Modular differential equations of W(Dn)-invariant Jacobi forms

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2024-10-05 DOI:10.1016/j.geomphys.2024.105339

Dmitrii Adler , Valery Gritsenko

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引用次数: 0

Abstract

We study rings of weak Jacobi forms invariant with respect to the action of the Weyl group for the root systems

C_{n}

and

D_{n}

, and provide an explicit construction of generators of such rings by using modular differential operators. The construction of generators in the form of a

D_{n}

-tower (

n \geq 2

) gives a simple proof that these graded rings of Jacobi forms are polynomial. We study in detail modular differential equations (MDEs), which are satisfied by generators of index 1. Interesting anomalies are noticed for the lattices

D_{4}

D_{8}

and

D_{12}

. In particular, some generators for these lattices satisfy the Kaneko–Zagier type MDEs of order 2 or MDEs of order 1 similar to the differential equation of the elliptic genus of three-dimensional Calabi–Yau manifolds.

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W(Dn)不变雅可比形式的模微分方程

我们研究了根系统 Cn 和 Dn 在韦尔群作用方面不变的弱雅各比形式环，并通过使用模微分算子明确构造了此类环的生成器。以 Dn 塔（n≥2）形式构造的生成器简单地证明了这些雅可比形式的分级环是多项式的。我们详细研究了指数为 1 的生成器所满足的模块微分方程（MDE）。我们注意到网格 D4、D8 和 D12 的有趣反常现象。特别是，这些网格的一些生成器满足阶数为 2 的 Kaneko-Zagier 型 MDE 或类似于三维 Calabi-Yau 流形的椭圆属微分方程的阶数为 1 的 MDE。

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity