A ($2+1$)-dimensional superfluid in a rapidly rotating trap forms an array of vortices, with collective excitations called Tkachenko modes. Du et al. [Phys. Rev. Res. 6, L012040 (2024)] argued from an effective field theory viewpoint that these excitations are described by a field theory living on a noncommutative space. We elucidate the microscopic origin of these noncommutative fields, and present a derivation of the effective field theory for this superfluid using a lowest Landau level projected coherent state path-integral approach. Aside from conceptual clarity, this approach makes quantitative predictions about the long-wavelength, low-energy behavior in terms of the microscopic parameters of the short-range interacting lowest Landau level superfluid, relevant to trapped Bose-Einstein condensate experiments.

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• Bose-Einstein condensates
• Landau levels
• Superfluidity
• Vortices in superfluids