Note 6
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>
To be continued
Tokyo April 26, 2009
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