1,829 research outputs found
Generating Second Order (Co)homological Information within AT-Model Context
In this paper we design a new family of relations between
(co)homology classes, working with coefficients in a field and starting
from an AT-model (Algebraic Topological Model) AT(C) of a finite cell
complex C These relations are induced by elementary relations of type
“to be in the (co)boundary of” between cells. This high-order connectivity
information is embedded into a graph-based representation model,
called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This
graph, having as nodes the different homology classes of C, is in turn,
computed from two generalized abstract cell complexes, called primal
and dual AT-segmentations of C. The respective cells of these two complexes
are connected regions (set of cells) of the original cell complex C,
which are specified by the integral operator of AT(C). In this work in
progress, we successfully use this model (a) in experiments for discriminating
topologically different 3D digital objects, having the same Euler
characteristic and (b) in designing a parallel algorithm for computing
potentially significant (co)homological information of 3D digital objects.Ministerio de EconomĂa y Competitividad MTM2016-81030-PMinisterio de EconomĂa y Competitividad TEC2012-37868-C04-0
Non-perturbative selection rules in F-theory
We discuss the structure of charged matter couplings in 4-dimensional
F-theory compactifications. Charged matter is known to arise from M2-branes
wrapping fibral curves on an elliptic or genus-one fibration Y. If a set of
fibral curves satisfies a homological relation in the fibre homology, a
coupling involving the states can arise without exponential volume suppression
due to a splitting and joining of the M2-branes. If the fibral curves only sum
to zero in the integral homology of the full fibration, no such coupling is
possible. In this case an M2-instanton wrapping a 3-chain bounded by the fibral
matter curves can induce a D-term which is volume suppressed. We elucidate the
consequences of this pattern for the appearance of massive U(1) symmetries in
F-theory and analyse the structure of discrete selection rules in the coupling
sector. The weakly coupled analogue of said M2-instantons is worked out to be
given by D1-F1 instantons. The generation of an exponentially suppressed F-term
requires the formation of half-BPS bound states of M2 and M5-instantons. This
effect and its description in terms of fluxed M5-instantons is discussed in a
companion paper.Comment: 49 pages, 9 figures; v2: references adde
On exact categories and applications to triangulated adjoints and model structures
We show that Quillen's small object argument works for exact categories under
very mild conditions. This has immediate applications to cotorsion pairs and
their relation to the existence of certain triangulated adjoint functors and
model structures. In particular, the interplay of different exact structures on
the category of complexes of quasi-coherent sheaves leads to a streamlined and
generalized version of recent results obtained by Estrada, Gillespie, Guil
Asensio, Hovey, J{\o}rgensen, Neeman, Murfet, Prest, Trlifaj and possibly
others.Comment: 38 pages; version 2: major revision, more explanation added at
several places, reference list updated and extended, misprints correcte
Deformation theory of representations of prop(erad)s
We study the deformation theory of morphisms of properads and props thereby
extending to a non-linear framework Quillen's deformation theory for
commutative rings. The associated chain complex is endowed with a Lie algebra
up to homotopy structure. Its Maurer-Cartan elements correspond to deformed
structures, which allows us to give a geometric interpretation of these
results.
To do so, we endow the category of prop(erad)s with a model category
structure. We provide a complete study of models for prop(erad)s. A new
effective method to make minimal models explicit, that extends Koszul duality
theory, is introduced and the associated notion is called homotopy Koszul.
As a corollary, we obtain the (co)homology theories of (al)gebras over a
prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex
is endowed with a canonical Lie algebra up to homotopy structure in general and
a Lie algebra structure only in the Koszul case. In particular, we explicit the
deformation complex of morphisms from the properad of associative bialgebras.
For any minimal model of this properad, the boundary map of this chain complex
is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this
paper provides a complete proof of the existence of a Lie algebra up to
homotopy structure on the Gerstenhaber-Schack bicomplex associated to the
deformations of associative bialgebras.Comment: Version 4 : Statement about the properad of (non-commutative)
Frobenius bialgebras fixed in Section 4. [82 pages
Hidden Selection Rules, M5-instantons and Fluxes in F-theory
We introduce a new approach to investigate the selection rules governing the
contributions of fluxed M5-instantons to the F-theory four-dimensional
effective action, with emphasis on the generation of charged matter F-terms.
The structure of such couplings is unraveled by exploiting the perturbative and
non-perturbative homological relations, introduced in our companion paper
arXiv:1506.06764, which encode the interplay between the self-dual 3-form flux
on the M5-brane, the background 4-form flux and certain fibral curves. The
latter are wrapped by time-like M2-branes representing matter insertions in the
instanton path integral. In particular, we clarify how fluxed M5-instantons
detect the presence of geometrically massive s which are responsible for
`hidden' selection rules. We discuss how for non-generic embeddings the
M5-instanton can probe `locally massless' symmetries if the rank of its
Mordell-Weil group is enhanced compared to that of the bulk. As a
phenomenological off-spring we propose a new type of non-perturbative
corrections to Yukawa couplings which may change the rank of the Yukawa matrix.
Along the way, we also gain new insights into the structure of massive
gauge fluxes in the stable degeneration limit.Comment: 42 pages; v2: references adde
A beginner's introduction to Fukaya categories
The goal of these notes is to give a short introduction to Fukaya categories
and some of their applications. The first half of the text is devoted to a
brief review of Lagrangian Floer (co)homology and product structures. Then we
introduce the Fukaya category (informally and without a lot of the necessary
technical detail), and briefly discuss algebraic concepts such as exact
triangles and generators. Finally, we mention wrapped Fukaya categories and
outline a few applications to symplectic topology, mirror symmetry and
low-dimensional topology. This text is based on a series of lectures given at a
Summer School on Contact and Symplectic Topology at Universit\'e de Nantes in
June 2011.Comment: 42 pages, 13 figure
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