We conjecture the factorized scattering description for OSP (m/2n)/OSP (m − 1/2n) supersphere sigma models and OSP (m/2n) Gross Neveu models. The non unitarity of these field theories translates into a lack of 'physical unitarity' of the S matrices, which are instead unitary with respect to the non-positive scalar product inherited from the orthosymplectic structure. Nevertheless, we find that formal thermodynamic Bethe ansatz calculations appear meaningful, reproduce the correct central charges, and agree with perturbative calculations. This paves the way to a more thorough study of these and other models with supergroup symmetries using the S matrix approach.

Abstract. Quantum wire networks have recently become of great interest. Here we deal with a novel nano material structure of a double gyroid wire network. We use methods of commutative and noncommutative geometry to describe this wire network. Its noncommutative geometry is closely related to noncommutative 3-tori as we discuss in detail. Mathematics Subject Classification (2010).

Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians.The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of A n type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an A k singularity.We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties.

This paper is a survey of the $\mathbb{Z}_2$-valued invariant of topological insulators used in condensed matter physics. The $\mathbb{Z}$-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The $\mathbb{Z}_2$ invariant is more mysterious, we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.Comment: 62 pages, 3 figure

As a step to understand general patterns of integrability in 1+1 quantum field theories with supergroup symmetry, we study in details the case of OSP (1/2). Our results include the solutions of natural generalizations of models with ordinary group symmetry: the U OSP (1/2) k WZW model with a current current perturbation, the U OSP (1/2) principal chiral model, and the U OSP (1/2) ⊗ U OSP (1/2)/U OSP (1/2) coset models perturbed by the adjoint. Graded parafermions are also discussed. A pattern peculiar to supergroups is the emergence of another class of models, whose simplest representative is the OSP (1/2)/OSP (0/2) sigma model, where the (non unitary) orthosymplectic symmetry is realized non linearly (and can be spontaneously broken). For most models, we provide an integrable lattice realization. We show in particular that integrable osp(1/2) spin chains with integer spin flow to U OSP (1/2) WZW models in the continuum limit, hence providing what is to our knowledge the first physical realization of a super WZW model.