1,650 research outputs found
On the complexity of optimal homotopies
In this article, we provide new structural results and algorithms for the
Homotopy Height problem. In broad terms, this problem quantifies how much a
curve on a surface needs to be stretched to sweep continuously between two
positions. More precisely, given two homotopic curves and
on a combinatorial (say, triangulated) surface, we investigate the problem of
computing a homotopy between and where the length of the
longest intermediate curve is minimized. Such optimal homotopies are relevant
for a wide range of purposes, from very theoretical questions in quantitative
homotopy theory to more practical applications such as similarity measures on
meshes and graph searching problems.
We prove that Homotopy Height is in the complexity class NP, and the
corresponding exponential algorithm is the best one known for this problem.
This result builds on a structural theorem on monotonicity of optimal
homotopies, which is proved in a companion paper. Then we show that this
problem encompasses the Homotopic Fr\'echet distance problem which we therefore
also establish to be in NP, answering a question which has previously been
considered in several different settings. We also provide an O(log
n)-approximation algorithm for Homotopy Height on surfaces by adapting an
earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the
planar setting
Non-Abelian vortices and monopoles in SO(N) theories
Non-Abelian BPS vortex solutions are constructed in N=2 theories with gauge
groups SO(N)\times U(1). The model has N_f flavors of chiral multiplets in the
vector representation of SO(N), and we consider a color-flavor locked vacuum in
which the gauge symmetry is completely broken, leaving a global SO(N)_{C+F}
diagonal symmetry unbroken. Individual vortices break this symmetry, acquiring
continuous non-Abelian orientational moduli. By embedding this model in
high-energy theories with a hierarchical symmetry breaking pattern such as
SO(N+2) --> SO(N)\times U(1) --> 1, the correspondence between non-Abelian
monopoles and vortices can be established through homotopy maps and flux
matching, generalizing the known results in SU(N) theories. We find some
interesting hints about the dual (non-Abelian) transformation properties among
the monopoles.Comment: LaTeX, 26 pages and 4 figure
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