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

Combinatorial Construction of Fullerene Structures

Combinatorial fullerene structures (buckyballs for short) have been introduced as a result of the recent discovery of fullerene molecules in laboratories. The stable forms of these materials appear to depend on the method of production as well as on energetic considerations. To understand the stable forms, one would have to examine the confirmed structures among the theoretical structures. As an alternate route to computerized enumeration (which appears to be expensive and not totally safe), we present a procedure that is geometrically transparent. Under certain conditions, our procedure is economical and complete. For example, in the case of buckyballs C„ with v < 84 satisfying the isolated pentagon rule, our procedure can be carried out by hand. To distinguish the inequivalent structures, we present a procedure that does not involve costly spectral computation. In particular, we show that Cgo and C70 are uniquely characterized as the IPR C„ for the two smallest permissible values of v. Some of our results can be used to study qualitative selection rules as well as the structure of hexagonal cylinders