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Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians

David R. Morrison

TL;DR

This work reframes the Candelas et al. calculation of rational curves on a generic quintic threefold in purely algebraic–geometric terms by developing a $q$-expansion principle for Yukawa couplings near maximally unipotent boundary points and anchoring these in variations of Hodge structure. It defines intrinsic, canonical coordinates and a mathematically normalized Yukawa coupling, then applies the framework to the quintic–mirror family ${\cal W}_{\sqrt[5]{\lambda}}$ to extract a $q$-expansion whose coefficients coincide with enumerative invariants, notably $n_1=2875$ and $n_2=609250$, thereby providing a rigorous bridge between period data, mirror symmetry, and counts of rational curves. The paper surveys substantial mathematical evidence for mirror symmetry, details the quintic–mirror construction via quotients and resolutions, and discusses the intriguing possibility of a deeper arithmetic structure—'mirror moonshine'—underpinning these enumerative predictions. Overall, it offers a mathematically natural, coordinate-intrinsic path from Hodge theory and period asymptotics to concrete enumerative consequences in Calabi–Yau geometry.

Abstract

We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new $q$-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the ``mirror symmetry'' phenomenon recently observed by string theorists.

Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians

TL;DR

This work reframes the Candelas et al. calculation of rational curves on a generic quintic threefold in purely algebraic–geometric terms by developing a -expansion principle for Yukawa couplings near maximally unipotent boundary points and anchoring these in variations of Hodge structure. It defines intrinsic, canonical coordinates and a mathematically normalized Yukawa coupling, then applies the framework to the quintic–mirror family to extract a -expansion whose coefficients coincide with enumerative invariants, notably and , thereby providing a rigorous bridge between period data, mirror symmetry, and counts of rational curves. The paper surveys substantial mathematical evidence for mirror symmetry, details the quintic–mirror construction via quotients and resolutions, and discusses the intriguing possibility of a deeper arithmetic structure—'mirror moonshine'—underpinning these enumerative predictions. Overall, it offers a mathematically natural, coordinate-intrinsic path from Hodge theory and period asymptotics to concrete enumerative consequences in Calabi–Yau geometry.

Abstract

We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new -expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the ``mirror symmetry'' phenomenon recently observed by string theorists.

Paper Structure

This paper contains 6 sections, 6 theorems, 84 equations, 1 figure.

Table of Contents

  1. Variations of Hodge structure arising from families of Calabi-Yau manifolds
  2. The asymptotic behavior of the periods
  3. The $q$-expansion of the Yukawa coupling
  4. Mirror symmetry
  5. The quintic-mirror family
  6. Mirror moonshine?

Key Result

Lemma 1

Let $\pi: {\cal X}\to C$ be a one-parameter family of varieties with $h^{n,0}=1$. Let $P\in B=\overline C-C$ be a boundary point at which the monodromy on $R^n\pi_*{\Bbb Z}$ is maximally unipotent and let $N$ be the logarithm of the monodromy. Then the image of $N^{n}$ is a $\Bbb Q$-vector space of

Figures (1)

  • Figure 1: The steps in the toroidal resolution.

Theorems & Definitions (6)

  • Lemma 1
  • Proposition 1
  • Corollary 1
  • Lemma 2
  • Corollary 2
  • Lemma 3: Essentially due to Friedman and Scattone fs