Chern Classes of the Moduli Stack of Curves

Bini, Gilberto

doi:10.4310/mrl.2005.v12.n5.a12

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…This formula could be thought of as a local version of the GrothendieckRiemann-Roch theorem applied to the forgetful map π n+1 : M g,n+1 → M g,n and a certain sheaf on M g,n+1 depending on ℓ (when ℓ = 0, it is the sheaf of sections of the trivial line bundle). When ℓ = 0 or ℓ = 1, our formula agrees modulo boundary terms with the one obtained by Bini [6] using the Grothendieck-Riemann-Roch theorem.…”

Section: Introductionsupporting

confidence: 66%

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A Local Families Index Formula for $${\overline{\partial}}$$ -Operators on Punctured Riemann Surfaces

Albin

Rochon

2009

Commun. Math. Phys.

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Abstract. Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of ∂-operators on the Teichmüller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space Mg,n in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf.

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Section: Introductionsupporting

confidence: 66%

“…When n > 0, a Grothendieck-Riemann-Roch formula was obtained for the sheaf ω ℓ π n+1 by Bini [6], (6.24)…”

Section: Theorem 1 the Local Family Index Of The Family Of Operatorsmentioning

confidence: 99%

A Local Families Index Formula for $${\overline{\partial}}$$ -Operators on Punctured Riemann Surfaces

Albin

Rochon

2009

Commun. Math. Phys.

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“…The preprint[105] also explains a similar computation, but we do not agree with their formula. Their formula fails in particular the checks that we perform here.…”

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confidence: 53%

Off-shell Partition Functions in 3d Gravity

Eberhardt¹

2022

Preprint

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We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of PSL(2, R) Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface Σ is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form Σ × S 1 , where Σ can have asymptotic boundaries. Most of these topologies do not admit a classical solution and are thus not amenable to a direct semiclassical path integral computation. We use an index theorem that expresses the partition function as an integral of characteristic classes over phase space. In the presence of n asymptotic boundaries, we use techniques from equivariant cohomology to localize the integral to a finitedimensional integral over M g,n , which we evaluate in low genus cases. Higher genus partition functions quickly become complicated since they depend in an oscillatory way on Newton's constant. There is a precise sense in which one can isolate the nonoscillatory part which we call the fake partition function. We establish that there is a topological recursion that computes the fake partition functions for arbitrary Riemann surfaces Σ. There is a scaling limit in which the model reduces to JT gravity and our methods give a novel way to compute JT partition functions via equivariant localization.

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“…, where π denotes the relative sheaf of differentials. Adapting Mumford's [28] well-known application of the Grothendieck-Riemann-Roch theorem to the universal family π , Bini [6,Theorem 2] gives an explicit formula for ch(T * M g,n…”

Section: Chern Classes Of the Tangent Bundlementioning

confidence: 99%

“…In particular, the Chern classes of T H g,G,ξ may be computed explicitly by combining the formula of Bini [6,Theorem 2] with the earlier results of this section.…”

Section: Interlude: Chern Classes Of the Tangent Bundlementioning

confidence: 99%

The $${\mathcal {H}}$$-tautological ring

Lian

2021

Sel. Math. New Ser.

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We extend the theory of tautological classes on moduli spaces of stable curves to the more general setting of moduli spaces of admissible Galois covers of curves, introducing the so-called $${\mathcal {H}}$$ H -tautological ring. The main new feature is the existence of restriction-corestriction morphisms remembering intermediate quotients of Galois covers, which are a rich source of new classes. In particular, our new framework includes classes of Harris–Mumford admissible covers on moduli spaces of curves, which are known in some (and speculatively many more) examples to lie outside the usual tautological ring. We give additive generators for the $${\mathcal {H}}$$ H -tautological ring and show that their intersections may be algorithmically computed, building on work of Schmitt-van Zelm. As an application, we give a method for computing integrals of Harris-Mumford loci against tautological classes of complementary dimension, recovering and giving a mild generalization of a recent quasi-modularity result of the author for covers of elliptic curves.

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Chern Classes of the Moduli Stack of Curves

Abstract: Abstract. Here we calculate the Chern classes of Mg,n, the moduli stack of stable n-pointed genus g curves. In particular, we prove that such classes lie in the tautological ring.

Cited by 5 publications

References 13 publications

A Local Families Index Formula for $${\overline{\partial}}$$ -Operators on Punctured Riemann Surfaces

A Local Families Index Formula for $${\overline{\partial}}$$ -Operators on Punctured Riemann Surfaces

Off-shell Partition Functions in 3d Gravity

The $${\mathcal {H}}$$-tautological ring

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