Homology and K-theory of the Bianchi groups

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant $K$-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the $K$-theory of their reduced $C^*$-algebras in terms of isomorphic images of the computed $K$-homology. We further find an application to Chen/Ruan orbifold cohomology. % {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I +++ (2011).

Loop groups in Yang-Mills theory

We consider the Yang-Mills equations with a matrix gauge group $G$ on the de Sitter dS$_4$, anti-de Sitter AdS$_4$ and Minkowski $R^{3,1}$ spaces. On all these spaces one can introduce a doubly warped metric in the form $d s^2 =-d u^2 + f^2 d v^2 +h^2 d s^2_{H^2}$, where $f$ and $h$ are the functions of $u$ and $d s^2_{H^2}$ is the metric on the two-dimensional hyperbolic space $H^2$. We show that in the adiabatic limit, when the metric on $H^2$ is scaled down, the Yang-Mills equations become the sigma-model equations describing harmonic maps from a two-dimensional manifold (dS$_2$, AdS$_2$ or $R^{1,1}$, respectively) into the based loop group $\Omega G=C^\infty (S^1, G)/G$ of smooth maps from the boundary circle $S^1=\partial H^2$ of $H^2$ into the gauge group $G$. From this correspondence and the implicit function theorem it follows that the moduli space of Yang-Mills theory with a gauge group $G$ in four dimensions is bijective to the moduli space of two-dimensional sigma model with $\Omega G$ as the target space. The sigma-model field equations can be reduced to equations of geodesics on $\Omega G$, solutions of which yield magnetic-type configurations of Yang-Mills fields. The group $\Omega G$ naturally acts on their moduli space.Comment: 8 pages; v3: clarifying remarks and references adde

A Review of the Genus \u3ci\u3eGryllus\u3c/i\u3e (Orthoptera: Gryllidae), With a New Species From Korea

Gryllus is the most widely distributed genus of the Tribe Gryllini, and may be the largest; it includes 69 described species occupying most of the New World, Africa, and Europe, and much of Asia. A new species from Korea significantly extends the known range of the genus