On the Language of Standard Discrete Planes and Surfaces

International audienceA standard discrete plane is a subset of Z^3 verifying the double Diophantine inequality mu =< ax+by+cz < mu + omega, with (a,b,c) != (0,0,0). In the present paper we introduce a generalization of this notion, namely the (1,1,1)-discrete surfaces. We first study a combinatorial representation of discrete surfaces as two-dimensional sequences over a three-letter alphabet and show how to use this combinatorial point of view for the recognition problem for these discrete surfaces. We then apply this combinatorial representation to the standard discrete planes and give a first attempt of to generalize the study of the dual space of parameters for the latter [VC00]

Geometric discretization of the Bianchi system

We introduce the dual Koenigs lattices, which are the integrable discrete analogues of conjugate nets with equal tangential invariants, and we find the corresponding reduction of the fundamental transformation. We also introduce the notion of discrete normal congruences. Finally, considering quadrilateral lattices "with equal tangential invariants" which allow for harmonic normal congruences we obtain, in complete analogy with the continuous case, the integrable discrete analogue of the Bianchi system together with its geometric meaning. To obtain this geometric meaning we also make use of the novel characterization of the circular lattice as a quadrilateral lattice whose coordinate lines intersect orthogonally in the mean.Comment: 26 pages, 7 postscript figure

DGD Gallery: Storage, sharing, and publication of digital research data

We describe a project, called the "Discretization in Geometry and Dynamics Gallery", or DGD Gallery for short, whose goal is to store geometric data and to make it publicly available. The DGD Gallery offers an online web service for the storage, sharing, and publication of digital research data.Comment: 19 pages, 8 figures, to appear in "Advances in Discrete Differential Geometry", ed. A. I. Bobenko, Springer, 201

Critical and Multicritical Semi-Random (1+d)-Dimensional Lattices and Hard Objects in d Dimensions

We investigate models of (1+d)-D Lorentzian semi-random lattices with one random (space-like) direction and d regular (time-like) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)-D Lorentzian surfaces, with fractal dimensions $d_F=k+1$, k=1,2,3,..., as well as a new model of (1+2)-D critical tetrahedral complexes, with fractal dimension $d_F=12/5$. Critical exponents and universal scaling functions follow from this solution. We finally establish a general connection between (1+d)-D Lorentzian lattices and directed-site lattice animals in (1+d) dimensions.Comment: 44 pages, 15 figures, tex, harvmac, epsf, references adde

Fixing All Moduli in a Simple F-Theory Compactification

We discuss a simple example of an F-theory compactification on a Calabi-Yau fourfold where background fluxes, together with nonperturbative effects from Euclidean D3 instantons and gauge dynamics on D7 branes, allow us to fix all closed and open string moduli. We explicitly check that the known higher order corrections to the potential, which we neglect in our leading approximation, only shift the results by a small amount. In our exploration of the model, we encounter interesting new phenomena, including examples of transitions where D7 branes absorb O3 planes, while changing topology to preserve the net D3 charge.Comment: 68 pages, 19 figures; v2: references adde