The fundamental curve counting problem asks: Given a smooth projective variety $X$, how many curves of genus $g$ in homology class $\beta \in H_2(X,\mathbb{Z})$ pass through specified constraints?

For Calabi-Yau threefolds, the canonical question counts rational curves (genus 0) in degree $d$ through $k$ points, expressed as Gromov-Witten invariants: $n_d = \langle pt, \ldots, pt\rangle_{0,d\beta_0}$

where $\beta_0$ is the hyperplane class.

The Gromov-Witten potential organizes curve counts into a generating function: $F(q) = \sum_{d=0}^\infty n_d q^d$

where $q = e^{2\pi i t}$ and $t$ is the complexified Kähler parameter. This function encodes all genus-zero curve-counting data.

For higher genus, the total potential is: $\mathcal{F}(q, \lambda) = \sum_{g=0}^\infty \sum_{d=0}^\infty n_{g,d} q^d \lambda^{2g-2}$

where $n_{g,d}$ counts genus $g$, degree $d$ curves.

The mirror principle states that for a mirror pair $(X, X^\vee)$: $F_{\text{GW}}(X; q) = F_{\text{period}}(X^\vee; z(q))$

where:

• Left side: Gromov-Witten generating function (quantum/A-model)
• Right side: Period integral generating function (classical/B-model)
• $z(q)$: Mirror map relating coordinates

This allows computation of $n_d$ by:

1. Solving Picard-Fuchs equations for periods on $X^\vee$
2. Computing the mirror map $q \mapsto z$
3. Expanding periods in powers of $q$ to extract coefficients

The instanton numbers $n_d$ are alternatively called:

• Gromov-Witten invariants (mathematical)
• Worldsheet instantons (physical)
• Rational curve counts (geometric)

In physics, these count contributions to the path integral from holomorphic maps $\mathbb{P}^1 \to X$ wrapping degree $d$ curves. The exponential weight $e^{-\text{Area}}$ makes higher degree curves contribute quantum corrections.