Floer Homology Language
Note 7
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 = dqi
dpi
dpi
Subset of C∞( R2n ) A
Element of A F 
Differential operator of R2n D(F)
D({F, G}) ≡ 
[D({F}, D({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
is seemed to be essential for mirror symmetry of language by explanation
8.
[References]
Tokyo
June 24, 2009