  We formulate a generalized pseudospin formalism for multiband superconductors
in the presence of an external perturbing electromagnetic field. Our theory
naturally captures the effects of quantum band geometric quantities and is
valid even for flat-band superconductors. As an interesting consequence of our
theory, we show that there is an interband pairing fluctuations induced by the
external field and mediated by the quantum band geometry. Surprisingly, this
interband fluctuation is independent of the band gap, which can be understood
from the geometric nature of such novel fluctuations. We derive the generalized
equation of motion for the multiband pseudospin and the self-consistency
equation. We present a formal solution to the pseudospin equation of motion in
powers of the perturbing electromagnetic field. As a simple illustration of our
theory, we calculate the Leggett modes for the two band case.
