RCFT with defects: Factorization and fundamental world sheets

It is known that for any full rational conformal field theory, the correlation functions that are obtained by the TFT construction satisfy all locality, modular invariance and factorization conditions, and that there is a small set of fundamental correlators to which all others are related via factorization - provided that the world sheets considered do not contain any non-trivial defect lines. In this paper we generalize both results to oriented world sheets with an arbitrary network of topological defect lines.Comment: 46 pages, several pictures. v2: typos correcte

Spectrum of periodic chain graphs with time-reversal non-invariant vertex coupling

We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly non-invariant with respect to time reversal. We discuss, in particular, the high-energy behavior of such systems and the limiting situations when one of the edges in the elementary cell of such a graph shrinks to zero. The spectrum depends on the topology and geometry of the graph. The probability that an energy belongs to the spectrum takes three different values reflecting the vertex parities and mirror symmetry, and the band patterns are influenced by commensurability of graph edge lengths.Comment: 31 pages, 16 figure

On the Connectivity of Cobordisms and Half-Projective TQFT's

We consider a generalization of the axioms of a TQFT, so called half-projective TQFT's, with an anomaly, $x^{\mu}$, in the composition law. $\mu$ is a coboundary on the cobordism categories with non-negative, integer values. The element $x$ of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be 0. This modification makes it possible to extend quantum-invariants, which vanish on $S^1\times S^2$, to non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be trivial all together). Under a few natural assumptions the notion of a half-projective TQFT is shown to be the only possible generalization. Based on separate work with Lyubashenko on connected TQFT's, we construct a large class of half-projective TQFT's with $x=0$. Their invariants vanish on $S^1\times S^2$, and they coincide with the Hennings invariant for non-semisimple Hopf algebras. Several toplogical tools that are relevant for vanishing properties of such TQFT's are developed. They are concerned with connectivity properties of cobordisms, as for example maximal non-separating surfaces. We introduce in particular the notions of ``interior'' homotopy and homology groups, and of coordinate graphs, which are functions on cobordisms with values in the morphisms of a graph category. For applications we will prove that half-projective TQFT's with $x=0$ vanish on cobordisms with infinite interior homology, and we argue that the order of divergence of the TQFT on a cobordism in the ``classical limit'' can be estimated by the rank of its maximal free interior group.Comment: 55 pages, Late

Self-adjoint local boundary problems on compact surfaces. I. Spectral flow

The paper deals with first order self-adjoint elliptic differential operators on a smooth compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. The first result is the computation of the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required. In the next paper of the series we generalize these results to families of such operators parametrized by points of an arbitrary compact space instead of an interval. The integer-valued spectral flow is replaced then by the family index taking values in the $K^1$-group of the base space.Comment: 46 pages. V4: minor changes; version accepted by the Journal of Geometric Analysi