203 research outputs found

    Tube algebras, excitations statistics and compactification in gauge models of topological phases

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    We consider lattice Hamiltonian realizations of (dd+1)-dimensional Dijkgraaf-Witten theory. In (2+1)d, it is well-known that the Hamiltonian yields point-like excitations classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalization of this strategy that is valid in any dimensions. We then apply the tube algebra approach to derive the algebraic structure of loop-like excitations in (3+1)d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1)d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an RR-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a manifold that is nn-times compactified can be expressed in terms of another model in nn-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.Comment: 71 page

    Nonperturbative Quantum Gravity

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    Asymptotic safety describes a scenario in which general relativity can be quantized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultraviolet fixed point, for which evidence has been found using renormalization group equations in the continuum. "Causal Dynamical Triangulations" (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point. We describe the formalism of CDT, its phase diagram, possible fixed points and the "quantum geometries" which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Ho\v{r}ava-Lifshitz gravitational models.Comment: Review, 146 pages, many figure

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Dilation, Transport, Visibility and Fault-Tolerant Algorithms

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    Connecting some points in the plane by a road network is equivalent to constructing a finite planar graph G whose vertex set contains a predefined set of vertices (i. e., the possible destinations in the road network). The dilation between two vertices p and q of graph G is defined as the Euclidean length of a shortest path in G from p to q, divided by the Euclidean distance from p to q. That is, given a point set P, the goal is to place some additional crossing vertices C such that there exists a planar graph G = (P ∪ C, E) whose dilation is small. Here, the dilation of G is defined as the maximum dilation between two vertices in G. We show that, except for some special point sets P, there is a lower bound Δ(P) > 1, depending on P, on the dilation of any finite graph containing P in its vertex set. The transportation problem is the problem of finding a transportation plan that minimizes the total transport cost. We are given a set of suppliers, and each supplier produces a fixed amount of some commodity, say, bread. Furthermore, there is a set of customers, and each customer has some demand of bread, such that the total demand equals the amount of bread the suppliers produce. The task is to assign each unit of bread produced to some customer, such that the total transportation cost becomes a minimum. A first idea is to assign each unit of bread to the client to which the transport cost of this unit is minimal. Clearly, this gives rise to a transportation plan which minimizes the total transportation cost. However, it is likely that not every customer will obtain the required amount of bread. Therefore, we need to use a different algorithm for distributing the supplier's bread. We show that if the bread produced by the suppliers is given by a continuous probability density function and the set of customers is discrete, then every optimal transport plan can be characterized by a unique additively weighted Voronoi diagram for the customers. When managing the construction process of a building by a digital model of the building, it is necessary to compute essential parts between walls of the building. Given two walls A and B, the essential part between A and B is the set of line segments s where one endpoint belongs to A, the other endpoint belongs to B, and s does not intersect A or B. We give an algorithm that computes, in linear time, the essential parts between A and B. Our algorithm is based on computing the visibility polygon of A and of B, and two shortest paths connecting points of A with points of B. We conclude the thesis by giving fault-tolerant algorithms for some fundamental geometric problems. We assume that a basic primitive operation used by an algorithm fails with some small probability p. Depending on the results of the primitive operations, it is possible that the algorithm will not work correctly. For example, one faulty comparison when executing a sorting algorithm can result in some numbers being placed far away from their true positions. An algorithm is called tolerant, if with high probability a good answer is given, if the error probability p is small. We provide tolerant algorithms that find the maximum of n numbers, search for a key in a sorted sequence of n keys, sort a set of n numbers, and solve Linear Programming in R2

    Collection of abstracts of the 24th European Workshop on Computational Geometry

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    International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop
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