  A platform that serves as an ideal playground for realizing ``high''
temperature superconductors are materials where the electrons' kinetic energy
is completely quenched, and interactions provide the only energy scale in the
problem for $T_c$. However, when the non-interacting bandwidth for a set of
isolated bands is small compared to the scale of the interactions, the problem
is inherently non-perturbative and requires going beyond the traditional
mean-field theory of superconductivity. In two spatial dimensions, $T_c$ is
controlled by the superconducting phase stiffness. Here we present a general
theoretical framework for computing the electromagnetic response for generic
model Hamiltonians, which controls the maximum possible superconducting phase
stiffness and thereby $T_c$, without resorting to any mean-field approximation.
Importantly, our explicit computations demonstrate that the contribution to the
phase stiffness arises from (i) ``integrating out'' the remote bands that
couple to the microscopic current operator, and (ii) the density-density
interactions projected onto the isolated narrow bands. Our framework can be
used to obtain an upper bound on the phase stiffness, and relatedly the
superconducting transition temperature, for a range of physically inspired
models involving both topological and non-topological narrow bands with
arbitrary density-density interactions. We discuss a number of salient aspects
of this formalism by applying it to a specific model of interacting flat bands
and compare it against the known $T_c$ from independent numerically exact
computations.
