# integral -- evaluate Hodge integrals

## Synopsis

• Usage:
integral(g, n, klp)
• Inputs:
• Outputs:

## 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.

• 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 .