Conformal Field Theories, Graphs and Quantum Algebras

This article reviews some recent progress in our understanding of the structure of Rational Conformal Field Theories, based on ideas that originate for a large part in the work of A. Ocneanu. The consistency conditions that generalize modular invariance for a given RCFT in the presence of various types of boundary conditions --open, twisted-- are encoded in a system of integer multiplicities that form matrix representations of fusion-like algebras. These multiplicities are also the combinatorial data that enable one to construct an abstract ``quantum'' algebra, whose $6j$- and $3j$-symbols contain essential information on the Operator Product Algebra of the RCFT and are part of a cell system, subject to pentagonal identities. It looks quite plausible that the classification of a wide class of RCFT amounts to a classification of ``Weak $C^*$- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001 --Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in Math., Birkhauser. References and comments adde

On the crossing relation in the presence of defects

The OPE of local operators in the presence of defect lines is considered both in the rational CFT and the $c>25$ Virasoro (Liouville) theory. The duality transformation of the 4-point function with inserted defect operators is explicitly computed. The two channels of the correlator reproduce the expectation values of the Wilson and 't Hooft operators, recently discussed in Liouville theory in relation to the AGT conjecture.Comment: TEX file with harvmac; v3: JHEP versio

Acts of terrorism and mass violence targeting schools : Analysis and implications for preparedness in the USA

To enhance the preparedness of US schools to acts of terrorism and mass violence, the landscape of threats against schools must first be understood. This includes exploring the global trends of acts of terrorism against schools, as well as looking specifically at the history of terrorism and acts of mass violence against schools domestically. This paper conducts a review of two databases in order to look at the trends in acts of terrorism and mass violence carried out against schools, and provides recommendations for domestic school preparedness based on this information

Schools and Terrorism: Global Trends, Impacts, and Lessons for Resilience

This study characterizes trends in the frequency and characteristics of terrorist attacks in child-serving educational institutions around the world, examining the specific vulnerabilies of children and schools with regard to terrorist violence, as well as the various impacts that violence has on children, communities, and societies. Following the analysis of available data on terrorist attacks against educational institutions, vulnerabilities, and impacts, the study concludes with a discussion of what still needs to be understood in the intersection of child vulnerability and terrorism, and provides recommendations for improving resilience to terrorist attacks against child-serving educational institutions. One would like to think that certain truths or values would be universally understood as rules of engagement, formally declared or otherwise. The sanctity of children's well-being should be unquestioned, regardless of the issues at stake in the larger conflict. Sadly, history shows that this understanding is neither universally shared nor uniformly valued. —Irwin Redlener, Americans at Risk, 2007 Since the violent attack on School Number One in Beslan, Russia in 2004, the perceived threat of massive terror attacks targeting schoolchildren has loomed in the public consciousness. In recent years, attacks against educational institutions worldwide have increasingly been reported and documented. The kidnapping of 276 schoolgirls by Boko Haram in Nigeria and the massacre of at least 150 students and staff in a Peshawar school by the Pakistani Taliban in 2014 are still fresh in the minds of the public. These attacks serve as reminders of the vulnerability that children in schools face, being “soft targets” whose symbolic value has the capacity of invoking mayhem at the largest possible scale. They also demonstrate the urgency with which this emerging trend in violence must be systemically recorded, analyzed, and mitigated. The following article will discuss the vulnerability of children with regard to terrorist violence, exploring the literature on what makes children and educational institutions particularly desirable targets. Trends in the frequency and characteristics of terrorist attacks against child-serving educational institutions around the world will be examined, paying particular attention to potential school level and gendered disparities. Finally, a critical analysis on what can be learned from the available data and what still needs to be researched in the nexus of child vulnerability and terrorism will be provided

Climate Change and Health on the U.S. Gulf Coast: Public Health Adaptation is Needed to Address Future Risks

The impacts of climate change on human health have been documented globally and in the United States. Numerous studies project greater morbidity and mortality as a result of extreme weather events and other climate-sensitive hazards. Public health impacts on the U.S. Gulf Coast may be severe as the region is expected to experience increases in extreme temperatures, sea level rise, and possibly fewer but more intense hurricanes. Through myriad pathways, climate change is likely to make the Gulf Coast less hospitable and more dangerous for its residents, and may prompt substantial migration from and into the region. Public health impacts may be further exacerbated by the concentration of people and infrastructure, as well as the region’s coastal geography. Vulnerable populations, including the very young, elderly, and socioeconomically disadvantaged may face particularly high threats to their health and well-being. This paper provides an overview of potential public health impacts of climate variability and change on the Gulf Coast, with a focus on the region’s unique vulnerabilities, and outlines recommendations for improving the region’s ability to minimize the impacts of climate-sensitive hazards. Public health adaptation aimed at improving individual, public health system, and infrastructure resilience is urgently needed to meet the challenges climate change may pose to the Gulf Coast in the coming decades

The Virtue of Defects in 4D Gauge Theories and 2D CFTs

We advance a correspondence between the topological defect operators in Liouville and Toda conformal field theories - which we construct - and loop operators and domain wall operators in four dimensional N=2 supersymmetric gauge theories on S^4. Our computation of the correlation functions in Liouville/Toda theory in the presence of topological defect operators, which are supported on curves on the Riemann surface, yields the exact answer for the partition function of four dimensional gauge theories in the presence of various walls and loop operators; results which we can quantitatively substantiate with an independent gauge theory analysis. As an interesting outcome of this work for two dimensional conformal field theories, we prove that topological defect operators and the Verlinde loop operators are different descriptions of the same operators.Comment: 59 pages, latex; v2 corrections to some formula

't Hooft Operators in Gauge Theory from Toda CFT

We construct loop operators in two dimensional Toda CFT and calculate with them the exact expectation value of certain supersymmetric 't Hooft and dyonic loop operators in four dimensional \Ncal=2 gauge theories with SU(N) gauge group. Explicit formulae for 't Hooft and dyonic operators in \Ncal=2^* and \Ncal=2 conformal SQCD with SU(N) gauge group are presented. We also briefly speculate on the Toda CFT realization of arbitrary loop operators in these gauge theories in terms of topological web operators in Toda CFT.Comment: 49 pages, LaTeX. Typos fixed, references adde

From conformal embeddings to quantum symmetries: an exceptional SU(4) example

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures (ps files) in such a way that they are still understood when converted to gray levels. Version 3: Several references have been adde

The charges of a twisted brane

The charges of the twisted D-branes of certain WZW models are determined. The twisted D-branes are labelled by twisted representations of the affine algebra, and their charge is simply the ground state multiplicity of the twisted representation. It is shown that the resulting charge group is isomorphic to the charge group of the untwisted branes, as had been anticipated from a K-theory calculation. Our arguments rely on a number of non-trivial Lie theoretic identities.Comment: 27 pages, 1 figure, harvmac (b

Graphs and Reflection Groups

It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the conditions satisfied by these graphs, we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots. Some ``non integrally laced " graphs are shown to be associated with subgroups of these reflection groups. The empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled, and the connections with recent developments in the context of ${\cal N}=2$ supersymmetric theories and topological field theories are discussed.Comment: 42 pages TEX file, harvmac and epsf macros, AMS fonts optional, uuencoded, 8 figures include