Generating Function
Symplectic Language Theory
TANAKA Akio
Note5
Generating Function
[Gromov-Witten invariant]
3-dimensional Calabi-Yau manifold X
Degree of holomorphic map
Riemann surface C
Genus of C g
Moduli space that is derived from regular curve from C to X
Virtual fundamental class of
Gromov-Witten invariant
[Generating function of Gromov-Witten invariant]
generating function F
F of Gromov-Witten invariant] FGW
Representation of FGW
Here , are variants that are corresponded with holomorphic map and Riemann surface's
genus of domain in respective generating functions.
[Explanation of generating function]
n-dimensional manifold M
Vector bundle over M, which has order m V
Function over V F
Darboux coordinate of M
m-number variant that is coordinate V's fiber direction λ→
iF(q→, x→)=(q→,δF/δq1 (q→ , λ→), ...,δF/δqn (q→ , λ→))
Lagrangian immersion (LF, iF)
F is called lagrangian immersion iF's generating function.
[Impression]
1
Word is given by F.
Meaning minimum is given by iF.
Meaning unit is given by q→.
Time unit is given by λ→ .
2
Language is given by 3-dimensional Calabi-Yau manifold.
Language has an universal that is expressed by Gromov-Witten invariant.
Word is given by generating function of Gromov-Witten invariant.
3
Language will also have an universal that is expressed by Donaldson-Thomas invariant.
4
Generating function's relationship between Gromov-Witten invariant and Donaldson-Thomas
invariant is called S-duality that will show us a deeper structure in language.
To be continued
Tokyo March 17, 2009
Tokyo March 17, 2009
Sekinan Research Field of language
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