Topological Sectors and a Dichotomy in Conformal Field Theory

Let A be a local conformal net of factors on the circle with the split property. We provide a topological construction of soliton representations of the tensor product of n copies of A, that restrict to true representations of subnet inviant under cyclic permutations (cyclic orbifold). We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the the cyclic orbifold has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the mu-index of A finite then A turns out to be strongly additive. By [KLM], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.Comment: 50 pages, Latex; minor correction

Solitons in Affine and Permutation Orbifolds

We consider properties of solitons in general orbifolds in the algebraic quantum field theory framework and constructions of solitons in affine and permutation orbifolds. Under general conditions we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to prove a number of conjectures: in the affine orbifold case we clarify the issue of ``fixed point resolutions''; in the permutation orbifold case we determine all irreducible representations of the orbifold, and we also determine the fusion rules in a nontrivial case, which imply an integral property of chiral data for any completely rational conformal net.Comment: Latex, 48 pages, minor style correction

Implicit body representations and the conscious body image

Recent studies have revealed that somatosensory processing relies on a class of implicit body representations showing large distortions of size and shape. The relation between these representations and the conscious body image remains unclear. Dissociations have been reported in the clinical literature on eating disorders between different body image measures, with larger and more consistent distortions found with depictive measures, in which participants compare their body to a visual depiction of a body, than metric measures, in which participants compare their body to some non-body standard. Here, we compared implicit body representations underlying position sense to the body image measured with both depictive and metric methods. The body image was measured using both a depictive method (template matching) in which participants judged whether their hand was wider or more slender than a shown hand picture, and a metric method (line length) in which participants judged whether different parts of the their hand were shorter or longer than a presented line. Consistent with previous findings, characteristic distortions were found for the implicit body representation underlying position sense. These distortions were also found in attenuated form for metric – but not depictive – body image measures. While replicating the basic dissociation between implicit body representations and the conscious body image, these results demonstrate that this dissociation is not absolute and specific tasks may utilise both to varying degrees depending on task demands. Metric measures may not be pure measures of body image, but some combination of visual and somatosensory body representations

An Analogue of the Kac-Wakimoto Formula and Black Hole Conditional Entropy

A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes-Stoermer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function the logarithm of an integer once the initial state or the final state is taken fixed.Comment: AMS-TeX v1.1c, minor grammatical correction

Facebook: The Newest Weight Loss Supplement?

This study sought to find a correlation between Facebook usage and exercise and diet habits among college-age females. A review of literature has shown the prevalence of eating disorders on college campuses and the pressures directly associated with these statistics. Eighty surveys were collected in order to measure both qualitative and quantitative data concerning the participants Facebook, eating, and dieting habits. Results revealed a strong drive among these females to stay thin through the use of diet and exercise habits, along with a correlation between viewing Facebook photos and the urge to workout. Further research on this subject should focus on the potential danger caused by extreme Facebook usage

Conformal Subnets and Intermediate Subfactors

Given an irreducible local conformal net A of von Neumann algebras on the circle and a finite-index conformal subnet B of A, we show that A is completely rational iff B is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [KLM] hold. Our proofs are based on an analysis of the net inclusion B in A; among other things we show that, for a fixed interval I, every von Neumann algebra R intermediate between B(I) and A(I) comes from an intermediate conformal net L between B and A with L(I)=R. We make use of a theorem of Watatani (type II case) and Teruya and Watatani (type III case) on the finiteness of the set I(N,M) of intermediate subfactors in an irreducible inclusion of factors N in M with finite Jones index [M:N]. We provide a unified proof of this result that gives in particular an explicit bound for the cardinality of I(N,M) which depends only on [M:N].Comment: 29 pages; AMS-Latex2

The Bisognano-Wichmann Theorem for Charged States and the Conformal Boundary of a Black Hole

Recent applications of Operator Algebras to Quantum Field Theory on a Curved Spacetime show that the incremental entropy associated with a quantum black hole, due the addition of a short range charge, is quantized proportionally to the logarithm of an integer. This talk first reviews the case of a Rindler black hole and then sketches the case of a spacetime with bifurcate Killing horizon and charges localizable on the horizon. An important tool is the construction of the conformal symmetries on the horizon.Comment: AMS-LaTeX, 7 pages. talk delivered at the conference on ``Mathematical Physics and Quantum Field Theory'', Berkeley, 11-13 June 1999. Electronic Journal of Differential Equations (to appear