# wittenTau -- Witten tau integrals

## Synopsis

• Usage:
wittenTau(g,a), wittenTau(a)
• Inputs:
• Outputs:

## 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