Floer Homology Language
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 = 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 8.
is seemed to be essential for mirror symmetry of languageby explanation 8.
Tokyo
June 24, 2009
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