668 research outputs found
On the Brauer groups of symmetries of abelian Dijkgraaf-Witten theories
Symmetries of three-dimensional topological field theories are naturally
defined in terms of invertible topological surface defects. Symmetry groups are
thus Brauer-Picard groups. We present a gauge theoretic realization of all
symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a
Dijkgraaf-Witten theory with gauge group a finite abelian group , and with
vanishing 3-cocycle, is generated by group automorphisms of , by
automorphisms of the trivial Chern-Simons 2-gerbe on the stack of -bundles,
and by partial e-m dualities.
We show that transmission functors naturally extracted from extended
topological field theories with surface defects give a physical realization of
the bijection between invertible bimodule categories of a fusion category and
braided auto-equivalences of its Drinfeld center. The latter provides the
labels for bulk Wilson lines; it follows that a symmetry is completely
characterized by its action on bulk Wilson lines.Comment: 21 pages, 9 figures. v2: Minor changes, typos corrected and
references added. v3: Typos correcte
Simplicial embeddings between multicurve graphs
We study some graphs associated to a surface, called k-multicurve graphs,
which interpolate between the curve complex and the pants graph. Our main
result is that, under certain conditions, simplicial embeddings between
multicurve graphs are induced by -injective embeddings of the
corresponding surfaces. We also prove the rigidity of the multicurve graphs.Comment: New introduction and some changes in Section 2, main results
unchanged. References added. 18 pages, 5 figure
Geometry of generated groups with metrics induced by their Cayley color graphs
Let be a group and let be a generating set of . In this article,
we introduce a metric on with respect to , called the cardinal
metric. We then compare geometric structures of and ,
where denotes the word metric. In particular, we prove that if is
finite, then and are not quasi-isometric in the case when
has infinite diameter and they are bi-Lipschitz equivalent
otherwise. We also give an alternative description of cardinal metrics by using
Cayley color graphs. It turns out that color-permuting and color-preserving
automorphisms of Cayley digraphs are isometries with respect to cardinal
metrics
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