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

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

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

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 2) As gluing local data : gerbe 1) Locally : Lie group structure

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

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

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

10. Intertwiner where

11. Example

12. Description (1) (1) Expressas via the isomorphism (2) Compute the star exponential function (3) Gluing and forand

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

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

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

16. Manifolds, vector bundle, etc = Gerbe

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

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

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

22. (2) (1) diverges (poles) has sign ambiguity for taking the square root Solution for a curve where (not defined for some ) ( 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 (A) is a solution of (B) Question: Describe this as a geometric object

24. ODE (A) Consider the Solution of (A) : Lemma 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-forms.t. (1) (2) (3) (k : not integer) No global geometric quantization E Line bundle over However : Locally OK glue infinitesimally connection

28. Monodromy appears!

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