345 research outputs found
The genus of curve, pants and flip graphs
This article is about the graph genus of certain well studied graphs in
surface theory: the curve, pants and flip graphs. We study both the genus of
these graphs and the genus of their quotients by the mapping class group. The
full graphs, except for in some low complexity cases, all have infinite genus.
The curve graph once quotiented by the mapping class group has the genus of a
complete graph so its genus is well known by a theorem of Ringel and Youngs.
For the other two graphs we are able to identify the precise growth rate of the
graph genus in terms of the genus of the underlying surface. The lower bounds
are shown using probabilistic methods.Comment: 26 pages, 9 figure
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
BPS Graphs: From Spectral Networks to BPS Quivers
We define "BPS graphs" on punctured Riemann surfaces associated with
theories of class . BPS graphs provide a bridge between
two powerful frameworks for studying the spectrum of BPS states: spectral
networks and BPS quivers. They arise from degenerate spectral networks at
maximal intersections of walls of marginal stability on the Coulomb branch.
While the BPS spectrum is ill-defined at such intersections, a BPS graph
captures a useful basis of elementary BPS states. The topology of a BPS graph
encodes a BPS quiver, even for higher-rank theories and for theories with
certain partial punctures. BPS graphs lead to a geometric realization of the
combinatorics of Fock-Goncharov -triangulations and generalize them in
several ways.Comment: 48 pages, 44 figure
Hierarchical hyperbolicity of graphs of multicurves
We show that many graphs naturally associated to a connected, compact,
orientable surface are hierarchically hyperbolic spaces in the sense of
Behrstock, Hagen and Sisto. They also automatically have the coarse median
property defined by Bowditch. Consequences for such graphs include a distance
formula analogous to Masur and Minsky's distance formula for the mapping class
group, an upper bound on the maximal dimension of quasiflats, and the existence
of a quadratic isoperimetric inequality. The hierarchically hyperbolic
structure also gives rise to a simple criterion for when such graphs are Gromov
hyperbolic.Comment: 27 pages, 4 figures. Minor changes from previous version. Addition of
appendix describing a hierarchically hyperbolic structure on the arc grap
Combinatorial Heegaard Floer homology and nice Heegaard diagrams
We consider a stabilized version of hat Heegaard Floer homology of a
3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed
3-manifolds). We give a combinatorial algorithm for constructing this
invariant, starting from a Heegaard decomposition for Y, and give a
combinatorial proof of its invariance properties
Network and Seiberg Duality
We define and study a new class of 4d N=1 superconformal quiver gauge
theories associated with a planar bipartite network. While UV description is
not unique due to Seiberg duality, we can classify the IR fixed points of the
theory by a permutation, or equivalently a cell of the totally non-negative
Grassmannian. The story is similar to a bipartite network on the torus
classified by a Newton polygon. We then generalize the network to a general
bordered Riemann surface and define IR SCFT from the geometric data of a
Riemann surface. We also comment on IR R-charges and superconformal indices of
our theories.Comment: 28 pages, 28 figures; v2: minor correction
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