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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 ki, the cotangent line classes ψi, and the Chern classes and Chern characters, λi and chi 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 λg formula [FP, Theorem 1],

Mg,n ψ1a1...ψnan λg= |B2g|(2g+n-3)!(22g-1-1) / (a1!...an!22g-1(2g)!),

where Bi 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 λg conjecture. Annals of Mathematics, 156 (2002), 97-124.

See also