Geometric and analytical aspects of anyons
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 happen to lie precisely in the lift of the diagonal \(D_n\). Using a (not necessarily canonical) desingularization of \({\mathbf X}^{(n)}\) one is led to an exotic exclusion principle: anyons can coalesce in either two or more pairs at a time or in groups of three or more at a time, thus not exactly two of them can occupy the same position but no other configurations is a priori excluded. For \(n \geq 4\) canonical desingularizations of the \({\mathbf X}^{(n)}\) are explicitly constructed.
scientific article; zbMATH DE number 854758
scientific article