Geometric and analytical aspects of anyons
type: Item
instance of: scholarly article
title: Geometric and analytical aspects of anyons
zbMATH Open document ID: 0849.32026
DOI: 10.1007/BF00670777
author: Garth A. Baker
author: S. B. Mulay
published in: International Journal of Theoretical Physics
publication date: 1996-03-12T00:00:00Z
review text: One considers systems of \(n \geq 2\) indistinguishable quantum particles constrained to a compact Riemann surface \(M\) and satisfying fractional statistics (anyons). Classically the configuration space \(C_n (M)\) of such a system is described by \(C_n (M) = (M^n \backslash \delta_n)/S_n\), where \(\delta_n \subset M^n\) is the diagonal, and \(S_n\) denotes the symmetric group on \(n\) elements. The geometry of \(C_n (M)\) has been the subject of extensive study for several decades, e.g. the fundamental group \(\pi_1 (C_n (M)) = B_n (M)\), the \(n \)-braid group of \(M\). To an irreducible unitary representation \(\chi_k : \pi_1 (C_n (M)) \to U(k)\) one sets up a Schrödinger picture with wave functions that are multivalued on \(C_n (M)\), but that are single valued on the covering \(\widetilde C_n (M, \chi_k)\) of \(C_n (M)\) corresponding to the normal subgroup \(\text{Ker} (\chi_k)\) of \(\pi_1 (C_n (M)) = B_n (M)\). The exclusion of the diagonal implies that the particles satisfy an exclusion principle. This is not justified for bosons, thus for anyons (which are a kind of interpolation between bosons and fermions) it is not natural and undesirable to exclude \(\delta_n\). In the underlying paper no a priori exclusion principle is supposed and the configuration space to be studied will be the entire symmetric product \(S_n (M) = M^n/S_n\). For concreteness only the case \(k = 1\) is considered, and then one may assume \(M = S^2\) (the 2-sphere) without loss of generality. \(S_n (M)\) can be identified with \(n\)-dimensional complex projective space \({\mathbf C} {\mathbf P}^n\) and \(\delta_n\) becomes the discriminant hypersurface \(D_n\). The counterpart of \(\widetilde C_n (M, \chi_k)\) becomes, after some reductions, a space \({\mathbf X}^{(n)}\) lying over \({\mathbf C} {\mathbf P}^n\). However, \({\mathbf X}^{(n)}\) has singularities, at least for \(n \geq 3\), which lead to obstructions to formulate a good Schrödinger theory on it. These singularities ha…
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 81V25
zbMATH DE Number: 854758
zbMATH Keywords: exclusion principle
zbMATH Keywords: desingularization
zbMATH Keywords: anyons
zbMATH Keywords: Riemann surface
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00670777
OpenAlex ID: W2080164125
description: scientific article; zbMATH DE number 854758