By Hossein Movasati

GMCD goals to unify automorphic types with topological string partition features and provides a scientific approach of learning sub-moduli of moduli areas utilizing the speculation of holomorphic/algebraic foliations. It goals to interchange the difference of Hodge constructions with yes vector fields whose suggestions are an enormous generalization of automorphic types. It goals to come back Hodge concept to its foundation that's the research of a number of integrals because of Abel, Gauss, Legendre, Poincare', Picard and so on.

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**Example text**

A complete description of the image of τ0 is not yet known. Now, ti ’s are well-defined holomorphic functions on H. 6. Since Γ is generated by M0 and M1 it is enough to explain them for these two elements. The functional equations of ti ’s with respect to the action of M0 and written in the τ0 -coordinate are the trivial equalities ti (τ0 ) = ti (τ0 + 1), i = 0, 1, . . , 6. 12). 8. Since we do not know the global behavior of τ0 , the above equalities must be interpreted in the following way: for any fixed branch of ti (τ0 ) 0 and such there is a path γ in the image of τ0 : H → C which connects τ0 to τ2τ+1 that the analytic continuation of ti ’s along the path γ satisfy the above equalities.

3) with k = t0−1 . We have g∗ η = t0 ω1 and ∂ −1 t06 ∂ = ∂z 5 t4 ∂t0 = t05 ∂ ∂t4 . ˜ 0 ,t4 ) such that From these two equalities we obtain a matrix S˜ = S(t [η, ∂ η ∂ 2 η ∂ 3 η tr ˜−1 , , ] = S [ω1 , ω2 , ω3 , ω4 ]tr , ∂ z ∂ z2 ∂ z3 where tr denotes the transpose of matrices, and the Gauss-Manin connection in the basis ωi , i = 1, 2, 3, 4 is: 40 3 Moduli of enhanced mirror quintics t4 t4 A˜ = d S˜ + S˜ · A( 5 ) · d( 5 ) · S˜−1 , t0 t0 which is the following matrix after doing explicit calculations: − 5t14 dt4 0 0 0 dt0 + −t 5t4 dt4 −2 5t4 dt4 0 0 0 dt0 + −t 5t4 dt4 −3 5t4 dt4 3t03 −25t03 t2 −15t 2 −t0 dt + 5t 5 t 0−5t 2 dt4 t 5 −t 0 dt0 + t 5 t −t 2 dt4 t 5 −t dt0 + t05 −t4 0 0 4 4 0 4 0 4 4 0 4 0 0 .

1) z = 0, 1, z ∈ k and G is the group G := {(ζ1 , ζ2 , · · · , ζ5 ) | ζi5 = 1, ζ1 ζ2 ζ3 ζ4 ζ5 = 1} acting on Wz coordinstewise. Such a resolution is described explicitly in [Mor93], however, for the purpose of the present text we do not need it. The moduli of mirror quintics is the punctured line P1 − {0, 1} with the coordinate system z. Note that for z = ∞ we have still a smooth mirror quintic and so there is no universal family of mirror quintics over the punctured line P1 − {0, 1}. Mirror quintic Calabi-Yau threefolds produced a great amount of excitement in mathematics when in 1991 Candelas et al.

### Gauss-Manin Connection in Disguise by Hossein Movasati

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