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# integral -- evaluate Hodge integrals

- Usage:
    `integral(g, n, klp)`
- Inputs:
  - g, an integer
  - n, an integer
- Outputs:
  - a rational number

## Description

This function computes top intersection numbers among tautological classes on the moduli space of curves. The tautological classes include products of the Mumford-Morita-Miller classes $k_i$, the cotangent line classes $\psi_i$, and the Chern classes and Chern characters, $\lambda_i$ and $ch_i$ of the Hodge bundle.

The function hodgeRing must be called previously with values of g and n at least as large as those to be used.,

## Examples

Here are a few examples illustrating the $\lambda_g$ formula [FP, Theorem 1], $$\int_{{\bar M}_{g,n}} \psi_1^{a_1}...\psi_n^{a_n} \lambda_g= |B_{2g}|(2g+n-3)!(2^{2g-1}-1) / (a_1!...a_n!2^{2g-1}(2g)!),$$ where $B_i$ represents the $i$-th Bernoulli number.

    i1 : R = hodgeRing (3, 3);

    i2 : integral (1, 1, lambda_1)
    warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
           : debug with expression   debug 1257   or with command line option   --debug 1257

          1
    o2 = --
         24

    o2 : R

    i3 : integral (2, 2, psi_1 * psi_2^2 * lambda_2)
    warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
           : debug with expression   debug 1257   or with command line option   --debug 1257

           7
    o3 = ----
         1920

    o3 : R

    i4 : integral (3, 3, psi_1 * psi_2^2 * psi_3^3 * lambda_3)
    warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
           : debug with expression   debug 1257   or with command line option   --debug 1257

           31
    o4 = -----
         16128

    o4 : R

Here are a few more examples.

    i5 : R = hodgeRing (4, 0);

    i6 : integral (2, 0, lambda_1^3)
    warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
           : debug with expression   debug 1257   or with command line option   --debug 1257
    warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
           : debug with expression   debug 1257   or with command line option   --debug 1257

           1
    o6 = ----
         2880

    o6 : R

    i7 : integral (3, 0, lambda_1^6)
    warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
           : debug with expression   debug 1257   or with command line option   --debug 1257

           1
    o7 = -----
         90720

    o7 : R

    i8 : integral (4, 0, lambda_1^9)
    warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
           : debug with expression   debug 1257   or with command line option   --debug 1257

            1
    o8 = ------
         113400

    o8 : R

## References

[FP] Faber, C. and Pandharipande, R., Hodge integrals, partition matrices, and the $\lambda_g$ conjecture. Annals of Mathematics, 156 (2002), 97-124.

## See also

- HodgeIntegrals -- Hodge integrals on the moduli space of curves
- hodgeRing -- create a ring containing algebraic classes on moduli spaces of curves
- wittenTau -- Witten tau integrals

## For the programmer

The object integral is a function closure.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/HodgeIntegrals.m2:629:0.