Floer Homology Language Note7 Quantization of Language

Floer Homology Language

TANAKA Akio

   

Note7

Quantization of Language

Theorem
1
(Barannikov, Kontsevich 1998)
<.,.>, ° defines structure of Frobenius manifold at neighborhood of H's origin.

2
(Kontsevich 2003)
There exists φk : EkΠ2(Γ(M;Ω(M))) → Π2CD(A, A), k = 2, ... .
 is L∞ map.



Explanation
1
(Local coordinates of Poisson structure)
{f, g} 

2
(Map)
{.,.} : C∞ × C∞  →C∞
The map  has next conditions.
(i)   {.,.} is R bilinear,{f, g} = - {g, f}.
(ii)  Jacobi law is satisfied.
(iii) {f, gh} = g{f, h} + h{f, g}

3
(Gerstenharber bracket)


4




5



6





7



8
(  )
Manifold     M= R2n
Coordinates     p, q
Differential form     w = dqidpi
Subset of C∞( R2n )        A
Element of A       F    

Differential operator of R2n      D(F)

D({F, G}) ≡ [D({F}, D({G}]


[Image 1]
Quantization of language is defined by theorem (Kontsevich 2003).

[Image 2]
Complex unit  is seemed to be essential for mirror symmetry of language by explanation 8.


[References]

Quantum Theory for language / Synopsis / Tokyo January 15, 2004
Mirror Theory / Tokyo June 5, 2004
For WITTGENSTEIN Ludwig / Position of Language / Tokyo December 10, 2005

Tokyo June 24, 2009

Sekinan Research Field of Language

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