94 research outputs found
Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions
There is a remarkable type of field of two-planes special to four dimensions
known as an Engel distributions. They are the only stable regular distributions
besides the contact, quasi-contact and line fields. If an arbitrary two-plane
field on a four-manifold is slightly perturbed then it will be Engel at generic
points. On the other hand, if a manifold admits an oriented Engel structure
then the manifold must be parallelizable and consequently the alleged Engel
distribution must have a degeneration loci -- a point set where the Engel
conditions fails. By a theorem of Zhitomirskii this locus is a finite union of
surfaces. We prove that these surfaces represent Chern classes associated to
the distribution.Comment: LaTeX, 15 page
Generalised ordinary vs fully simple duality for -point functions and a proof of the Borot--Garcia-Failde conjecture
We study a duality for the -point functions in VEV formalism that we call
the ordinary vs fully simple duality. It provides an ultimate generalisation
and a proper context for the duality between maps and fully simple maps
observed by Borot and Garcia-Failde. Our approach allows to transfer the
algebraicity properties between the systems of -point functions related by
this duality, and gives direct tools for the analysis of singularities. As an
application, we give a proof of a recent conjecture of Borot and Garcia-Failde
on topological recursion for fully simple maps.Comment: 25 pages; minor correction
Explicit closed algebraic formulas for Orlov-Scherbin -point functions
We derive a new explicit formula in terms of sums over graphs for the
-point correlation functions of general formal weighted double Hurwitz
numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric
type (also known as Orlov-Scherbin partition functions). Notably, we use the
change of variables suggested by the associated spectral curve, and our formula
turns out to be a polynomial expression in a certain small set of formal
functions defined on the spectral curve.Comment: 35 pages; statements regarding the -deformations were
corrected; further minor change
KP integrability through the swap relation
We discuss a universal relation that we call the swap relation, which
plays a prominent role in the theory of topological recursion, Hurwitz theory,
and free probability theory. We describe in a very precise and detailed way the
interaction of the swap relation and KP integrability. As an application,
we prove a recent conjecture that relates some particular instances of
topological recursion to the Mironov-Morozov-Semenoff matrix integrals.Comment: 27 pages; minor correction
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