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Floer Homology Language Note2 Supersymmetric Harmonic Oscillator 2009

 Monday 13 July 2015


Floer Homology Language

   

Note2

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
is Euclid space E.
Vector field over E      v 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 = D+ thE.
DE,t = Q + Q*

Q* = 

6
Supersymmetric harmonic oscillator is seemed as <meaning minimum>.
On <meaning minimum>, refer to the next.
From Cell to Manifold / Cell Theory / Tokyo June 2, 2007
Word and Meaning Minimum / Energy Distance Theory / Tokyo September 22, 2008
Amplitude of Meaning Minimum / Complex Manifold Deformation Theory / Tokyo December 17, 2008
Generating Function / Symplectic Language Theory / Tokyo March 17, 2009

7
Distance of supersymmetric harmonic oscillator is seemed as <distance> of <meaning minimum>.
On <distance>, refer to the next.
Distance Theory / Tokyo May 5, 2004
I first clearly declared <distance> on language in this theory. 



[References]
Below papers are partially developed on <distance> from Distance Theory.
Actual Language and Imaginary Language / Tokyo September 23, 2004
Distance / Distance Theory Algebraically Supplemented / Tokyo Ocotber 26, 2007
Finsler Manifold and Distance / Energy Distance Theory / Tokyo November 7, 2008
Distance of Word / Complex Manifold Deformation Theory / Tokyo November 30, 2008
To be continued
Tokyo May 6, 2009

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