For the quintic threefold $X_5 \subset \mathbb{P}^4$ and its mirror $\tilde{X}_5$, the quantum cohomology prepotential equals the classical prepotential on the mirror: $F_{\text{GW}}^{(0)}(X_5; t) = F_{\text{period}}(\tilde{X}_5; z(t))$

where $z(t)$ is the mirror map determined by: $t = \frac{\omega_1(z)}{\omega_0(z)}$

and $\omega_i$ are period integrals satisfying the Picard-Fuchs equation.

This theorem was proven independently by Givental (1996) and Lian-Liu-Yau (1997) using different techniques.

For a Calabi-Yau threefold $X$, the Gopakumar-Vafa invariants $n_\beta^g$ extracted from Gromov-Witten invariants via: $\sum_{g,\beta} N_{g,\beta} q^\beta \lambda^{2g-2} = \sum_{\beta,g,m} \frac{n_\beta^g}{m}\left(2\sin\frac{m\lambda}{2}\right)^{2g-2} q^{m\beta}$

are integers: $n_\beta^g \in \mathbb{Z}$.

Furthermore, the invariants are conjecturally non-negative for all $g$ and $\beta$.

The number of degree $d$ rational curves in $\mathbb{P}^n$ through $dn+1$ general points is: $N_d^{\mathbb{P}^n} = \frac{(dn+1)!}{(d!)^{n+1}}$

This can be derived from the quantum cohomology of $\mathbb{P}^n$ or via localization on the moduli space of stable maps.

The genus zero Gromov-Witten invariants of a Calabi-Yau threefold satisfy the WDVV equations. Combined with initial data (e.g., three-point functions), these equations recursively determine all higher point invariants.

For the quintic, knowledge of:

• $n_1 = 2875$ (degree 1)
• WDVV constraints
• Divisor axiom

suffices to compute all $n_d$ recursively (though mirror symmetry provides explicit formulas).

For a toric Fano manifold $X_\Sigma$ and its Landau-Ginzburg mirror $(M, W)$: $I_X(q) = \int_\Gamma e^{W/z} \Omega$

where $I_X$ is the J-function encoding quantum cohomology and the integral is an oscillatory integral over a Lefschetz thimble $\Gamma$.

This isomorphism of D-modules (quantum differential equation = Picard-Fuchs equation) was proven by Givental and others using Fourier-Laplace transform techniques.

The generating function of Gromov-Witten invariants satisfies infinite recursion relations called Virasoro constraints: $\mathcal{L}_n \langle\tau\rangle = 0$

for operators $\mathcal{L}_n$ forming a Virasoro algebra representation. These constraints dramatically reduce the data needed to specify all invariants.

For Calabi-Yau threefolds, the constraints are particularly strong, determining higher genus from genus zero in many cases.