Deformation Quantizationsand

Gerbes

Yoshiaki Maeda(Keio University)

Joint work with H.Omori, N.Miyazaki, A.Yoshioka

Seminar at Hanoi , April 5, 2007

Answer : NOT CLEAR !

Motivation (Question)What is the complex version of the Metaplectic group

Weyl algebra

where

= the algebra over

with the generatorssuch that

Set of quadratic forms

Lemma

forms a real Lie algebra

forms a complex Lie algebra

Construct a “group” for these Lie algebras

Idea: star exponential function

for

Question: Give a rigorous meaning for the star exponential functions for

Theorem 1

=

Theorem 2

dose not give a classical geometric object

2) As gluing local data : gerbe

1) Locally : Lie group structure

Ordering problem

Lemma ( As linear space )

Realizing the algebraic structure

(uniquely)

Product (

for

where

Weyl product

product

anti- product

-product) on

Proposition

gives an associative

(noncommutative) algebra for every

(1)

(2) is isomorphic to

(3) There is an intertwiner (algebraic isomorphism)

Intertwiner

where

Example

Description (1)

(1) Express as

via the isomorphism

(2) Compute the star exponential function

(3) Gluing and

for and

Star exponential functions for quadratic functions

Evolution Equation(1)

Evolution Equation (2)

in

in

Solution for

set of entire functions on

Theorem The equation (2) is solved in

i.e.

Explicit form for and

where

Twisted Cayley transformation

(1) depends on and 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

Manifolds, vector bundle, etc

=

Gerbe

Description (2)

View an element as a set

Infinitesimal Intertwiner

where

at

Geometric setting

1) Fibre bundle :

3) Connection(horizontal subspacce):

2) Tangent space:

Tangent space and Horizontal spaces

Parallel sections

: curve in

: parallel section along

e.g. is a parallel section through

Extend this to

Extended parallel sections

Parallel section for

curve in

where

where

(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)

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

ODE (A)

Consider the Solution of (A) :

Lemma

solution through

trivial solution

ODE (B)

Solution :

(Negative) Propositon

: cannot be a fibre bundle over

(no local triviality)

Problem: moving branching points

Painleve equations: without moving branch point

Infinitesimal Geometry

(1) Tangent space for For

(2) Horizontal space at

(3) Parallel section : multi-valued section

Geometric Quantization for non-integral 2-form

On : consider 2-form

s.t.

(1)

(2)

(3)

(k : not integer)

No global geometric quantizationE

Line bundle over

However : Locally OK

glue infinitesimally

connection

Monodromy appears!

Infinitesimal Geometry

(2) Tangent space

(3) connection(Horizontal space)

Objects :

Requirement:

Accept multi-valued parallel sections

Gluing infinitasimally

(1) Local structure