The B-Model and Variations of Hodge Structure - Core Definitions

The B-model topological string theory provides the mirror counterpart to the A-model, encoding complex geometric data through variations of Hodge structure. This framework connects period integrals, differential equations, and special geometry in a unified mathematical package.

DefinitionVariation of Hodge Structure

A variation of Hodge structure (VHS) of weight $n$ over a base $B$ consists of:

A local system $\mathbb{V}$ of finite-dimensional $\mathbb{Q}$ -vector spaces
A Hodge filtration $F^\bullet$ on $\mathcal{V} = \mathbb{V} \otimes \mathcal{O}_B$ varying holomorphically
A flat connection $\nabla: \mathcal{V} \to \Omega^1_B \otimes \mathcal{V}$ (Gauss-Manin connection)

satisfying:

Griffiths transversality: $\nabla(F^p) \subset \Omega^1_B \otimes F^{p-1}$
Polarization: A non-degenerate pairing compatible with the Hodge decomposition

For a family $\pi: \mathcal{X} \to B$ of Calabi-Yau manifolds, the cohomology $R^n\pi_*\mathbb{Q}$ forms a variation of Hodge structure. The Hodge filtration varies as we move in the moduli space, with its motion governed by Griffiths transversality.

DefinitionPeriod Map

The period map $\Phi: B \to \Gamma \backslash \mathcal{D}$ associates to each point in the base the Hodge structure of the corresponding fiber. The period domain $\mathcal{D}$ is a homogeneous space: $\mathcal{D} = G_\mathbb{R}/H$

where $G$ is the group preserving the polarization and $H$ preserves a reference Hodge structure.

For Calabi-Yau threefolds, the relevant period domain for $H^3$ is: $\mathcal{D} = \{F^3 \subset \mathbb{V}_\mathbb{C} : Q(F^3, \bar{F}^3) = 0, \, Q(F^3, \bar{F}^2) > 0\}$

where $Q$ is the intersection form on $H^3$ .

The B-Model Correlation Functions

Unlike the A-model which counts curves, the B-model computes correlation functions from complex geometry.

DefinitionB-Model Observables

B-model observables are elements of the Dolbeault cohomology: $\mathcal{O}^{(p,q)} \in H^q(X, \Omega^p_X)$

The genus zero B-model correlation function is: $\langle\mathcal{O}_1 \cdots \mathcal{O}_k\rangle^B_0 = \int_{X} \mathcal{O}_1 \wedge \cdots \wedge \mathcal{O}_k \wedge \Omega$

where $\Omega$ is the holomorphic volume form.

These correlators are purely topological, independent of the Kähler structure, depending only on complex structure. This is the mirror property to the A-model's dependence only on symplectic (Kähler) structure.

Remark

The B-model topological string partition function has genus expansion: $Z_B = \exp\left(\sum_{g=0}^\infty \lambda^{2g-2} F^B_g\right)$

The genus zero part $F^B_0$ is the prepotential on complex structure moduli space, related to the Yukawa coupling: $C_{ijk} = \partial_i\partial_j\partial_k F^B_0$

Gauss-Manin Connection

The Gauss-Manin connection governs how cohomology varies in families.

DefinitionGauss-Manin Connection

For a family $\pi: \mathcal{X} \to B$ , the Gauss-Manin connection is the flat connection: $\nabla_{\text{GM}}: R^n\pi_*\mathbb{C} \to \Omega^1_B \otimes R^n\pi_*\mathbb{C}$

In terms of period integrals $\Pi_i(t) = \int_{\gamma_i} \Omega_t$ : $\nabla_{\text{GM}} \Pi = d\Pi - \omega \Pi$

where $\omega$ is the connection matrix.

The flatness condition $\nabla^2_{\text{GM}} = 0$ implies that periods satisfy differential equations (the Picard-Fuchs equations). These equations are the computational heart of B-model calculations.