{"Title": "Weil-Petersson volumes $V_{g,n}$ of the moduli spaces of hyperbolic surfaces", "Parameters": {"g": {"display": "$g$", "title": "genus", "type": "Z", "constraints": "$g\\geq0$"}, "n": {"display": "$n$", "title": "number of labelled cusps", "type": "Z", "constraints": "$n\\geq0$, $2g-2+n>0$"}}, "Data properties": {"type": "R", "digits": "100", "rigour": "proven", "complete": "no", "complete-note": "it holds every stable pair with $0\\leq g\\leq3$ and $0\\leq n\\leq8$, giving the first four genera with up to eight labelled cusps, including the closed surfaces when $g\\geq2$", "rigour details": "The coefficient of $\\pi^{6g-6+2n}$ is obtained in exact rational arithmetic from Theorem 1.1 and equation (3) of CITE{LX}. Multiplication by the power of $\\pi$ uses Arb ball arithmetic with a 64-bit guard beyond the precision needed for 100 significant decimal digits. The smallest measured relative accuracy is 390 bits, more than 117 decimal digits. The value $V_{0,3}=1$ is stored exactly.\n\nThe independent checker uses the set-partition transform in CITE{LX}, Lemma 2.2, and a separate implementation of the pure cotangent-class recurrence with Python fractions. It compares every rational coefficient, all 17 constant terms from Appendix B of CITE{Do} within this range, and every genus-zero coefficient against Zograf's scalar recurrence. A separate 512-bit MPFI computation checks every volume and the full uncertainty interval represented by each stored decimal. The reproducible generator, checks and numerical evidence are attached."}, "Numbers": {"0": {"3": {"number": "1", "comment": "The thrice-punctured sphere has a unique hyperbolic structure, and its moduli space has volume $1$."}, "4": "19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446", "5": "974.0909103400243723644033268870511124972758567268542169146785938997085545682719619012186723475299255", "6": "78192.98774412476087845784808373015177453369228392512476320752991537297157784448403127000794435581321", "7": "8723122.847440881037220203746591620022430273660928702021613210707584844917083645274821936871306055974", "8": "1247292101.130784684147534348793595103236150390208761843431936436638047918858975283398010091880818901"}, "1": {"1": "0.8224670334241132182362075833230125946094749506033992188677791146850037352016004369168144503098793527", "2": "24.35227275850060930911008317217627781243189641817135542286696484749271386420679904753046680868824814", "3": "1495.494301117140235380341356792653176015671710348294735908340736086368308865878109887677747569646701", "4": "139428.6918750370458269726789324077529570224427916147984479510655158443002519110572389655730275835028", "5": "17429982.70302074093700809196986204398609561258357940712580283977539404280690370180811949159183217659", "6": "2732194472.090147331462757217456415878542842548306426640954964041010170842829341182961448241947061345", "7": "514937788550.1345965564161117465020544628800860736899832387596314578529001111420177555966762431069098", "8": "113374151839310.5370271758572190810276418570614307313256778939206998620178322540365056404195134339513"}, "2": {"0": "19.13876635358244944087936855419169243115294153124305763216031299158149919084308295391968545996869290", "1": "1433.163538885659615660045258402584417209469816029121864954128390817397698714114128047023257726815306", "2": "153543.7778409944531381563232823573437823553120329793005791464118175979608198750819688754389600462992", "3": "21367577.29159783400895482955493443201175821744612269635954253632814491945580341944970256638783674421", "4": "3655405666.724411976966699282802160862940018618210036420583492055848755305630792204843525119824888210", "5": "741729128222.7063320588063175614170411931111958808436312363456714958824893486397174874465106902160395", "6": "174089883630542.8687540530440308934948447689720150460685910980264056455462952175662856792838989733509", "7": "46390079170961834.32584205295641160152235475236546258250825639795642826257049480086004097940921716753", "8": "13834123058880853697.68999808313112730863173587959500203067657187489404489851438013104786412392820937"}, "3": {"0": "134909.2211327896628611937091443298427377890517834393893483551010317859623797946959831934124292658003", "1": "20764963.34676300829623633044500201850655642596978386739978269243927490134559483449125865369247219308", "2": "3859604421.279218840930310083855726196593562210385937588470541058953237782280841254803713169152271635", "3": "840428602082.7202126215963989700162039965076824953187132910186677036659463475047679549652438357385638", "4": "209753766802193.5108271573271504902686046611862866977832655152293267739280425548929639190238388178477", "5": "59019285882055650.28469598366915842363648876368141465166365369321675275164062800277235805006133818302", "6": "18481467614260465167.96841963588755781843778046147304260564698673456221647349731734124220013507223470", "7": "6374104811376373204762.226121254490799449201156365475865142781072021909124677830677888397289351695300", "8": "2400646570201089687467099.492416429735758926424473693452826320880100476635074294410769523187064028314"}}, "Definition": "The Weil-Petersson volume $V_{g,n}$ is the orbifold symplectic volume of the moduli space of complete oriented hyperbolic surfaces of curvature $-1$, genus $g$, and $n$ labelled cusps CITE{LX}.", "Tags": ["volume", "period"], "Keywords": ["Weil-Petersson volumes", "Mirzakhani volumes", "moduli spaces of curves", "punctured Riemann surfaces", "kappa class intersection numbers"], "Comments": {"orbifold": "Integration includes the automorphism weights of the Deligne-Mumford stack. Thus $V_{1,1}=\\pi^2/12$ and $V_{2,0}=43\\pi^6/2160$. Omitting the generic involution weight doubles these two values to $\\pi^2/6$ and $43\\pi^6/1080$, respectively CITE{Do}.", "labelling": "The cusps are ordered, and no factor of $n!$ divides the volume. A cusp has boundary length zero, so these values are the constant terms $V_{g,n}(0,\\ldots,0)$ of the bordered-surface volume polynomials CITE{Do}.", "classes": "The class $\\kappa_1$ is $\\pi_*(\\psi_{n+1}^{\\,2})$, where $\\pi:\\overline{\\mathcal M}_{g,n+1}\\longrightarrow\\overline{\\mathcal M}_{g,n}$ forgets the last marked point and stabilizes the curve, and $\\psi_{n+1}$ is its cotangent line class CITE{LX}. Its top powers reduce to HREF{T458}[intersection numbers of cotangent line classes] by the set-partition formula in CITE{LX}, Lemma 2.2.", "normalisation": "The Weil-Petersson symplectic form is normalized by $[\\omega_{\\mathrm{WP}}]=2\\pi^2\\kappa_1$, where $\\kappa_1$ is the logarithmic first kappa class CITE{LX}."}, "Formulas": {"volume": "For $m=3g-3+n$, $\\displaystyle V_{g,n}=\\frac{1}{m!}\\int_{\\mathcal M_{g,n}}\\omega_{\\mathrm{WP}}^m=\\frac{(2\\pi^2)^m}{m!}\\int_{\\overline{\\mathcal M}_{g,n}}\\kappa_1^m$ CITE{LX}.", "small": "$V_{0,3}=1$, $V_{0,4}=2\\pi^2$, $V_{0,5}=10\\pi^4$, $V_{1,1}=\\pi^2/12$, $V_{1,2}=\\pi^4/4$, and $V_{2,0}=43\\pi^6/2160$ CITE{Do}."}, "Display properties": {"number-header": "$V_{g,n}$"}, "Similar tables": [{"table": "HREF{T458}[intersection numbers of cotangent line classes]", "relation": "determine the kappa class integrals appearing in the Weil-Petersson volumes through a set-partition transform"}], "Links": {"author-paper": {"title": "Liu and Xu: Recursion formulae of higher Weil-Petersson volumes", "url": "https://archive.ymsc.tsinghua.edu.cn/pacm_download/460/10620-Kefeng_Liu_27.pdf"}}, "References": {"LX": {"bib": "K. Liu and H. Xu, Recursion formulae of higher Weil-Petersson volumes, International Mathematics Research Notices 2009 (2009), no. 5, 835-859.", "arxiv": "0708.0565", "doi": "10.1093/imrn/rnn148"}, "Do": {"bib": "N. Do, Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes, preprint (2011), Sections 2.3 and 3.1 and Appendix B.", "arxiv": "1103.4674"}}, "Programs": {"sage": {"language": "Sage", "code": "# Download the attached generate.py into the current directory.\nfrom generate import volume\nvolume(2, 0, digits=100)"}}}