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Derived Category Language 1 Category theoretic mirror symmetry conjecture / References added

 19/05/2016 14:15

Derived Category Language 1 Category theoretic mirror symmetry conjecture / References added

 

 

1.
Fukaya category is
Fuk(X, ω).

2.
General symplectic manifold is  .
3.

Derived category of A_∞   is   .

4.
Conjecture

When

   and  

have physical mirror relation, 

there exists the next triangle category's equivalence

 

[References 1]

1. Symplectic Language Theory ​ Note 3 Mirror Symmetry Conjecture on Rational Curve
    / Symplectic Language Theory​
2. Homological Mirror Symmetry Conjecture by KONTSEVICH 
   / Symplectic Language Theory​
3. Supersymmetric Harmonic Oscillator 
   / Floer Homology Language​
4. Symplectic Language Theory / sekinanlogos
5. Floer Homology Language / sekinanlogos

 

[References 2]

1. Bridge across mathematics and physics
2. Kontsevich's conjecture Category theoretic mirror symmetry conjecture

Tokyo

18 May 2016

SRFL Theory

Read more: https://srfl-theory.webnode.page/news/derived-category-language-1-category-theoretic-mirror-symmetry-conjecture-references-added/

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