12,978 research outputs found
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
Causal Dynamics of Discrete Surfaces
We formalize the intuitive idea of a labelled discrete surface which evolves
in time, subject to two natural constraints: the evolution does not propagate
information too fast; and it acts everywhere the same.Comment: In Proceedings DCM 2013, arXiv:1403.768
Symmetric groups and checker triangulated surfaces
We consider triangulations of surfaces with edges painted three colors so
that edges of each triangle have different colors. Such structures arise as
Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate
pairs of permutations determined up to a common conjugation. The topic of these
notes is links of such combinatorial structures with infinite symmetric groups
and their representations.Comment: 20p., 5 fi
Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory
Classification and construction of symmetry protected topological (SPT)
phases in interacting boson and fermion systems have become a fascinating
theoretical direction in recent years. It has been shown that the (generalized)
group cohomology theory or cobordism theory can give rise to a complete
classification of SPT phases in interacting boson/spin systems. Nevertheless,
the construction and classification of SPT phases in interacting fermion
systems are much more complicated, especially in 3D. In this work, we revisit
this problem based on the equivalent class of fermionic symmetric local unitary
(FSLU) transformations. We construct very general fixed point SPT wavefunctions
for interacting fermion systems. We naturally reproduce the partial
classifications given by special group super-cohomology theory, and we show
that with an additional (the so-called
obstruction free subgroup of ) structure, a complete
classification of SPT phases for three-dimensional interacting fermion systems
with a total symmetry group can be obtained for
unitary symmetry group . We also discuss the procedure of deriving a
general group super-cohomology theory in arbitrary dimensions.Comment: 48 pages, 35 figures, published versio
Moduli of products of stable varieties
We study the moduli space of a product of stable varieties over the field of
complex numbers, as defined via the minimal model program. Our main results
are: (a) taking products gives a well-defined morphism from the product of
moduli spaces of stable varieties to the moduli space of a product of stable
varieties, (b) this map is always finite \'etale, and (c) this map very often
is an isomorphism. Our results generalize and complete the work of Van Opstall
in dimension 1. The local results rely on a study of the cotangent complex
using some derived algebro-geometric methods, while the global ones use some
differential-geometric input.Comment: 26 pages, suggestions and comments are welcome
- …