Let C be a smooth, connected, complex, projective curve of genus g and let D₁,D₂ be divisors of degree k₁,k₂ respectively. Let S be the decomposable ruled surface given by the total space of the following P¹-bundle over C:

p_C: P(O_C ⊕ O_C(-D₁))→C.

Let C₀ be the locus of (1:0) in S ≅ P(O_C ⊕ O_C(-D₁)). Then let X be the threefold given by the total space of the following P¹-bundle over S:

p_S:P(O_s⊕O_s(-E)) → S

where E=aC₀+p_C^-1(D₂). This determines an H_a-bundle over C where H_a is a Hirzebruch surface.

In this thesis we determine the equivariant Gromov-Witten partition function for all "section classes'' of the form s+m₁f₁ + m₂f₂ where s is a section of the map X → C and f₁,f₂ are fiber classes, in the case that a=0,-1. A class is Calabi-Yau if K_X⋅β=0. For a=0, the partition function of Calabi-Yau section classes is given by

[Special characters omitted]

where v₁,v₂ count the number of fibers and φ=2sinu/2. In the case that a=-1 the partition functions of Calabi-Yau section classes satisfy the following relations

Z(g|k₁,k₂)=Z(g|k₁-2,k₂+1)

Z(g|k₁,k₂)=I₁² Z(g|k₁-3,k₂)+v²v₂ Z(g|k₁-4,k₂)

Z(g|k₁,k₂)=-φ²v₁²v₂^-2Z(g-1|k₁,k₂)

+6φ⁴v₁²v₂^-1Z(g-2|k₁,k₂)+(256φ⁸v₁²v₂+27φ⁸v₁⁴v₂^-2)Z(g-4|k₁,k₂)

which allow us to compute all the Calabi-Yau section class invariants from the following base cases:

g=0 1 2 3

k₁=0 0 4 -φ²v₁²v₂^-2 12φ⁴v₁²v₂^-1+φ⁴v₁⁴v₂^-4

1 φ^-2 0 φ²v₁²v₂^-1 16φ⁴v₁²-φ⁴v₁⁴v₂^-3

2 0 0 8φ²v₁² 64φ⁴v₁²v₂+φ⁴v₁⁴v₂^-2

3 0 3v₁² 16φ²v₁²v₂ -φ⁴v₁⁴v₂^-1

As a corollary, we establish the Gromov-Witten/Donaldson-Thomas/Stable Pairs correspondence for the Calabi-Yau section class partition functions for these families of non-toric threefolds.