Wieler solenoids, Cuntz-Pimsner algebras and K-theory

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the groupoids of Deaconu-Renault and Putnam-Spielberg, extending results of Thomsen. The Deaconu-Renault groupoid and the explicit Morita equivalence leads to a Cuntz-Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz-Pimsner algebras are then studied using the Cuntz-Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions we characterize the KMS weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples

Open-string vertex algebras, tensor categories and operads

We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are ``open-string-theoretic,'' ``noncommutative'' generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the ``meromorphic center,'' inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center. This result gives us a general method for constructing open-string vertex algebras. Besides obvious examples obtained from associative algebras and vertex (super)algebras, we give a nontrivial example constructed from the minimal model of central charge c=1/2. We establish an equivalence between the associative algebras in the braided tensor category of modules for a suitable vertex operator algebra and the grading-restricted conformal open-string vertex algebras containing a vertex operator algebra isomorphic to the given vertex operator algebra. We also give a geometric and operadic formulation of the notion of grading-restricted conformal open-string vertex algebra, we prove two isomorphism theorems, and in particular, we show that such an algebra gives a projective algebra over what we call the ``Swiss-cheese partial operad.''Comment: 53 page