Hurwitz numbers, matrix models and enumerative geometry (0709.1458v2)

Abstract: 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.

Hurwitz Numbers, Matrix Models, and Enumerative Geometry: Insights and Conjectures

The paper "Hurwitz numbers, matrix models and enumerative geometry" by Bouchard and Mariño embarks on a sophisticated exploration of enumerative geometry, focusing specifically on Hurwitz numbers. These are the quintessential objects of paper that count branched covers of Riemann surfaces, interfacing with modern enumerative geometry through connections with Gromov-Witten theory and Hodge integrals. The authors propose a novel conjectural recursion, leveraging advancements in type B topological string theory on the mirrors of toric Calabi-Yau manifolds.

Conjectural Recursion and Topological String Theory

The main thrust of the paper is the introduction of a new recursion solution for Hurwitz numbers at all genera. This conjecture emerges from a refined understanding of type B topological string theory on mirrors of toric Calabi-Yau manifolds—specifically highlighted through a relationship with mirror symmetry and the topological vertex. The conjecture aims to solve for Hodge integrals with three Hodge class insertions through a recursive method akin to known techniques in the field. Although presented without a formal proof, the conjecture is supported by empirical evidence and aligned with theoretical expectations.

Connection with Matrix Models

The authors draw parallels between their conjectural framework and the well-established results in matrix models. Matrix models have long elucidated the structure of spectral curves, where asymptotic expansions in the matrix size $N$ yield the genus $g$ amplitudes—$F_g$ for closed surfaces and $W_g$ for open surfaces. The recursive formalism employed in matrix models, and adapted for topological B-model string theory, offers a compelling template for handling Hurwitz numbers.

Numerical Results and Evidence

The paper details several specific numerical results obtained from this new recursive formula. These include low-genus examples where the recursion reliably reproduces Hurwitz numbers and estimates various Hodge integrals. Such consistency bolsters confidence in the conjecture's robustness and its applicability across broader scenarios.

Implications and Future Directions

The implications of this work are multifold. The conjectural recursion provides a potentially comprehensive tool for calculating Hurwitz numbers, thus enriching the theoretical landscape of enumerative geometry. It unifies disparate mathematical frameworks, suggesting that insights from string theory and matrix models may have broader implications on classical problems in algebraic geometry.

The discussion hints at future explorations in linking this formalism to established structures, like the cut-and-join equations for Hurwitz numbers, and possibly offering a novel perspective on the symplectic geometry of toric Calabi-Yau manifolds. The authors usher in a promising research trajectory that may find applications in theoretical physics, particularly concerning string theory's integrative aspects.

Conclusion

Bouchard and Mariño furnish a rigorous yet conjectural framework marrying traditional algebraic geometry with cutting-edge theoretical physics. Their new recursion conjecture for Hurwitz numbers not only aligns with contemporary approaches in Gromov-Witten theory but also stands as a testament to the deep interconnections between different domains within mathematics and physics. Future research will be crucial in substantiating and expanding upon these seminal ideas, potentially unveiling novel insights into the fabric of enumerative geometry and beyond.