The A-Model and Gromov-Witten Theory - Main Theorem

The foundational theorems of Gromov-Witten theory establish the existence and properties of virtual fundamental classes and the axioms governing invariants. These results make the theory rigorous and computable.

TheoremVirtual Fundamental Class Existence

For any smooth projective variety $X$ and data $(g, k, \beta)$ where $g$ is genus, $k$ is the number of marked points, and $\beta \in H_2(X,\mathbb{Z})$ , the moduli space $\overline{M}_{g,k}(X,\beta)$ of stable maps admits a virtual fundamental class: $[\overline{M}_{g,k}(X,\beta)]^{\text{vir}} \in A_{\text{vdim}}(\overline{M}_{g,k}(X,\beta))$

of dimension: $\text{vdim} = (1-g)(\dim X - 3) + \int_\beta c_1(TX) + k$

This class is functorial with respect to morphisms and satisfies gluing axioms under degeneration.

The virtual class construction uses either perfect obstruction theories (Li-Tian) or derived algebraic geometry (Toen-Vezzosi). Both approaches yield the same class and provide deformation-invariance.

Remark

For Calabi-Yau threefolds with $c_1(X) = 0$ , the virtual dimension simplifies to: $\text{vdim} = (1-g) \cdot 0 + k = k$

independent of the curve class $\beta$ . This dimensional independence is special to Calabi-Yau threefolds and reflects the balance between deformations and obstructions.

TheoremGromov Compactness

Let $(f_n: C_n \to X)$ be a sequence of stable maps with bounded genus and fixed homology class $\beta$ . Then there exists a subsequence converging to a stable map $f: C \to X$ in the moduli space $\overline{M}_{g,k}(X,\beta)$ .

The limit may have:

Components contracted to points
Nodes where the curve has degenerated
Redistribution of genus among components

This convergence is in the Gromov topology, which allows bubbling of spheres.

The compactness theorem justifies the stable map compactification and ensures that moduli spaces are proper, allowing integration of cohomology classes to define invariants.

TheoremKontsevich's Formula

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

This formula can be derived from the virtual localization formula applied to the torus action on $\mathbb{P}^n$ .

Kontsevich's formula, proved in 1995, was among the first rigorous results in Gromov-Witten theory and demonstrated the power of the virtual class construction.

TheoremString Equation

The string equation relates Gromov-Witten invariants with different numbers of marked points: $\langle\alpha_1, \ldots, \alpha_k, 1\rangle_{g,\beta} = \sum_{i=1}^k \langle\alpha_1, \ldots, \psi_i\alpha_i, \ldots, \alpha_k\rangle_{g,\beta}$

where $\psi_i$ is the first Chern class of the universal cotangent line bundle at the $i$ -th marked point, and $1$ is the unit in cohomology.

The string equation is one of several axioms (including divisor, dilaton, and topological recursion relations) that characterize Gromov-Witten invariants and provide computational tools.

ExampleApplication to Recursive Computation

Using the string equation and divisor axiom together allows reduction of higher invariants to lower ones. For the quintic threefold, these axioms constrain the structure enough that mirror symmetry predictions can be verified recursively.

These theorems form the axiomatic foundation of Gromov-Witten theory, enabling both abstract structural results and concrete enumerative calculations.