17,591 research outputs found
Discrete Riemann Surfaces and the Ising model
We define a new theory of discrete Riemann surfaces and present its basic
results. The key idea is to consider not only a cellular decomposition of a
surface, but the union with its dual. Discrete holomorphy is defined by a
straightforward discretisation of the Cauchy-Riemann equation. A lot of
classical results in Riemann theory have a discrete counterpart, Hodge star,
harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of
holomorphic forms with prescribed holonomies. Giving a geometrical meaning to
the construction on a Riemann surface, we define a notion of criticality on
which we prove a continuous limit theorem. We investigate its connection with
criticality in the Ising model. We set up a Dirac equation on a discrete
universal spin structure and we prove that the existence of a Dirac spinor is
equivalent to criticality
Moduli in N=1 heterotic/F-theory duality
The moduli in a 4D N=1 heterotic compactification on an elliptic CY, as well
as in the dual F-theoretic compactification, break into "base" parameters which
are even (under the natural involution of the elliptic curves), and "fiber" or
twisting parameters; the latter include a continuous part which is odd, as well
as a discrete part. We interpret all the heterotic moduli in terms of
cohomology groups of the spectral covers, and identify them with the
corresponding F-theoretic moduli in a certain stable degeneration. The argument
is based on the comparison of three geometric objects: the spectral and cameral
covers and the ADE del Pezzo fibrations. For the continuous part of the
twisting moduli, this amounts to an isomorphism between certain abelian
varieties: the connected component of the heterotic Prym variety (a modified
Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the
discrete part generalizes the matching of heterotic 5brane / F-theoretic 3brane
impurities.Comment: Latex, 26 pages. Acknowledgements adde
Non-birational twisted derived equivalences in abelian GLSMs
In this paper we discuss some examples of abelian gauged linear sigma models
realizing twisted derived equivalences between non-birational spaces, and
realizing geometries in novel fashions. Examples of gauged linear sigma models
with non-birational Kahler phases are a relatively new phenomenon. Most of our
examples involve gauged linear sigma models for complete intersections of
quadric hypersurfaces, though we also discuss some more general cases and their
interpretation. We also propose a more general understanding of the
relationship between Kahler phases of gauged linear sigma models, namely that
they are related by (and realize) Kuznetsov's `homological projective duality.'
Along the way, we shall see how `noncommutative spaces' (in Kontsevich's sense)
are realized physically in gauged linear sigma models, providing examples of
new types of conformal field theories. Throughout, the physical realization of
stacks plays a key role in interpreting physical structures appearing in GLSMs,
and we find that stacks are implicitly much more common in GLSMs than
previously realized.Comment: 54 pages, LaTeX; v2: typo fixe
Discrete curvature approximations and segmentation of polyhedral surfaces
The segmentation of digitized data to divide a free form surface into patches is one of the key steps required to perform a reverse engineering process of an object. To this end, discrete curvature approximations are introduced as the basis of a segmentation process that lead to a decomposition of digitized data into areas that will help the construction of parametric surface patches. The approach proposed relies on the use of a polyhedral representation of the object built from the digitized data input. Then, it is shown how noise reduction, edge swapping techniques and adapted remeshing schemes can participate to different preparation phases to provide a geometry that highlights useful characteristics for the segmentation process. The segmentation process is performed with various approximations of discrete curvatures evaluated on the polyhedron produced during the preparation phases. The segmentation process proposed involves two phases: the identification of characteristic polygonal lines and the identification of polyhedral areas useful for a patch construction process. Discrete curvature criteria are adapted to each phase and the concept of invariant evaluation of curvatures is introduced to generate criteria that are constant over equivalent meshes. A description of the segmentation procedure is provided together with examples of results for free form object surfaces
Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation
We revisit two NP-hard geometric partitioning problems - convex decomposition
and surface approximation. Building on recent developments in geometric
separators, we present quasi-polynomial time algorithms for these problems with
improved approximation guarantees.Comment: 21 pages, 6 figure
- …