Grothendieck's Dessins d'Enfants in a Web of Dualities. III

Abstract

We identify the dessin partition function with the partition function of the Laguerre unitary ensemble (LUE). Combined with the result due to Cunden et al on the relationship between the LUE correlators and strictly monotone Hurwitz numbers introduced by Goulden et al, we then establish connection of dessin counting to strictly monotone Hurwitz numbers. We also introduce a correction factor for the dessin/LUE partition function, which plays an important role in showing that the corrected dessin/LUE partition function is a tau-function of the Toda lattice hierarchy. As an application, we use the approach of Dubrovin and Zhang for the computation of the dessin correlators. In physicists' terminology, we establish dualities among dessin counting, generalized Penner model, and P 1 -topological sigma model. Contents 1. Introduction 1 2. Review on dessins, LUE, and one-dimensional Toda chain 8 3. Proofs of Theorem 1 and Corollary 1 13 4. Calculating dessin correlators via P 1 Frobenius manifold 16 5. Grothendieck's dessin counting and monotone Hurwitz numbers 22 6. Concluding remarks 25 A. A Reflection Formula for the Barnes G-Function 26 B. The Penner model and the generalized Penner model 27 References 29

GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back

“…This Frobenius manifold is often called an NLS Frobenius manifold [11,12,18], and will be discussed in details in the next of the article-series. Our proof given in Section 3 will be a relatively more direct one, which is similar to the one given recently in [69] for a result for Grothendieck's dessins d'enfant/Laguerre Unitary Ensemble (LUE).…”

“…More precisely, firstly, the free energy in higher genera gives rise to the topological tau-function when the jet variables are subjected to the topological solution of the principal hierarchy. Secondly, by construction the DZ hierarchy is quasi-trivial, namely, it is obtained from the principal hierarchy under a quasi-Miura transformation, which is given by the higher genera free energy in terms of jets; and this is interesting, because the quasi-Miura transformation could be substituted by any monotone solution to the principal hierarchy and makes it become a solution to the DZ hierarchy [21,23,29,33,34,69]. In [69], it is suggested to interpret this as a universality class of Dubrovin [19,20,22,24].…”

“…Secondly, by construction the DZ hierarchy is quasi-trivial, namely, it is obtained from the principal hierarchy under a quasi-Miura transformation, which is given by the higher genera free energy in terms of jets; and this is interesting, because the quasi-Miura transformation could be substituted by any monotone solution to the principal hierarchy and makes it become a solution to the DZ hierarchy [21,23,29,33,34,69]. In [69], it is suggested to interpret this as a universality class of Dubrovin [19,20,22,24]. A particular dense subset of monotone solutions to the principal hierarchy can be obtained by performing time shifts in the topological solution, but there are also other interesting monotone solutions.…”

“…According to the above discussion, their quasi-trivial transformations can be obtained from the free energy of GW invariants of P 1 . For example, in [69], the dessins/LUE solution to the Toda lattice hierarchy is considered and it is shown that this solution can be obtained by the application of quasi-triviality.…”

“…Similarly as we do for the LUE case in [69], we proceed with the higher genera by using quasi-triviality. According to [33,34], the following quasi-trivial map…”