Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Completed Cycles Leaky Hurwitz Numbers

Davide Accadia, Maksim Karev, Danilo Lewański

Abstract

We introduce $(r+1)$-completed cycles $k$-leaky Hurwitz numbers and prove piecewise polynomiality as well as establishing their chamber polynomiality structure and their wall crossing formulae. For $k=0$ the results recover previous results of Shadrin-Spitz-Zvonkine. The specialization for $r=1$ recovers Hurwitz numbers that are close to the ones studied by Cavalieri-Markwig-Ranganathan and Cavalieri-Markwig-Schmitt. The ramifications differ by a lower order torus correction, natural from the Fock space perspective, not affecting the genus zero enumeration, nor the enumeration for leaky parameter values $k = \pm 1$ in all genera.

Completed Cycles Leaky Hurwitz Numbers

Abstract

We introduce -completed cycles -leaky Hurwitz numbers and prove piecewise polynomiality as well as establishing their chamber polynomiality structure and their wall crossing formulae. For the results recover previous results of Shadrin-Spitz-Zvonkine. The specialization for recovers Hurwitz numbers that are close to the ones studied by Cavalieri-Markwig-Ranganathan and Cavalieri-Markwig-Schmitt. The ramifications differ by a lower order torus correction, natural from the Fock space perspective, not affecting the genus zero enumeration, nor the enumeration for leaky parameter values in all genera.

Paper Structure

This paper contains 16 sections, 9 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Acknowledgements
  4. Semi-infinite wedge formalism and VEVs computation
  5. Computation of VEVs
  6. Leaky Hurwitz numbers as VEVs
  7. Geometric and tropical interpretation
  8. VEVs computation and strong piecewise polynomiality
  9. Wall crossing formulae
  10. Computation of $T_1^K$
  11. Computation of $T_2^K$
  12. Wall crossing formulae in genus zero
  13. One-part leaky Hurwitz numbers
  14. Vanishing in genus zero
  15. Cut-and-join operators
  16. ...and 1 more sections

Key Result

Proposition 3.5

For any subsets $I, M \subset [l(\mu)]$, $J,N \subset [l(\nu)]$ and $K,L \subset [s]$ such that $I \cap M=J \cap N=K \cap L=\emptyset$, we have:

Figures (2)

  • Figure 1: A VEV-computation tree. We compute the VEV of operators $\mathcal{O}_2,\mathcal{O}_1,\mathcal{O}_{-3}$ with the commutation relation $[\mathcal{O}_{e_1},\mathcal{O}_{e_2}] = p(e_1,e_2)\mathcal{O}_{e_1 + e_2}$, where $p$ is some function. In each node, its right son is the corresponding passing term, and its left son is the corresponding canceling term. The edges connecting a vertex with the corresponding canceling term is marked by the value of the function $p$ that appears as the factor. To compute the commutator we have to take the sum over all essential paths of products of all the labels written on the edges of the path, and the label of the bottom-most vertex of the path. In the depicted case, there is only one essential path, so the VEV in question equals $p(1,-3)p(2,-2)\langle \mathcal{O}_0 \rangle$.
  • Figure 2: A coarse VEV-computation tree constructed in the assumption $k > 0$ and the both $\mu$ and $\nu$ have one part only. Notice, that the same coarse VEV-computation tree works for, say, $\mu = (5), k = 1, \nu = (4)$, and, say, $\mu = (6), k = 3, \nu = (3)$.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Proposition 3.5
  • Theorem 5.1
  • Remark 5.2
  • Definition 5.3
  • Remark 5.4
  • ...and 18 more