7,467 research outputs found
Combed 3-Manifolds with Concave Boundary, Framed Links, and Pseudo-Legendrian Links
We provide combinatorial realizations, according to the usual objects/moves
scheme, of the following three topological categories: (1) pairs (M,v) where M
is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field,
up to homotopy; here possibly the boundary of M is non-empty and v may be
tangent to the boundary, but only in a concave fashion, and homotopy should
preserve tangency type; (2) framed links L in M, up to framed isotopy; (3)
triples (M,v,L), with (M,v) as above and L transversal to v, up to
pseudo-Legendrian isotopy (transversality-preserving simultaneous homotopy of v
and isotopy of L). All realizations are based on the notion of branched
standard spine, and build on results previously obtained. Links are encoded by
means of diagrams on branched spines, where the diagram is smooth with respect
to the branching. Several motivations for being interested in combinatorial
realizations of the topological categories considered in this paper are given
in the introduction. The encoding of links is suitable for the comparison of
the framed and the pseudo-Legendrian categories, and some applications are
given in connection with contact structures, torsion and finite-order
invariants. An estension of Trace's notion of winding number of a knot diagram
is introduced and discussed.Comment: 38 pages, 33 figure
A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries
We propose that, up to invertible topological orders, 2+1D fermionic
topological orders without symmetry and 2+1D fermionic/bosonic topological
orders with symmetry are classified by non-degenerate unitary braided
fusion categories (UBFC) over a symmetric fusion category (SFC); the SFC
describes a fermionic product state without symmetry or a fermionic/bosonic
product state with symmetry , and the UBFC has a modular extension. We
developed a simplified theory of non-degenerate UBFC over a SFC based on the
fusion coefficients and spins . This allows us to obtain a list
that contains all 2+1D fermionic topological orders (without symmetry). We find
explicit realizations for all the fermionic topological orders in the table.
For example, we find that, up to invertible
fermionic topological orders, there
are only four fermionic topological orders with one non-trivial topological
excitation: (1) the
fractional quantum Hall state, (2) a Fibonacci bosonic topological order
stacking with a fermionic product state, (3) the time-reversal
conjugate of the previous one, (4) a primitive fermionic topological order that
has a chiral central charge , whose only topological excitation has
a non-abelian statistics with a spin and a quantum dimension
. We also proposed a categorical way to classify 2+1D invertible
fermionic topological orders using modular extensions.Comment: 23 pages, 8 table
A trace for bimodule categories
We study a 2-functor that assigns to a bimodule category over a finite
k-linear tensor category a k-linear abelian category. This 2-functor can be
regarded as a category-valued trace for 1-morphisms in the tricategory of
finite tensor categories. It is defined by a universal property that is a
categorification of Hochschild homology of bimodules over an algebra. We
present several equivalent realizations of this 2-functor and show that it has
a coherent cyclic invariance.
Our results have applications to categories associated to circles in
three-dimensional topological field theories with defects. This is made
explicit for the subclass of Dijkgraaf-Witten topological field theories.Comment: 49 pages; typos correcte
Etale realization on the A^1-homotopy theory of schemes
We compare Friedlander's definition of the etale topological type for
simplicial schemes to another definition involving realizations of
pro-simplicial sets. This can be expressed as a notion of hypercover descent
for etale homotopy. We use this result to construct a homotopy invariant
functor from the category of simplicial presheaves on the etale site of schemes
over S to the category of pro-spaces. After completing away from the
characteristics of the residue fields of S, we get a functor from the
Morel-Voevodsky A^1-homotopy category of schemes to the homotopy category of
pro-spaces
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