On the integrability of N=2 supersymmetric massive theories

In this paper we propose a criteria to establish the integrability of N=2 supersymmetric massive theories.The basic data required are the vacua and the spectrum of Bogomolnyi solitons, which can be neatly encoded in a graph (nodes=vacua and links= Bogomolnyi solitons). Integrability is then equivalent to the existence of solutions of a generalized Yang-Baxter equation which is built up from the graph (graph-Yang-Baxter equation). We solve this equation for two general types of graphs: circular and daisy, proving, in particular, the inte- grability of the following Landau-Ginzburg superpotentials: A_n(t_1), A_n(t_2), D_n(\tau),E_6(t_7), E_8(t_16). For circular graphs the solutions are intertwiners of the affine Hopf algebra $\tilde(U)_q(A^{(1)}_1)$, while for daisy graphs the solution corresponds to a susy generalization of the Boltzmann weights of the chiral Potts model in the trigonometric regime. A chiral Potts like solution is conjectured for the more tricky case $D_n(t_2)$. The scattering theory of circular models, for instance $A_n(t_1)$ or $D_n(\tau)$, is Toda like. The physical spectrum of daisy models, as $A_n(t_2), E_6(t_7)$ or $E_8(t_16)$, is given by confined states of radial solitons. The scattering theory of the confined states is again Toda like. Bootstrap factors for the confined solitons are given by fusing the susy chiral Potts S-matrices of the elementary constituents, i.e. the radial solitons of the daisy graph.Comment: 26 pages, Latex (this version replaces a previously corrupted one; epic.sty macro needed, available from hep-th in compressed form epic.sty.tar.Z

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure

Fixed parameter tractability of crossing minimization of almost-trees

We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1-page book crossing number, the 2-page book crossing number, and the minimum number of crossed edges in 1-page and 2-page book drawings.Comment: Graph Drawing 201

Boundary Partitions in Trees and Dimers

Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities, and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE_2, SLE_8, and SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor change