[Paper Review] Lines on Calabi Yau complete intersections, mirror symmetry, and Picard Fuchs equations

Anatoly Libgober, Jeremy Teitelbaum|ArXiv.org|Jan 4, 1993

Algebraic Geometry and Number TheoryMathematics4 references20 citations

TL;DR

This paper verifies mirror symmetry predictions for Calabi–Yau complete intersections in projective space by computing Yukawa couplings via Picard–Fuchs equations, confirming that the q-expansion coefficient of the triple derivative of the holomorphic 3-form matches the number of lines (degree-1 rational curves) on generic complete intersections. The key result confirms the predicted integer counts for lines on two cubic threefolds in ℙ⁵ and other complete intersections, with exact agreement across multiple mirror families.

ABSTRACT

A relation between the number of rational curves of fixed degree on Calabi Yau threefolds and the Picard Fuchs equations, which was suggested as part of the study of mirror symmetry, is verified in the case of complete intersection of two cubics and lines.

Motivation & Objective

• To verify that mirror symmetry predicts the correct number of rational curves of degree 1 (lines) on Calabi–Yau complete intersections in projective space.
• To establish that the coefficient of q in the q-expansion of the Yukawa coupling κ_sss equals the number of lines on a generic complete intersection.
• To confirm that the monodromy at infinity is maximally unipotent and that the Euler characteristic of the mirror family satisfies χ(V) = −χ(W(V)) for the resolved quotient of V_λ by G_81.
• To extend the method of asymptotic normalization of the coordinate s to other complete intersection types beyond the quintic, using hypergeometric Picard–Fuchs equations.

Proposed method

• Uses the asymptotic normalization of the coordinate s defined by s(z) ∼ log(z) − ∑d_i log(d_i) as z → 0, ensuring consistency with physical normalization in mirror symmetry.
• Applies the Picard–Fuchs differential equation to compute the holomorphic solution F₀ and the logarithmic solution F₁, which together define the coordinate s = F₁/F₀.
• Computes the Yukawa coupling via the Wronskian W = F₀ΘF₁ − F₁ΘF₀, expressing κ_sss in terms of F₀, W, and s.
• Employs the mirror map q = exp(2πis) to expand κ_sss in powers of q, with the coefficient of q^d related to the number of rational curves of degree d.
• Uses group action (G₈₁) on V_λ to compute the Euler characteristic of the quotient resolution via the formula ∑_g χ(V_λ^g / G₈₁), verifying χ(V_λ) = −χ(W(V_λ)).
• Applies Schubert calculus to compute line counts on complete intersections by intersecting cycles in Grassmannians, such as 4Ω₄,₆ for four quadrics in ℙ⁷.

Experimental results

Research questions

• RQ1 Does the coefficient of q in the q-expansion of the Yukawa coupling κ_sss equal the number of lines on a generic Calabi–Yau complete intersection of two cubics in ℙ⁵?
• RQ2 Is the Euler characteristic of the mirror family W(V_λ) related to that of V_λ by χ(V_λ) = −χ(W(V_λ))?
• RQ3 Does the asymptotic normalization s(z) ∼ log(z) − ∑d_i log(d_i) yield consistent integer counts for rational curves across different complete intersection types?
• RQ4 Are the Picard–Fuchs equations for remaining Calabi–Yau complete intersections hypergeometric with parameters matching the degrees of defining hypersurfaces?

Key findings

• The coefficient of q in the q-expansion of κ_sss for the mirror family W(V_λ) is 1053, matching the number of lines on a generic complete intersection of two cubics in ℙ⁵.
• The Euler characteristic of V_λ is −144, and the resolution of V_λ/G₈₁ has Euler characteristic 144, confirming χ(V_λ) = −χ(W(V_λ)).
• The number of lines on the complete intersection of four quadrics in ℙ⁷ is 512, computed via Schubert calculus as the 4-fold self-intersection of 4Ω₄,₆.
• The number of lines on the complete intersection of a cubic and two quadrics in ℙ⁶ is 720, computed as the intersection (18Ω₂,₅ + 27Ω₃,₄)(4Ω₃,₅)².
• For all studied complete intersection types, the asymptotic normalization s(z) ∼ log(z) − ∑d_i log(d_i) leads to integral values of n_d in the expansion κ_sss = 9 + ∑(n_d d³ q^d)/(1−q^d).
• The predicted number of rational curves of degree 10 on the complete intersection of two cubics in ℙ⁵ is 512,045,241,907,209,106,828,608, confirming the method's consistency.