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Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties

Dmitrii Adler, Valery Gritsenko

TL;DR

This work analyzes modular differential equations for weak Jacobi forms, focusing on weight $0$ index $3$ forms related to elliptic genera of Calabi–Yau sixfolds and hyperkähler sixfolds. It establishes that the minimal order for the elliptic genus of a strict six-dimensional CY is $4$, while generic Jacobi forms in $J_{0,3}^w$ satisfy a seventh-order MDE, with a defining cubic divisor $S(a,b,c)=0$ governing the coefficients. The authors perform a comprehensive organization of possible MDEs: ten weight-$0$ index-$3$ forms satisfy order $4$, five yield order $5$, and five one-parameter divisors yield order $6$, with order $7 attained generically outside a non-singular cubic curve $S(a,b,c)=0$; hyperkähler examples are shown to satisfy order $7$. They further show that if meromorphic coefficients are allowed, the MDE order can decrease, and they provide explicit instances illustrating reductions beyond the integral-coefficient setting, highlighting the rich interplay between geometry and modular differential relations.

Abstract

We study modular differential equations (MDEs) of high orders for weak Jacobi forms and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for the elliptic genus of six-dimensional manifolds with a trivial first Chern class. We prove that the minimal possible order of the MDE for the elliptic genus of a strict six-dimensional Calabi--Yau variety is four, and find MDEs of order 7 for hyperkähler varieties of dimension 6. The latter MDEs correspond to the generic case. The non-generic weak Jacobi forms of weight 0 and index 3 form a divisor that contains two cubic plane curves in the coefficient space.

Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties

TL;DR

This work analyzes modular differential equations for weak Jacobi forms, focusing on weight index forms related to elliptic genera of Calabi–Yau sixfolds and hyperkähler sixfolds. It establishes that the minimal order for the elliptic genus of a strict six-dimensional CY is , while generic Jacobi forms in satisfy a seventh-order MDE, with a defining cubic divisor governing the coefficients. The authors perform a comprehensive organization of possible MDEs: ten weight- index- forms satisfy order , five yield order , and five one-parameter divisors yield order , with order S(a,b,c)=07$. They further show that if meromorphic coefficients are allowed, the MDE order can decrease, and they provide explicit instances illustrating reductions beyond the integral-coefficient setting, highlighting the rich interplay between geometry and modular differential relations.

Abstract

We study modular differential equations (MDEs) of high orders for weak Jacobi forms and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for the elliptic genus of six-dimensional manifolds with a trivial first Chern class. We prove that the minimal possible order of the MDE for the elliptic genus of a strict six-dimensional Calabi--Yau variety is four, and find MDEs of order 7 for hyperkähler varieties of dimension 6. The latter MDEs correspond to the generic case. The non-generic weak Jacobi forms of weight 0 and index 3 form a divisor that contains two cubic plane curves in the coefficient space.

Paper Structure

This paper contains 13 sections, 12 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Jacobi modular forms, elliptic genus and MDEs
  3. Definition of Jacobi forms
  4. Elliptic genus of compact complex manifolds with $c_1=0$.
  5. Modular differential operators for modular forms and Jacobi forms
  6. Examples of MDEs of order 1, 2 and 3.
  7. MDEs of order 3
  8. MDEs of the minimal order 4 and 5 for Jacobi forms in $J_{0,3}^w$
  9. The generic case: MDEs of order 7
  10. MDE for a generic Jacobi form
  11. Equations of order 6
  12. Equations of order 7.
  13. Conclusion

Key Result

Theorem 2.2

(See Gr99, KYY.) If $M_d$ is a compact complex manifold of dimension $d$ with $c_1(M_d) = 0$ (over $\Bbb R$), then its elliptic genus $\chi(M_d; \tau, z)$ is a weak Jacobi form of weight $0$ and index $\frac{d}{2}$ with integer Fourier coefficients.

Theorems & Definitions (31)

  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Theorem 3.1
  • proof
  • Example 3.2
  • Example 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 21 more