Deformation Quantizations and Gerbes

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Deformation Quantizationsand Gerbes

• Yoshiaki Maeda
• (Keio University)

Joint work with H.Omori, N.Miyazaki, A.Yoshioka
Seminar at Hanoi , April 5, 2007
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Motivation (Question)
What is the complex version of the Metaplectic
group
Answer NOT CLEAR !
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Weyl algebra
the algebra over
with the generators
such that
where
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Set of quadratic forms
Lemma
forms a real Lie algebra
forms a complex Lie algebra

Construct a group for these Lie algebras
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Idea star exponential function
for
Question Give a rigorous meaning for the star
exponential functions for
Theorem 1

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Theorem 2
dose not give a classical geometric object
1) Locally Lie group structure
2) As gluing local data gerbe
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Ordering problem
( As linear space )
Lemma
(uniquely)
Realizing the algebraic structure
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Product (
-product) on
for
where
Weyl product
product
anti-
product
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Proposition
gives an associative
(1)
(noncommutative) algebra for every
(2)
is isomorphic to
(3)
There is an intertwiner (algebraic isomorphism)
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Intertwiner
where
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Example
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Description (1)
(1) Express
as
via the isomorphism
(2) Compute the star exponential function
(3) Gluing
and
for
and
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Star exponential functions for quadratic functions
Evolution Equation(1)
in
Evolution Equation (2)
in
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Solution for
set of entire functions on
Theorem
The equation (2) is solved in
i.e.
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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 a complex functions on
has an ambiguity for choosing the sign
Multi-valued
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Manifolds, vector bundle, etc

Gerbe
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Description (2)
View an element
as a set
Infinitesimal Intertwiner
at
where
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Geometric setting
1) Fibre bundle
2) Tangent space
3) Connection(horizontal subspacce)
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Tangent space and Horizontal spaces
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Parallel sections
curve in
parallel section along
e.g.
is a parallel section through
Extend this to
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Extended parallel sections
Parallel section for
curve in
where
where
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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)
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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
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ODE (A)
Lemma
Consider the Solution of (A)
solution through
trivial solution
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ODE (B)
Solution
(Negative) Propositon

cannot be a fibre bundle over
(no local triviality)
Problem moving branching points
Painleve equations without moving branch point
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Infinitesimal Geometry
(1) Tangent space for
For
(2) Horizontal space at
(3) Parallel section multi-valued section
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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
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Monodromy appears!
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Infinitesimal Geometry
Objects
(1) Local structure
(2) Tangent space
(3) connection(Horizontal space)
Gluing infinitasimally
Requirement
Accept multi-valued parallel sections