421 research outputs found

    Group field theories for all loop quantum gravity

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    Group field theories represent a 2nd quantized reformulation of the loop quantum gravity state space and a completion of the spin foam formalism. States of the canonical theory, in the traditional continuum setting, have support on graphs of arbitrary valence. On the other hand, group field theories have usually been defined in a simplicial context, thus dealing with a restricted set of graphs. In this paper, we generalize the combinatorics of group field theories to cover all the loop quantum gravity state space. As an explicit example, we describe the GFT formulation of the KKL spin foam model, as well as a particular modified version. We show that the use of tensor model tools allows for the most effective construction. In order to clarify the mathematical basis of our construction and of the formalisms with which we deal, we also give an exhaustive description of the combinatorial structures entering spin foam models and group field theories, both at the level of the boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic

    Bar constructions for topological operads and the Goodwillie derivatives of the identity

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    We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper20.abs.html Version 3: Reference to Salvatore added (see footnote 3, page 834

    Deformations of bordered Riemann surfaces and associahedral polytopes

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    We consider the moduli space of bordered Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of associahedra and multiplihedra.Comment: 25 pages, 31 figure

    From Composite Indicators to Partial Orders: Evaluating Socio-Economic Phenomena Through Ordinal Data

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    In this paper we present a new methodology for the statistical evaluation of ordinal socio-economic phenomena, with the aim of overcoming the issues of the classical aggregative approach based on composite indicators. The proposed methodology employs a benchmark approach to evaluation and relies on partially ordered set (poset) theory, a branch of discrete mathematics providing tools for dealing with multidimensional systems of ordinal data. Using poset theory and the related Hasse diagram technique, evaluation scores can be computed without performing any variable aggregation into composite indicators. This way, ordinal scores need not be turned into numerical values, as often done in evaluation studies, inconsistently with the real nature of the phenomena at hand. We also face the problem of \u201cweighting\u201d evaluation dimensions, to account for their different relevance, and show how this can be handled in pure ordinal terms. A specific focus is devoted to the binary variable case, where the methodology can be specialized in a very effective way. Although the paper is mainly methodological, all of the basic concepts are illustrated through real examples pertaining to material deprivation
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