The B-Model and Variations of Hodge Structure - Key Properties

The B-model exhibits rich mathematical structure connecting complex geometry, differential equations, and special Kähler geometry. These properties make it computationally tractable and provide the framework for mirror symmetry.

Griffiths Transversality and Horizontality

The variation of Hodge structure satisfies fundamental differential constraints that control how Hodge filtrations can vary.

DefinitionGriffiths Transversality

For a variation of Hodge structure with filtration $F^p \subset \mathcal{V}$ , the Gauss-Manin connection satisfies: $\nabla(F^p) \subset \Omega^1_B \otimes F^{p-1}$

This means the connection lowers Hodge index by at most one.

For Calabi-Yau threefolds, Griffiths transversality implies that the infinitesimal variation of the holomorphic 3-form $\Omega$ under complex structure deformations lands in $H^{2,1}$ : $\nabla_i\Omega \in H^{2,1}(X) \oplus H^{3,0}(X)$

This constrains the period map and ensures it takes values in the period domain.

Remark

The horizontality condition for periods states that parallel transport preserves the integral lattice $H^n(X,\mathbb{Z})$ . Combined with Griffiths transversality, this determines the monodromy representation: $\rho: \pi_1(B) \to \text{Aut}(H^n, Q)$

where $Q$ is the intersection pairing.

Special Geometry

The complex structure moduli space carries a special Kähler metric derived from the variation of Hodge structure.

DefinitionSpecial Coordinates

Special coordinates on the moduli space are coordinates $t^i$ such that in these coordinates:

The metric is Kähler with potential $K = -\log(i\int \Omega \wedge \bar{\Omega})$
The connection is flat
The Yukawa coupling $C_{ijk} = \partial_i\partial_j\partial_k F$ is totally symmetric

where $F$ is the prepotential.

The prepotential satisfies homogeneity: $F(\lambda X) = \lambda^2 F(X)$ for projective coordinates $X^I$ . In affine coordinates $t^i = X^i/X^0$ : $F(t) = \frac{1}{2}(X^0)^{-2}\left(c_{ijk}t^it^jt^k + \text{corrections}\right)$

where $c_{ijk}$ are classical intersection numbers.

Yukawa Couplings and WDVV

The Yukawa coupling encodes the quantum product structure via the B-model.

DefinitionYukawa Coupling

The Yukawa coupling is the third derivative of the prepotential: $C_{ijk}(t) = \int_X \phi_i \wedge \phi_j \wedge \phi_k \wedge \Omega$

where $\phi_i \in H^{2,1}(X)$ are basis elements dual to complex structure parameters $t^i$ .

These couplings satisfy the WDVV equations from associativity of the quantum product. For the B-model, these are classical equations without instanton corrections, unlike the A-model.

Period Integrals and Special Functions

Period integrals can often be expressed in terms of classical special functions.

ExampleHypergeometric Periods

For one-parameter families, periods are typically hypergeometric functions: $\Pi(z) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{z^n}{n!}$

The Picard-Fuchs operator annihilating $\Pi$ is the hypergeometric differential operator.

Limiting Mixed Hodge Structures

At boundary points of moduli space, Hodge structures degenerate to mixed Hodge structures.

DefinitionLimiting MHS

Near a boundary divisor, the limiting Hodge filtration satisfies: $F^p_{\lim} = \lim_{t \to 0} F^p(t)$

exists and defines a mixed Hodge structure with weight filtration determined by monodromy: $W_k = \ker(N^{k+1})$

where $N = \log T$ for monodromy $T$ .

The monodromy weight filtration controls how periods blow up near singular points, with logarithmic growth determined by the nilpotency of $N$ . This structure is essential for understanding conifold transitions and other geometric degenerations.