Decorated super-Teichmüller space
Decorated super-Teichmüller space
复制标题
装修超级Teichmüller空间
DOI：
10.4310/jdg/1552442609
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发表时间：
2015
影响因子：
2.5
通讯作者：
A. Zeitlin
中科院分区：
文献类型：
--
作者：
R. Penner;A. Zeitlin
我们在曲面$F$的超Teichmueller空间$ST(F)$上引入主丛$S T(F)$的坐标，并在通常的Teichmueller空间$T(F)$上用$S\geq1$穿孔扩展装饰丛$\tilde T(F)=T(F)上的λ长度坐标。实际上，$PSL(2，{\mathbb R})$的Fuchsian子群在Minkowski空间${\mathbb R}^{2，1}$上的作用被$OSP(1|2)$在超Minkowski空间${\mathbb R}^{2，1|2}$上的超Fuchsian子群所取代，其中$OSP(1|2)$表示正交辛Lie超群，其长度由${\mathbb R}中适当的迷向量三元组的费米子不变量扩展.在玻色子的情况下，现在在偶坐标和奇坐标上都有类似的托勒密变换，并且在推广了Weil-Petersson Kaehler形式的$S T(F)上有不变的偶二形式。这最终解决了大约30年前尤里·伊万诺维奇·马宁在莫斯科的研讨会上提出的寻找装饰Teichmueller理论的超相似的问题，并为$S T(F)$的超模在${Mathbb R}^{2，1|2}$中提供了一个自然的几何解释。
We introduce coordinates for a principal bundle $S\tilde T(F)$ over the super Teichmueller space $ST(F)$ of a surface $F$ with $s\geq 1$ punctures that extend the lambda length coordinates on the decorated bundle $\tilde T(F)=T(F)\times {\mathbb R}_+^s$ over the usual Teichmueller space $T(F)$. In effect, the action of a Fuchsian subgroup of $PSL(2,{\mathbb R})$ on Minkowski space ${\mathbb R}^{2,1}$ is replaced by the action of a super Fuchsian subgroup of $OSp(1|2)$ on the super Minkowski space ${\mathbb R}^{2,1|2}$, where $OSp(1|2)$ denotes the orthosymplectic Lie supergroup, and the lambda lengths are extended by fermionic invariants of suitable triples of isotropic vectors in ${\mathbb R}^{2,1|2}$. As in the bosonic case, there is the analogue of the Ptolemy transformation now on both even and odd coordinates as well as an invariant even two-form on $S\tilde T(F)$ generalizing the Weil-Petersson Kaehler form. This finally solves a problem posed in Yuri Ivanovitch Manin's Moscow seminar some thirty years ago to find the super analogue of decorated Teichmueller theory and provides a natural geometric interpretation in ${\mathbb R}^{2,1|2}$ for the super moduli of $S\tilde T(F)$.