A Construction of Solutions to Reflection Equations for Interaction-Round-a-Face Models

We present a procedure in which known solutions to reflection equations for interaction-round-a-face lattice models are used to construct new solutions. The procedure is particularly well-suited to models which have a known fusion hierarchy and which are based on graphs containing a node of valency $1$. Among such models are the Andrews-Baxter-Forrester models, for which we construct reflection equation solutions for fixed and free boundary conditions.Comment: 9 pages, LaTe

The Appalachian Model Teaching Consortium: Preparing Teachers for Rural Appalachia

The Appalachian Model Teacher Consortium is a partnership involving Radford University, Wytheville Community College, and the Grayson County (Virginia) School System. Its purpose is to prepare highly qualified teachers for rural southwest Virginia. The model was developed in response to the growing teacher shortage facing school districts in rural southwest Virginia. Poorer, more rural districts often have weaker tax bases that provide limited, and at times inadequate, financial support for their school districts. This lack of local resources often results in lower salaries and benefits when compared to many districts that compete for the shrinking pool of potential teachers. Additionally, rural communities are often geographically isolated areas and have fewer amenities that attract young teachers from outside the district. The Appalachian Model Teacher Consortium attempts to naturalize shortages by recruiting potential teachers from the local area, and providing incentives for them to stay and teach in their home community

Integrable Boundaries, Conformal Boundary Conditions and A-D-E Fusion Rules

The $sl(2)$ minimal theories are labelled by a Lie algebra pair $(A,G)$ where $G$ is of $A$-$D$-$E$ type. For these theories on a cylinder we conjecture a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph $A\otimes G$. The cylinder partition functions are given by fusion rules arising from the graph fusion algebra of $A\otimes G$. We further conjecture that, for each conformal boundary condition, an integrable boundary condition exists as a solution of the boundary Yang-Baxter equation for the associated lattice model. The theory is illustrated using the $(A_4,D_4)$ or 3-state Potts model.Comment: 4 pages, REVTe

Algebras in Higher Dimensional Statistical Mechanics - the Exceptional Partition (MEAN Field) Algebras

We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of Q \in \C, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models (in the continuous $Q$ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is non-semi-simple iff $Q$ is a non-negative integer less than $n$. We determine the dimension of the key irreducible representation in every specialization.Comment: 4 page

Identification of a Core Amino Acid Motif within the α Subunit of GABAARs that Promotes Inhibitory Synaptogenesis and Resilience to Seizures

The fidelity of inhibitory neurotransmission is dependent on the accumulation of γ-aminobutyric acid type A receptors (GABAARs) at the appropriate synaptic sites. Synaptic GABAARs are constructed from α(1-3), β(1-3), and γ2 subunits, and neurons can target these subtypes to specific synapses. Here, we identify a 15-amino acid inhibitory synapse targeting motif (ISTM) within the α2 subunit that promotes the association between GABAARs and the inhibitory scaffold proteins collybistin and gephyrin. Using mice in which the ISTM has been introduced into the α1 subunit (Gabra1-2 mice), we show that the ISTM is critical for axo-axonic synapse formation, the efficacy of GABAergic neurotransmission, and seizure sensitivity. The Gabra1-2 mutation rescues seizure-induced lethality in Gabra2-1 mice, which lack axo-axonic synapses due to the deletion of the ISTM from the α2 subunit. Taken together, our data demonstrate that the ISTM plays a critical role in promoting inhibitory synapse formation, both in the axonic and somatodendritic compartments

The Blob Algebra and the Periodic Temperley-Lieb Algebra

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the `blob' algebra (the reason for the name will become obvious shortly!). We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. Here we complete the analysis, using results from the study of the blob algebra.Comment: 12 page

BPS kinks in the Gross-Neveu model

We find the exact spectrum and degeneracies for the Gross-Neveu model in two dimensions. This model describes N interacting Majorana fermions; it is asymptotically free, and has dynamical mass generation and spontaneous chiral symmetry breaking. We show here that the spectrum contains 2^{N/2} kinks for any $N$. The unusual \sqrt{2} in the number of kinks for odd $N$ comes from restrictions on the allowed multi-kink states. These kinks are the BPS states for a generalized supersymmetry where the conserved current is of dimension N/2; the N=3 case is the {\cal N}=1 supersymmetric sine-Gordon model, for which the spectrum consists of 2\sqrt{2} kinks. We find the exact S matrix for these kinks, and the exact free energy for the model.Comment: 22 pages, 5 figure