Abstract. The trace or the 0th Hochschild-Mitchell homology of a linear category C may be regarded as a kind of decategorification of C. We compute the traces of the two versionsU and U * of categorified quantum sl 2 introduced by the third author. The trace of U is isomorphic to the split Grothendieck group K 0 (U ), and the higher Hochschild-Mitchell homology ofU is zero. The trace of U * is isomorphic to the idempotented integral form of the current algebra U(sl 2 [t]).

We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.

No abstract

Due to their unique electrical properties, graphene nanoribbons (GNRs) show great promise as the building blocks of novel electronic devices. However, these properties are strongly dependent on the geometry of the edges of the graphene devices. Thus far only zigzag and armchair edges have been extensively studied. However, several other self passivating edge reconstructions are possible, and were experimentally observed. Here we utilize the Nonequilibrium Green's Function (NEGF) technique in conjunction with tight binding methods to model quantum transport through armchair, zigzag, and several other self-passivated edge reconstructions. In addition we consider the experimentally relevant cases of mixed edges, where random combinations of possible terminations exist on a given GNR boundary. We find that transport through GNR's with self-passivating edge reconstructions is governed by the sublattice structure of the edges, in a manner similar to their parent zigzag or armchair configurations. Furthermore, we find that the reconstructed armchair GNR's have a larger band gap energy than pristine armchair edges and are more robust against edge disorder. These results offer novel insights into the transport in GNRs with realistic edges and are thus of paramount importance in the development of GNR based devices.Comment: J. Phys. Chem. C, 201

No abstract

Abstract. We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many hairy graph cohomology classes out of non-hairy classes by a mechanism which we call the waterfall mechanism. By this mechanism we can construct many previously unknown classes and provide a first glimpse at the tentative global structure of the hairy graph cohomology.

We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the homology of standard Kontsevich’s graph complex. This result may have applications in theory of multi-vector fields $T_{\textrm{poly}}^{\geq 1}$ of degree at least one, and to the hairy graph complex that computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. We also develop a theory of graph complexes with arbitrary edge types.

The ability to interferometrically detect inertial rotations via the Sagnac effect has been a strong stimulus for the development of atom interferometry because of the potential 10^{10} enhancement of the rotational phase shift in comparison to optical Sagnac gyroscopes. Here we analyze ballistic transport of matter waves in a one dimensional chain of N coherently coupled quantum rings in the presence of a rotation of angular frequency, \Omega. We show that the transmission probability, T, exhibits zero transmission stop gaps as a function of the rotation rate interspersed with regions of rapidly oscillating finite transmission. With increasing N, the transition from zero transmission to the oscillatory regime becomes an increasingly sharp function of \Omega with a slope \partialT/\partial \Omega N^2. The steepness of this slope dramatically enhances the response to rotations in comparison to conventional single ring interferometers such as the Mach-Zehnder and leads to a phase sensitivity well below the standard quantum limit