Dilogarithm Identities in Conformal Field Theory and Group Homology

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all $2 \times 2$ real matrices viewed as a {\it discrete} group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic $K$-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all $2 \times 2$ real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with the summary of a number of open conjectures on the mathematical side.Comment: 20 pages, 2 figures not include

Moduli Stacks of Bundles on Local Surfaces

We give an explicit groupoid presentation of certain stacks of vector bundles on formal neighborhoods of rational curves inside algebraic surfaces. The presentation involves a M\"obius type action of an automorphism group on a space of extensions.Comment: submitted upon invitation to the 2011 Mirror Symmetry and Tropical Geometry Conference (Cetraro, Italy) volume of the Springer Lecture Notes in Mathematic

K-theory and K-cohomology of certain group varieties

SIGLETIB Hannover: RO 8278(90-026) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Comparison Between Algebraic and Topological K-Theory for Banach Algebras and C*-Algebras

For a Banach algebra, one can define two kinds of K-theory: topological K-theory, which satisfies Bott periodicity, and algebraic K-theory, which usually does not. It was discovered, starting in the early 80’s, that the “comparison map ” from algebraic to topological K-theory is a surprisingly rich object. About the same time, it was also found that the algebraic (as opposed to topological) K-theory of operator algebras does have some direct applications in operator theory. This article will summarize what is known about these applications and the comparison map. 1 Some Problems in Operator Theory 1.1 Toeplitz operators and K-Theory The connection between operator theory and K-theory has very old roots, although it took a long time for the connection to be understood. We begin with an example. Think of S1 as the unit circle in the complex plane and let H ⊂ L2(S1) be the Hilbert space H2 of functions all of whose negative Fourier coefficients vanish. In other words, if we identify functions with their formal Fourier expansions, H = n=0 cnz n with n=