Enumerative Applications - Key Properties

The enumerative applications of mirror symmetry exhibit remarkable mathematical properties that both constrain the invariants and provide computational tools. These properties reveal deep structure in curve-counting problems.

Integrality and Positivity

Despite being defined through integration over moduli spaces with rational coefficients, Gromov-Witten invariants of Calabi-Yau threefolds have special arithmetic properties.

DefinitionBPS Invariants

The BPS invariants (Gopakumar-Vafa invariants) $n_\beta^g \in \mathbb{Z}$ are integers defined by reorganizing Gromov-Witten invariants: $\sum_{g,\beta} N_{g,\beta}^{\text{GW}} q^\beta \lambda^{2g-2} = \sum_{\beta,g,m} \frac{n_\beta^g}{m}(2\sin(m\lambda/2))^{2g-2} q^{m\beta}$

These integers count BPS states in string theory and are conjecturally non-negative.

The integrality of $n_\beta^g$ is proven in many cases but remains conjectural in full generality. When proven, it provides strong constraints on Gromov-Witten invariants.

Modularity and Special Functions

For certain Calabi-Yau manifolds, the generating functions exhibit modularity properties.

Remark

For elliptic Calabi-Yau threefolds $X \to B$ with elliptic fibration, the generating function: $F(q, \tau) = \sum_{n,d} n_{d,n} q^d e^{2\pi i n\tau}$

is related to modular forms. Here $\tau$ parametrizes the elliptic fiber and $q$ the base.

This modularity connects enumerative geometry to number theory and has been exploited to compute invariants recursively.

Recursion Relations

Gromov-Witten invariants satisfy powerful recursion relations that allow computation of higher degree from lower degree invariants.

DefinitionWDVV Equations for Enumerative Geometry

The WDVV equations provide quadratic recursion relations: $\sum_{d_1+d_2=d} n_{d_1} \cdot n_{d_2} \cdot P(d_1, d_2) = Q(d)$

where $P$ and $Q$ are polynomials determined by intersection theory. These allow recursive computation of all $n_d$ from initial data.

For the quintic, the WDVV equations combined with divisor axioms determine all $n_d$ from $n_1 = 2875$ .

Multiple Cover Formula

Curves can wrap fundamental curves multiple times, leading to wall multiple cover contributions.

DefinitionMultiple Cover Formula

If a curve $C$ appears with multiplicity, its contribution to $n_{kd}$ includes covers: $n_{kd} = n_d + \text{multiple cover corrections}$

The Aspinwall-Morrison formula gives: $n_{kd} \geq \frac{1}{k^3} n_d$

for degree $kd$ rational curves including $k$ -fold covers of degree $d$ curves.

Understanding multiple cover contributions is essential for extracting primitive curve counts from Gromov-Witten invariants.

Growth Rates

The curve counts exhibit super-exponential growth.

Remark

For Calabi-Yau threefolds, the generating function has finite radius of convergence: $F(q) = \sum n_d q^d, \quad |q| < r$

typically $r = e^{-2\pi}$ or related to discriminant points. The growth rate: $n_d \sim C \cdot d^{-\alpha} \cdot \beta^d$

for constants $C, \alpha, \beta$ determined by geometry.

This rapid growth makes direct computation beyond small degrees impossible, highlighting mirror symmetry's computational advantage.

Descendant Invariants

Beyond primary curve counts, descendant invariants involve additional insertions of $\psi$ -classes.

DefinitionGravitational Descendants

Descendant Gromov-Witten invariants are: $\langle\tau_{k_1}(\alpha_1), \ldots, \tau_{k_n}(\alpha_n)\rangle_{g,\beta} = \int_{[\overline{M}_{g,n}(X,\beta)]^{\text{vir}}} \psi_1^{k_1}\text{ev}_1^*(\alpha_1) \cdots$

where $\psi_i$ are cotangent line classes and $\tau_k(\alpha) = \psi^k \cup \alpha$ .

These refine primary invariants and satisfy Givental's formalism connecting all genera.

Mirror symmetry extends to descendants, with the B-model computing descendant structure through derivatives of periods.

These properties make enumerative mirror symmetry a rich mathematical theory with connections to number theory, integrable systems, and representation theory.