The A-Model and Gromov-Witten Theory - Examples and Constructions

Explicit computation of Gromov-Witten invariants requires both sophisticated theoretical machinery and concrete geometric insight. Classical examples illustrate the general theory and provide benchmarks for mirror symmetry predictions.

ExampleProjective Space $\mathbb{P}^n$

For $\mathbb{P}^n$ , the genus zero Gromov-Witten invariants counting degree $d$ curves through $dn+1$ points are: $N_d = \frac{(dn+1)!}{(d!)^{n+1}}$

For $\mathbb{P}^2$ , this gives $N_1 = 1$ (one line through 3 points), $N_2 = 1$ (one conic through 5 points), $N_3 = 12$ (twelve cubics through 8 points).

These classical numbers from enumerative geometry were known in the 19th century. Gromov-Witten theory provides a modern framework that extends to arbitrary genus and target spaces.

Toric Calabi-Yau Manifolds

Toric geometry provides systematic methods for computing Gromov-Witten invariants via localization.

DefinitionToric Localization

For a toric variety $X$ with torus action, Gromov-Witten invariants can be computed by localizing to the fixed loci: $\int_{[\overline{M}_{g,k}(X,\beta)]^{\text{vir}}} \alpha = \sum_F \int_{[F]} \frac{\iota_F^*\alpha}{e(N_F)}$

where the sum is over fixed loci $F$ and $e(N_F)$ is the equivariant Euler class of the normal bundle.

This reduces integration over high-dimensional moduli spaces to finite sums over combinatorially describable fixed point sets.

ExampleLocal $\mathbb{P}^2$

The local $\mathbb{P}^2$ is the total space of $\mathcal{O}_{\mathbb{P}^2}(-3)$ , a non-compact Calabi-Yau threefold. Gromov-Witten invariants count curves in $\mathbb{P}^2$ weighted by contributions from the normal bundle. Toric localization gives: $N_d = \frac{1}{d^3} \sum_{k|d} k^2 \mu(d/k) N_k^{\mathbb{P}^2}$

where $\mu$ is the Möbius function.

Mirror Symmetry Calculations

The most powerful method for computing Gromov-Witten invariants uses mirror symmetry to translate the problem to period calculations on the mirror manifold.

ExampleQuintic Calculation via Mirror Symmetry

For the quintic $X_5$ , the number $n_d$ of rational curves of degree $d$ is computed from the mirror quintic periods:

Solve the Picard-Fuchs equation for periods $\Pi(z)$
Compute the mirror map $q(z) = \exp(2\pi i t(z))$ where $t = \Pi_1/\Pi_0$
Invert to get $z(q)$ and expand $\Pi(z(q))$ in powers of $q$
Extract coefficients which give $n_d$

This yields: $n_1 = 2875$ , $n_2 = 609250$ , $n_3 = 317206375$ .

Relative Gromov-Witten Theory

When $X$ has a divisor $D$ , we can count curves with prescribed tangency conditions along $D$ .

DefinitionRelative Invariants

Relative Gromov-Witten invariants $\langle\alpha_1, \ldots, \alpha_k | \mu_1, \ldots, \mu_l\rangle_{g,\beta}$ count curves meeting cycles $\alpha_i$ generally and having contact orders $\mu_j$ with the divisor $D$ .

The relative moduli space $\overline{M}_{g,k|l}(X,D,\beta)$ parametrizes stable maps with marked points on $X$ and contact points on $D$ .

Relative invariants provide tools for degenerating $X$ and computing absolute invariants via degeneration formulas.

Degeneration and Gluing

The degeneration formula expresses invariants of $X$ in terms of invariants of simpler pieces when $X$ degenerates.

Remark

When $X$ degenerates to $X_0 \cup_D X_1$ (union along a divisor $D$ ), Gromov-Witten invariants of $X$ decompose: $\langle\alpha\rangle_X = \sum_{\mu} \langle\alpha_0 | \mu\rangle_{X_0} \cdot \langle\mu | \alpha_1\rangle_{X_1}$

where the sum is over partitions corresponding to ways curves can split across the node.

This technique reduces computations on complicated varieties to simpler building blocks, enabling recursive calculations of invariants.