this section we present an algebraic version of this result. But first a definition. Definition 4.1. Let A be a Poisson deformation of O X . A quantization of A is an associative deformation B , such that the first order brackets satisfy {− , −} B = {− , −} A . Recalling Example 2.2, we see that this definition captures the essence of deformation quantization, namely quantizing a Poisson bracket on O X . Amnon Yekutieli (BGU) Deformation Quantization 14 / 33

  42. 4. Deformation Quantization Theorem 4.2. ([Ye1]) Let K be a field containing R , and let X be a smooth affine algebraic variety over K . There is a canonical bijection quant : { Poisson deformations of O X } gauge equivalence → { associative deformations of O X } ≃ , gauge equivalence which is a quantization as defined above. Amnon Yekutieli (BGU) Deformation Quantization 15 / 33

  43. 4. Deformation Quantization Theorem 4.2. ([Ye1]) Let K be a field containing R , and let X be a smooth affine algebraic variety over K . There is a canonical bijection quant : { Poisson deformations of O X } gauge equivalence → { associative deformations of O X } ≃ , gauge equivalence which is a quantization as defined above. Amnon Yekutieli (BGU) Deformation Quantization 15 / 33

  44. 4. Deformation Quantization Theorem 4.2. ([Ye1]) Let K be a field containing R , and let X be a smooth affine algebraic variety over K . There is a canonical bijection quant : { Poisson deformations of O X } gauge equivalence → { associative deformations of O X } ≃ , gauge equivalence which is a quantization as defined above. Amnon Yekutieli (BGU) Deformation Quantization 15 / 33

  45. 4. Deformation Quantization Theorem 4.2. ([Ye1]) Let K be a field containing R , and let X be a smooth affine algebraic variety over K . There is a canonical bijection quant : { Poisson deformations of O X } gauge equivalence → { associative deformations of O X } ≃ , gauge equivalence which is a quantization as defined above. Amnon Yekutieli (BGU) Deformation Quantization 15 / 33

  46. 4. Deformation Quantization By “canonical” I mean that this quantization map commutes with étale morphisms X ′ → X . Actually our result in [Ye1] is stronger – it holds for a wider class of varieties, not just affine varieties. However all these cases are subsumed in Corollary 6.2 below. Theorem 4.2 is a consequence of the following more general result. Amnon Yekutieli (BGU) Deformation Quantization 16 / 33

  47. 4. Deformation Quantization By “canonical” I mean that this quantization map commutes with étale morphisms X ′ → X . Actually our result in [Ye1] is stronger – it holds for a wider class of varieties, not just affine varieties. However all these cases are subsumed in Corollary 6.2 below. Theorem 4.2 is a consequence of the following more general result. Amnon Yekutieli (BGU) Deformation Quantization 16 / 33

  48. 4. Deformation Quantization By “canonical” I mean that this quantization map commutes with étale morphisms X ′ → X . Actually our result in [Ye1] is stronger – it holds for a wider class of varieties, not just affine varieties. However all these cases are subsumed in Corollary 6.2 below. Theorem 4.2 is a consequence of the following more general result. Amnon Yekutieli (BGU) Deformation Quantization 16 / 33

  49. � � 4. Deformation Quantization Theorem 4.3. ([Ye1]) Let K be a field containing R , and let X be a smooth algebraic variety over K . Then there is a diagram T poly , X D poly , X � Mix ( D poly , X ) Mix ( T poly , X ) where: ◮ Mix ( T poly , X ) and Mix ( D poly , X ) are sheaves of DG Lie algebras on X, called mixed resolutions ; ◮ the vertical arrows are DG Lie algebra quasi-isomorphisms; ◮ and the horizontal arrow is an L ∞ quasi-isomorphism. Amnon Yekutieli (BGU) Deformation Quantization 17 / 33

  50. � � 4. Deformation Quantization Theorem 4.3. ([Ye1]) Let K be a field containing R , and let X be a smooth algebraic variety over K . Then there is a diagram T poly , X D poly , X � Mix ( D poly , X ) Mix ( T poly , X ) where: ◮ Mix ( T poly , X ) and Mix ( D poly , X ) are sheaves of DG Lie algebras on X, called mixed resolutions ; ◮ the vertical arrows are DG Lie algebra quasi-isomorphisms; ◮ and the horizontal arrow is an L ∞ quasi-isomorphism. Amnon Yekutieli (BGU) Deformation Quantization 17 / 33

  51. � � 4. Deformation Quantization Theorem 4.3. ([Ye1]) Let K be a field containing R , and let X be a smooth algebraic variety over K . Then there is a diagram T poly , X D poly , X � Mix ( D poly , X ) Mix ( T poly , X ) where: ◮ Mix ( T poly , X ) and Mix ( D poly , X ) are sheaves of DG Lie algebras on X, called mixed resolutions ; ◮ the vertical arrows are DG Lie algebra quasi-isomorphisms; ◮ and the horizontal arrow is an L ∞ quasi-isomorphism. Amnon Yekutieli (BGU) Deformation Quantization 17 / 33

  52. � � 4. Deformation Quantization Theorem 4.3. ([Ye1]) Let K be a field containing R , and let X be a smooth algebraic variety over K . Then there is a diagram T poly , X D poly , X � Mix ( D poly , X ) Mix ( T poly , X ) where: ◮ Mix ( T poly , X ) and Mix ( D poly , X ) are sheaves of DG Lie algebras on X, called mixed resolutions ; ◮ the vertical arrows are DG Lie algebra quasi-isomorphisms; ◮ and the horizontal arrow is an L ∞ quasi-isomorphism. Amnon Yekutieli (BGU) Deformation Quantization 17 / 33

  53. � � 4. Deformation Quantization Theorem 4.3. ([Ye1]) Let K be a field containing R , and let X be a smooth algebraic variety over K . Then there is a diagram T poly , X D poly , X � Mix ( D poly , X ) Mix ( T poly , X ) where: ◮ Mix ( T poly , X ) and Mix ( D poly , X ) are sheaves of DG Lie algebras on X, called mixed resolutions ; ◮ the vertical arrows are DG Lie algebra quasi-isomorphisms; ◮ and the horizontal arrow is an L ∞ quasi-isomorphism. Amnon Yekutieli (BGU) Deformation Quantization 17 / 33

  54. � � 4. Deformation Quantization Theorem 4.3. ([Ye1]) Let K be a field containing R , and let X be a smooth algebraic variety over K . Then there is a diagram T poly , X D poly , X � Mix ( D poly , X ) Mix ( T poly , X ) where: ◮ Mix ( T poly , X ) and Mix ( D poly , X ) are sheaves of DG Lie algebras on X, called mixed resolutions ; ◮ the vertical arrows are DG Lie algebra quasi-isomorphisms; ◮ and the horizontal arrow is an L ∞ quasi-isomorphism. Amnon Yekutieli (BGU) Deformation Quantization 17 / 33

  55. � � 4. Deformation Quantization Theorem 4.3. ([Ye1]) Let K be a field containing R , and let X be a smooth algebraic variety over K . Then there is a diagram T poly , X D poly , X � Mix ( D poly , X ) Mix ( T poly , X ) where: ◮ Mix ( T poly , X ) and Mix ( D poly , X ) are sheaves of DG Lie algebras on X, called mixed resolutions ; ◮ the vertical arrows are DG Lie algebra quasi-isomorphisms; ◮ and the horizontal arrow is an L ∞ quasi-isomorphism. Amnon Yekutieli (BGU) Deformation Quantization 17 / 33

  56. 4. Deformation Quantization The mixed resolutions combine the commutative Čech resolution associated to an affine open covering of X , and the Grothendieck sheaf of jets. An L ∞ quasi-isomorphism is a generalization of a DG Lie algebra quasi-isomorphism. Theorem 4.3 is proved using the Formality Theorem of Kontsevich [Ko1] and formal geometry. More on the proof of Theorem 4.3 in Appendices B and C. Remark 4.4. Calaque and Van den Bergh [CV] proved (using ideas of Tamarkin and Halbout) that a global quantization map exists over Q . According to Kontsevich, changing the local formality isomorphism (i.e. changing the Drinfeld associator) can have a nontrivial effect on the global quantization process. This phenomenon is very intriguing and should be studied. Amnon Yekutieli (BGU) Deformation Quantization 18 / 33

  57. 4. Deformation Quantization The mixed resolutions combine the commutative Čech resolution associated to an affine open covering of X , and the Grothendieck sheaf of jets. An L ∞ quasi-isomorphism is a generalization of a DG Lie algebra quasi-isomorphism. Theorem 4.3 is proved using the Formality Theorem of Kontsevich [Ko1] and formal geometry. More on the proof of Theorem 4.3 in Appendices B and C. Remark 4.4. Calaque and Van den Bergh [CV] proved (using ideas of Tamarkin and Halbout) that a global quantization map exists over Q . According to Kontsevich, changing the local formality isomorphism (i.e. changing the Drinfeld associator) can have a nontrivial effect on the global quantization process. This phenomenon is very intriguing and should be studied. Amnon Yekutieli (BGU) Deformation Quantization 18 / 33

  58. 4. Deformation Quantization The mixed resolutions combine the commutative Čech resolution associated to an affine open covering of X , and the Grothendieck sheaf of jets. An L ∞ quasi-isomorphism is a generalization of a DG Lie algebra quasi-isomorphism. Theorem 4.3 is proved using the Formality Theorem of Kontsevich [Ko1] and formal geometry. More on the proof of Theorem 4.3 in Appendices B and C. Remark 4.4. Calaque and Van den Bergh [CV] proved (using ideas of Tamarkin and Halbout) that a global quantization map exists over Q . According to Kontsevich, changing the local formality isomorphism (i.e. changing the Drinfeld associator) can have a nontrivial effect on the global quantization process. This phenomenon is very intriguing and should be studied. Amnon Yekutieli (BGU) Deformation Quantization 18 / 33

  59. 4. Deformation Quantization The mixed resolutions combine the commutative Čech resolution associated to an affine open covering of X , and the Grothendieck sheaf of jets. An L ∞ quasi-isomorphism is a generalization of a DG Lie algebra quasi-isomorphism. Theorem 4.3 is proved using the Formality Theorem of Kontsevich [Ko1] and formal geometry. More on the proof of Theorem 4.3 in Appendices B and C. Remark 4.4. Calaque and Van den Bergh [CV] proved (using ideas of Tamarkin and Halbout) that a global quantization map exists over Q . According to Kontsevich, changing the local formality isomorphism (i.e. changing the Drinfeld associator) can have a nontrivial effect on the global quantization process. This phenomenon is very intriguing and should be studied. Amnon Yekutieli (BGU) Deformation Quantization 18 / 33

  60. 4. Deformation Quantization The mixed resolutions combine the commutative Čech resolution associated to an affine open covering of X , and the Grothendieck sheaf of jets. An L ∞ quasi-isomorphism is a generalization of a DG Lie algebra quasi-isomorphism. Theorem 4.3 is proved using the Formality Theorem of Kontsevich [Ko1] and formal geometry. More on the proof of Theorem 4.3 in Appendices B and C. Remark 4.4. Calaque and Van den Bergh [CV] proved (using ideas of Tamarkin and Halbout) that a global quantization map exists over Q . According to Kontsevich, changing the local formality isomorphism (i.e. changing the Drinfeld associator) can have a nontrivial effect on the global quantization process. This phenomenon is very intriguing and should be studied. Amnon Yekutieli (BGU) Deformation Quantization 18 / 33

  61. 4. Deformation Quantization The mixed resolutions combine the commutative Čech resolution associated to an affine open covering of X , and the Grothendieck sheaf of jets. An L ∞ quasi-isomorphism is a generalization of a DG Lie algebra quasi-isomorphism. Theorem 4.3 is proved using the Formality Theorem of Kontsevich [Ko1] and formal geometry. More on the proof of Theorem 4.3 in Appendices B and C. Remark 4.4. Calaque and Van den Bergh [CV] proved (using ideas of Tamarkin and Halbout) that a global quantization map exists over Q . According to Kontsevich, changing the local formality isomorphism (i.e. changing the Drinfeld associator) can have a nontrivial effect on the global quantization process. This phenomenon is very intriguing and should be studied. Amnon Yekutieli (BGU) Deformation Quantization 18 / 33

  62. 4. Deformation Quantization The mixed resolutions combine the commutative Čech resolution associated to an affine open covering of X , and the Grothendieck sheaf of jets. An L ∞ quasi-isomorphism is a generalization of a DG Lie algebra quasi-isomorphism. Theorem 4.3 is proved using the Formality Theorem of Kontsevich [Ko1] and formal geometry. More on the proof of Theorem 4.3 in Appendices B and C. Remark 4.4. Calaque and Van den Bergh [CV] proved (using ideas of Tamarkin and Halbout) that a global quantization map exists over Q . According to Kontsevich, changing the local formality isomorphism (i.e. changing the Drinfeld associator) can have a nontrivial effect on the global quantization process. This phenomenon is very intriguing and should be studied. Amnon Yekutieli (BGU) Deformation Quantization 18 / 33

  63. 5. Twisted Deformations of Algebraic Varieties 5. Twisted Deformations of Algebraic Varieties What can be done in general, when the the variety X is not affine? Can we still make use of Theorem 4.3? In the paper [Ko3] Kontsevich suggests that in general the deformation quantization of a Poisson bracket might have to be a stack of algebroids. This is a generalization of the notion of sheaf of algebras. Actually stacks of algebroids appeared earlier, under the name sheaves of twisted modules, in the work of Kashiwara [Ka]. See also [DP], [PS], [KS]. Amnon Yekutieli (BGU) Deformation Quantization 19 / 33

  64. 5. Twisted Deformations of Algebraic Varieties 5. Twisted Deformations of Algebraic Varieties What can be done in general, when the the variety X is not affine? Can we still make use of Theorem 4.3? In the paper [Ko3] Kontsevich suggests that in general the deformation quantization of a Poisson bracket might have to be a stack of algebroids. This is a generalization of the notion of sheaf of algebras. Actually stacks of algebroids appeared earlier, under the name sheaves of twisted modules, in the work of Kashiwara [Ka]. See also [DP], [PS], [KS]. Amnon Yekutieli (BGU) Deformation Quantization 19 / 33

  65. 5. Twisted Deformations of Algebraic Varieties 5. Twisted Deformations of Algebraic Varieties What can be done in general, when the the variety X is not affine? Can we still make use of Theorem 4.3? In the paper [Ko3] Kontsevich suggests that in general the deformation quantization of a Poisson bracket might have to be a stack of algebroids. This is a generalization of the notion of sheaf of algebras. Actually stacks of algebroids appeared earlier, under the name sheaves of twisted modules, in the work of Kashiwara [Ka]. See also [DP], [PS], [KS]. Amnon Yekutieli (BGU) Deformation Quantization 19 / 33

  66. 5. Twisted Deformations of Algebraic Varieties 5. Twisted Deformations of Algebraic Varieties What can be done in general, when the the variety X is not affine? Can we still make use of Theorem 4.3? In the paper [Ko3] Kontsevich suggests that in general the deformation quantization of a Poisson bracket might have to be a stack of algebroids. This is a generalization of the notion of sheaf of algebras. Actually stacks of algebroids appeared earlier, under the name sheaves of twisted modules, in the work of Kashiwara [Ka]. See also [DP], [PS], [KS]. Amnon Yekutieli (BGU) Deformation Quantization 19 / 33

  67. 5. Twisted Deformations of Algebraic Varieties I will use the term twisted associative deformation, and present an approach that treats the Poisson case as well. This approach was suggested to us by Kontsevich. A similar point of view is taken in [BGNT]. Here I will explain only a naive definition of twisted deformations. A more sophisticated definition, involving gerbes, may be found in Appendix D. The fact that the two definitions agree follows from our work on central extensions of gerbes and obstructions classes [Ye5]. Amnon Yekutieli (BGU) Deformation Quantization 20 / 33

  68. 5. Twisted Deformations of Algebraic Varieties I will use the term twisted associative deformation, and present an approach that treats the Poisson case as well. This approach was suggested to us by Kontsevich. A similar point of view is taken in [BGNT]. Here I will explain only a naive definition of twisted deformations. A more sophisticated definition, involving gerbes, may be found in Appendix D. The fact that the two definitions agree follows from our work on central extensions of gerbes and obstructions classes [Ye5]. Amnon Yekutieli (BGU) Deformation Quantization 20 / 33

  69. 5. Twisted Deformations of Algebraic Varieties I will use the term twisted associative deformation, and present an approach that treats the Poisson case as well. This approach was suggested to us by Kontsevich. A similar point of view is taken in [BGNT]. Here I will explain only a naive definition of twisted deformations. A more sophisticated definition, involving gerbes, may be found in Appendix D. The fact that the two definitions agree follows from our work on central extensions of gerbes and obstructions classes [Ye5]. Amnon Yekutieli (BGU) Deformation Quantization 20 / 33

  70. 5. Twisted Deformations of Algebraic Varieties I will use the term twisted associative deformation, and present an approach that treats the Poisson case as well. This approach was suggested to us by Kontsevich. A similar point of view is taken in [BGNT]. Here I will explain only a naive definition of twisted deformations. A more sophisticated definition, involving gerbes, may be found in Appendix D. The fact that the two definitions agree follows from our work on central extensions of gerbes and obstructions classes [Ye5]. Amnon Yekutieli (BGU) Deformation Quantization 20 / 33

  71. 5. Twisted Deformations of Algebraic Varieties Let U ⊂ X be an affine open set, and let C := Γ( U , O X ) . Suppose A is an associative or Poisson deformation of the K -algebra C . One may assume that A = C [[ � ]] , and it is either endowed with a Poisson bracket {− , −} , or with a star product ⋆ . In either case A becomes a pronilpotent Lie algebra, and � A is a Lie subalgebra. In the Poisson case the Lie bracket is {− , −} , and in the associative case the Lie bracket is the commutator [ a , b ] := a ⋆ b − b ⋆ a . Let us denote the corresponding pronilpotent group by IG ( A ) := exp ( � A ) , and call it the group of inner gauge transformations of A . Amnon Yekutieli (BGU) Deformation Quantization 21 / 33

  72. 5. Twisted Deformations of Algebraic Varieties Let U ⊂ X be an affine open set, and let C := Γ( U , O X ) . Suppose A is an associative or Poisson deformation of the K -algebra C . One may assume that A = C [[ � ]] , and it is either endowed with a Poisson bracket {− , −} , or with a star product ⋆ . In either case A becomes a pronilpotent Lie algebra, and � A is a Lie subalgebra. In the Poisson case the Lie bracket is {− , −} , and in the associative case the Lie bracket is the commutator [ a , b ] := a ⋆ b − b ⋆ a . Let us denote the corresponding pronilpotent group by IG ( A ) := exp ( � A ) , and call it the group of inner gauge transformations of A . Amnon Yekutieli (BGU) Deformation Quantization 21 / 33

  73. 5. Twisted Deformations of Algebraic Varieties Let U ⊂ X be an affine open set, and let C := Γ( U , O X ) . Suppose A is an associative or Poisson deformation of the K -algebra C . One may assume that A = C [[ � ]] , and it is either endowed with a Poisson bracket {− , −} , or with a star product ⋆ . In either case A becomes a pronilpotent Lie algebra, and � A is a Lie subalgebra. In the Poisson case the Lie bracket is {− , −} , and in the associative case the Lie bracket is the commutator [ a , b ] := a ⋆ b − b ⋆ a . Let us denote the corresponding pronilpotent group by IG ( A ) := exp ( � A ) , and call it the group of inner gauge transformations of A . Amnon Yekutieli (BGU) Deformation Quantization 21 / 33

  74. 5. Twisted Deformations of Algebraic Varieties Let U ⊂ X be an affine open set, and let C := Γ( U , O X ) . Suppose A is an associative or Poisson deformation of the K -algebra C . One may assume that A = C [[ � ]] , and it is either endowed with a Poisson bracket {− , −} , or with a star product ⋆ . In either case A becomes a pronilpotent Lie algebra, and � A is a Lie subalgebra. In the Poisson case the Lie bracket is {− , −} , and in the associative case the Lie bracket is the commutator [ a , b ] := a ⋆ b − b ⋆ a . Let us denote the corresponding pronilpotent group by IG ( A ) := exp ( � A ) , and call it the group of inner gauge transformations of A . Amnon Yekutieli (BGU) Deformation Quantization 21 / 33

  75. 5. Twisted Deformations of Algebraic Varieties Let U ⊂ X be an affine open set, and let C := Γ( U , O X ) . Suppose A is an associative or Poisson deformation of the K -algebra C . One may assume that A = C [[ � ]] , and it is either endowed with a Poisson bracket {− , −} , or with a star product ⋆ . In either case A becomes a pronilpotent Lie algebra, and � A is a Lie subalgebra. In the Poisson case the Lie bracket is {− , −} , and in the associative case the Lie bracket is the commutator [ a , b ] := a ⋆ b − b ⋆ a . Let us denote the corresponding pronilpotent group by IG ( A ) := exp ( � A ) , and call it the group of inner gauge transformations of A . Amnon Yekutieli (BGU) Deformation Quantization 21 / 33

  76. 5. Twisted Deformations of Algebraic Varieties Let U ⊂ X be an affine open set, and let C := Γ( U , O X ) . Suppose A is an associative or Poisson deformation of the K -algebra C . One may assume that A = C [[ � ]] , and it is either endowed with a Poisson bracket {− , −} , or with a star product ⋆ . In either case A becomes a pronilpotent Lie algebra, and � A is a Lie subalgebra. In the Poisson case the Lie bracket is {− , −} , and in the associative case the Lie bracket is the commutator [ a , b ] := a ⋆ b − b ⋆ a . Let us denote the corresponding pronilpotent group by IG ( A ) := exp ( � A ) , and call it the group of inner gauge transformations of A . Amnon Yekutieli (BGU) Deformation Quantization 21 / 33

  77. 5. Twisted Deformations of Algebraic Varieties The group IG ( A ) acts on the deformation A by gauge equivalences. We denote this action by Ad. In the Poisson case the gauge transformation Ad ( g ) , for g ∈ IG ( A ) , can be viewed as a formal hamiltonian flow. In the associative case the intrinsic exponential function � 1 exp ( a ) = i ! a i , i ≥ 0 for a ∈ � A , allows us to identify the group IG ( A ) with the multiplicative subgroup { g ∈ A | g ≡ 1 mod � } . Under this identification the operation Ad ( g ) is just conjugation by the invertible element g . Amnon Yekutieli (BGU) Deformation Quantization 22 / 33

  78. 5. Twisted Deformations of Algebraic Varieties The group IG ( A ) acts on the deformation A by gauge equivalences. We denote this action by Ad. In the Poisson case the gauge transformation Ad ( g ) , for g ∈ IG ( A ) , can be viewed as a formal hamiltonian flow. In the associative case the intrinsic exponential function � 1 exp ( a ) = i ! a i , i ≥ 0 for a ∈ � A , allows us to identify the group IG ( A ) with the multiplicative subgroup { g ∈ A | g ≡ 1 mod � } . Under this identification the operation Ad ( g ) is just conjugation by the invertible element g . Amnon Yekutieli (BGU) Deformation Quantization 22 / 33

  79. 5. Twisted Deformations of Algebraic Varieties The group IG ( A ) acts on the deformation A by gauge equivalences. We denote this action by Ad. In the Poisson case the gauge transformation Ad ( g ) , for g ∈ IG ( A ) , can be viewed as a formal hamiltonian flow. In the associative case the intrinsic exponential function � 1 exp ( a ) = i ! a i , i ≥ 0 for a ∈ � A , allows us to identify the group IG ( A ) with the multiplicative subgroup { g ∈ A | g ≡ 1 mod � } . Under this identification the operation Ad ( g ) is just conjugation by the invertible element g . Amnon Yekutieli (BGU) Deformation Quantization 22 / 33

  80. 5. Twisted Deformations of Algebraic Varieties The group IG ( A ) acts on the deformation A by gauge equivalences. We denote this action by Ad. In the Poisson case the gauge transformation Ad ( g ) , for g ∈ IG ( A ) , can be viewed as a formal hamiltonian flow. In the associative case the intrinsic exponential function � 1 exp ( a ) = i ! a i , i ≥ 0 for a ∈ � A , allows us to identify the group IG ( A ) with the multiplicative subgroup { g ∈ A | g ≡ 1 mod � } . Under this identification the operation Ad ( g ) is just conjugation by the invertible element g . Amnon Yekutieli (BGU) Deformation Quantization 22 / 33

  81. 5. Twisted Deformations of Algebraic Varieties The above can be sheafified: to a deformation A of O X we assign the sheaf of groups IG ( A ) . Let us fix an affine open covering { U 0 , . . . , U m } of X . We write U i , j ,... := U i ∩ U j ∩ · · · . Amnon Yekutieli (BGU) Deformation Quantization 23 / 33

  82. 5. Twisted Deformations of Algebraic Varieties The above can be sheafified: to a deformation A of O X we assign the sheaf of groups IG ( A ) . Let us fix an affine open covering { U 0 , . . . , U m } of X . We write U i , j ,... := U i ∩ U j ∩ · · · . Amnon Yekutieli (BGU) Deformation Quantization 23 / 33

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