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.
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
|
[LX] Liu, K. and Xu, H. An effective recursion formula for computing intersection numbers. Available at http://front.math.ucdavis.edu/0710.5322
The object wittenTau is a method function.