Noncommutative Distance Theory
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 ( Y ) ∈ Z ( Σ )
Axiom 1 Z ( Σ* ) = Z ( Σ )* Σ* is reverse orientation of Σ. Z ( Σ )* is dual space of Z ( Σ ).
Axiom 2 Z ( Σ1⋃Σ2) = Z ( Σ1 ) ⊗ Z ( Σ2 )
Axiom 3 Z ( Y ) = Z ( Y2 ) ◌ Z ( Y1 )
Axiom 4 Z ( 0 ) = ℂ
Axiom 5 Z ( Σ×I ) = idZ( Σ )
2
In TQFT, when Z ( Y1 ) = Z ( Y2 ) and the both are connected, Y 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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