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In this talk we will review our physical proposal, with D. Persson and R. Volpato, to understand the genus zero property of monstrous moonshine. The latter was proven by Borcherds in 1992 by brute-force computation but has since resisted a conceptual understanding. We embed the Monster VOA of Frenkel-Lepowsky-Meurman into a heterotic string compactification and use physical arguments, i.e. T-dualities and decompactification limits, to understand the genus zero property. We find that the Monster Lie algebra (a Generalized, or Borcherds-Kac-Moody algebra) acts as a sort of "algebra of BPS states". We also sketch an analogous proposal, with S. Harrison and R. Volpato, for Conway moonshine using the type II string. Along the way we construct a new super Borcherds-Kac-Moody algebra on which the Conway group acts faithfully and prove its (twisted) denominator identities, which should be identified with BPS state counts in our string theory. 

Already in their early papers, Kosterlitz and Thouless envisaged the melting of solids by the unbinding of the topological defects associated with translational order: dislocations. Later it was realized that the resulting phases have translational symmetry but rotational rigidity: they are liquid crystals.

We consider the topological melting of solids as a zero-temperature quantum phase transition. In a generalization of particle-vortex duality, the Goldstone modes of the solid, phonons, map onto gauge bosons which mediate long-range interactions between dislocations. The phase transition is achieved by a Bose-Einstein condensation of dislocations, restoring translational symmetry and destroying shear rigidity. The dual gauge fields become massive due to the Anderson-Higgs mechanism. In this sense, the liquid crystal is a "stress superconductor".

We have developed this dual gauge field theory both in 2+1D, where dislocations are particle-like and phonons are vector bosons, and 3+1D where dislocations are string-like and phonons are Kalb-Ramond gauge fields. Focussing mostly on the theoretical formalism, I will discuss the relevance to recent experiments on helium monolayers, which show evidence for a quantum hexatic phase.

References:

2+1D : Physics Reports 683, 1 (2017)

3+1D : Physical Review B 96, 1651115 (2017)