199 research outputs found
Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups
We extend each higher Johnson homomorphism to a crossed homomorphism from the
automorphism group of a finite-rank free group to a finite-rank abelian group.
We also extend each Morita homomorphism to a crossed homomorphism from the
mapping class group of once-bounded surface to a finite-rank abelian group.
This improves on the author's previous results [Algebr. Geom. Topol. 7
(2007):1297-1326]. To prove the first result, we express the higher Johnson
homomorphisms as coboundary maps in group cohomology. Our methods involve the
topology of nilpotent homogeneous spaces and lattices in nilpotent Lie
algebras. In particular, we develop a notion of the "polynomial straightening"
of a singular homology chain in a nilpotent homogeneous space.Comment: 34 page
Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction
Besides its usual interpretation as a system of indistinguishable
particles moving on the circle, the trigonometric Sutherland system can be
viewed alternatively as a system of distinguishable particles on the circle or
on the line, and these 3 physically distinct systems are in duality with
corresponding variants of the rational Ruijsenaars-Schneider system. We explain
that the 3 duality relations, first obtained by Ruijsenaars in 1995, arise
naturally from the Kazhdan-Kostant-Sternberg symplectic reductions of the
cotangent bundles of the group U(n) and its covering groups
and , respectively. This geometric interpretation
enhances our understanding of the duality relations and simplifies Ruijsenaars'
original direct arguments that led to their discovery.Comment: 34 pages, minor additions and corrections of typos in v
Generic canonical form of pairs of matrices with zeros
We consider a family of pairs of m-by-p and m-by-q matrices, in which some
entries are required to be zero and the others are arbitrary, with respect to
transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that
almost all of these pairs reduce to the same pair (C, D) from this family,
except for pairs whose arbitrary entries are zeros of a certain polynomial. The
polynomial and the pair (C D) are constructed by a combinatorial method based
on properties of a certain graph.Comment: 13 page
Particle Topology, Braids, and Braided Belts
Recent work suggests that topological features of certain quantum gravity
theories can be interpreted as particles, matching the known fermions and
bosons of the first generation in the Standard Model. This is achieved by
identifying topological structures with elements of the framed Artin braid
group on three strands, and demonstrating a correspondence between the
invariants used to characterise these braids (a braid is a set of
non-intersecting curves, that connect one set of points with another set of
points), and quantities like electric charge, colour charge, and so on. In
this paper we show how to manipulate a modified form of framed braids to yield
an invariant standard form for sets of isomorphic braids, characterised by a
vector of real numbers. This will serve as a basis for more complete
discussions of quantum numbers in future work.Comment: 21 pages, 16 figure
Classification and analysis of two dimensional abelian fractional topological insulators
We present a general framework for analyzing fractionalized, time reversal
invariant electronic insulators in two dimensions. The framework applies to all
insulators whose quasiparticles have abelian braiding statistics. First, we
construct the most general Chern-Simons theories that can describe these
states. We then derive a criterion for when these systems have protected
gapless edge modes -- that is, edge modes that cannot be gapped out without
breaking time reversal or charge conservation symmetry. The systems with
protected edge modes can be regarded as fractionalized analogues of topological
insulators. We show that previous examples of 2D fractional topological
insulators are special cases of this general construction. As part of our
derivation, we define the concept of "local Kramers degeneracy" and prove a
local version of Kramers theorem.Comment: 19 pages, 2 figures, added reference, corrected typo
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