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Analyzing complex versions of Metaplectic group, Weyl algebra, and star exponential functions with rigorous mathematical treatments. Examining geometric objects, evolution equations, and gerbes utilizing algebraic structures and parallel sections.
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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
Motivation (Question) What is the complex version of the Metaplectic group Answer : NOT CLEAR !
Weyl algebra = the algebra over with the generators such that where
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 1) Locally : Lie group structure 2)As gluing local data : gerbe
Ordering problem ( As linear space ) Lemma (uniquely) Realizing the algebraic structure
Product ( -product) on for where Weyl product product anti- product
Proposition gives an associative (1) (noncommutative) algebra for every (2) is isomorphic to (3) There is an intertwiner (algebraic isomorphism)
Intertwiner where
Description (1) (1) Express as via the isomorphism (2) Compute the star exponential function and (3) Gluing for and
Star exponential functions for quadratic functions Evolution Equation(1) in Evolution Equation (2) 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 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
Manifolds, vector bundle, etc = Gerbe
Description (2) View an element as a set Infinitesimal Intertwiner at where
Geometric setting 1) Fibre bundle : 2) Tangent space: 3) Connection(horizontal subspacce):
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
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)
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
ODE (A) Lemma Consider the Solution of (A) : 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) (k : not integer) (2) (3) No global geometric quantization Line bundle over E However : Locally OK glue infinitesimally connection
Infinitesimal Geometry Objects : (1) Local structure (2) Tangent space (3) connection(Horizontal space) Gluing infinitasimally Requirement: Accept multi-valued parallel sections
