1,063 research outputs found
On the M\"obius Function and Topology of General Pattern Posets
We introduce a formal definition of a pattern poset which encompasses several
previously studied posets in the literature. Using this definition we present
some general results on the M\"obius function and topology of such pattern
posets. We prove our results using a poset fibration based on the embeddings of
the poset, where embeddings are representations of occurrences. We show that
the M\"obius function of these posets is intrinsically linked to the number of
embeddings, and in particular to so called normal embeddings. We present
results on when topological properties such as Cohen-Macaulayness and
shellability are preserved by this fibration. Furthermore, we apply these
results to some pattern posets and derive alternative proofs of existing
results, such as Bj\"orner's results on subword order.Comment: 28 Page
Nexus solitons in the center vortex picture of QCD
It is very plausible that confinement in QCD comes from linking of Wilson
loops to finite-thickness vortices with magnetic fluxes corresponding to the
center of the gauge group. The vortices are solitons of a gauge-invariant QCD
action representing the generation of gluon mass. There are a number of other
solitonic states of this action. We discuss here what we call nexus solitons,
in which for gauge group SU(N), up to N vortices meet a a center, or nexus,
provided that the total flux of the vortices adds to zero (mod N). There are
fundamentally two kinds of nexuses: Quasi-Abelian, which can be described as
composites of Abelian imbedded monopoles, whose Dirac strings are cancelled by
the flux condition; and fully non-Abelian, resembling a deformed sphaleron.
Analytic solutions are available for the quasi-Abelian case, and we discuss
variational estimates of the action of the fully non-Abelian nexus solitons in
SU(2). The non-Abelian nexuses carry Chern-Simons number (or topological charge
in four dimensions). Their presence does not change the fundamentals of
confinement in the center-vortex picture, but they may lead to a modified
picture of the QCD vacuum.Comment: LateX, 24 pages, 2 .eps figure
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