The B-Model and Variations of Hodge Structure - Applications

The B-model framework extends far beyond mirror symmetry, connecting to arithmetic geometry, mathematical physics, and representation theory. These applications demonstrate the fundamental role of variations of Hodge structure in modern mathematics.

Mirror Symmetry Dictionary

The B-model provides one half of the mirror symmetry correspondence, with precise dictionary entries.

TheoremB-Model/A-Model Correspondence

For a mirror pair $(X, X^\vee)$ of Calabi-Yau threefolds:

Complex structure moduli $\mathcal{M}_c(X)$ mirror Kähler moduli $\mathcal{M}_K(X^\vee)$
B-model correlators on $X$ equal A-model correlators on $X^\vee$
Picard-Fuchs equations for periods on $X$ equal quantum differential equations on $X^\vee$

Specifically: $F^B_0(X; t_c) = F^A_0(X^\vee; t_K)$

after appropriate identification of parameters via the mirror map.

This correspondence allows computation of difficult A-model invariants (Gromov-Witten numbers) using classical B-model geometry (period integrals).

Computing Gromov-Witten Invariants

The primary application is computing enumerative invariants via period calculations.

ExampleQuintic Curve Counting

The number $n_d$ of rational curves on the quintic is extracted from periods:

Solve Picard-Fuchs for fundamental period $\omega_0(z)$
Compute mirror map $q(z) = \exp(2\pi i t(z))$ where $t = \omega_1/\omega_0$
Invert to get $z(q)$ and expand
The prepotential $F(t) = \frac{1}{(2\pi i)^3}\log(\omega_0(z(q)))$ gives: $F(q) = \frac{5}{2}t^3 + \sum_{d=1}^\infty n_d \text{Li}_3(q^d)$

where $\text{Li}_3$ is the trilogarithm.

This method computed $n_1 = 2875$ , $n_2 = 609250$ , ..., later verified rigorously.

Arithmetic Applications

Periods of algebraic varieties defined over number fields have arithmetic significance.

TheoremPeriods and L-Functions

For a Calabi-Yau variety $X$ over a number field $K$ , the periods: $\Pi_\gamma = \int_\gamma \Omega$

lie in algebraic extensions of $K$ . Special values of the L-function $L(H^n(X), s)$ at integer points are conjecturally related to these periods by: $L(H^n(X), n+1) \sim \text{periods} \times \text{regulator}$

This extends the Birch-Swinnerton-Dyer conjecture to higher-dimensional varieties.

Topological String Theory

The B-model computes partition functions in topological string theory.

Remark

The genus $g$ topological string amplitude on $X$ is: $F_g(t) = \int_{\overline{M}_g(X)} \lambda_g^3 \psi_1^{d_1} \cdots$

where $\lambda_g$ is the Hodge bundle. For $g=0$ , this reduces to the prepotential. Higher genus uses BCOV recursion.

The all-genus partition function: $Z(t, g_s) = \exp\left(\sum_{g=0}^\infty g_s^{2g-2} F_g(t)\right)$

has modular properties and connects to topological M-theory.

Gauged Linear Sigma Models

Variations of Hodge structure describe the low-energy effective theory of gauged linear sigma models (GLSMs).

TheoremGLSM/VHS Correspondence

The classical VHS on complex structure moduli of a Calabi-Yau corresponds to the quantum Kähler geometry of the GLSM describing the mirror. The Picard-Fuchs operator equals the quantum differential equation: $\mathcal{L}_{\text{PF}} \Pi = 0 \quad \leftrightarrow \quad \mathcal{L}_{\text{QDE}} \mathcal{I} = 0$

where $\mathcal{I}$ is the J-function encoding quantum cohomology.

Hodge Loci and Special Subvarieties

Periods detect special geometric features.

Remark

Hodge loci are subvarieties of moduli space where additional Hodge classes appear: $\mathcal{HL}^{p,q} = \{[X] : h^{p,q}(X) > h^{p,q}(\text{general})\}$

These loci correspond to enhanced symmetries or special geometric properties. The Hodge conjecture predicts they are algebraic and correspond to algebraic cycles.

These applications show how B-model techniques bridge pure mathematics and theoretical physics, making abstract Hodge theory computational.