6,152 research outputs found
Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
String-net models allow us to systematically construct and classify 2+1D
topologically ordered states which can have gapped boundaries. We can use a
simple ideal string-net wavefunction, which is described by a set of F-matrices
[or more precisely, a unitary fusion category (UFC)], to study all the
universal properties of such a topological order. In this paper, we describe a
finite computational method -- Q-algebra approach, that allows us to compute
the non-Abelian statistics of the topological excitations [or more precisely,
the unitary modular tensor category (UMTC)], from the string-net wavefunction
(or the UFC). We discuss several examples, including the topological phases
described by twisted gauge theory (i.e., twisted quantum double ).
Our result can also be viewed from an angle of holographic bulk-boundary
relation. The 1+1D anomalous topological orders, that can appear as edges of
2+1D topological states, are classified by UFCs which describe the fusion of
quasiparticles in 1+1D. The 1+1D anomalous edge topological order uniquely
determines the 2+1D bulk topological order (which are classified by UMTC). Our
method allows us to compute this bulk topological order (i.e., the UMTC) from
the anomalous edge topological order (i.e., the UFC).Comment: 32 pages, 8 figures, reference updated, some refinement
Comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering
Comparing large covariance matrices has important applications in modern
genomics, where scientists are often interested in understanding whether
relationships (e.g., dependencies or co-regulations) among a large number of
genes vary between different biological states. We propose a computationally
fast procedure for testing the equality of two large covariance matrices when
the dimensions of the covariance matrices are much larger than the sample
sizes. A distinguishing feature of the new procedure is that it imposes no
structural assumptions on the unknown covariance matrices. Hence the test is
robust with respect to various complex dependence structures that frequently
arise in genomics. We prove that the proposed procedure is asymptotically valid
under weak moment conditions. As an interesting application, we derive a new
gene clustering algorithm which shares the same nice property of avoiding
restrictive structural assumptions for high-dimensional genomics data. Using an
asthma gene expression dataset, we illustrate how the new test helps compare
the covariance matrices of the genes across different gene sets/pathways
between the disease group and the control group, and how the gene clustering
algorithm provides new insights on the way gene clustering patterns differ
between the two groups. The proposed methods have been implemented in an
R-package HDtest and is available on CRAN.Comment: The original title dated back to May 2015 is "Bootstrap Tests on High
Dimensional Covariance Matrices with Applications to Understanding Gene
Clustering
On the singular hyperbolicity of star flows
We prove for a generic star vector field that, if for every chain
recurrent class of all singularities in have the same index, then
the chain recurrent set of is singular hyperbolic. We also prove that every
Lyapunov stable chain recurrent class of is singular hyperbolic. As a
corollary, we prove that the chain recurrent set of a generic 4-dimensional
star flow is singular hyperbolic.Comment: 29 pages, version to appear in J. Mod. Dy
A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
Topological orders are new phases of matter beyond Landau symmetry breaking.
They correspond to patterns of long-range entanglement. In recent years, it was
shown that in 1+1D bosonic systems there is no nontrivial topological order,
while in 2+1D bosonic systems the topological orders are classified by a pair:
a modular tensor category and a chiral central charge. In this paper, we
propose a partial classification of topological orders for 3+1D bosonic
systems: If all the point-like excitations are bosons, then such topological
orders are classified by unitary pointed fusion 2-categories, which are
one-to-one labeled by a finite group and its group 4-cocycle up to group automorphisms. Furthermore, all such 3+1D
topological orders can be realized by Dijkgraaf-Witten gauge theories.Comment: An important new result "Untwisted sector of dimension reduction is
the Drinfeld center of E" is added in Sec. IIIC; other minor refinements and
improvements; 23 pages, 10 figure
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