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HodgeIntegrals :: wittenTau

wittenTau -- Witten tau integrals

Synopsis

Description

The Witten tau coefficients are top intersection numbers of cotangent line classes on the moduli space of curves. The integral of ψ1d1ψ2d2...ψndn on the moduli space of stable n-pointed curves of genus g is denoted:

Mg,n ψ1d1...ψndn = <τd0τd1...τdn> = <τ0a0τ1a1...τkak>.

The list {a0,a1,...,ak} is the argument for wittenTau. These integrals are computed recursively using the string equation, dilation equation, and an effective genus recursion formula of Liu and Xu [LX].

The genus is an optional parameter. If it is omitted, the genus is automatically calculated.

Examples

Here are some examples illustrating the well-known formula that is a result of Witten's conjecture:

M0,n ψ1a1...ψnan = ((n-3)!)/(a1!...an!)

i1 : wittenTau (0,{3})

o1 = 1

o1 : QQ
i2 : wittenTau (0,{4, 1, 1})

o2 = 3

o2 : QQ
i3 : wittenTau (0,{5, 0, 2})

o3 = 6

o3 : QQ

Here are some additional examples in higher genus.

i4 : wittenTau (1,{0,1})

      1
o4 = --
     24

o4 : QQ
i5 : wittenTau (3,{0,0,0,0,0,1})

o5 = 0

o5 : QQ
i6 : wittenTau (5,{0,0,0,0,0,3})

       41873
o6 = ---------
     255467520

o6 : QQ

References

[LX] Liu, K. and Xu, H. An effective recursion formula for computing intersection numbers. Available at http://front.math.ucdavis.edu/0710.5322

See also

Ways to use wittenTau :