Symmetry-forced rigidity of frameworks on surfaces

A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to frameworks that are symmetric with respect to some point group but are otherwise generic, and secondly to frameworks in Euclidean 3-space that are constrained to lie on 2-dimensional algebraic varieties. We combine these two settings and consider the rigidity of symmetric frameworks realised on such surfaces. First we establish necessary conditions for a framework to be symmetry-forced rigid for any group and any surface by setting up a symmetry-adapted rigidity matrix for such frameworks and by extending the methods in Jordán et al. (2012) to this new context. This gives rise to several new symmetry-adapted rigidity matroids on group-labelled quotient graphs. In the cases when the surface is a sphere, a cylinder or a cone we then also provide combinatorial characterisations of generic symmetry-forced rigid frameworks for a number of symmetry groups, including rotation, reflection, inversion and dihedral symmetry. The proofs of these results are based on some new Henneberg-type inductive constructions on the group-labelled quotient graphs that correspond to the bases of the matroids in question. For the remaining symmetry groups in 3-space—as well as for other types of surfaces—we provide some observations and conjectures

Non periodic Ishibashi states: the su(2) and su(3) affine theories

We consider the su(2) and su(3) affine theories on a cylinder, from the point of view of their discrete internal symmetries. To this end, we adapt the usual treatment of boundary conditions leading to the Cardy equation to take the symmetry group into account. In this context, the role of the Ishibashi states from all (non periodic) bulk sectors is emphasized. This formalism is then applied to the su(2) and su(3) models, for which we determine the action of the symmetry group on the boundary conditions, and we compute the twisted partition functions. Most if not all data relevant to the symmetry properties of a specific model are hidden in the graphs associated with its partition function, and their subgraphs. A synoptic table is provided that summarizes the many connections between the graphs and the symmetry data that are to be expected in general.Comment: 19 pages, 3 figure

Symmetry adapted Assur decompositions

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs, and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure

From modular invariants to graphs: the modular splitting method

We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3)_k exceptional cases at levels 5 and 9.Comment: 28 pages, 7 figures. Version 2: updated references. Typos corrected. su(2) example has been removed to shorten the paper. Dual annular matrices for the rejected exceptional su(3) diagram are determine