We calculate four-point correlation functions of two weight-2 and two weight-3 1 2 -BPS operators in N = 4 SYM in the large N limit in supergravity approximation. By the AdS/CFT conjecture, these operators are dual to AdS 5 supergravity scalar fields s 2 and s 3 with mass m 2 = −4 and m 2 = −3 respectively. This is the first nontrivial four-point function of mixed-weight operators of lowest conformal dimensions.We show that the supergravity-induced four-point function splits into a "free" and a "quantum" part, where the quantum contribution obeys non-trivial constraints coming from the insertion procedure in the gauge theory, in particular, it depends on only one function of the conformal cross-ratios. *
We consider various aspects of Kitaev's toric code model on a plane in the C * -algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher-Haag-Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of D(Z 2 ), i.e. Drinfel'd's quantum double of the group algebra of Z 2 .
We prove Haag duality for conelike regions in the ground state representation corresponding to the translational invariant ground state of Kitaev's quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localised outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localised in disjoint regions commute.As an application we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher-Haag-Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
We consider charge superselection sectors of two-dimensional quantum spin models corresponding to cone localisable charges, and prove that the number of equivalence classes of such charges is bounded by the Kosaki-Longo index of an inclusion of certain observable algebras. To demonstrate the power of this result we apply the theory to the toric code on a 2D infinite lattice. For this model we can compute the index of this inclusion, and conclude that there are four distinct irreducible charges in this model, in accordance with the analysis of the toric code model on compact surfaces. We also give a sufficient criterion for the non-degeneracy of the charge sectors, in the sense that Verlinde's matrix S is invertible. *
Abstract. We prove that Haag duality holds for cones in the toric code model. That is, for a cone , the algebra R of observables localized in and the algebra R c of observables localized in the complement c generate each other's commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if 1 ⊂ 2 are two cones whose boundaries are well separated, there is a Type I factor N such that R 1 ⊂ N ⊂ R 2 . We demonstrate this by explicitly constructing N .
Mathematics Subject Classification (2010). 81R15 (46L60, 81T05, 82B20).
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