Monday, 30 April 2018

Noncommutative Distance Theory Note 4 Atiyah’s Axiomatic System



Note 4
Atiyah’s Axiomatic System


1
ATIYAH Michael’s axiomatic system of topological quantum field theory, abbreviated TQFT, is shown below.
Oriented smooth compact d-dimensional manifold     Σ
Finite dimensional complex vector space     Z ( Σ )
d + 1 dimensional manifold that has boundary     Y
Functor     Z    
Z (  Z ( Σ )
Axiom 1   Z ( Σ* ) = Z ( Σ )*   Σ* is reverse orientation of ΣZ ( Σ )* is dual space of Z ( Σ ).
Axiom 2   Z ( Σ1Σ2) = Z ( Σ1  Z ( Σ2 )
Axiom 3   Z ( ) = Z ( Y2  Z ( Y1 )
Axiom 4   Z ( 0 ) = 
Axiom 5   Z ( Σ×) = idZ( Σ )
2
In TQFT, when Z ( Y1 ) = Z ( Y2 ) and the both are connected, becomes a compact manifold.
3
The generated manifold has meridian α and longitude β.
α is oriented by Σ.
β is seemed to be time development by axiom 5.
4
On algebra, α is corresponded to monodromy and β is corresponded to Frobenius automorphism.
5
From algebraic number field K’s Galois extension L/K and K’s prime ideal p, Frobenius automorphism is defined.
6
From prime ideal, prime number is considered for the roots of space that has orientation which shows the distance.


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