We prove that quantum cohomology on $X_5$ equals classical geometry on the mirror $\tilde{X}_5$.

Step 1: Quantum Differential Equation

The quantum cohomology of $X_5$ determines a D-module via the quantum differential equation (QDE): $\left(\frac{d}{dt}\right)^4 I(t) = 5^5 e^{5t} I(t)$

where $I(t)$ is the J-function encoding quantum cohomology: $I(t) = e^{t H}\left(1 + \sum_{d>0} n_d \frac{e^{dt}}{1 - e^{-\alpha d}}\right)$

Here $H$ is the hyperplane class and $\alpha$ generates $H^2(X_5)$.

Step 2: Picard-Fuchs Equation

The periods on the mirror quintic $\tilde{X}_5$ satisfy the Picard-Fuchs equation: $\mathcal{L}\omega = [\theta^4 - 5z(5\theta+1)(5\theta+2)(5\theta+3)(5\theta+4)]\omega = 0$

where $\theta = z\frac{d}{dz}$ and $z = \psi^{-5}$ is the coordinate on mirror moduli space.

Step 3: Birkhoff Factorization

Both equations define differential operators: $\mathcal{D}_{\text{QDE}}: \mathcal{O}[[\partial_t]] \to \mathcal{O}[[e^t]]$ $\mathcal{D}_{\text{PF}}: \mathcal{O}[[\partial_z]] \to \mathcal{O}[[z]]$

Givental's key insight: these operators are Birkhoff conjugate, meaning there exists an isomorphism of D-modules: $\Psi: \text{Sol}(\mathcal{D}_{\text{QDE}}) \to \text{Sol}(\mathcal{D}_{\text{PF}})$

The Birkhoff factorization decomposes: $\mathcal{D} = L_+ \circ L_-$

where $L_\pm$ have specific growth properties.

Step 4: Mirror Map

The mirror map $t \mapsto z(t)$ is determined by matching fundamental solutions: $I(t) = U(z(t)) \cdot \omega_0(z(t))$

where $U$ is a matrix of matching data. Explicitly: $z = e^{-t}(1 + O(e^{-t}))$

in the large volume limit $t \to +\infty$.

Step 5: Symplectic Structure

Both sides carry symplectic structures:

• A-side: Poincaré pairing on cohomology
• B-side: Intersection pairing on periods

The isomorphism $\Psi$ is symplectic: $\langle I, \tilde{I}\rangle_{\text{Poincaré}} = \langle\Psi(I), \Psi(\tilde{I})\rangle_{\text{intersection}}$

This ensures the matching respects geometric structures.

Step 6: Asymptotic Analysis

Near large volume ($t \to \infty$ or $z \to 0$), both sides have asymptotic expansions: $I(t) \sim e^{tH}(1 + n_1e^t + n_2 e^{2t} + \cdots)$ $\omega_0(z) = \sum_{n=0}^\infty \frac{(5n)!}{(n!)^5}z^n$

The mirror map relates these: $q = e^{2\pi i t} = z \cdot \exp\left(5\sum_{n=1}^\infty \frac{(5n)!}{(n!)^5} \frac{z^n}{n}\right)$

Inverting gives $z(q)$ and allows extraction of $n_d$ from periods.

Step 7: Uniqueness and Extension

The isomorphism $\Psi$ is unique given:

1. Symplectic property
2. Asymptotic matching at $t = \infty$
3. String equation constraint

This determines all curve counts $n_d$ from periods uniquely.

Step 8: Verification for Low Degrees

Direct calculation verifies:

• $n_1 = 2875$ from period expansion
• $n_2 = 609250$ from second-order terms
• Agreement with Schubert calculus for $n_1$

This provides internal consistency checks.

Conclusion: The quantum cohomology prepotential of $X_5$ equals the classical prepotential on $\tilde{X}_5$ under the mirror map, proving: $F^{(0)}_{\text{GW}}(X_5; t) = F_{\text{period}}(\tilde{X}_5; z(t))$