3,865 research outputs found
Regular patterns, substitudes, Feynman categories and operads
We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theorem (exploiting Power's General Coherence Result) which allows to reduce to the case where the structure maps are identity-on-objects functors and strict monoidal. Second, we subsume the Getzler and Kaufmann-Ward hereditary axioms into the notion of Guitart exactness, a general condition ensuring compatibility between certain left Kan extensions and a given monad, in this case the free-symmetric-monoidal-category monad. Finally we set up a biadjunction between substitudes and what we call pinned symmetric monoidal categories, from which the results follow as a consequence of the fact that the hereditary map is precisely the counit of this biadjunction
Open-closed TQFTs extend Khovanov homology from links to tangles
We use a special kind of 2-dimensional extended Topological Quantum Field
Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov
homology from links to arbitrary tangles, not necessarily even. For every plane
diagram of an oriented tangle, we construct a chain complex whose homology is
invariant under Reidemeister moves. The terms of this chain complex are modules
of a suitable algebra A such that there is one action of A or A^op for every
boundary point of the tangle. We give examples of such algebras A for which our
tangle homology theory reduces to the link homology theories of Khovanov, Lee,
and Bar-Natan if it is evaluated for links. As a consequence of the Cardy
condition, Khovanov's graded theory can only be extended to tangles if the
underlying field has finite characteristic. In all cases in which the algebra A
is strongly separable, i.e. for Bar-Natan's theory in any characteristic and
for Lee's theory in characteristic other than 2, we also provide the required
algebraic operation for the composition of oriented tangles. Just as Khovanov's
theory for links can be recovered from Lee's or Bar-Natan's by a suitable
spectral sequence, we provide a spectral sequence in order to compute our
tangle extension of Khovanov's theory from that of Bar-Natan's or Lee's theory.
Thus, we provide a tangle homology theory that is locally computable and still
strong enough to recover characteristic p Khovanov homology for links.Comment: 56 pages, LaTeX2e with xypic and pstricks macro
Matrix geometries and fuzzy spaces as finite spectral triples
A class of real spectral triples that are similar in structure to a
Riemannian manifold but have a finite-dimensional Hilbert space is defined and
investigated, determining a general form for the Dirac operator. Examples
include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are
investigated in detail, and it is shown that the fuzzy analogues correspond to
two spinor fields on the commutative sphere. In some cases it is necessary to
add a mass mixing matrix to the commutative Dirac operator to get a precise
agreement for the eigenvalues.Comment: 39 pages, final versio
- …