wittenTau -- Witten tau integrals

Synopsis

• Usage:

  wittenTau(g,a), wittenTau(a)

• Inputs:
  • g, an integer
  • a, a list
• Outputs:
  • a rational number

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

See also

• HodgeIntegrals -- Hodge integrals on the moduli space of curves
• hodgeRing -- create a ring containing algebraic classes on moduli spaces of curves
• integral -- evaluate Hodge integrals