Communications in Number Theory and Physics

Volume 8 (2014)

Number 3

Invariants of spectral curves and intersection theory of moduli spaces of complex curves

Pages: 541 – 588



B. Eynard (Institut de Physique Théorique, CE Saclay, Gif-sur-Yvette, France; and Département de Mathématiques, Université of Genève, Switzerland)


To any spectral curve $\mathcal{S}$, we associate a topological class $\hat{\Lambda} (\mathcal{S})$ in a moduli space $\mathcal{M}^{\mathfrak{b}}_{g,n}$ of “$\mathfrak{b}$-colored” stable Riemann surfaces of given topology (genus $g, n$ boundaries), whose integral coincides with the topological recursion invariants $W_{g,n}(\mathcal{S})$ of the spectral curve $\mathcal{S}$. This formula can be viewed as a generalization of the ELSV formula (whose spectral curve is the Lambert function and the associated class is the Hodge class), or Mariño-Vafa formula (whose spectral curve is the mirror curve of the framed vertex, and the associated class is the product of three Hodge classes), but for an arbitrary spectral curve. In other words, to a B-model (i.e., a spectral curve) we systematically associate a mirror A-model (integral in a moduli space of “colored” Riemann surfaces).We find that the mirror map, i.e., the relationship between the A-model moduli and B-model moduli, is realized by the Laplace transform.

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