Enumerative Geometry 10 Gromov Witten

Recap

  • Moduli space of curves parametrizes genus \(g\) curve with \(n\) distinct point.
  • Moduli space of stable curves \(\overline{\mathcal{M}_{g,n}} (X)\) parametrizes genus \(g\) stable curve with \(n\) distinct point.
  • Moduli space of stable maps \(\overline{\mathcal{M}_{g,n}} (X, \beta)\) parametrizes stable maps to \(X\) with homology image with a genus \(g\) curve with \(n\) distinct points.

Gromov Witten Invariant

  • Rational Nodal Curve in \(\mathbb{P}^2\)
  • Invariant \(\langle{\gamma_1,\ldots,\gamma_n}\rangle^X_{g, \beta}=\int_{[\overline{\mathcal{M}_{g,n}} (X, \beta)]_{vir}} ev^*_1(\gamma_1),\ldots, ev^*_n(\gamma_n)\)
  • The integral may not be a integer.
  • Application to Physics String theory.
  • Virtual fundamental class
  • Nonzero if incidence conditions correspond to dimension of virtual class.
    • \[vdim = 3d-1 + \int 1\]
  • Property
    • Divisor \(\langle{\gamma_1,\ldots,\gamma_{n-1}}, D\rangle^X_{g, \beta}=\int_\beta) D \cdot \langle{\gamma_1,\ldots,\gamma_{n-1}}\rangle^X_{g, \beta}\)
    • Degree \(0\)
    • Fundamental Class

WDVV

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