SLIDE 15 Coaction on Appell functions
- A similar construction can be applied to generalised Hypergeometrics.
Let’s consider Appell F3, depending on two variables.
F3(α, α0, β, β0, γ; x, y) = Γ(γ) Γ(β)Γ(β0)Γ(γ − β − β0)× Z 1 Z 1v uβ1vβ01(1 − u − v)γββ01(1 − xu)α(1 − yv)α0 du dv
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ϕab = d log(u) ∧ d log(v) = du∧dv
uv
ϕbc = d log(1 − xu) ∧ d log(v) = − x du∧dv
(1−xu)v
ϕcd = d log(1 − xu) ∧ d log(1 − yv) =
xy du∧dv (1−xu)(1−yv)
ϕad = d log(u) ∧ d log(1 − yv) = − y du∧dv
u(1−yv)
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Φ =ua✏vb✏(1 − xu)c✏(1 − yv)d✏(1 − u − v)g✏
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Pss R
S ΦϕT
ab bc cd ad abg
1 ab✏2
bcg
1 bc✏2
cdg
1 cd✏2
adg
1 ad✏2
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a b d c g
1 x
1 Re(u)
1 y
1 Re(v)
Z
γ
ω with ω = Φϕ
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period matrix: basis of 2-forms:
∆ ✓Z
γ
ω ◆ ≡ X
i
Z
γ
ωi ⊗ Z
γi
ω
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- The coaction can be obtained from
- The result for the coaction can be fully expressed in terms of Appell F3.