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{"Title": "Weil--Petersson volume polynomials $V_{g,n}(L_1,\\dots,L_n)$", "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 $L_1,\\ldots,L_n$ CITE{Do}.", "Parameters": {"g": {"type": "Z", "title": "genus", "display": "$g$", "constraints": "$g\\geq0$"}, "n": {"type": "Z", "title": "number of labelled boundaries", "display": "$n$", "constraints": "$n\\geq0$, with $2g-2+n>0$"}}, "Tags": ["volume", "period", "polynomial"], "Keywords": ["Weil-Petersson volume polynomials", "Mirzakhani volume polynomials", "moduli spaces of curves", "bordered Riemann surfaces", "hyperbolic surfaces", "kappa class intersection numbers"], "Comments": {"normalization": "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 CITE{Do}. Volume is the integral of $\\omega_{\\mathrm{WP}}^m/m!$.", "orbifold": "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 CITE{Do}. 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 CITE{DN}.", "boundaries": "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 HREF{T459}[Weil-Petersson volumes of cusped surfaces].", "classes": "The class $\\psi_i$ is the first Chern class of the cotangent line at the $i$th marked point. Its top products are the HREF{T458}[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 CITE{LX}.", "variables": "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."}, "Formulas": {"intersection": "For $m=3g-3+n$, $\\displaystyle V_{g,n}(L)=\\frac{1}{m!}\\int_{\\overline{\\mathcal M}_{g,n}}\\left(2p^2\\kappa_1+\\frac12\\sum_{i=1}^{n}L_i^2\\psi_i\\right)^m$ CITE{Do}.", "degree": "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 CITE{Do}.", "top-degree": "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 HREF{T458}[cotangent-class intersection numbers] CITE{Do}.", "string": "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 $i^2=-1$ CITE{DN}.", "dilaton": "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 $i^2=-1$ CITE{DN}."}, "Similar tables": [{"table": "HREF{T459}[Weil-Petersson volumes of cusped surfaces]", "relation": "give the specialization in which all boundary lengths vanish and the formal variable is evaluated at pi"}, {"table": "HREF{T458}[intersection numbers of cotangent line classes]", "relation": "determine the highest length-degree coefficients directly and all remaining coefficients through the kappa-class transform"}], "Links": {"DN": {"title": "Do and Norbury: Weil-Petersson volumes and cone surfaces", "url": "https://arxiv.org/abs/math/0603406"}}, "References": {"Do": {"bib": "N. Do, Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes, preprint (2011), Theorems 13, 18 and 22 and Appendix B.", "arxiv": "1103.4674"}, "LX": {"bib": "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": "0708.0565", "doi": "10.1093/imrn/rnn148"}}, "Programs": {"sage": {"language": "Sage", "code": "# Download the attached generate.py into the current directory.\nfrom generate import volume\nvolume(1, 2)"}}, "Data properties": {"type": "Q[]", "rigour": "exact", "complete": "no", "complete-note": "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", "rigour details": "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 CITE{LX}, Lemma 2.2, to powers of $\\kappa_1$, and uses the volume coefficient formula in CITE{Do}. No floating-point approximation of $\\pi$ enters the stored polynomials.\n\nA 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 CITE{Do}. 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 HREF{T458}, and all eight constant terms held in HREF{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."}, "Display properties": {"number-header": "$V_{g,n}(L_1,\\ldots,L_n)$"}, "Numbers": {"0": {"3": {"number": "1", "comment": "The hyperbolic pair of pants is determined by its three boundary lengths, so its moduli space has volume $1$."}, "4": "1/2*L1^2 + 1/2*L2^2 + 1/2*L3^2 + 1/2*L4^2 + 2*p^2", "5": "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": {"1": {"number": "1/48*L1^2 + 1/12*p^2", "comment": "The factor $1/2$ from the generic elliptic involution is included."}, "2": "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", "3": "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"}, "2": {"0": "43/2160*p^6", "1": "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"}}}