Hurwitz numbers, matrix models and enumerative geometry

By V. Bouchard, Marcos Mariño. 2007

186 citations recorded by the source corpus.

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/15811ca22ce69ca4d4d991c8ecd5fcfd1e866270

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In Geometry and Topology

1. Marcos Mariño
2. B. Eynard
3. Sara Pasquetti
4. A. Klemm
5. N. Orantin
6. V. Bouchard
7. Nick Scott
8. Paul T. Norbury

In Mathematical Physics

1. B. Eynard
2. Marcos Mariño
3. N. Orantin
4. V. Bouchard
5. Sara Pasquetti
6. A. Klemm
7. Nick Scott
8. Paul T. Norbury

In Nuclear and High Energy Physics

1. Marcos Mariño
2. B. Eynard
3. N. Orantin
4. A. Klemm
5. V. Bouchard
6. Sara Pasquetti
7. A. Morozov
8. Nick Scott

Citations in the corpus

• Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture (2012, background): “20) looks very similar to the ELSV formula, and indeed it reduces to the ELSV formula for the spectral curve (C, x(z) = z − ln z, y(z) = z,B(z1, z2) = dz1⊗dz2/(z1−z2)(2)) which appears in the study of simple Hurwitz numbers [11, 8, 18, 15].”
• Invariants of spectral curves and intersection theory of moduli spaces of complex curves (2011, methodology): “More generally, it was initially proposed by Mariño [21] and refined by Bouchard– Klemm–Mariño–Pasquetti (BKMP) [4], that if we choose the spectral curve to be the mirror curve of a toric Calabi–Yau 3–fold X, then the W (g) n ’s should be generating functions for the Gromov–Witten invariants of X,”
• Remodeling the B-Model (2007, background): “This is a nice mathematical consequence of the formalism proposed in this paper, which is studied in [16].”
• Gromov–Witten invariants of ℙ1 and Eynard–Orantin invariants (2011, methodology): “Important examples of the Eynard-Orantin invariants, using different choices of (C, x, y), store intersection numbers over the moduli space of curves [5]; simple Hurwitz numbers [1, 3, 6]; a count of lattice points in the moduli space of curves [17]; and conjecturally local Gromov-Witten invariants ”
• Algebra of differential operators associated with Young diagrams (2012, background): “ntial operators of infinitely many variables. The differential operators W(∆), corresponding to the Young diagrams ∆, are closely related to the Hurwitz numbers, matrix integrals and integrable systems [23, 5, 20, 22, 19]. We prove that the Schur functions form a complete set of the common eigenfuncti”