Théories cohomologiques des champs et espaces de modules

Mg,n
Mg,n
P
F/C
C
C MCC
C MC
C

C
Pn
Mg,n
g n
g n C g
n x1, . . . , xn∈C
C xi
(C, x1, . . . , xn) (D, y1,...yn)
f:C→D f(xi) = yi1≤i≤n
C
(C)
2g−2 + n > 0Mg,n
g n
Mg,n
Mg,n
Mg,n = 3g−3 + n
g= 0
0P1(C)
(P1(C), x1, . . . , xn)
P1(C)
xi
n≥3

n= 3 3 P1(C)
M0,3
n > 3
P1(C)
M0,n '(P1(C)\S)n−3\D S
P1(C)D
(n−3) P1(C)\S
M0,n n−3
M1,1H/ (Z,2) H
C1
Mg,n
g n
g
n x1,...xn∈C
C C0C
C C C0
2g−2 + n > 0Mg,n
Mg,n
g n
Mg,n
Mg,n
C1∈ Mg1,n1+1 C2∈ Mg2,n2+1
n1+ 1 n2+ 1
s(C1, C2)∈Mg,n g=g1+g2n=n1+n2
s:Mg1,n1+1 × Mg2,n2+1 → Mg,n
Mg−1,n+2
n+ 1 n+ 2
q:Mg−1,n+2 → Mg,n

C
C
n+ 1
p:Mg,n+1 → Mg,n
SnMg,n
Mg,n
Mg,n C
(C)
CH•(Mg,n)
Mg,n
AC1∈A
η=ηµν eµ⊗eνA
ηµν eµ⊗eν
(A, η, 1)
Ωg,n :A⊗n→H•(Mg,n)
n g 3g−3 + n≥0
Ωg,n Sng n
Ω0,3(1⊗u⊗v) = η(u, v)u, v ∈A
q∗Ωg,n(v1⊗. . .⊗vn) = ηµν Ωg−1,n+2(v1⊗. . .⊗vn⊗eµ⊗eν)v1, . . . , vn∈
A
s∗Ωg,n(v1⊗. . . ⊗vn) = ηµν Ωg1,n1+1(v1⊗. . . ⊗vn1⊗eµ)×Ωg2,n2+1(vn1+1 ⊗. . . ⊗
vn⊗eν)v1, . . . , vn∈A
p∗Ωg,n(v1⊗. . . ⊗vn) = Ωg,n+1(v1⊗. . . ⊗vn⊗1)v1, . . . , vn∈A
µ ν s q p
SnMg,n

Mg,n
Mg,n
(C, x1, . . . , xn)∈ Mg,n
C xiLi→ Mg,n i ψ
ψi:= 1(Li)∈H2(Mg,n).
ψn+1 p:
Mg,n+1 → Mg,n κ
κm:= p∗(ψm+1
n+1 )∈H2m(Mg,n).
ψ κ
X
C→X g
n p1, . . . , pn∈X
piγi∈Zki(X)
γi
X g γ1, . . . , γn
Mg,n
Ωg,n
A
η(v1, v2·v3)=Ω0,3(v1⊗v2⊗v3)
A(eµ)µA
η
eµ·eν=δµν θ−1
µeµ