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{
"Title": "$\\lambda_g$ Hodge integrals on $\\overline{\\mathcal M}_{g,n}$",
"Definition": "These are top intersections of $\\lambda_g$, the top Chern class of the Hodge bundle, with cotangent line classes on $\\overline{\\mathcal M}_{g,n}$ CITE{FP}.",
"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 with $\\sum_i d_i=2g-3+n$"
}
},
"References": {
"FP": {
"bib": "C. Faber and R. Pandharipande, Hodge integrals, partition matrices, and the $\\lambda_g$ conjecture, Annals of Mathematics 157 (2003), 97\u2013124, Theorem 1 and equations (4)\u2013(5).",
"arxiv": "math/9908052",
"doi": "10.4007/annals.2003.157.97"
}
},
"Links": {
"M2": {
"title": "Macaulay2 HodgeIntegrals: integral",
"url": "https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/HodgeIntegrals/html/_integral.html"
}
},
"Tags": [
"algebra",
"combinatorics"
],
"Data properties": {
"type": "Q",
"rigour": "exact",
"complete": "no",
"complete-note": "it holds every top-degree profile with $1\\leq g\\leq5$ and $1\\leq n\\leq5$, covering the first five positive genera with up to five labelled marked points; genus-zero values, for which $\\lambda_0=1$, are described by the genus-zero formula",
"rigour details": "Every entry is an exact rational number. The generator inverts the formal series $\\sin(t/2)/(t/2)$ coefficient by coefficient over $\\mathbb Q$ and multiplies by an exact integer multinomial coefficient, using CITE{lambda-g} and CITE{sine}.\n\nA separate checker obtains Bernoulli numbers by a rational recurrence and evaluates CITE{bernoulli} independently. It agrees on all 232 entries. Additional checks cover three published Macaulay2 examples CITE{M2}, the five even Bernoulli numbers in HREF{Bernoulli_numbers}, a 512-bit Arb calculation from even zeta values, all 155 applicable string equations and 122 applicable dilaton equations within the stored range, and an independent enumeration of the parameter range. All twelve genus-one entries also agree with boundary reduction to genus-zero cotangent intersections, computed by the string equation; four of these genus-zero values are present in the companion table HREF{T458}. The checker tests known answers and deliberately incorrect controls before comparing data. The generator, checker and numerical evidence are attached, and the same checks are applied to the stored values."
},
"Display properties": {
"number-header": "$I_g(d_1,\\ldots,d_n)$"
},
"Numbers": {
"1": {
"1": {
"0": {
"number": "1/24",
"comment": "The integral $\\int_{\\overline{\\mathcal M}_{1,1}}\\lambda_1=1/24$ includes the generic elliptic involution."
}
},
"2": {
"0,1": "1/24"
},
"3": {
"0,0,2": "1/24",
"0,1,1": "1/12"
},
"4": {
"0,0,0,3": "1/24",
"0,0,1,2": "1/8",
"0,1,1,1": "1/4"
},
"5": {
"0,0,0,0,4": "1/24",
"0,0,0,1,3": "1/6",
"0,0,0,2,2": "1/4",
"0,0,1,1,2": "1/2",
"0,1,1,1,1": "1"
}
},
"2": {
"1": {
"2": "7/5760"
},
"2": {
"0,3": "7/5760",
"1,2": "7/1920"
},
"3": {
"0,0,4": "7/5760",
"0,1,3": "7/1440",
"0,2,2": "7/960",
"1,1,2": "7/480"
},
"4": {
"0,0,0,5": "7/5760",
"0,0,1,4": "7/1152",
"0,0,2,3": "7/576",
"0,1,1,3": "7/288",
"0,1,2,2": "7/192",
"1,1,1,2": "7/96"
},
"5": {
"0,0,0,0,6": "7/5760",
"0,0,0,1,5": "7/960",
"0,0,0,2,4": "7/384",
"0,0,0,3,3": "7/288",
"0,0,1,1,4": "7/192",
"0,0,1,2,3": "7/96",
"0,0,2,2,2": "7/64",
"0,1,1,1,3": "7/48",
"0,1,1,2,2": "7/32",
"1,1,1,1,2": "7/16"
}
},
"3": {
"1": {
"4": "31/967680"
},
"2": {
"0,5": "31/967680",
"1,4": "31/193536",
"2,3": "31/96768"
},
"3": {
"0,0,6": "31/967680",
"0,1,5": "31/161280",
"0,2,4": "31/64512",
"0,3,3": "31/48384",
"1,1,4": "31/32256",
"1,2,3": "31/16128",
"2,2,2": "31/10752"
},
"4": {
"0,0,0,7": "31/967680",
"0,0,1,6": "31/138240",
"0,0,2,5": "31/46080",
"0,0,3,4": "31/27648",
"0,1,1,5": "31/23040",
"0,1,2,4": "31/9216",
"0,1,3,3": "31/6912",
"0,2,2,3": "31/4608",
"1,1,1,4": "31/4608",
"1,1,2,3": "31/2304",
"1,2,2,2": "31/1536"
},
"5": {
"0,0,0,0,8": "31/967680",
"0,0,0,1,7": "31/120960",
"0,0,0,2,6": "31/34560",
"0,0,0,3,5": "31/17280",
"0,0,0,4,4": "31/13824",
"0,0,1,1,6": "31/17280",
"0,0,1,2,5": "31/5760",
"0,0,1,3,4": "31/3456",
"0,0,2,2,4": "31/2304",
"0,0,2,3,3": "31/1728",
"0,1,1,1,5": "31/2880",
"0,1,1,2,4": "31/1152",
"0,1,1,3,3": "31/864",
"0,1,2,2,3": "31/576",
"0,2,2,2,2": "31/384",
"1,1,1,1,4": "31/576",
"1,1,1,2,3": "31/288",
"1,1,2,2,2": "31/192"
}
},
"4": {
"1": {
"6": "127/154828800"
},
"2": {
"0,7": "127/154828800",
"1,6": "127/22118400",
"2,5": "127/7372800",
"3,4": "127/4423680"
},
"3": {
"0,0,8": "127/154828800",
"0,1,7": "127/19353600",
"0,2,6": "127/5529600",
"0,3,5": "127/2764800",
"0,4,4": "127/2211840",
"1,1,6": "127/2764800",
"1,2,5": "127/921600",
"1,3,4": "127/552960",
"2,2,4": "127/368640",
"2,3,3": "127/276480"
},
"4": {
"0,0,0,9": "127/154828800",
"0,0,1,8": "127/17203200",
"0,0,2,7": "127/4300800",
"0,0,3,6": "127/1843200",
"0,0,4,5": "127/1228800",
"0,1,1,7": "127/2150400",
"0,1,2,6": "127/614400",
"0,1,3,5": "127/307200",
"0,1,4,4": "127/245760",
"0,2,2,5": "127/204800",
"0,2,3,4": "127/122880",
"0,3,3,3": "127/92160",
"1,1,1,6": "127/307200",
"1,1,2,5": "127/102400",
"1,1,3,4": "127/61440",
"1,2,2,4": "127/40960",
"1,2,3,3": "127/30720",
"2,2,2,3": "127/20480"
},
"5": {
"0,0,0,0,10": "127/154828800",
"0,0,0,1,9": "127/15482880",
"0,0,0,2,8": "127/3440640",
"0,0,0,3,7": "127/1290240",
"0,0,0,4,6": "127/737280",
"0,0,0,5,5": "127/614400",
"0,0,1,1,8": "127/1720320",
"0,0,1,2,7": "127/430080",
"0,0,1,3,6": "127/184320",
"0,0,1,4,5": "127/122880",
"0,0,2,2,6": "127/122880",
"0,0,2,3,5": "127/61440",
"0,0,2,4,4": "127/49152",
"0,0,3,3,4": "127/36864",
"0,1,1,1,7": "127/215040",
"0,1,1,2,6": "127/61440",
"0,1,1,3,5": "127/30720",
"0,1,1,4,4": "127/24576",
"0,1,2,2,5": "127/20480",
"0,1,2,3,4": "127/12288",
"0,1,3,3,3": "127/9216",
"0,2,2,2,4": "127/8192",
"0,2,2,3,3": "127/6144",
"1,1,1,1,6": "127/30720",
"1,1,1,2,5": "127/10240",
"1,1,1,3,4": "127/6144",
"1,1,2,2,4": "127/4096",
"1,1,2,3,3": "127/3072",
"1,2,2,2,3": "127/2048",
"2,2,2,2,2": "381/4096"
}
},
"5": {
"1": {
"8": "73/3503554560"
},
"2": {
"0,9": "73/3503554560",
"1,8": "73/389283840",
"2,7": "73/97320960",
"3,6": "511/291962880",
"4,5": "511/194641920"
},
"3": {
"0,0,10": "73/3503554560",
"0,1,9": "73/350355456",
"0,2,8": "73/77856768",
"0,3,7": "73/29196288",
"0,4,6": "511/116785152",
"0,5,5": "511/97320960",
"1,1,8": "73/38928384",
"1,2,7": "73/9732096",
"1,3,6": "511/29196288",
"1,4,5": "511/19464192",
"2,2,6": "511/19464192",
"2,3,5": "511/9732096",
"2,4,4": "2555/38928384",
"3,3,4": "2555/29196288"
},
"4": {
"0,0,0,11": "73/3503554560",
"0,0,1,10": "73/318504960",
"0,0,2,9": "73/63700992",
"0,0,3,8": "73/21233664",
"0,0,4,7": "73/10616832",
"0,0,5,6": "511/53084160",
"0,1,1,9": "73/31850496",
"0,1,2,8": "73/7077888",
"0,1,3,7": "73/2654208",
"0,1,4,6": "511/10616832",
"0,1,5,5": "511/8847360",
"0,2,2,7": "73/1769472",
"0,2,3,6": "511/5308416",
"0,2,4,5": "511/3538944",
"0,3,3,5": "511/2654208",
"0,3,4,4": "2555/10616832",
"1,1,1,8": "73/3538944",
"1,1,2,7": "73/884736",
"1,1,3,6": "511/2654208",
"1,1,4,5": "511/1769472",
"1,2,2,6": "511/1769472",
"1,2,3,5": "511/884736",
"1,2,4,4": "2555/3538944",
"1,3,3,4": "2555/2654208",
"2,2,2,5": "511/589824",
"2,2,3,4": "2555/1769472",
"2,3,3,3": "2555/1327104"
},
"5": {
"0,0,0,0,12": "73/3503554560",
"0,0,0,1,11": "73/291962880",
"0,0,0,2,10": "73/53084160",
"0,0,0,3,9": "73/15925248",
"0,0,0,4,8": "73/7077888",
"0,0,0,5,7": "73/4423680",
"0,0,0,6,6": "511/26542080",
"0,0,1,1,10": "73/26542080",
"0,0,1,2,9": "73/5308416",
"0,0,1,3,8": "73/1769472",
"0,0,1,4,7": "73/884736",
"0,0,1,5,6": "511/4423680",
"0,0,2,2,8": "73/1179648",
"0,0,2,3,7": "73/442368",
"0,0,2,4,6": "511/1769472",
"0,0,2,5,5": "511/1474560",
"0,0,3,3,6": "511/1327104",
"0,0,3,4,5": "511/884736",
"0,0,4,4,4": "2555/3538944",
"0,1,1,1,9": "73/2654208",
"0,1,1,2,8": "73/589824",
"0,1,1,3,7": "73/221184",
"0,1,1,4,6": "511/884736",
"0,1,1,5,5": "511/737280",
"0,1,2,2,7": "73/147456",
"0,1,2,3,6": "511/442368",
"0,1,2,4,5": "511/294912",
"0,1,3,3,5": "511/221184",
"0,1,3,4,4": "2555/884736",
"0,2,2,2,6": "511/294912",
"0,2,2,3,5": "511/147456",
"0,2,2,4,4": "2555/589824",
"0,2,3,3,4": "2555/442368",
"0,3,3,3,3": "2555/331776",
"1,1,1,1,8": "73/294912",
"1,1,1,2,7": "73/73728",
"1,1,1,3,6": "511/221184",
"1,1,1,4,5": "511/147456",
"1,1,2,2,6": "511/147456",
"1,1,2,3,5": "511/73728",
"1,1,2,4,4": "2555/294912",
"1,1,3,3,4": "2555/221184",
"1,2,2,2,5": "511/49152",
"1,2,2,3,4": "2555/147456",
"1,2,3,3,3": "2555/110592",
"2,2,2,2,4": "2555/98304",
"2,2,2,3,3": "2555/73728"
}
}
},
"Keywords": [
"lambda_g conjecture",
"Faber-Pandharipande formula",
"Hodge integrals",
"moduli of curves",
"tautological intersection numbers"
],
"Comments": {
"stack": "The Deligne-Mumford stack $\\overline{\\mathcal M}_{g,n}$ parametrizes stable complex curves of genus $g$ with $n$ ordered marked points. Integration includes the automorphism weights of the moduli stack. In particular, $I_1(0)=1/24$ CITE{M2}; removing the factor $1/2$ contributed by the generic elliptic involution would give $1/12$. The marked points remain labelled, so no factor of $n!$ or of the number of permutations of an exponent profile divides an entry.",
"profiles": "Markings are ordered, but the integrals are symmetric in their exponents, so profiles are written in non-decreasing order. The profile includes every zero: $0,0,2$ denotes $(0,0,2)$. A permuted profile such as $(2,0,0)$ has the same integral and is indexed by $0,0,2$. The top Chern class $\\lambda_g$ has complex codimension $g$, so the top-degree constraint is $g+\\sum_i d_i=3g-3+n$.",
"hodge-bundle": "The class $\\lambda_g=c_g(\\mathbb E)$ is the top Chern class of the rank-$g$ Hodge bundle $\\mathbb E$, whose fiber over a stable curve $C$ is $H^0(C,\\omega_C)$, where $\\omega_C$ is the dualizing sheaf CITE{FP}. The class $\\psi_i=c_1(L_i)$ is the first Chern class of the cotangent line bundle at the $i$th marked point.",
"one-point": "Write $b_0=1$ and $b_g=I_g(2g-2)$ for $g\\geq1$. In identities, an integral with a negative exponent is taken to be zero; unstable moduli spaces do not supply entries.",
"genus-zero-comment": "In genus zero, $\\lambda_0=1$, so the Hodge integrals become the HREF{T458}[intersection numbers of cotangent line classes]. The stored entries begin in genus one; the genus-zero values are given by CITE{genus-zero}."
},
"Formulas": {
"integral": "$\\displaystyle I_g(d_1,\\ldots,d_n)=\\int_{\\overline{\\mathcal M}_{g,n}}\\lambda_g\\prod_{i=1}^n\\psi_i^{d_i}$ for $2g-2+n>0$ and $\\sum_i d_i=2g-3+n$ CITE{FP}.",
"lambda-g": "For $g\\geq1$, $n\\geq1$ and $\\sum_i d_i=2g-3+n$, $\\displaystyle I_g(d_1,\\ldots,d_n)=\\frac{(2g-3+n)!}{d_1!\\cdots d_n!}\\,b_g$, where $b_g=I_g(2g-2)$ CITE{FP}.",
"bernoulli": "For $g\\geq1$, $\\displaystyle b_g=\\left(1-2^{1-2g}\\right)\\frac{|B_{2g}|}{(2g)!}$, where the HREF{Bernoulli_numbers}[Bernoulli numbers] are defined by $t/(e^t-1)=\\sum_{k\\geq0}B_k t^k/k!$ CITE{FP}.",
"sine": "$\\displaystyle\\sum_{g\\geq0}b_g t^{2g}=\\frac{t/2}{\\sin(t/2)}$ as a formal power series, with $b_0=1$ CITE{FP}.",
"genus-zero": "For $n\\geq3$ and $\\sum_i d_i=n-3$, $\\displaystyle I_0(d_1,\\ldots,d_n)=\\frac{(n-3)!}{d_1!\\cdots d_n!}$ CITE{FP}.",
"string": "For a stable pair $(g,n)$ and nonnegative exponents satisfying $\\sum_i d_i=2g-2+n$, $\\displaystyle I_g(0,d_1,\\ldots,d_n)=\\sum_{j=1}^n I_g(d_1,\\ldots,d_j-1,\\ldots,d_n)$.",
"dilaton": "For a stable pair $(g,n)$ and $\\sum_i d_i=2g-3+n$, $\\displaystyle I_g(1,d_1,\\ldots,d_n)=(2g-2+n)I_g(d_1,\\ldots,d_n)$."
},
"Similar tables": [
{
"table": "HREF{Bernoulli_numbers}[Bernoulli numbers]",
"relation": "determine the one-point integrals through the Faber-Pandharipande formula"
},
{
"table": "HREF{T458}[intersection numbers of cotangent line classes]",
"relation": "integrate the same cotangent classes without a Hodge-class insertion; in genus zero the two constructions coincide"
}
],
"Programs": {
"sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.rational_field import QQ\nfrom sage.arith.misc import bernoulli\nfrom math import factorial\n\ndef lambda_g_integral(g, d):\n n = len(d)\n if g < 0 or min(d, default=0) < 0 or 2*g-2+n <= 0:\n raise ValueError('A stable pair and nonnegative exponents are required')\n if sum(d) != 2*g-3+n:\n raise ValueError('The exponent profile must have degree 2*g-3+n')\n b = QQ(1) if g == 0 else ((1-QQ(1)/QQ(2**(2*g-1)))\n * abs(bernoulli(2*g))/QQ(factorial(2*g)))\n result = b*QQ(factorial(sum(d)))\n for exponent in d:\n result /= QQ(factorial(exponent))\n return result\n\nprint(lambda_g_integral(3, [1,2,3])) # 31/16128\n"
}
}
}