Generalized moonshine II: Borcherds products

The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. The Lie algebras that we construct conjecturally appear in an orbifold conformal field theory with symmetries given by the monster simple group. We introduce vector-valued modular functions attached to families of modular functions of different levels, and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half-planes. We recast this result using the Borcherds-Harvey-Moore singular theta lift, and show that the vector-valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps. For each element of the monster simple group, we construct an infinite dimensional Lie algebra, such that its denominator formula is an infinite product expansion of the automorphic function arising from that element's McKay-Thompson series. These Lie algebras have the unusual property that their simple roots and all root multiplicities are known. We show that under certain hypotheses, characters of groups acting on these Lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group.Comment: v3: final version, minor corrections and explanations added, 41 page

Monstrous Moonshine for integral group rings

We propose a conjecture that is a substantial generalization of the genus zero assertions in both Monstrous Moonshine and Modular Moonshine. Our conjecture essentially asserts that if we are given any homomorphism to the complex numbers from a representation ring of a group ring for a subgroup of the monster, we obtain a hauptmodul by applying this homomorphism to a self-dual integral form of the moonshine module. We reduce this conjecture to the genus-zero problem for "quasi-replicable" functions, by applying Borcherds's integral form of the Goddard-Thorn no-ghost theorem together with some analysis of the Laplacian on an integral form of the Monster Lie algebra. We prove our conjecture for cyclic subgroups of the monster generated by elements in class 4A, and we explicitly determine the multiplicities for a decomposition of the integral moonshine module into indecomposable modules of the integral group rings for these groups.Comment: 29 page

High-Field MAS Dynamic Nuclear Polarization Using Radicals Created by γ-Irradiation

High-field magic angle spinning dynamic nuclear polarization (MAS DNP) is often used to enhance the sensitivity of solid-state nuclear magnetic resonance (ssNMR) experiments by transferring spin polarization from electron spins to nuclear spins. Here, we demonstrate that γ-irradiation induces the formation of stable radicals in inorganic solids, such as fused quartz and borosilicate glasses as well as organic solids such as glucose, cellulose, and a urea/polyethylene polymer. The radicals were then used to polarize 29Si or 1H spins in the core of some of these materials. Significant MAS DNP enhancements (ε) greater than 400 and 30 were obtained for fused quartz and glucose, respectively. For other samples negligible ε were obtained likely due to low concentrations of radicals or the presence of abundant quadrupolar spins. These results demonstrate that ionizing radiation is a promising alternative method for generating stable radicals suitable for high-field MAS DNP experiments

Elimination of TFA-Mediated Cleavage in Distributed Drug Discovery

Distributed Drug Discovery (D3) is a multi-disciplinary approach to the discovery of new drugs, which target neglected diseases or conditions common to developing-world countries. As part of a continuing effort to improve D3 methodology, two approaches for eliminating the final step TFA-mediated resin cleavage are proposed for investigation. Cleavage under basic conditions (saponification) and mild acid conditions (dilute HCl/hexafluoroisopropanol or dilute HCl/trifluoroethanol) represent improvements in safety and convenience to the undergraduate student researcher. Previous studies have shown that saponification provides yields comparable to the traditional TFA cleavage but recovery is not as convenient. Further improvements in the saponification workup will be evaluated by analyzing the effectiveness of simple trituration with acetone compared to use of a strong anion-exchange resin or drying reagents to isolate the free acid from the salt. Different trituration procedural modifications have been made and are being tested. Results have shown that in the presence of methanol, esterification will occur when the acid is liberated from the salt using HCl. To counter this problem, the samples are first evaporated to remove methanol and then the pH is adjusted with HCl. It was shown that using acetic acid did not result in pH levels low enough to guarantee complete protonation of the carboxylate. Through the use of a Bill-Board, an apparatus that holds six reaction vessels, several procedural modifications can be carried out simultaneously. Analysis is conducted by liquid chromatography coupled with a mass spectrometer and with nuclear magnetic resonance spectroscopy. Further studies will be carried out to assess the efficiency and practicality of using mild acidic conditions for cleavage using HCl/hexafluoroisopropanol or dilute HCl/trifluoroethanol. Both saponification and mild acid cleavage would represent improvements in safety and convenience to the undergraduate student researcher