Weil-Petersson volumes $V_{g,n}$ of the moduli spaces of hyperbolic surfaces

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Numbers

$g$

$n$

$V_{g,n}$

comment: The thrice-punctured sphere has a unique hyperbolic structure, and its moduli space has volume $1$.

19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446

974.0909103400243723644033268870511124972758567268542169146785938997085545682719619012186723475299255

78192.98774412476087845784808373015177453369228392512476320752991537297157784448403127000794435581321

8723122.847440881037220203746591620022430273660928702021613210707584844917083645274821936871306055974

1247292101.130784684147534348793595103236150390208761843431936436638047918858975283398010091880818901

0.8224670334241132182362075833230125946094749506033992188677791146850037352016004369168144503098793527

24.35227275850060930911008317217627781243189641817135542286696484749271386420679904753046680868824814

1495.494301117140235380341356792653176015671710348294735908340736086368308865878109887677747569646701

139428.6918750370458269726789324077529570224427916147984479510655158443002519110572389655730275835028

17429982.70302074093700809196986204398609561258357940712580283977539404280690370180811949159183217659

2732194472.090147331462757217456415878542842548306426640954964041010170842829341182961448241947061345

514937788550.1345965564161117465020544628800860736899832387596314578529001111420177555966762431069098

113374151839310.5370271758572190810276418570614307313256778939206998620178322540365056404195134339513

19.13876635358244944087936855419169243115294153124305763216031299158149919084308295391968545996869290

1433.163538885659615660045258402584417209469816029121864954128390817397698714114128047023257726815306

153543.7778409944531381563232823573437823553120329793005791464118175979608198750819688754389600462992

21367577.29159783400895482955493443201175821744612269635954253632814491945580341944970256638783674421

3655405666.724411976966699282802160862940018618210036420583492055848755305630792204843525119824888210

741729128222.7063320588063175614170411931111958808436312363456714958824893486397174874465106902160395

174089883630542.8687540530440308934948447689720150460685910980264056455462952175662856792838989733509

46390079170961834.32584205295641160152235475236546258250825639795642826257049480086004097940921716753

13834123058880853697.68999808313112730863173587959500203067657187489404489851438013104786412392820937

134909.2211327896628611937091443298427377890517834393893483551010317859623797946959831934124292658003

20764963.34676300829623633044500201850655642596978386739978269243927490134559483449125865369247219308

3859604421.279218840930310083855726196593562210385937588470541058953237782280841254803713169152271635

840428602082.7202126215963989700162039965076824953187132910186677036659463475047679549652438357385638

209753766802193.5108271573271504902686046611862866977832655152293267739280425548929639190238388178477

59019285882055650.28469598366915842363648876368141465166365369321675275164062800277235805006133818302

18481467614260465167.96841963588755781843778046147304260564698673456221647349731734124220013507223470

6374104811376373204762.226121254490799449201156365475865142781072021909124677830677888397289351695300

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 [1].

Parameters

$g$

— genus ($g\geq0$)

$n$

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

Formulas

(1)

For $m=3g-3+n$, $\displaystyle V_{g,n}=\frac{1}{m!}\int_{\mathcal M_{g,n}}\omega_{\mathrm{WP}}^m=\frac{(2\pi^2)^m}{m!}\int_{\overline{\mathcal M}_{g,n}}\kappa_1^m$ [1].

(2)

$V_{0,3}=1$, $V_{0,4}=2\pi^2$, $V_{0,5}=10\pi^4$, $V_{1,1}=\pi^2/12$, $V_{1,2}=\pi^4/4$, and $V_{2,0}=43\pi^6/2160$ [2].

Comments

(3)

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 [2].

(4)

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 [2].

(5)

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 [1]. Its top powers reduce to intersection numbers of cotangent line classes by the set-partition formula in [1], Lemma 2.2.

(6)

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 [1].

Programs

(P1)

Sage

# Download the attached generate.py into the current directory.
from generate import volume
volume(2, 0, digits=100)

References

[1]

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

[2]

N. Do, Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes, preprint (2011), Sections 2.3 and 3.1 and Appendix B. (arXiv)

Links

[3]

Liu and Xu: Recursion formulae of higher Weil-Petersson volumes

Similar tables

intersection numbers of cotangent line classes — determine the kappa class integrals appearing in the Weil-Petersson volumes through a set-partition transform

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (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 $g\geq2$)

How they were obtained:

The coefficient of $\pi^{6g-6+2n}$ is obtained in exact rational arithmetic from Theorem 1.1 and equation (3) of [1]. 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.

The independent checker uses the set-partition transform in [1], 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 [2] 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.