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