WDVV Equations (Associativity of Quantum Cohomology)

Theorem7.2WDVV Equations

The genus-0 Gromov-Witten potential $F_0(t) = \sum_{k \geq 3} \frac{1}{k!} \sum_A \mathrm{GW}_{0,k,A}(t, \ldots, t) \cdot q^A$ satisfies the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations: for all indices $\alpha, \beta, \gamma, \delta$ , $\sum_{\mu, \nu} \frac{\partial^3 F_0}{\partial t^\alpha \partial t^\beta \partial t^\mu} \eta^{\mu\nu} \frac{\partial^3 F_0}{\partial t^\nu \partial t^\gamma \partial t^\delta} = \sum_{\mu, \nu} \frac{\partial^3 F_0}{\partial t^\alpha \partial t^\gamma \partial t^\mu} \eta^{\mu\nu} \frac{\partial^3 F_0}{\partial t^\nu \partial t^\beta \partial t^\delta},$ where $\eta^{\mu\nu}$ is the inverse of the Poincaré pairing $\eta_{\mu\nu} = \int_M e_\mu \cup e_\nu$ .

The WDVV equations express the associativity of the quantum product. They arise geometrically from the factorization of the boundary of the moduli space $\overline{\mathcal{M}}_{0,4} \cong \mathbb{CP}^1$ .

Proof

Consider the forgetful map $\pi: \overline{\mathcal{M}}_{0,k}(M, A) \to \overline{\mathcal{M}}_{0,4}$ that forgets all but 4 marked points. The boundary of $\overline{\mathcal{M}}_{0,4} \cong \mathbb{CP}^1$ consists of three points, corresponding to three ways a 4-pointed sphere can degenerate into two components. The identity $[\text{boundary}_1] = [\text{boundary}_2]$ in $H^2(\overline{\mathcal{M}}_{0,4})$ pulls back via the Gromov-Witten theory to the WDVV equations, using the splitting axiom for GW invariants. $\square$

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ExampleWDVV for $\mathbb{CP}^2$

For $\mathbb{CP}^2$ , with $H^*(\mathbb{CP}^2) = \mathbb{Z}[H]/(H^3)$ , the WDVV equations and the initial condition $N_1 = 1$ recursively determine all $N_d$ . This is Kontsevich's celebrated formula for the number of rational curves.