Symplectic Language Theory
TANAKA Akio
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).
Mirror Theory papers in early stage of Sekinan Linguistic Field
To be continued
Tokyo April 26, 2009
Sekinan Research Field of language