Floer Homology Language
Note 2
Supersymmetric Harmonic Oscillator
1
Riemannian manifold X
Vector field over X v
All the zero points of v Compact subset of X
Tangent vector bundle of X TX
Clifford module added Z/Z2 degree WX
2
Definition of WX
Here exterior product from left is expressed by
.
.
Cotangent vector bundle of X T*X
X has Riemannian metric.
TX
T*X
3
Vector field v
Exterior differentiation d
Formally conjugate operator d*
TX
T*X
Differential form corresponded with v
Triplex (WX, DX, hX,v) is defined by the next.
4
X is Euclid space E.
Vector field over E v : E →T*E
TE = E × E
hE = hE,v
Triplex (WE, DE, hE) is called supersymmetric harmonic oscillator.
5
Distance of vector field |q|
= d(|q|2/2)
Supersymmetric harmonic oscillator over E is expressed by DE,t = DE + thE.
DE,t = Q + Q*
Q =
Q* =
6
Supersymmetric harmonic oscillator is seemed as <meaning minimum>.
On <meaning minimum>, refer to the next.
17, 2008
7
Distance of supersymmetric harmonic oscillator is seemed as <distance> of
<meaning minimum>.
On <distance>, refer to the next.
I first clearly declared <distance> on language in this theory.
[References]
Below papers are partially developed on <distance> from ![]()
Distance Theory.
To be continued
Tokyo May 6, 2009
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