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

Engineering of Tripeptide-Stabilized Gold Nanoclusters with Inherent Photosensitizing Property for Bioimaging and Photodynamic Therapy

Gold nanoclusters have the characteristics of small size, unique optical properties, and eco-friendly synthesis that make them promising candidates for biomedical applications, especially for bioimaging. However, their inherent photochemical activity and therapeutic efficiency are largely unknown and remain unexplored. Here, we report a simple one-step green synthesis procedure for the preparation of two tripeptide-stabilized silver-doped gold nanoclusters (TPGNCs) and their photodynamic therapeutic effect on cancer cells and simultaneous imaging. The custom-designed tripeptides were used for the preparation of silver-doped gold nanoclusters with enhanced fluorescence emission. These TPGNCs exhibited strong red fluorescence with high quantum yield, large Stokes shift, good photostability, and excellent biocompatibility toward normal cells. TPGNCs imparted minimum dark toxicity toward breast cancer cells. These TPGNCs exhibited appreciable photosensitization to generate ROS within the cancer cells triggering loss of mitochondrial membrane potential, leading to apoptotic cell death. The photosensitizing ability of TPGNCs will be a new avenue in the area of photoinduced cancer therapy with negligible side effects