# HodgeIntegrals -- Hodge integrals on the moduli space of curves

## Description

HodgeIntegrals is a package for evaluating intersection numbers on the Deligne-Mumford moduli space of $n$-pointed stable curves of genus $g$, often denoted ${\bar M}_{g,n}$. This package evaluates integrals of the form $$\int_{{\bar M}_{g,n}} \psi_1^{e_1} ... \psi_n^{e_n} k_1^{f_1} ... k_b^{f_b} \lambda_1^{h_1} ... \lambda_g^{h_g},$$ where the values of $\psi_i$, $k_i$, and $\lambda_i$ are defined as follows:

• $\psi_i$ is the first Chern class of the $i$-th cotangent line bundle $L_i$, whose value at a fixed curve $(C; p_1,...,p_n)$ is the cotangent space to $C$ at $p_i$.
• $k_j$ is the pushforward of $\psi_i^{j+1}$ via the forgetful morphism which forgets the $i$-th marked point.
• $\lambda_i$ is the $i$-th Chern class of the Hodge bundle $E$, whose value at a fixed curve $(C; p_1,...,p_n)$ is $H^0(C,K_C)$, or the space of differential one-forms on $C$.

A good introduction to ${\bar M}_{g,n}$ and related spaces can be found in the textbook [HM]. Two good references for the algebraic classes $\psi_i$, $k_i$, and $\lambda_i$, as well as their properties, are [AC] and [M].

This package is modelled after Carel Faber's Maple program KaLaPs, available for download [F]. For more details on how this package works, please read [Y].

## References

[AC] Arbarello, E. and Cornalba, M. Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom. 5. (1996), no. 4, 705--749.

[F] Faber, Carel. Maple program for calculating intersection numbers on moduli spaces of curves. Available at http://math.stanford.edu/~vakil/programs/index.html.

[HM] Harris J., and Morrison, I. Moduli of Curves, Graduate Texts in Mathematics 187. Springer-Verlag, New York, 1996. ISBN: 0387984291.

[V] Vakil, R. The moduli space of curves and Gromov-Witten theory. Enumerative invariants in algebraic geometry and string theory (Behrend and Manetti eds.), Lecture Notes in Mathematics 1947, Springer, Berlin, 2008.

[Y] Yang, S., Intersection numbers on ${\bar M}_{g,n}$.

## Contributors

The following person has generously contributed code or worked on our code.

## Certification

Version 1.2.1 of this package was accepted for publication in volume 2 of The Journal of Software for Algebra and Geometry: Macaulay2 on 2010-04-17, in the article Intersection numbers on Mbar_{g,n}. That version can be obtained from the journal or from the Macaulay2 source code repository.

## Version

This documentation describes version 1.2.1 of HodgeIntegrals.

## Source code

The source code from which this documentation is derived is in the file HodgeIntegrals.m2.

## Exports

• Functions and commands
• hodgeRing -- create a ring containing algebraic classes on moduli spaces of curves
• integral -- evaluate Hodge integrals
• wittenTau -- Witten tau integrals
• Methods
• "wittenTau(List)" -- see wittenTau -- Witten tau integrals
• "wittenTau(ZZ,List)" -- see wittenTau -- Witten tau integrals
• Other things
• ch -- Chern character of the Hodge bundle
• kappa -- Miller-Morita-Mumford classes
• lambda -- Chern class of the Hodge bundle
• psi -- cotangent line class

## For the programmer

The object HodgeIntegrals is .