1,025,739 research outputs found
Complex Geometry and Supersymmetry
I stress how the form of sigma models with (2, 2) supersymmetry differs
depending on the number of manifest supersymmetries. The differences correspond
to different aspects/formulations of Generalized K\"ahler Geometry.Comment: 9 pages, Proceedings of the Corfu Summer Institute 2011 School and
Workshops on Elementary Particle Physics and Gravity September 4-18, 2011
Corfu, Greec
Complex geometry of quantum cones
The algebras obtained as fixed points of the action of the cyclic group
on the coordinate algebra of the quantum disc are studied. These can be
understood as coordinate algebras of quantum or non-commutative cones. The
following observations are made. First, contrary to the classical situation,
the actions of are free and the resulting algebras are homologically
smooth. Second, the quantum cone algebras admit differential calculi that have
all the characteristics of calculi on smooth complex curves. Third, the
corresponding volume forms are exact, indicating that the constructed algebras
describe manifolds with boundaries.Comment: 6 pages; submitted to the proceedings of Corfu Summer Institute 201
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
- …