The B-Model and Variations of Hodge Structure - Main Theorem

The foundational theorems governing variations of Hodge structure provide the mathematical framework for the B-model and establish its computational power. These results connect geometry, analysis, and algebra.

TheoremSchmid's Nilpotent Orbit Theorem

Let $\mathcal{V}$ be a variation of Hodge structure over the punctured disk $\Delta^* = \{0 < |z| < 1\}$ with unipotent monodromy $T$ . Then near $z=0$ , the period map has an asymptotic expansion: $\Phi(z) \sim \exp(N\log z + constant) \cdot (\text{holomorphic})$

where $N = \frac{1}{2\pi i}\log T$ is nilpotent. The limiting Hodge filtration $F_{\lim}$ together with the monodromy weight filtration $W_\bullet$ defines a limiting mixed Hodge structure.

This theorem describes precisely how Hodge structures degenerate at boundary points of moduli spaces. The nilpotent orbit approximation becomes exact in a suitable metric and coordinate system.

Remark

For Calabi-Yau threefolds, the monodromy at a conifold point is maximally unipotent with $N^2 = 0$ but $N \neq 0$ . At large complex structure limits, $N^4 \neq 0$ , reflecting the different geometry of the degeneration.

TheoremCattani-Kaplan-Schmid Global Torelli

For generic Calabi-Yau threefolds, the period map: $\Phi: \mathcal{M}_c \to \Gamma \backslash \mathcal{D}$

is generically injective when restricted to the complement of certain loci. This means the periods determine the complex structure up to finite ambiguity.

Global Torelli theorems assert that geometry is determined by Hodge theory. For Calabi-Yau threefolds this holds generically but can fail when the manifold has large automorphism groups.

TheoremBCOV Holomorphic Anomaly Equations

The higher genus B-model free energies $F_g^B$ for $g \geq 1$ satisfy recursion relations: $\bar{\partial}_i F_g = \frac{1}{2}\bar{C}_{ij}^k\left(D_k F_{g-1} + \sum_{r=1}^{g-1} D_k F_r \cdot D_k F_{g-r}\right)$

where $\bar{C}_{ij}^k$ is the anti-holomorphic Yukawa coupling and $D_k$ are covariant derivatives.

These allow recursive computation of all $F_g$ from $F_0$ (the prepotential) and $F_1$ (determined by the holomorphic anomaly).

The BCOV equations arose from topological string theory but have purely mathematical formulations and proofs using special geometry.

TheoremDeligne's Theorem on Absolute Hodge Classes

For a variation of Hodge structure over a variety defined over a number field, absolute Hodge classes (those Hodge classes that remain Hodge under all embeddings into $\mathbb{C}$ ) are algebraic.

This extends the Hodge conjecture to families and has implications for arithmetic aspects of mirror symmetry.

ExampleApplication to Period Calculations

For the mirror quintic, Schmid's theorem describes the behavior of periods near $z=0$ (large complex structure) and near $z=\infty$ (conifold point). The asymptotic expansions: $\Pi_0(z) \sim \sum_{n=0}^\infty a_n z^n$ $\Pi_1(z) \sim \log(z)\Pi_0(z) + \text{regular}$

encode the monodromy and allow analytic continuation around singular points.

These theorems provide both existence results and computational tools, making the B-model a powerful framework for calculating Hodge-theoretic invariants.