Note 6
Homology Structure of Word
§ 1
1
Compact manifold in small diameter M
Inner product space h (M)
Map h (M) 
k→ h (M)
k→ h (M)
2
A model and B model by Witten, E.
A model M as symplectic structure
B model M as complex structure
3
(A model)
(Definition)
H kA (M) = H k(M;C)
H *A (M) = 
H kA (M)
H kA (M)
Inner product <. , .>A <u, v>A = 
( 
; cup product)
( ; cup product)
mA, 02(u, v) = 
4
(Theorem)
( H *A (M), mA, 02, <. , .>A ) is Frobenius algebra.
5
Oriented 2-dimensional manifold with genus g Σg
6
J (Σg) = {JΣg | Smooth complex structure over Σg }
Integer over 0 k
Different k-number points over Σg z1, ..., zk (Gathered points are expressed by 
. )
. )
Diff (Σg, 
) = {ψ : Σg → Σg | ψ(
) = 
, ψ is differential homeomorphism. }
) = {ψ : Σg → Σg | ψ() = , ψ is differential homeomorphism. }
Quotient space M g,k = J (Σg) / Diff (Σg, 
)
)
Rieman surface of genus g with k marked points (Σ, 
)
)
(Σ, 
) 
M g,k
) M g,k
Autmorphism group Aut (Σ, 
) = { ψ : Σ → Σ | ψ is biregular.ψ(
) = 
}
) = { ψ : Σ → Σ | ψ is biregular.ψ() = }
Compactification of M g,k CM g,k
§2
１
Symplectic manifold M
Differential 2-form over M wM
Well-formed almost complex structure with wM JM
β
H2(M; Z)
H2(M; Z)
(
is pseudoholomorphic.)
(Σ,
, φ) 
, φ)
2
[(Σ,
, φ)] 
, φ)]
Evaluation map ev[(Σ,
, φ)]=
, φ)]=
3
Forgetting map fg : 
Enlarged Forgetting map fg : 
3
(Definition)
(Gromov-Witten invariant)
(ev, fg)*
Gromov-Witten invariant is expressed by GWg,k(M, wM; β)
4
(Theorem)
Sumset 
is compact.
is compact.
5
(Associative law)
(Theorem)
[Image]
Meaning minimum of word is identified with 
.
.
Word is identified with 
Commutativity of meaning minimums in word guaranteed by theorem of associative law.
[References]
September 10, 2007
22, 2008
Tokyo December 17, 2008
Tokyo June 16, 2009
[Related Note / June 18, 2009]
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