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Deformation Quantizations and Gerbes

Deformation Quantizations and Gerbes. Yoshiaki Maeda (Keio University). Joint work with H.Omori, N.Miyazaki, A.Yoshioka. Seminar at Hanoi , April 5, 2007. Motivation (Question). What is the complex version of the Metaplectic group. Answer : NOT CLEAR !. Weyl algebra.

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Deformation Quantizations and Gerbes

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  1. Deformation Quantizationsand Gerbes Yoshiaki Maeda (Keio University) Joint work with H.Omori, N.Miyazaki, A.Yoshioka Seminar at Hanoi , April 5, 2007

  2. Motivation (Question) What is the complex version of the Metaplectic group Answer : NOT CLEAR !

  3. Weyl algebra = the algebra over with the generators such that where

  4. Set of quadratic forms Lemma forms a real Lie algebra forms a complex Lie algebra Construct a “group” for these Lie algebras

  5. Idea: star exponential function for Question:Give a rigorous meaning for the star exponential functions for Theorem 1 =

  6. Theorem 2 dose not give a classical geometric object 1) Locally : Lie group structure 2)As gluing local data : gerbe

  7. Ordering problem ( As linear space ) Lemma (uniquely) Realizing the algebraic structure

  8. Product ( -product) on for where Weyl product product anti- product

  9. Proposition gives an associative (1) (noncommutative) algebra for every (2) is isomorphic to (3) There is an intertwiner (algebraic isomorphism)

  10. Intertwiner where

  11. Example

  12. Description (1) (1) Express as via the isomorphism (2) Compute the star exponential function and (3) Gluing for and

  13. Star exponential functions for quadratic functions Evolution Equation(1) in Evolution Equation (2) in

  14. Solution for set of entire functions on Theorem The equation (2) is solved in i.e.

  15. Explicit form for and where Twisted Cayley transformation Remarks: (1) depends on and there are some on which is not defined (2) can be viewed as acomplex functionson has an ambiguity for choosing the sign Multi-valued

  16. Manifolds, vector bundle, etc = Gerbe

  17. Description (2) View an element as a set Infinitesimal Intertwiner at where

  18. Geometric setting 1) Fibre bundle : 2) Tangent space: 3) Connection(horizontal subspacce):

  19. Tangent space and Horizontal spaces

  20. Parallel sections : curve in : parallel section along e.g. is a parallel section through Extend this to

  21. Extended parallel sections Parallel section for curve in where where

  22. Solution for a curve where (1) (not defined for some diverges (poles) ) (2) has sign ambiguity for taking the square root ( multi-valued function as a complex function)

  23. Toy models Phase space for ODEs: (A) (B) ( or ) Solution spaces for (A) and (B) is a solution of (B) is a solution of (A) Question: Describe this as a geometric object

  24. ODE (A) Lemma Consider the Solution of (A) : solution through trivial solution

  25. ODE (B) Solution : (Negative) Propositon : cannot be a fibre bundle over (no local triviality) Problem: moving branching points Painleve equations: without moving branch point

  26. Infinitesimal Geometry (1) Tangent space for For (2) Horizontal space at (3) Parallel section : multi-valued section

  27. Geometric Quantization for non-integral 2-form On : consider 2-form s.t. (1) (k : not integer) (2) (3) No global geometric quantization Line bundle over E However : Locally OK glue infinitesimally connection

  28. Monodromy appears!

  29. Infinitesimal Geometry Objects : (1) Local structure (2) Tangent space (3) connection(Horizontal space) Gluing infinitasimally Requirement: Accept multi-valued parallel sections

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