# 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 $\psi_1^{d_1}\psi_2^{d_2}...\psi_n^{d_n}$ on the moduli space of stable $n$-pointed curves of genus $g$ is denoted: $$\int_{{\bar M}_{g,n}} \psi_1^{d_1}...\psi_n^{d_n} = <\tau_{d_0}\tau_{d_1}...\tau_{d_n}> = <\tau_0^{a_0}\tau_1^{a_1}...\tau_k^{a_k}>.$$ The list $\{a_0,a_1,...,a_k\}$ 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: $$\int_{{\bar M}_{0,n}} \psi_1^{a_1}...\psi_n^{a_n} = \frac{(n-3)!}{a_1!...a_n!}$$

 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