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