Symplectic Language Theory Note6 Homological Mirror Symmetry Conjecture by KONTSEVICH

Symplectic Language Theory

TANAKA Akio

   

Note6

Homological Mirror Symmetry Conjecture by KONTSEVICH

1
R       Commutative ring over C
C       R module that has degree
(ΠC)k = Ck+1
BC     Free coassociative coalgebra
EC     Free coassociative cocommutative coalgebra
BkΠC  BΠC that has k number tensor product
EkΠC  EΠC that has k number tensor product
mk : BkΠC → ΠC
lk   : EkΠC → ΠC

2
                         Coderivative
A∞-algebra             = 0 at (BΠC, mk) (k>0)
Weak A∞-algebra     = 0 at (BΠC, mk) (k≥0)
L∞-algebra             = 0 at (EΠC, mk) (k>0)
Weak L∞-algebra     = 0 at (EΠC,  mk) (k≥0)

3
M(C)                     Complex structure's moduli space over compact manifold c    
Unobstructed         Weak A∞-algebra that satisfies M(C)     

4 
M       Symplectic manifold
M†  

          Complex manifold that is mirror of M
L        Lagrangian submanifold of M that Weak A∞-algebra  is unobstructed            
FL      Object of M†  's analitic coherent sheaf's category

(Conjecture)
For L there exists FL. FL's infinite small transformation's moduli space is coefficient to M(L).  

5
[b]     Element of M(L)
[b] defines A∞-algebra.
[b] defines chain complex's boundary map m1b
Cohomologyy of m1 b is called Floer cohomology.
Floer cohomology is expressed by HF((L, b), (L, b)) 

6 (Impression)
Word is seemed as L.
For L there exist language FL and M(L).
Mirror theory on language is supposed by the existence of FL and M(L).
<References>
Mirror Theory papers in early stage of Sekinan Linguistic Field

To be continued

Tokyo April 26, 2009

Sekinan Research Field of language

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