Loop groups and diffeomorphism groups of the circle as colimits

We show that loop groups and the universal cover of $\mathrm{Diff}_+(S^1)$ can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of $S^1$. Analogous results hold for based loop groups and for the based diffeomorphism group of $S^1$. These results continue to hold for the corresponding centrally extended groups. We use the above results to construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie algebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group.Comment: updated to published versio

What Chern-Simons theory assigns to a point

In this note, we answer the questions "What does Chern-Simons theory assign to a point?" and "What kind of mathematical object does Chern-Simons theory assign to a point?". Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group $\Omega G$ that we locally normal representations. We define the fusion product of such representations and we prove that, modulo certain conjectures, the Drinfel'd centre of that representation category of $\Omega G$ is equivalent to the category of positive energy representations of the free loop group $LG$. The above mentioned conjectures are known to hold when the gauge group is abelian or of type $A_1$. Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: they are tensor categories that are equivalent to their bicommutant inside $\mathrm{Bim}(R)$, the category of bimodules over a hyperfinite $\mathit{III}_1$ factor. We prove that, modulo certain conjectures, the category of locally normal representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type $A_n$. Our work builds on the formalism of coordinate free conformal nets, developed jointly with A. Bartels and C. Douglas.Comment: Minor changes. Added a summary diagram at the end. Replaced the terminology "positive energy representations of the based loop group" by "locally normal representations of the based loop group

A Periodicity Theorem for the Octahedron Recurrence

We investigate a variant of the octahedron recurrence which lives in a 3-dimensional lattice contained in [0,n] x [0,m] x R. Generalizing results of David Speyer math.CO/0402452, we give an explicit non-recursive formula for the values of this recurrence in terms of perfect matchings. We then use it to prove that the octahedron recurrence is periodic of period n+m. This result is reminiscent of Fomin and Zelevinsky's theorem about the periodicity of Y-systems.Comment: 22 pages, (a few pictures added, section 3 has been reorganized

Students' Perceptions of their Rights in Portugal

The present study analyses young Portuguese students’perceptions of their rights at school and at home. The sample consisted of 294 male and female students, in Year 7 to Year 9, from several regions in the country. The assessment instrument used was the ‘Children’s Rights Scale’ (Hart, 1993; Hart et al., 1996) and results were related to a number of other questions regarding the following independent variables: (1) school variables (school year, school achievement, teacher’s support, violence at school); (2) family variables (parental authority, family cohesion, parents’ education); (3) personal variables (age, gender, professional interests). Significant relations were found between the students’ perceptions of their rights and these independent variables. In general, Portuguese students declare that their rights exist and are very important. Students who have a greater degree of personal limitations, or who live in worse school or family contexts, are the ones who indicate less importance and perceive less existence of rights at school and at home. The study gives directions to efforts to strengthen projects for the promotion of students’ rights by including training programs for teachers and parents, in a close net connection among school, family and significant social agents.This study was financed by JNICT/FCT and by the PRAXIS XXI Program for the period from 1997 to 2002 and developed at the Center for Research in Education of the Faculty of Sciences of the University of Lisbon (CIEFCUL)R. Ernesto Vasconcelos, Edif. C1–2º, 1749–016 Lisbon, Portugal

Loop quantum cosmology and the Wheeler-De Witt equation

We present some results concerning the large volume limit of loop quantum cosmology in the flat homogeneous and isotropic case. We derive the Wheeler-De Witt equation in this limit. Looking for the action from which this equation can also be obtained, we then address the problem of the modifications to be brought to the Friedman's equation of motion and to the equation of motion of the scalar field, in the classical limit.Comment: 17 pages; 5 figure