Let $C$ be a smooth, connected, complex, projective curve of genus $g$ and let $D_1,D_2$ be divisors of degree $k_1,k_2$ respectively. Let $S$ be the decomposable ruled surface given by the total space of the following $\mathbb{P}^1$-bundle over $C$: $$p_C: \PP(\mathcal{O}_C \oplus \mathcal{O}_C(-D_1))\rightarrow C.$$ Let $C_0$ be the locus of $(1:0)$ in $S \cong \PP(\mathcal{O}_C \oplus \mathcal{O}_C(-D_1))$. Then let $X$ be the threefold given by the total space of the following $\PP^1$-bundle over $S$:

$$p_S:\mathbb{P}(\mathcal{O}_S \oplus \mathcal{O}_S(-E)) \rightarrow S$$

where $E=aC_0+p_C^{-1}(D_2)$. This determines an $\mathcal{H}_a$-bundle over $C$ where $\mathcal{H}_a$ is a Hirzebruch surface.

In this thesis we determine the equivariant Gromov-Witten partition function for all ``section classes'' of the form $s+m_1 f_1 + m_2 f_2$ where $s$ is a section of the map $X \rightarrow C$ and $f_1,f_2$ are fiber classes, in the case that $a=0,-1$. A class is Calabi-Yau if $K_X \cdot \beta=0$. For $a=0$, the partition function of Calabi-Yau section classes is given by

$$Z(g|k_1,k_2)=\left{ \begin{array}{cc}4^g \phi^{2g-2} v_1^{\frac{g-1+k_1}{2}} v_2^{\frac{g-1+k_2}{2}} & (g-1) \equiv k_1 \equiv k_2 \mod 2 \ 0 & \text{otherwise} \end{array}\right.$$

where $v_1,v_2$ count the number of fibers and $\phi=2\sin{\frac{u}{2}}$. In the case that $a=-1$ the partition functions of Calabi-Yau section classes satisfy the following relations

$$Z(g|k_1,k_2)=Z(g|k_1-2,k_2+1)$$

$$Z(g|k_1,k_2)=v_1^2 Z(g|k_1-3,k_2)+v_1^2v_2 Z(g|k_1-4,k_2)$$

$$Z(g|k_1,k_2)=-\phi^2v_1^2v_2^{-2}Z(g-1|k_1,k_2)$$ $$+6\phi^4v_1^2v_2^{-1}Z(g-2|k_1,k_2)+(256\phi^8v_1^2v_2+27\phi^8v_1^4v_2^{-2})Z(g-4|k_1,k_2)$$

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

$$\begin{array}{ccccccc}

& & & g=0 & 1 & 2 & 3 \vspace{0.2cm}\

& k_1=0 & \hspace{0.5cm} & 0 & 4 & -\phi^2 v_1^2 v_2^{-2} & 12 \phi^4 v_1^2 v_2^{-1} + \phi^4 v_1^4 v_2^{-4} \

& 1 & & \phi^{-2} & 0 & \phi^2 v_1^2 v_2^{-1} & 16 \phi^4 v_1^2 - \phi^4 v_1^4 v_2^{-3} \

& 2 & & 0 & 0 & 8 \phi^2 v_1^2 & 64 \phi^4 v_1^2 v_2 + \phi^4 v_1^4 v_2^{-2} \

& 3 & & 0 & 3 v_1^2 & 16 \phi^2 v_1^2 v_2 & -\phi^4 v_1^4 v_2^{-1} \end{array}\vspace{0.5cm}$$

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.