Superselection Sectors With Braid Group Statistics and Exchange Algebras Ii: Geometric Aspects and Conformal Covariance

Section: Framework Assumptions and Resultsmentioning

confidence: 99%

The Spin-Statistics Theorem for Anyons and Plektons in d = 2+1

Mund

2008

“…It is quite obvious that the braided tensor structure on Rep A provided by the above constructions is independent, up to equivalence, of the choice of the point ∞ ∈ S 1 . For an approach to the representation theory of QFTs on S 1 that does not rely on cutting the circle see [21]. The latter, however, seems less suited for the analysis of G−Loc A for non-trivial G since the g-localized endomorphisms of A ∞ do not extend to endomorphisms of the global algebra A univ of [21] if g = e.…”

Section: Remarkmentioning

confidence: 99%

“…In Section 3 we will mostly be interested in finite groups, but we will also comment on infinite compact groups. ✷ The subsequent considerations are straightforward generalizations of the well known theory [16,20,21] for G = {e}. Since modifications of the latter are needed throughout -and also in the interest of the non-expert reader -we prefer to develop the case for non-trivial G from scratch.…”

mentioning

confidence: 99%

Conformal Orbifold Theories and Braided Crossed G-Categories

Mueger

2005

The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G − LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant.Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence Rep A G ≃ (G−LocA) G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A G . In the opposite direction we have G− LocA ≃ Rep A G ⋊ S, where S ⊂ Rep A G is the full subcategory of representations of A G contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44,48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A ≃ Vect C ) this allows to classify the possible categories G−LocA and to clarify the rôle of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds. * Supported by NWO.

“…All that was done for quantum field theories on Minkowski spacetime in dimensions d ≥ 3. Other groups of researchers have been able to follow the same route in various directions, especially in the direction of conformal quantum field theory in two dimensions, and besides the crucial results of Fredenhagen, Rehren and Schroer [24], the main success was obtained by Kawahigashi and Longo in [34], where they have been able to completely classify theories with central charge less than one.…”

Section: Introductionmentioning

confidence: 99%

Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors

Brunetti

Ruzzi

2008

127

In which is developed a new form of superselection sectors of topological origin. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C * -algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts' cohomological analysis to the case where 1-cocycles bear non trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1-cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1-cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part resembles much what in literature are known as geometric phases. Indeed, by the very geometrical origin of the 1-cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations.