Narrow Results

We extend our analysis of the gauging of rigid symmetries in bosonic two-dimensional sigma models with Wess–Zumino terms in the action to the case of world-sheets with defects. A structure that permits a non-anomalous coupling of such sigma models to world-sheet gauge fields of arbitrary topology is analyzed, together with obstructions to its existence, and the classification of its inequivalent choices.

In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face. The construction allows internal degrees of freedom in the defect curves by introducing a second gauge-group from which edges of the curve are labeled in the state-sum construction. Edges incident with the defect, but not lying in it, have states lying in a set with commuting actions of the two gauge-groups. We determine the appropriate generalizations of the 2-cocycles specifying twistings of defect-free 2D Dijkgraaf-Witten theory. Examples arising by restriction of group 2-cocycles, and constructed from characters of the 2-dimensional gauge group are presented.

We derive a general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2D TQFTs without defects (cf. Lauda and Pfeiffer [State-sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications16 (2007) 1121–1163, doi: 10.1142/S0218216507005725]). From the extended Pachner moves (Crane and Yetter [Moves on filtered PL manifolds and stratified PL spaces, arXiv:1404.3142]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [On 2-dimensional Dijkgraaf–Witten theory with defects, to appear in J. Knots Theory Ramifications], we require that triangulations be flaglike, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flaglikeness.

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf–Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocycle condition.

We present an inverse scattering approach to defects in classical integrable field theories. Integrability is proved systematically by constructing the generating function of the infinite set of modified integrals of motion. The contribution of the defect to all orders is explicitely identified in terms of a defect matrix. The underlying geometric picture is that those defects correspond to Bäcklund transformations localized at a given point. A classification of defect matrices as well as the corresponding defect conditions is performed. The method is applied to a collection of well-known integrable models and previous results are recovered (and extended) directly as special cases. Finally, a brief discussion of the classical r-matrix approach in this context shows the relation to inhomogeneous lattice models and the need to resort to lattice regularizations of integrable field theories with defects.

We analyze some aspects of the kinematic theory of non-uniformly defective elastic crystals. Concentrating on the problem of identifying continuous defective lattices possessing the given defectiveness, as defined by the dislocation density tensor, we investigate the relation between the dislocation density tensor and the Lie algebra of vector fields associated with a defective lattice.

In this paper, we discuss a new way to get a quantum holonomy around topological defects in $C_{60}$ fullerenes. For this, we use a Kaluza–Klein extra dimension approach. Furthermore, we discuss how an extra dimension could promote the formation of new freedom degrees which would open a discussion about a possible qubits computation.

A symbolic calculus named as the R-calculus is built to revise the defects of axiomatic systems mechanically when some counterexamples are found. The R-calculus consists of the rules of logical connective symbols and logical quantifier symbols of first order languages. The concept of reachability, soundness and completeness of the R-calculus are introduced. The basic theorem of software testing based on the R-calculus is also introduced to help mechanizing model checking.