论文标题
不可还原的度量图和Weil-Petersson卷
Irreducible metric maps and Weil-Petersson volumes
论文作者
论文摘要
我们考虑在属属$ g $的表面上的地图，所有学位的所有顶点至少为三个，而分配给边缘的真实长度至少为正面。特别是，我们研究了具有固定属$ g $和固定数字$ n $ face的$α_1，\ ldots，α_n$和$β$ irreducibility限制的固定属$ n $面孔的家族，这大致要求所有违约周期至少具有至少$β$β$ $β$。利用对具有不可约束约束的离散图的列举的最新结果，我们计算了该图的$ v_ {g，n}^{（β）}（α_1，\ ldots，α_n）$，这些图自然来自lebesgue的lebesgue测量。它被证明是$β，α_1，\ ldots，α_n$ of度量$ 6G-6+2N $的均质多项式，并满足弦乐和Dilaton方程。出乎意料的是，对于$ g = 0,1 $和$β=2π$卷$ v_ {g，n}^{（2π）} $与weil-petersson卷$ v_ {g，n}^{ $ l_i = \ sqrt {α_i^2-4π^2} $，$ i = 1，\ ldots，n $。对于属$ g \ geq 2 $，卷之间的身份失败了，但是我们为两种类型的卷提供了明确的生成功能，表明它们密切相关。最后，我们讨论了通过双曲线多面体进行徒解释的可能性。
We consider maps on a surface of genus $g$ with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus $g$ and fixed number $n$ of faces with circumferences $α_1,\ldots,α_n$ and a $β$-irreducibility constraint, which roughly requires that all contractible cycles have length at least $β$. Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume $V_{g,n}^{(β)}(α_1,\ldots,α_n)$ of this family of maps that arises naturally from the Lebesgue measure on the edge lengths. It is shown to be a homogeneous polynomial in $β, α_1,\ldots, α_n$ of degree $6g-6+2n$ and to satisfy string and dilaton equations. Surprisingly, for $g=0,1$ and $β=2π$ the volume $V_{g,n}^{(2π)}$ is identical, up to powers of two, to the Weil-Petersson volume $V_{g,n}^{\mathrm{WP}}$ of hyperbolic surfaces of genus $g$ and $n$ geodesic boundary components of length $L_i = \sqrt{α_i^2 - 4π^2}$, $i=1,\ldots,n$. For genus $g\geq 2$ the identity between the volumes fails, but we provide explicit generating functions for both types of volumes, demonstrating that they are closely related. Finally we discuss the possibility of bijective interpretations via hyperbolic polyhedra.
