String theory under geometric limits

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String theory is described classically as a sigma model with a target space (the string background) which is typically a lorentzian manifold with extra geometric data satisfying the field equations of some (super)gravity theory. Perturbative string theory on such a background is described by a two-dimensional conformal field theory with a BRST differential, whose cohomology can be identified with an instance of Feigin's semi-infinite cohomology of some extension of the Virasoro algebra. Some classical string backgrounds admit geometric limits resulting in a different background of the same string theory or a background of a different kind of string theory. The aim of the project to study the effect of geometric limits on the BRST cohomology. The concrete departing string theory and the limits to consider in this project will follow after some initial exploration.