Gromov-Witten Theory

Gromov-Witten theory is the mathematical framework for counting pseudo-holomorphic curves in symplectic manifolds. It produces deformation invariants with deep connections to enumerative geometry, string theory, and integrable systems.

Structure of the Theory

Definition7.7Quantum Cohomology

The quantum cohomology $QH^*(M)$ is a deformation of the ordinary cohomology ring $H^*(M)$ by Gromov-Witten invariants. The quantum product $\alpha *_q \beta$ is defined by $\langle \alpha *_q \beta, \gamma \rangle = \sum_{A \in H_2(M)} \mathrm{GW}_{0,3,A}(\alpha, \beta, \gamma) \cdot q^A$ , where $q^A = e^{\int_A \omega}$ . This product is associative (WDVV equations).

ExampleQuantum Cohomology of $\mathbb{CP}^n$

$QH^*(\mathbb{CP}^n) = \mathbb{C}[H, q] / (H^{n+1} - q)$ , where $H$ is the hyperplane class. The relation $H^{n+1} = q$ encodes the fact that there is exactly one line through two general points in $\mathbb{CP}^n$ (the class $A = [\text{line}]$ gives $\mathrm{GW}_{0,3,[\text{line}]}(H^a, H^b, H^c) = 1$ when $a + b + c = 2n + 1$ ).

Enumerative Applications

RemarkKontsevich's Formula

The number $N_d$ of rational curves of degree $d$ through $3d - 1$ general points in $\mathbb{CP}^2$ satisfies the recursion: $N_d = \sum_{\substack{d_1 + d_2 = d \\ d_1, d_2 > 0}} N_{d_1} N_{d_2} \left[d_1^2 d_2^2 \binom{3d-4}{3d_1-2} - d_1^3 d_2 \binom{3d-4}{3d_1-1}\right].$ This gives $N_1 = 1$ , $N_2 = 1$ , $N_3 = 12$ , $N_4 = 620$ , etc. The recursion follows from the WDVV associativity relations in quantum cohomology.

Definition7.8Mirror Symmetry Prediction

For a Calabi-Yau threefold $M$ , the genus-0 Gromov-Witten potential $F_0 = \sum_d N_d q^d$ is predicted by mirror symmetry to equal a period integral on the mirror manifold $\check{M}$ . This prediction has been verified in numerous cases (quintic threefold by Givental and Lian-Liu-Yau).