L2 dimensions of spaces of braid-invariant harmonic forms

Abstract

Let X be a Riemannian manifold endowed with a co-compact isometric action of an infinite discrete group. We consider L 2 spaces of harmonic vector-valued forms on the product manifold X N , which are invariant with respect to an action of the braid group B N , and compute their von Neumann dimensions (the braided L 2 -Betti numbers).

Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations

“…For further discussions of anyons in mathematical physics literature (including the discrete setting), see e.g. [13,15,17,19,33,[36][37][38][39][40][41] and the references therein. We also refer to the paper [8] which deals with a Fock representation of the commutations relations identified by a sequence of self-adjoint operators in a Hilbert space which have norm ≤ 1 and which satisfy the braid relations.…”