计算 $${\mathbb {C}\mathbb {P}}^1$ 上的微分函数

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Alexandr Buryak, Paolo Rossi
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引用次数: 0

摘要

我们给出了在\(\mathbb{C}\mathbb{P}^1\)上具有两个零点和任意数量无残差极点的微分的明确公式,以及在\(\mathbb{C}\mathbb{P}^1\)上具有一个零点、两个无约束残差极点和任意数量无残差极点的微分的明确公式,这些公式都是根据它们的零点和极点的阶来计算的。在 \(\textrm{PGL}(2,\mathbb {C})\) 的作用下,这些是 \(\mathbb{C}\mathbb{P}^1\) 上唯一两个在极点子集上具有消失残差条件的有限微分族。第一个数族通过简单的积分与三重赫维兹数相关,我们展示了它与\(textrm{SL}_2(\mathbb {C})\)的表示理论和无分散KP层次方程的联系。第二个族有一个非常简单的产生数列,我们利用曲线和微分模空间的交集理论通过令人惊讶的计算恢复了它。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Counting meromorphic differentials on \({\mathbb {C}\mathbb {P}}^1\)

Counting meromorphic differentials on \({\mathbb {C}\mathbb {P}}^1\)

We give explicit formulas for the number of meromorphic differentials on \(\mathbb{C}\mathbb{P}^1\) with two zeros and any number of residueless poles and for the number of meromorphic differentials on \(\mathbb{C}\mathbb{P}^1\) with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on \(\mathbb{C}\mathbb{P}^1\) with vanishing residue conditions at a subset of poles, up to the action of \(\textrm{PGL}(2,\mathbb {C})\). The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of \(\textrm{SL}_2(\mathbb {C})\) and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials.

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来源期刊
Letters in Mathematical Physics
Letters in Mathematical Physics 物理-物理:数学物理
CiteScore
2.40
自引率
8.30%
发文量
111
审稿时长
3 months
期刊介绍: The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.
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