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{
"Title": "Intersection numbers of $\\psi$ classes on $\\overline{\\mathcal M}_{g,n}$",
"Definition": "The entries are top intersection numbers of cotangent line classes on the Deligne-Mumford stack $\\overline{\\mathcal M}_{g,n}$ of stable genus-$g$ curves with $n$ ordered marked points CITE{LX}.",
"Parameters": {
"g": {
"type": "Z",
"title": "genus",
"display": "$g$",
"constraints": "$g\\geq0$"
},
"n": {
"type": "Z",
"title": "number of marked points",
"display": "$n$",
"constraints": "$n\\geq0$, with $2g-2+n>0$"
},
"d": {
"type": "Symbolic",
"title": "exponent profile",
"display": "$(d_1,\\ldots,d_n)$",
"constraints": "a comma-separated non-decreasing list of $n$ nonnegative integers satisfying $\\sum_i d_i=3g-3+n$"
}
},
"Tags": [
"algebra",
"combinatorics"
],
"Data properties": {
"type": "Q",
"rigour": "exact",
"complete": "no",
"complete-note": "it holds every stable pair with $0\\leq g\\leq5$ and $1\\leq n\\leq5$, and every non-decreasing nonnegative exponent profile of total degree $3g-3+n$ for each pair; these bounds give a compact reference for low-genus intersections with up to five marked points",
"rigour details": "Every entry is computed in exact rational arithmetic by the Dijkgraaf-Verlinde-Verlinde recurrence in CITE{LX}, together with the string and dilaton equations and the initial values $\\langle\\tau_0^3\\rangle_0=1$ and $\\langle\\tau_1\\rangle_1=1/24$. The attached generator uses Sage rational numbers throughout.\n\nThe separate checker implements Theorem 1.1 of CITE{LX} with Python fractions and an independent genus-zero multinomial formula. It derives the genus-one normalization from genus zero. All 494 entries agree with that implementation. The checks also cover the published examples in CITE{M2}, every applicable string and dilaton equation within the stored range, all genus-zero and one-point entries, and a separate enumeration of the complete range. The same checks are repeated on values read back from the database."
},
"Display properties": {
"number-header": "$\\langle\\tau_{d_1}\\cdots\\tau_{d_n}\\rangle_g$"
},
"Numbers": {
"0": {
"3": {
"0,0,0": "1"
},
"4": {
"0,0,0,1": "1"
},
"5": {
"0,0,0,0,2": "1",
"0,0,0,1,1": "2"
}
},
"1": {
"1": {
"1": "1/24"
},
"2": {
"0,2": "1/24",
"1,1": "1/24"
},
"3": {
"0,0,3": "1/24",
"0,1,2": "1/12",
"1,1,1": "1/12"
},
"4": {
"0,0,0,4": "1/24",
"0,0,1,3": "1/8",
"0,0,2,2": "1/6",
"0,1,1,2": "1/4",
"1,1,1,1": "1/4"
},
"5": {
"0,0,0,0,5": "1/24",
"0,0,0,1,4": "1/6",
"0,0,0,2,3": "7/24",
"0,0,1,1,3": "1/2",
"0,0,1,2,2": "2/3",
"0,1,1,1,2": "1",
"1,1,1,1,1": "1"
}
},
"2": {
"1": {
"4": "1/1152"
},
"2": {
"0,5": "1/1152",
"1,4": "1/384",
"2,3": "29/5760"
},
"3": {
"0,0,6": "1/1152",
"0,1,5": "1/288",
"0,2,4": "11/1440",
"0,3,3": "29/2880",
"1,1,4": "1/96",
"1,2,3": "29/1440",
"2,2,2": "7/240"
},
"4": {
"0,0,0,7": "1/1152",
"0,0,1,6": "5/1152",
"0,0,2,5": "1/90",
"0,0,3,4": "17/960",
"0,1,1,5": "5/288",
"0,1,2,4": "11/288",
"0,1,3,3": "29/576",
"0,2,2,3": "5/72",
"1,1,1,4": "5/96",
"1,1,2,3": "29/288",
"1,2,2,2": "7/48"
},
"5": {
"0,0,0,0,8": "1/1152",
"0,0,0,1,7": "1/192",
"0,0,0,2,6": "89/5760",
"0,0,0,3,5": "83/2880",
"0,0,0,4,4": "17/480",
"0,0,1,1,6": "5/192",
"0,0,1,2,5": "1/15",
"0,0,1,3,4": "17/160",
"0,0,2,2,4": "7/48",
"0,0,2,3,3": "109/576",
"0,1,1,1,5": "5/48",
"0,1,1,2,4": "11/48",
"0,1,1,3,3": "29/96",
"0,1,2,2,3": "5/12",
"0,2,2,2,2": "7/12",
"1,1,1,1,4": "5/16",
"1,1,1,2,3": "29/48",
"1,1,2,2,2": "7/8"
}
},
"3": {
"1": {
"7": "1/82944"
},
"2": {
"0,8": "1/82944",
"1,7": "5/82944",
"2,6": "77/414720",
"3,5": "503/1451520",
"4,4": "607/1451520"
},
"3": {
"0,0,9": "1/82944",
"0,1,8": "1/13824",
"0,2,7": "17/69120",
"0,3,6": "103/193536",
"0,4,5": "37/48384",
"1,1,7": "5/13824",
"1,2,6": "77/69120",
"1,3,5": "503/241920",
"1,4,4": "607/241920",
"2,2,5": "17/5760",
"2,3,4": "1121/241920",
"3,3,3": "583/96768"
},
"4": {
"0,0,0,10": "1/82944",
"0,0,1,9": "7/82944",
"0,0,2,8": "11/34560",
"0,0,3,7": "251/322560",
"0,0,4,6": "251/193536",
"0,0,5,5": "37/24192",
"0,1,1,8": "7/13824",
"0,1,2,7": "119/69120",
"0,1,3,6": "103/27648",
"0,1,4,5": "37/6912",
"0,2,2,6": "179/34560",
"0,2,3,5": "167/17280",
"0,2,4,4": "407/34560",
"0,3,3,4": "1057/69120",
"1,1,1,7": "35/13824",
"1,1,2,6": "539/69120",
"1,1,3,5": "503/34560",
"1,1,4,4": "607/34560",
"1,2,2,5": "119/5760",
"1,2,3,4": "1121/34560",
"1,3,3,3": "583/13824",
"2,2,2,4": "53/1152",
"2,2,3,3": "205/3456"
},
"5": {
"0,0,0,0,11": "1/82944",
"0,0,0,1,10": "1/10368",
"0,0,0,2,9": "167/414720",
"0,0,0,3,8": "1061/967680",
"0,0,0,4,7": "251/120960",
"0,0,0,5,6": "547/193536",
"0,0,1,1,9": "7/10368",
"0,0,1,2,8": "11/4320",
"0,0,1,3,7": "251/40320",
"0,0,1,4,6": "251/24192",
"0,0,1,5,5": "37/3024",
"0,0,2,2,7": "149/17280",
"0,0,2,3,6": "2567/138240",
"0,0,2,4,5": "463/17280",
"0,0,3,3,5": "2393/69120",
"0,0,3,4,4": "61/1440",
"0,1,1,1,8": "7/1728",
"0,1,1,2,7": "119/8640",
"0,1,1,3,6": "103/3456",
"0,1,1,4,5": "37/864",
"0,1,2,2,6": "179/4320",
"0,1,2,3,5": "167/2160",
"0,1,2,4,4": "407/4320",
"0,1,3,3,4": "1057/8640",
"0,2,2,2,5": "311/2880",
"0,2,2,3,4": "2941/17280",
"0,2,3,3,3": "3043/13824",
"1,1,1,1,7": "35/1728",
"1,1,1,2,6": "539/8640",
"1,1,1,3,5": "503/4320",
"1,1,1,4,4": "607/4320",
"1,1,2,2,5": "119/720",
"1,1,2,3,4": "1121/4320",
"1,1,3,3,3": "583/1728",
"1,2,2,2,4": "53/144",
"1,2,2,3,3": "205/432",
"2,2,2,2,3": "193/288"
}
},
"4": {
"1": {
"10": "1/7962624"
},
"2": {
"0,11": "1/7962624",
"1,10": "7/7962624",
"2,9": "149/39813120",
"3,8": "947/92897280",
"4,7": "1781/92897280",
"5,6": "487/18579456"
},
"3": {
"0,0,12": "1/7962624",
"0,1,11": "1/995328",
"0,2,10": "23/4976640",
"0,3,9": "971/69672960",
"0,4,8": "341/11612160",
"0,5,7": "527/11612160",
"0,6,6": "487/9289728",
"1,1,10": "7/995328",
"1,2,9": "149/4976640",
"1,3,8": "947/11612160",
"1,4,7": "1781/11612160",
"1,5,6": "487/2322432",
"2,2,8": "53/460800",
"2,3,7": "16243/58060800",
"2,4,6": "1081/2322432",
"2,5,5": "533/967680",
"3,3,6": "1997/3317760",
"3,4,5": "719/829440",
"4,4,4": "175/165888"
},
"4": {
"0,0,0,13": "1/7962624",
"0,0,1,12": "1/884736",
"0,0,2,11": "7/1244160",
"0,0,3,10": "431/23224320",
"0,0,4,9": "431/9953280",
"0,0,5,8": "31/414720",
"0,0,6,7": "649/6635520",
"0,1,1,11": "1/110592",
"0,1,2,10": "23/552960",
"0,1,3,9": "971/7741440",
"0,1,4,8": "341/1290240",
"0,1,5,7": "527/1290240",
"0,1,6,6": "487/1032192",
"0,2,2,9": "17/97200",
"0,2,3,8": "3457/7257600",
"0,2,4,7": "5797/6451200",
"0,2,5,6": "3559/2903040",
"0,3,3,7": "134867/116121600",
"0,3,4,6": "44921/23224320",
"0,3,5,5": "829/362880",
"0,4,4,5": "257/92160",
"1,1,1,10": "7/110592",
"1,1,2,9": "149/552960",
"1,1,3,8": "947/1290240",
"1,1,4,7": "1781/1290240",
"1,1,5,6": "487/258048",
"1,2,2,8": "53/51200",
"1,2,3,7": "16243/6451200",
"1,2,4,6": "1081/258048",
"1,2,5,5": "533/107520",
"1,3,3,6": "1997/368640",
"1,3,4,5": "719/92160",
"1,4,4,4": "175/18432",
"2,2,2,7": "4907/1382400",
"2,2,3,6": "7897/1036800",
"2,2,4,5": "7597/691200",
"2,3,3,5": "1837/129600",
"2,3,4,4": "177/10240",
"3,3,3,4": "185251/8294400"
},
"5": {
"0,0,0,0,14": "1/7962624",
"0,0,0,1,13": "5/3981312",
"0,0,0,2,12": "269/39813120",
"0,0,0,3,11": "337/13934592",
"0,0,0,4,10": "431/6967296",
"0,0,0,5,9": "235/1990656",
"0,0,0,6,8": "229/1327104",
"0,0,0,7,7": "649/3317760",
"0,0,1,1,12": "5/442368",
"0,0,1,2,11": "7/124416",
"0,0,1,3,10": "431/2322432",
"0,0,1,4,9": "431/995328",
"0,0,1,5,8": "31/41472",
"0,0,1,6,7": "649/663552",
"0,0,2,2,10": "3211/12441600",
"0,0,2,3,9": "270559/348364800",
"0,0,2,4,8": "47587/29030400",
"0,0,2,5,7": "36767/14515200",
"0,0,2,6,6": "135803/46448640",
"0,0,3,3,8": "245491/116121600",
"0,0,3,4,7": "77303/19353600",
"0,0,3,5,6": "126449/23224320",
"0,0,4,4,6": "77303/11612160",
"0,0,4,5,5": "22823/2903040",
"0,1,1,1,11": "5/55296",
"0,1,1,2,10": "23/55296",
"0,1,1,3,9": "971/774144",
"0,1,1,4,8": "341/129024",
"0,1,1,5,7": "527/129024",
"0,1,1,6,6": "2435/516096",
"0,1,2,2,9": "17/9720",
"0,1,2,3,8": "3457/725760",
"0,1,2,4,7": "5797/645120",
"0,1,2,5,6": "3559/290304",
"0,1,3,3,7": "134867/11612160",
"0,1,3,4,6": "44921/2322432",
"0,1,3,5,5": "829/36288",
"0,1,4,4,5": "257/9216",
"0,2,2,2,8": "23/3456",
"0,2,2,3,7": "9407/580608",
"0,2,2,4,6": "156683/5806080",
"0,2,2,5,5": "19291/604800",
"0,2,3,3,6": "115541/3317760",
"0,2,3,4,5": "104203/2073600",
"0,2,4,4,4": "2827/46080",
"0,3,3,3,5": "107591/1658880",
"0,3,3,4,4": "328621/4147200",
"1,1,1,1,10": "35/55296",
"1,1,1,2,9": "149/55296",
"1,1,1,3,8": "947/129024",
"1,1,1,4,7": "1781/129024",
"1,1,1,5,6": "2435/129024",
"1,1,2,2,8": "53/5120",
"1,1,2,3,7": "16243/645120",
"1,1,2,4,6": "5405/129024",
"1,1,2,5,5": "533/10752",
"1,1,3,3,6": "1997/36864",
"1,1,3,4,5": "719/9216",
"1,1,4,4,4": "875/9216",
"1,2,2,2,7": "4907/138240",
"1,2,2,3,6": "7897/103680",
"1,2,2,4,5": "7597/69120",
"1,2,3,3,5": "1837/12960",
"1,2,3,4,4": "177/1024",
"1,3,3,3,4": "185251/829440",
"2,2,2,2,6": "2471/23040",
"2,2,2,3,5": "13783/69120",
"2,2,2,4,4": "5609/23040",
"2,2,3,3,4": "26017/82944",
"2,3,3,3,3": "134233/331776"
}
},
"5": {
"1": {
"13": "1/955514880"
},
"2": {
"0,14": "1/955514880",
"1,13": "1/106168320",
"2,12": "49/955514880",
"3,11": "307/1672151040",
"4,10": "29/61931520",
"5,9": "2351/2627665920",
"6,8": "2291/1751777280",
"7,7": "173/116785152"
},
"3": {
"0,0,15": "1/955514880",
"0,1,14": "1/95551488",
"0,2,13": "29/477757440",
"0,3,12": "1571/6688604160",
"0,4,11": "109/167215104",
"0,5,10": "2507/1839366144",
"0,6,9": "2315/1051066368",
"0,7,8": "2443/875888640",
"1,1,13": "1/10616832",
"1,2,12": "49/95551488",
"1,3,11": "307/167215104",
"1,4,10": "29/6193152",
"1,5,9": "2351/262766592",
"1,6,8": "2291/175177728",
"1,7,7": "865/58392576",
"2,2,11": "257/99532800",
"2,3,10": "35521/4180377600",
"2,4,9": "11827/597196800",
"2,5,8": "12491/364953600",
"2,6,7": "195983/4379443200",
"3,3,9": "213631/8360755200",
"3,4,8": "10723/199065600",
"3,5,7": "1276397/15328051200",
"3,6,6": "1178689/12262440960",
"4,4,7": "15727/154828800",
"4,5,6": "424987/3065610240",
"5,5,5": "41873/255467520"
},
"4": {
"0,0,0,16": "1/955514880",
"0,0,1,15": "11/955514880",
"0,0,2,14": "17/238878720",
"0,0,3,13": "659/2229534720",
"0,0,4,12": "659/743178240",
"0,0,5,11": "1853/919683072",
"0,0,6,10": "26233/7357464576",
"0,0,7,9": "26233/5255331840",
"0,0,8,8": "2443/437944320",
"0,1,1,14": "11/95551488",
"0,1,2,13": "319/477757440",
"0,1,3,12": "17281/6688604160",
"0,1,4,11": "1199/167215104",
"0,1,5,10": "2507/167215104",
"0,1,6,9": "2315/95551488",
"0,1,7,8": "2443/79626240",
"0,2,2,12": "4309/1194393600",
"0,2,3,11": "5399/418037760",
"0,2,4,10": "27577/836075520",
"0,2,5,9": "3761/59719680",
"0,2,6,8": "733/7962624",
"0,2,7,7": "41531/398131200",
"0,3,3,10": "71143/1672151040",
"0,3,4,9": "1229/12386304",
"0,3,5,8": "23879/139345920",
"0,3,6,7": "249869/1114767360",
"0,4,4,8": "58333/278691840",
"0,4,5,7": "90151/278691840",
"0,4,6,6": "27749/74317824",
"0,5,5,6": "12295/27869184",
"1,1,1,13": "11/10616832",
"1,1,2,12": "539/95551488",
"1,1,3,11": "3377/167215104",
"1,1,4,10": "319/6193152",
"1,1,5,9": "2351/23887872",
"1,1,6,8": "2291/15925248",
"1,1,7,7": "865/5308416",
"1,2,2,11": "2827/99532800",
"1,2,3,10": "390731/4180377600",
"1,2,4,9": "130097/597196800",
"1,2,5,8": "12491/33177600",
"1,2,6,7": "195983/398131200",
"1,3,3,9": "2349941/8360755200",
"1,3,4,8": "117953/199065600",
"1,3,5,7": "1276397/1393459200",
"1,3,6,6": "1178689/1114767360",
"1,4,4,7": "172997/154828800",
"1,4,5,6": "424987/278691840",
"1,5,5,5": "41873/23224320",
"2,2,2,10": "13081/99532800",
"2,2,3,9": "117871/298598400",
"2,2,4,8": "27629/33177600",
"2,2,5,7": "14237/11059200",
"2,2,6,6": "295693/199065600",
"2,3,3,8": "747709/696729600",
"2,3,4,7": "2821883/1393459200",
"2,3,5,6": "1924831/696729600",
"2,4,4,6": "4699193/1393459200",
"2,4,5,5": "92599/23224320",
"3,3,3,7": "1455581/557383680",
"3,3,4,6": "807977/185794560",
"3,3,5,5": "1790449/348364800",
"3,4,4,5": "624443/99532800",
"4,4,4,4": "84659/11059200"
},
"5": {
"0,0,0,0,17": "1/955514880",
"0,0,0,1,16": "1/79626240",
"0,0,0,2,15": "79/955514880",
"0,0,0,3,14": "2453/6688604160",
"0,0,0,4,13": "659/557383680",
"0,0,0,5,12": "213481/73574645760",
"0,0,0,6,11": "41057/7357464576",
"0,0,0,7,10": "26233/3065610240",
"0,0,0,8,9": "55549/5255331840",
"0,0,1,1,15": "11/79626240",
"0,0,1,2,14": "17/19906560",
"0,0,1,3,13": "659/185794560",
"0,0,1,4,12": "659/61931520",
"0,0,1,5,11": "1853/76640256",
"0,0,1,6,10": "26233/613122048",
"0,0,1,7,9": "26233/437944320",
"0,0,1,8,8": "2443/36495360",
"0,0,2,2,13": "41/8294400",
"0,0,2,3,12": "638977/33443020800",
"0,0,2,4,11": "493/9289728",
"0,0,2,5,10": "15461/139345920",
"0,0,2,6,9": "85643/477757440",
"0,0,2,7,8": "93/409600",
"0,0,3,3,11": "22867/334430208",
"0,0,3,4,10": "8117/46448640",
"0,0,3,5,9": "557771/1672151040",
"0,0,3,6,8": "543521/1114767360",
"0,0,3,7,7": "85559/154828800",
"0,0,4,4,9": "8117/19906560",
"0,0,4,5,8": "32707/46448640",
"0,0,4,6,7": "85559/92897280",
"0,0,5,5,7": "25271/23224320",
"0,0,5,6,6": "279967/222953472",
"0,1,1,1,14": "11/7962624",
"0,1,1,2,13": "319/39813120",
"0,1,1,3,12": "17281/557383680",
"0,1,1,4,11": "1199/13934592",
"0,1,1,5,10": "2507/13934592",
"0,1,1,6,9": "2315/7962624",
"0,1,1,7,8": "2443/6635520",
"0,1,2,2,12": "4309/99532800",
"0,1,2,3,11": "5399/34836480",
"0,1,2,4,10": "27577/69672960",
"0,1,2,5,9": "3761/4976640",
"0,1,2,6,8": "733/663552",
"0,1,2,7,7": "41531/33177600",
"0,1,3,3,10": "71143/139345920",
"0,1,3,4,9": "1229/1032192",
"0,1,3,5,8": "23879/11612160",
"0,1,3,6,7": "249869/92897280",
"0,1,4,4,8": "58333/23224320",
"0,1,4,5,7": "90151/23224320",
"0,1,4,6,6": "27749/6193152",
"0,1,5,5,6": "12295/2322432",
"0,2,2,2,11": "10781/49766400",
"0,2,2,3,10": "496843/696729600",
"0,2,2,4,9": "55181/33177600",
"0,2,2,5,8": "15887/5529600",
"0,2,2,6,7": "124657/33177600",
"0,2,3,3,9": "716929/334430208",
"0,2,3,4,8": "210113/46448640",
"0,2,3,5,7": "811817/116121600",
"0,2,3,6,6": "8994029/1114767360",
"0,2,4,4,7": "991661/116121600",
"0,2,4,5,6": "540991/46448640",
"0,2,5,5,5": "31967/2322432",
"0,3,3,3,8": "16250413/2786918400",
"0,3,3,4,7": "2557091/232243200",
"0,3,3,5,6": "8368379/557383680",
"0,3,4,4,6": "170407/9289728",
"0,3,4,5,5": "503369/23224320",
"0,4,4,4,5": "43921/1658880",
"1,1,1,1,13": "11/884736",
"1,1,1,2,12": "539/7962624",
"1,1,1,3,11": "3377/13934592",
"1,1,1,4,10": "319/516096",
"1,1,1,5,9": "2351/1990656",
"1,1,1,6,8": "2291/1327104",
"1,1,1,7,7": "865/442368",
"1,1,2,2,11": "2827/8294400",
"1,1,2,3,10": "390731/348364800",
"1,1,2,4,9": "130097/49766400",
"1,1,2,5,8": "12491/2764800",
"1,1,2,6,7": "195983/33177600",
"1,1,3,3,9": "2349941/696729600",
"1,1,3,4,8": "117953/16588800",
"1,1,3,5,7": "1276397/116121600",
"1,1,3,6,6": "1178689/92897280",
"1,1,4,4,7": "172997/12902400",
"1,1,4,5,6": "424987/23224320",
"1,1,5,5,5": "41873/1935360",
"1,2,2,2,10": "13081/8294400",
"1,2,2,3,9": "117871/24883200",
"1,2,2,4,8": "27629/2764800",
"1,2,2,5,7": "14237/921600",
"1,2,2,6,6": "295693/16588800",
"1,2,3,3,8": "747709/58060800",
"1,2,3,4,7": "2821883/116121600",
"1,2,3,5,6": "1924831/58060800",
"1,2,4,4,6": "4699193/116121600",
"1,2,4,5,5": "92599/1935360",
"1,3,3,3,7": "1455581/46448640",
"1,3,3,4,6": "807977/15482880",
"1,3,3,5,5": "1790449/29030400",
"1,3,4,4,5": "624443/8294400",
"1,4,4,4,4": "84659/921600",
"2,2,2,2,9": "33131/4976640",
"2,2,2,3,8": "1667/92160",
"2,2,2,4,7": "56647/1658880",
"2,2,2,5,6": "772639/16588800",
"2,2,3,3,7": "68093/1548288",
"2,2,3,4,6": "210041/2867200",
"2,2,3,5,5": "52373/604800",
"2,2,4,4,5": "438577/4147200",
"2,3,3,3,6": "7515827/79626240",
"2,3,3,4,5": "2259953/16588800",
"2,3,4,4,4": "1379693/8294400",
"3,3,3,3,5": "6990221/39813120",
"3,3,3,4,4": "3556993/16588800"
}
}
},
"Comments": {
"stack-convention": "Integration includes the automorphism weights of the moduli stack. In particular, $\\langle\\tau_1\\rangle_1=1/24$ CITE{M2}; removing the factor $1/2$ contributed by the generic elliptic involution would give $1/12$. No factor of $n!$ or of the number of permutations of an exponent profile divides the entries.",
"profiles": "The profile is written as a comma-separated list, including every zero: $0,0,2$ denotes $(0,0,2)$. Sorting only chooses a representative of the symmetric integral. In the formulas, a bracket with a negative exponent or the wrong total degree is zero. Unstable moduli spaces do not supply entries.",
"notation": "The notation $\\tau_d$ records a power $\\psi^d$ at one marked point; these intersection numbers are also called Witten correlators CITE{Afandi}. The multiplicity list used by Macaulay2 CITE{M2} is different from an exponent profile: the input $\\{a_0,\\ldots,a_k\\}$ means $\\tau_0^{a_0}\\cdots\\tau_k^{a_k}$.",
"cotangent-lines": "The fiber of the cotangent line bundle $L_i$ at a pointed curve $(C,p_1,\\ldots,p_n)$ is the cotangent space of $C$ at $p_i$. The class $\\psi_i=c_1(L_i)$ has complex codimension one CITE{Afandi}."
},
"Formulas": {
"integral": "$\\displaystyle\\langle\\tau_{d_1}\\cdots\\tau_{d_n}\\rangle_g=\\int_{\\overline{\\mathcal M}_{g,n}}\\prod_{i=1}^n\\psi_i^{d_i}$ CITE{LX}.",
"genus-zero": "For $n\\geq3$ and $\\sum_i d_i=n-3$, $\\displaystyle\\langle\\tau_{d_1}\\cdots\\tau_{d_n}\\rangle_0=\\frac{(n-3)!}{\\prod_{i=1}^n d_i!}$ CITE{Afandi}.",
"one-point": "For $g\\geq1$, $\\displaystyle\\langle\\tau_{3g-2}\\rangle_g=\\frac{1}{24^g g!}$ CITE{Afandi}.",
"string": "The string equation is $\\displaystyle\\left\\langle\\tau_0\\prod_{i=1}^{n}\\tau_{d_i}\\right\\rangle_g=\\sum_{j=1}^{n}\\left\\langle\\tau_{d_j-1}\\prod_{i\\ne j}\\tau_{d_i}\\right\\rangle_g$ for a stable pair $(g,n)$ CITE{Afandi}.",
"dilaton": "The dilaton equation is $\\displaystyle\\left\\langle\\tau_1\\prod_{i=1}^{n}\\tau_{d_i}\\right\\rangle_g=(2g-2+n)\\left\\langle\\prod_{i=1}^{n}\\tau_{d_i}\\right\\rangle_g$ for a stable pair $(g,n)$ CITE{Afandi}."
},
"Programs": {
"macaulay2": {
"language": "Macaulay2",
"code": "loadPackage \"HodgeIntegrals\";\n-- Three insertions of tau_5, so g = 5 and n = 3.\nwittenTau(5,{0,0,0,0,0,3})\n-- 41873/255467520"
}
},
"Similar tables": [
{
"table": "HREF{Genus-zero_Gromov-Witten_invariants_of_the_projective_plane}[genus-zero Gromov-Witten invariants of the projective plane]",
"relation": "are intersection invariants on moduli spaces of stable maps to the projective plane, a related enumerative setting"
}
],
"Links": {
"M2": {
"title": "Macaulay2 HodgeIntegrals: wittenTau",
"url": "https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/HodgeIntegrals/html/_witten__Tau.html"
}
},
"References": {
"LX": {
"bib": "K. Liu and H. Xu, An effective recursion formula for computing intersection numbers, preprint (2007).",
"arxiv": "0710.5322"
},
"Afandi": {
"bib": "A. Afandi, An Ehrhart theory for tautological intersection numbers, preprint (2022), Section 3.3.",
"arxiv": "2209.14131"
}
},
"Keywords": [
"psi classes",
"Witten correlators",
"Witten-Kontsevich intersection numbers",
"cotangent line classes",
"descendant intersection numbers",
"tautological intersection numbers"
]
}