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We construct the effective field theory for the A-phase of superfluid $^3$Heup to the next-to-leading order in the derivative expansion. To this end, wegauge the internal global symmetries of the theory on the curved space byintroducing the background gauge fields and spatial metric so as to expose ahidden local symmetry known as the nonrelativistic diffeomorphism. Thenonrelativistic diffeomorphism is particularly useful to yield an additionalconstraint on the effective field theory and reveal a universal expression forthe Hall viscosity in the A-phase. We find it five orders of magnitude largerthan that in the B-phase under a magnetic field so that its experimentalobservation is more feasible by measuring the induced elliptic polarization ofsound waves.

Hall viscosity in the A-phase of superfluid $^3$He