Weil-Petersson volume polynomials $V_{g,n}(L_1,\dots,L_n)$

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Polynomials

$g$

$n$

$V_{g,n}(L_1,\ldots,L_n)$

comment: The hyperbolic pair of pants is determined by its three boundary lengths, so its moduli space has volume $1$.

1/2*L1^2 + 1/2*L2^2 + 1/2*L3^2 + 1/2*L4^2 + 2*p^2

1/8*L1^4 + 1/2*L1^2*L2^2 + 1/8*L2^4 + 1/2*L1^2*L3^2 + 1/2*L2^2*L3^2 + 1/8*L3^4 + 1/2*L1^2*L4^2 + 1/2*L2^2*L4^2 + 1/2*L3^2*L4^2 + 1/8*L4^4 + 1/2*L1^2*L5^2 + 1/2*L2^2*L5^2 + 1/2*L3^2*L5^2 + 1/2*L4^2*L5^2 + 1/8*L5^4 + 3*L1^2*p^2 + 3*L2^2*p^2 + 3*L3^2*p^2 + 3*L4^2*p^2 + 3*L5^2*p^2 + 10*p^4

1/48*L1^2 + 1/12*p^2

comment: The factor $1/2$ from the generic elliptic involution is included.

1/192*L1^4 + 1/96*L1^2*L2^2 + 1/192*L2^4 + 1/12*L1^2*p^2 + 1/12*L2^2*p^2 + 1/4*p^4

1/1152*L1^6 + 1/192*L1^4*L2^2 + 1/192*L1^2*L2^4 + 1/1152*L2^6 + 1/192*L1^4*L3^2 + 1/96*L1^2*L2^2*L3^2 + 1/192*L2^4*L3^2 + 1/192*L1^2*L3^4 + 1/192*L2^2*L3^4 + 1/1152*L3^6 + 1/24*L1^4*p^2 + 1/8*L1^2*L2^2*p^2 + 1/24*L2^4*p^2 + 1/8*L1^2*L3^2*p^2 + 1/8*L2^2*L3^2*p^2 + 1/24*L3^4*p^2 + 13/24*L1^2*p^4 + 13/24*L2^2*p^4 + 13/24*L3^2*p^4 + 14/9*p^6

43/2160*p^6

1/442368*L1^8 + 29/138240*L1^6*p^2 + 139/23040*L1^4*p^4 + 169/2880*L1^2*p^6 + 29/192*p^8

Definition

The polynomial $V_{g,n}(L)$ is the Weil-Petersson orbifold volume of the moduli space of oriented hyperbolic surfaces of curvature $-1$, genus $g$, and $n$ labelled geodesic boundaries of lengths $L_1,\ldots,L_n$ [1]. The stored polynomials use $p$ as a formal stand-in for $\pi$.

Parameters

$g$

— genus ($g\geq0$)

$n$

— number of labelled boundaries ($n\geq0$, with $2g-2+n>0$)

Formulas

(1)

For $m=3g-3+n$, $\displaystyle V_{g,n}(L)=\frac{1}{m!}\int_{\overline{\mathcal M}_{g,n}}\left(2p^2\kappa_1+\frac12\sum_{i=1}^{n}L_i^2\psi_i\right)^m$ [1].

(2)

The polynomial is symmetric in $L_1,\ldots,L_n$, even in every variable, and homogeneous of total degree $6g-6+2n$ when $p$ has degree one [1].

(3)

The homogeneous part of degree $2m$ in the lengths is $\displaystyle\frac{1}{2^m}\sum_{d_1+\cdots+d_n=m}\frac{\langle\tau_{d_1}\cdots\tau_{d_n}\rangle_g}{d_1!\cdots d_n!}\prod_{i=1}^{n}L_i^{2d_i}$, where the brackets are cotangent-class intersection numbers [1].

(4)

For $2g-2+n>0$, $\displaystyle V_{g,n+1}(L_1,\ldots,L_n,2p\,i)=\sum_{j=1}^{n}\int_0^{L_j}t\,V_{g,n}(L_1,\ldots,L_{j-1},t,L_{j+1},\ldots,L_n)\,dt$, where $i^2=-1$ [3].

(5)

For $2g-2+n>0$, $\displaystyle\frac{\partial V_{g,n+1}}{\partial L_{n+1}}(L_1,\ldots,L_n,2p\,i)=2p\,i(2g-2+n)V_{g,n}(L_1,\ldots,L_n)$, where $i^2=-1$ [3].

Comments

(6)

Write $m=3g-3+n$. The Weil-Petersson form is normalized by $\omega_{\mathrm{WP}}=\sum_{j=1}^{m}d\ell_j\wedge d\tau_j$ in Fenchel-Nielsen coordinates, where $\ell_j$ and $\tau_j$ are the length and twist coordinates of the internal pants curves [1]. Volume is the integral of $\omega_{\mathrm{WP}}^m/m!$.

(7)

Automorphism weights are included: $V_{1,1}(L_1)=(L_1^2+4p^2)/48$ and $V_{2,0}=43p^6/2160$. Omitting the generic involution weight gives $(L_1^2+4p^2)/24$ and $43p^6/1080$, respectively [1]. The torus normalization used here is the orbifold convention of Do and Norbury; Mirzakhani's original formulas use the doubled torus volume with compensating factors in the recursion [3].

(8)

The boundaries are ordered, and no factor of $n!$ divides the volume. A zero boundary length denotes a cusp. Setting every length to zero gives the Weil-Petersson volumes of cusped surfaces.

(9)

The class $\psi_i$ is the first Chern class of the cotangent line at the $i$th marked point. Its top products are the cotangent-class intersection numbers. The logarithmic class $\kappa_1=f_*(\psi_{n+1}^2)$ is defined using the map $f:\overline{\mathcal M}_{g,n+1}\to\overline{\mathcal M}_{g,n}$ that forgets the last marked point and stabilizes the curve [2].

(10)

Entries lie in $\mathbb Q[L_1,\ldots,L_n,p]$, with the variables written in that order and $p$ representing $\pi$ formally. Substituting the numerical constant $\pi$ for $p$ gives the volume as a function of the boundary lengths. For $n=0$, an entry is a polynomial in $p$ alone.

Programs

(P1)

Sage

# Download the attached generate.py into the current directory.
from generate import volume
volume(1, 2)

References

[1]

N. Do, Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes, preprint (2011), Theorems 13, 18 and 22 and Appendix B. (arXiv)

[2]

K. Liu and H. Xu, Recursion formulae of higher Weil-Petersson volumes, International Mathematics Research Notices 2009 (2009), no. 5, 835-859, Theorem 1.2 and Lemma 2.2. (arXiv) (doi)

Links

[3]

Do and Norbury: Weil-Petersson volumes and cone surfaces

Similar tables

Weil-Petersson volumes of cusped surfaces — give the specialization in which all boundary lengths vanish and the formal variable is evaluated at pi

intersection numbers of cotangent line classes — determine the highest length-degree coefficients directly and all remaining coefficients through the kappa-class transform

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every stable pair with $2g-2+n\leq3$, corresponding to surfaces decomposable into at most three pairs of pants; these first three values of negative Euler characteristic give compact polynomials with at most five labelled boundaries)

How they were obtained:

Every coefficient is computed exactly using rational arithmetic. The generator evaluates pure cotangent-class intersections by the DVV recurrence, applies the set-partition transform of [2], Lemma 2.2, to powers of $\kappa_1$, and uses the volume coefficient formula in [1]. No floating-point approximation of $\pi$ enters the stored polynomials.

A separate implementation of Mirzakhani's geometric recursion uses exact kernel moments and agrees on all 61 monomial coefficients of the eight entries. All eight polynomials also agree with Appendix B of [1]. The checks cover symmetry, homogeneity, positivity, every applicable string and dilaton identity within the range, the genus-zero leading multinomial coefficients, the top-degree coefficients held in T458, and all eight constant terms held in T459. The constant-term comparison uses 512-bit interval arithmetic and the full uncertainty intervals of the stored 100-digit decimals. The generator, independent checks and numerical evidence are attached.