  We first apply functional-integral approach to a multiband Hubbard model near
the critical pairing temperature, and derive a generic effective action that is
quartic in the fluctuations of the pairing order parameter. Then we consider
time-reversal-symmetric systems with uniform (i.e., at both low-momentum and
low-frequency) pairing fluctuations in a unit cell, and derive the
corresponding time-dependent Ginzburg-Landau (TDGL) equation. In addition to
the conventional intraband contribution that depends on the derivatives of the
Bloch bands, we show that the kinetic coefficients of the TDGL equation have a
geometric contribution that is controlled by both the quantum-metric tensor of
the underlying Bloch states and their band-resolved quantum-metric tensors.
Furthermore we show that thermodynamic properties such as London penetration
depth, GL coherence length, GL parameter and upper critical magnetic field have
an explicit dependence on quantum geometry.
