Supersymmetric Harmonic Oscillator

sekinanlogoshome

Floer Homology Language

TANAKA Akio

   

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     W_X

2
Definition of W_X




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     vExterior differentiation     dFormally conjugate operator        d*
TX  T*XDifferential form corresponded with v     Triplex (W_X, D_X, h_X,v) is defined by the next.



4
X is Euclid space E.Vector field over E      v : E →T*E  TE = E × E
h_E = h_E,vTriplex (W_E, D_E, h_E) is called supersymmetric harmonic oscillator.

5
Distance of vector field       |q|
= d(|q|²/2)
Supersymmetric harmonic oscillator over E is expressed by D_E,t = D_E + th_E.
D_E,t = Q + 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}}