scholarly journals Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

2014 ◽  
Vol 57 (4) ◽  
pp. 749-764 ◽  
Author(s):  
Renzo Cavalieri ◽  
Steffen Marcus
Keyword(s):  
Moduli Space ◽  
Moduli Space Of Curves ◽  
Intersection Numbers ◽  
Hurwitz Numbers ◽  
Suggestive Evidence ◽  
Wall Crossing ◽  
Double Ramification Cycle ◽  
Correction Terms

AbstractWe describe doubleHurwitz numbers as intersection numbers on the moduli space of curves Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

scholarly journals Transitive Factorizations in the Hyperoctahedral Group

2008 ◽  
Vol 60 (2) ◽  
pp. 297-312
Author(s):  
G. Bini ◽  
I. P. Goulden ◽  
D. M. Jackson
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Point Of View ◽  
Moduli Space Of Curves ◽  
Reflection Groups ◽  
Hurwitz Numbers ◽  
Hyperoctahedral Group ◽  
Geometric Point ◽  
Enumeration Problem

AbstractThe classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type A to other finite reflection groups and, in particular, to type B. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an S2-symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type B case that is studied here.

scholarly journals Blobbed topological recursion: properties and applications

2016 ◽  
Vol 162 (1) ◽  
pp. 39-87 ◽  
Author(s):  
GAËTAN BOROT ◽  
SERGEY SHADRIN
Keyword(s):  
Initial Data ◽  
Matrix Model ◽  
Moduli Space ◽  
Graphical Representation ◽  
Infinitesimal Transformation ◽  
Moduli Space Of Curves ◽  
Free Energies ◽  
Intersection Numbers ◽  
Topological Recursion ◽  
Variational Formulas

AbstractWe study the set of solutions (ωg,n)g⩾0,n⩾1 of abstract loop equations. We prove that ωg,n is determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms in n variables (φg,n)2g−2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,n in terms of φg,n; (2) a graphical representation of ωg,n in terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n; (4) a definition for the free energies ωg,0 = Fg respecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

On the Moduli Space of a Spherical Polygonal Linkage

1999 ◽  
Vol 42 (3) ◽  
pp. 307-320 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Level Sets ◽  
Morse Function ◽  
Great Circle ◽  
Winding Number ◽  
Wall Crossing ◽  
Polygonal Linkage ◽  
Spherical Linkages ◽  
Wall Crossing Formula

AbstractWe give a “wall-crossing” formula for computing the topology of the moduli space of a closed n-gon linkage on 𝕊2. We do this by determining the Morse theory of the function ρn on the moduli space of n-gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first (n − 1) side-lengths are fixed. We obtain a Morse function on the (n − 2)-torus with level sets moduli spaces of n-gon linkages. The critical points of ρn are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ρn at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

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