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HodgeIntegrals :: integral

integral -- evaluate Hodge integrals

Synopsis

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

For the programmer

The object integral is a function closure.