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Big quantum cohomology of Fano complete intersections

Xiaowen Hu

TL;DR

This work develops a program to compute the full genus 0 quantum cohomology of smooth Fano complete intersections by exploiting monodromy symmetry to reduce the WDVV equations to a tractable system involving ambient data and a primitive sector. A central conjecture—the double root recursion—provides a pathway to reconstruct higher-length primitive invariants from ambient invariants, supported by a concrete algorithm and verification at the first nontrivial level (four primitive insertions). The paper proves non-semisimplicity of the ambient Frobenius structure in non-exceptional cases, derives reconstruction theorems for cubic hypersurfaces and odd intersections of two quadrics, and gives explicit formulas for key leading terms such as F^{(1)} and F^{(2)}(0), including special cases with gcd conditions and a closed form when (n-1)/a(n,d)=2. It also connects these algebraic structures to geometric data, notably via the Fano variety of lines in cubic hypersurfaces, and outlines a concrete Macaulay2 implementation to carry out the computations. The results advance the numerical and structural understanding of the full (big) quantum cohomology beyond ambient insertions, with implications for mirror symmetry and derived-category relationships.

Abstract

For smooth complete intersections in the projective spaces, we use the deformation invariance of Gromov-Witten invariants and results in classical invariant theory to study the symmetric reduction of the WDVV equation by the monodromy groups. We propose a double root recursion conjecture for genus 0 invariants of non-exceptional Fano complete intersections other than the cubic hypersurfaces and the odd dimensional intersection of two quadrics. Based on it we develop an algorithm to compute the genus 0 invariants of any length and with any number of primitive insertions. The effectiveness of our algorithm is the main conjecture of this paper. We prove the conjecture at the first non-trivial order, which amounts to the computation of genus zero Gromov-Witten invariants with exactly 4 primitive insertions. We show a reconstruction theorem for the odd dimensional intersection of two quadrics. There are some 4-point and 8-point genus 0 Gromov-Witten invariants for cubic hypersurfaces that cannot be computed by the monodromy group method. We compute them by studying the cohomology ring structure of the Fano variety of lines and by Zinger's reduced genus 1 Gromov-Witten invariants. Then we get a reconstruction theorem for cubic hypersurfaces. At the end of the paper, we give numerical examples and some conjectural closed formulae.

Big quantum cohomology of Fano complete intersections

TL;DR

This work develops a program to compute the full genus 0 quantum cohomology of smooth Fano complete intersections by exploiting monodromy symmetry to reduce the WDVV equations to a tractable system involving ambient data and a primitive sector. A central conjecture—the double root recursion—provides a pathway to reconstruct higher-length primitive invariants from ambient invariants, supported by a concrete algorithm and verification at the first nontrivial level (four primitive insertions). The paper proves non-semisimplicity of the ambient Frobenius structure in non-exceptional cases, derives reconstruction theorems for cubic hypersurfaces and odd intersections of two quadrics, and gives explicit formulas for key leading terms such as F^{(1)} and F^{(2)}(0), including special cases with gcd conditions and a closed form when (n-1)/a(n,d)=2. It also connects these algebraic structures to geometric data, notably via the Fano variety of lines in cubic hypersurfaces, and outlines a concrete Macaulay2 implementation to carry out the computations. The results advance the numerical and structural understanding of the full (big) quantum cohomology beyond ambient insertions, with implications for mirror symmetry and derived-category relationships.

Abstract

For smooth complete intersections in the projective spaces, we use the deformation invariance of Gromov-Witten invariants and results in classical invariant theory to study the symmetric reduction of the WDVV equation by the monodromy groups. We propose a double root recursion conjecture for genus 0 invariants of non-exceptional Fano complete intersections other than the cubic hypersurfaces and the odd dimensional intersection of two quadrics. Based on it we develop an algorithm to compute the genus 0 invariants of any length and with any number of primitive insertions. The effectiveness of our algorithm is the main conjecture of this paper. We prove the conjecture at the first non-trivial order, which amounts to the computation of genus zero Gromov-Witten invariants with exactly 4 primitive insertions. We show a reconstruction theorem for the odd dimensional intersection of two quadrics. There are some 4-point and 8-point genus 0 Gromov-Witten invariants for cubic hypersurfaces that cannot be computed by the monodromy group method. We compute them by studying the cohomology ring structure of the Fano variety of lines and by Zinger's reduced genus 1 Gromov-Witten invariants. Then we get a reconstruction theorem for cubic hypersurfaces. At the end of the paper, we give numerical examples and some conjectural closed formulae.

Paper Structure

This paper contains 72 sections, 111 theorems, 805 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations and Conventions
  4. Quick recap of Gromov-Witten invariants
  5. Genus zero generating functions
  6. Quantum cohomology of complete intersections with ambient insertions
  7. The monodromy group and its invariants
  8. The reduction of the WDVV equation by the monodromy group action
  9. Even dimensions
  10. Odd dimensions
  11. Incorporating the Euler vector field
  12. F(l)(0) as ratios
  13. An extension problem of Frobenius manifolds
  14. Structure of Frobenius algebras and reconstruction theorems
  15. F(1) and non-semisimplicity of Mamb
  16. ...and 57 more sections

Key Result

Lemma 1.2

Let $V$ be a subspace of $H^{*}(X)$. Suppose there is another subspace $V^{\bot}$, such that Then choosing a pure degree basis of $V$, says $\gamma_1,\dots,\gamma_l$, the generating function satisfies the WDVV equation

Theorems & Definitions (249)

  • Conjecture 1.1: Double root recursion
  • Lemma 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 239 more