[Paper Review] Rational curves on Calabi-Yau manifolds: verifying predictions of Mirror Symmetry

Sheldon Katz|ArXiv.org|Jan 27, 1993

Geometric and Algebraic Topology7 references17 citations

TL;DR

This paper verifies mirror symmetry predictions for Gromov-Witten invariants counting rational curves on Calabi-Yau hypersurfaces in weighted projective spaces using intersection theory and the Schubert computer algebra system. It computes and confirms 62 additional predictions—specifically for lines and conics of low degree on Calabi-Yau manifolds in $$\mathbb{P}(k,1^n)$$—extending the total number of verified predictions from 3 to 65, with all results matching theoretical physics predictions.

ABSTRACT

Mirror symmetry, a phenomenon in superstring theory, has recently been used to give tentative calculations of several numbers in algebraic geometry. In this paper, the numbers of lines and conics on various hypersurfaces which satisfy certain incidence properties are calculated, and shown to agree with the numbers predicted by Greene, Morrison, and Plesser using mirror symmetry in every instance. This increases the number of verified predictions from 3 to 65. Calculations are performed using the Maple package {\sc schubert} written by Katz and Strømme.

Motivation & Objective

• To verify mirror symmetry predictions for the number of rational curves on Calabi-Yau hypersurfaces in weighted projective spaces.
• To extend the number of verified predictions beyond the three known cases (lines, conics, and twisted cubics on the quintic threefold) by computing higher-degree invariants.
• To demonstrate the consistency of physics-based predictions with rigorous algebraic geometry via intersection-theoretic computation.
• To provide explicit numerical results for Gromov-Witten invariants of lines and conics satisfying incidence conditions on Calabi-Yau hypersurfaces of dimension up to 10.
• To validate the use of the Schubert package for computing enumerative invariants in weighted projective spaces.

Proposed method

• Express the number of rational curves as intersection numbers on Grassmannians and flag varieties using the moduli space of curves.
• Model weighted $r$-planes in $\mathbb{P}(k,1^n)$ as sections of a bundle $\mathbb{C} \oplus S^k(Q)$ over the Grassmannian $G(r+1,n)$.
• Compute Gromov-Witten invariants via top-degree intersection numbers of Chern classes of vector bundles, such as $c_{2k+5}(F)$ for conics.
• Incorporate incidence conditions with linear subspaces by pulling back tautological classes and using pushforwards via morphisms from pointed curve moduli spaces.
• Use the Schubert computer algebra system to evaluate the intersection-theoretic expressions numerically.
• Cross-validate results with classical enumerative geometry for specific cases, such as weighted lines in $\mathbb{P}(4,1^4)$.

Experimental results

Research questions

• RQ1 Do mirror symmetry predictions for Gromov-Witten invariants of rational curves on Calabi-Yau hypersurfaces in weighted projective spaces hold for higher-degree curves?
• RQ2 Can intersection theory on Grassmannians and flag varieties be used to compute the number of lines and conics meeting specified linear subspaces on Calabi-Yau manifolds?
• RQ3 Do the predicted numbers of rational curves in dimensions up to 10 agree with explicit algebraic geometry computations?
• RQ4 Is the Schubert package effective for computing Gromov-Witten invariants in weighted projective spaces with non-trivial orbifold singularities?
• RQ5 Can the relations between invariants for lines, as suggested by conformal field theory, be given a mathematical proof in the context of Calabi-Yau hypersurfaces?

Key findings

• The number of weighted lines in a weighted sextic in $\mathbb{P}(2,1^4)$ is computed and confirmed to match mirror symmetry predictions.
• The number of weighted lines in a weighted octic in $\mathbb{P}(4,1^4)$ is computed via classical enumerative geometry and agrees with mirror symmetry predictions.
• For Calabi-Yau hypersurfaces of dimension up to 10, the Gromov-Witten invariants $n^a_b(d)$ counting lines meeting specified linear subspaces are computed and match theoretical predictions.
• The Gromov-Witten invariants for conics on the same Calabi-Yau hypersurfaces are computed and verified to be consistent with mirror symmetry, including values such as $n^1_1(2) = 12607965435718224000$ for dimension 8.
• The paper confirms 62 new predictions, increasing the total number of verified mirror symmetry predictions from 3 to 65.
• A mathematical proof sketch is provided for the conformal field theory relations between line invariants, supporting the consistency of the predictions.