Noncommutative Distance Theory Note 4 Atiyah’s Axiomatic System

Noncommutative Distance Theory

Note 4

Atiyah’s Axiomatic System

TANAKA Akio

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 ) = id_{Z( 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.

Tokyo December 22, 2007

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