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Explore the intricate relationship between dualities in string theory and the compactifiability of moduli spaces in this mathematical physics lecture. Begin by examining self-dualities in string theory and understand how they act on the field content and spectrum of theoretical frameworks. Investigate the concept of compactifiability for moduli spaces of massless fields, defined as the condition requiring finite volume or growth rates no faster than Euclidean space. Discover how compactifiability generically implies the existence of non-trivial dualities through compelling evidence from string theory examples. Learn to connect compactifiability conditions to spectral properties, specifically how objects charged under duality groups must transform in semisimple representations. Examine bottom-up arguments supporting compactifiability and understand how it relates to the finiteness of massless states in one-dimensional compactifications, particularly in supersymmetric scenarios. Gain insights into cutting-edge research connecting geometric properties of moduli spaces with fundamental dualities in theoretical physics, drawing from recent developments in the field.
