The A-Model and Gromov-Witten Theory - Core Definitions

The A-model topological string theory on a Calabi-Yau manifold provides a mathematically rigorous framework for counting holomorphic curves. This theory lies at the heart of mirror symmetry's enumerative predictions and connects symplectic geometry with algebraic topology.

DefinitionGromov-Witten Invariants

Let $X$ be a smooth projective variety and $\beta \in H_2(X,\mathbb{Z})$ a curve class. The Gromov-Witten invariant $\langle\alpha_1, \ldots, \alpha_k\rangle_{g,\beta}$ counts genus $g$ curves in class $\beta$ passing through cycles dual to $\alpha_i \in H^*(X)$ :

$\langle\alpha_1, \ldots, \alpha_k\rangle_{g,\beta} = \int_{[\overline{M}_{g,k}(X,\beta)]^{\text{vir}}} \text{ev}_1^*(\alpha_1) \cup \cdots \cup \text{ev}_k^*(\alpha_k)$

where $\overline{M}_{g,k}(X,\beta)$ is the moduli space of stable maps and $[\overline{M}_{g,k}(X,\beta)]^{\text{vir}}$ is its virtual fundamental class.

The virtual fundamental class has the expected dimension even when the moduli space has excess dimension due to deformations. For genus zero curves on a Calabi-Yau threefold $X$ , the expected dimension is: $\dim[\overline{M}_{0,k}(X,\beta)]^{\text{vir}} = \int_\beta c_1(TX) + \dim X - 3 + k = k$

since $c_1(X) = 0$ for Calabi-Yau manifolds.

DefinitionStable Maps

A stable map $f: C \to X$ consists of:

A nodal curve $C$ of genus $g$ with $k$ marked points
A morphism $f: C \to X$

satisfying the stability condition: every component of $C$ contracted by $f$ has at least 3 special points (nodes or marked points).

The moduli space $\overline{M}_{g,k}(X,\beta)$ compactifies the space of smooth curves by allowing nodal curves. The boundary consists of degenerate curves where components have separated or where the genus has dropped via node formation.

Remark

The compactification is essential for defining invariants. Without allowing stable degenerations, moduli spaces are non-compact and integration would be ill-defined. The Gromov compactness theorem guarantees that sequences of holomorphic curves converge to stable maps.

The A-Model Topological String

The A-model is a topological twist of the $(2,2)$ superconformal field theory with target space $X$ . Observables depend only on the Kähler structure, not the complex structure.

DefinitionA-Model Correlation Functions

The A-model correlation function in genus $g$ is: $\langle\mathcal{O}_{\alpha_1} \cdots \mathcal{O}_{\alpha_k}\rangle_g = \sum_{\beta} q^\beta \langle\alpha_1, \ldots, \alpha_k\rangle_{g,\beta}$

where $q^\beta = e^{-\int_\beta \omega}$ with $\omega$ the Kähler form, and $\mathcal{O}_\alpha$ are operators corresponding to cohomology classes $\alpha \in H^*(X)$ .

The generating function for genus zero Gromov-Witten invariants is the A-model prepotential: $F^{(0)}_A(t) = \sum_{\beta, k} \frac{q^\beta}{k!} \langle t, \ldots, t\rangle_{0,\beta}$

where $t = \sum_i t^i \alpha_i$ encodes all cohomology insertions.

ExampleGenus Zero Invariants on the Quintic

For the quintic threefold $X_5$ , the degree $d$ rational curve invariants $n_d$ appear in the prepotential as: $F^{(0)}_A(t) = \frac{1}{6}t^3 + \sum_{d=1}^\infty n_d e^{dt}$

The first few values are $n_1 = 2875$ , $n_2 = 609250$ , determined by mirror symmetry.

The A-model provides a symplectic-geometric interpretation of mirror symmetry, where the quantum cohomology ring structure encodes curve-counting information essential for the mirror correspondence.