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Open Hurwitz numbers and the mKP hierarchy

Alexandr Buryak, Ran J. Tessler, Mikhail Troshkin

TL;DR

This work defines open Hurwitz numbers with a boundary-weight parameter $N$ and shows that their generating functions, after a simple rescaling, form a tau-sequence for the modified KP (mKP) hierarchy. The authors prove that the open partition functions $\tau^o_N$ yield KP tau-functions and that consecutive members are related by forward Bäcklund–Darboux transformations, linking open data to the well-established closed KP structure. The approach combines an open-cut refinement with the cut-and-join formalism and an explicit fusion with the closed tau-function $\tau^c$, generalizing the integrable framework for Hurwitz-type invariants to open surfaces. This provides a natural Hurwitz-type open analog of ABT17's open intersection-number refinements and strengthens the open-closed integrable structure in enumerative geometry.

Abstract

We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter $N$ taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to $\mathbb{CP}^1$. We prove that the resulting sequence of partition functions, depending on $N\in\mathbb{Z}$, is a tau-sequence of the mKP hierarchy, or in other words it is a sequence of tau-functions of the KP hierarchy where each tau-function is obtained from the previous one by a Bäcklund-Darboux transformation. Our result is motivated by a previous observation of Alexandrov and the first two authors that the refined intersection numbers on the moduli spaces of Riemann surfaces with boundary give a tau-sequence of the mKP hierarchy.

Open Hurwitz numbers and the mKP hierarchy

TL;DR

This work defines open Hurwitz numbers with a boundary-weight parameter and shows that their generating functions, after a simple rescaling, form a tau-sequence for the modified KP (mKP) hierarchy. The authors prove that the open partition functions yield KP tau-functions and that consecutive members are related by forward Bäcklund–Darboux transformations, linking open data to the well-established closed KP structure. The approach combines an open-cut refinement with the cut-and-join formalism and an explicit fusion with the closed tau-function , generalizing the integrable framework for Hurwitz-type invariants to open surfaces. This provides a natural Hurwitz-type open analog of ABT17's open intersection-number refinements and strengthens the open-closed integrable structure in enumerative geometry.

Abstract

We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to . We prove that the resulting sequence of partition functions, depending on , is a tau-sequence of the mKP hierarchy, or in other words it is a sequence of tau-functions of the KP hierarchy where each tau-function is obtained from the previous one by a Bäcklund-Darboux transformation. Our result is motivated by a previous observation of Alexandrov and the first two authors that the refined intersection numbers on the moduli spaces of Riemann surfaces with boundary give a tau-sequence of the mKP hierarchy.

Paper Structure

This paper contains 16 sections, 23 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Closed Hurwitz numbers
  3. Open Hurwitz numbers
  4. Definition
  5. Open partition function as an mKP tau-sequence
  6. KP hierarchy
  7. KP hierarchy in the Lax form
  8. Adjoint operators
  9. Fock space formalism
  10. Sato Grassmannian
  11. Cut-and-join operator
  12. Bäcklund--Darboux transformations and mKP hierarchy
  13. Bäcklund--Darboux transformations in the Lax form
  14. Example: multi-soliton solutions
  15. Bäcklund--Darboux transformations in fermionic form
  16. ...and 1 more sections

Key Result

Theorem 3.2

Define $\widetilde{\tau}_N:=\left.\tau^o_N\right|_{q_2\mapsto e^{N\beta_2}q_2}$. Then the sequence $(\widetilde{\tau}_N)_{N\in\mathbb Z}$ is a tau-sequence of the mKP hierarchy.

Figures (2)

  • Figure 1: Open simple ramified covering
  • Figure 2: Cutting procedure

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • ...and 39 more