Quantum Configuration Spaces of Extended Objects, Diffeomorphism Group Representations and Exotic Statistics

The pseudogroup of local solutions in Chapter 3 defines another pseudogroup by taking its centralizer inside the diffeomorphism group Diff(M) of a manifold M. These two pseudogroups define a Lie group structure on M.

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Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Entropy ◽

10.3390/e21030250 ◽

2019 ◽

Vol 21 (3) ◽

pp. 250

Author(s):

Frédéric Barbaresco ◽

Jean-Pierre Gazeau

Keyword(s):

Fourier Analysis ◽

Heat Equation ◽

Harmonic Analysis ◽

Lie Groups ◽

Coherent States ◽

Geometric Theory ◽

Plancherel Formula ◽

Group Representations ◽

Modern Development ◽

Xxist Century

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

Journal of High Energy Physics ◽

10.1007/jhep10(2020)054 ◽

2020 ◽

Vol 2020 (10) ◽

Cited By ~ 1

Author(s):

Song He ◽

Zhenjie Li ◽

Prashanth Raman ◽

Chi Zhang

Keyword(s):

Large Class ◽

Configuration Space ◽

Moduli Space ◽

Cluster Algebras ◽

Configuration Spaces ◽

Minkowski Sum ◽

Canonical Forms ◽

Infinite Class ◽

Special Cases ◽

Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

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