First of all, in some sense it is not that interesting of a statement. Whether gravity is a force or not, or whether light is a particle or a wave -- all that depends on definitions of the word force, particle, wave, etc. And nowadays that is the realm of philosophy instead of physics.
Ultimately physicists care not about the words but rather about the equations. Only equations allow them to predict the outcome of experiments (at least probabilistically). A physicists' ultimate dream is to find a single set of equations that predicts every experiment. Words are just used to describe what the equations do, to colleagues and laypeople alike.
So yes, our best model to describe gravitational phenomena is general relativity which describes these phenomena in terms of the bending of spacetime.
If you want to go back to a "force" description of general relativity you can do so as it's a free world. The description of all the other forces starts with a fixed background spacetime, so to do the same with gravity you'll have to contort the elegant covariant equations of general relativity into an ugly mess by expanding around such a background. And then your resulting "theory" will predict exactly the same experimental outcomes as general relativity. (Around flat space it is not so difficult, it produces Newton's law of gravity to the first approximation.)
All that just does not seem worthwhile. More importantly, the discovery of general relativity (and of the importance of general covariance) led to a paradigm shift in physics: all reasonable physicists now believe that any fundamental theory will not ultimately be a set of equations on a fixed background spacetime but rather a set of equations from which spacetime itself should somehow emerge. Very different, therefore, from quantum field theory which needs a fixed background and describes all the other forces in the standard model.
This was my attempt at addressing the "not like the other forces". For the quantum dynamics see the sibling comment https://news.ycombinator.com/item?id=41179419 . And thank you for teaching me about the "permanent" and "immanant" of a matrix, but I still have no idea what that comment is supposed to mean - sorry.
I'm out of my depth here, but I'm imagining something like a path integral[0] between n particles. For example, with two fermions, A and B, you'd get a fraction f of A moving to B's location, and vice versa, giving:
d[AB] = f * ([B] - [A])
The fraction is real as f^2 is the action A -> B -> A which shouldn't get a phase or A would self-destruct. If you had a bunch of anyons circling around in a magnetic field[1], then f could be a root of unity or something more complicated. You can generalize the boundary operator
d[123...n] = Σ(-1)^k * [123...k-1,k+1...n]
to be
d[123...n] = Σ χ(k) * [123...k-1,k+1...n]
where χ(k) is the character of your group. For example, if you have n particles interacting in a circle 1 -> 2 -> 3 -> ... -> n -> 1, then χ(k) = e^2πik/n. This is where the immanant comes from.
Ultimately physicists care not about the words but rather about the equations. Only equations allow them to predict the outcome of experiments (at least probabilistically). A physicists' ultimate dream is to find a single set of equations that predicts every experiment. Words are just used to describe what the equations do, to colleagues and laypeople alike.
So yes, our best model to describe gravitational phenomena is general relativity which describes these phenomena in terms of the bending of spacetime.
If you want to go back to a "force" description of general relativity you can do so as it's a free world. The description of all the other forces starts with a fixed background spacetime, so to do the same with gravity you'll have to contort the elegant covariant equations of general relativity into an ugly mess by expanding around such a background. And then your resulting "theory" will predict exactly the same experimental outcomes as general relativity. (Around flat space it is not so difficult, it produces Newton's law of gravity to the first approximation.)
All that just does not seem worthwhile. More importantly, the discovery of general relativity (and of the importance of general covariance) led to a paradigm shift in physics: all reasonable physicists now believe that any fundamental theory will not ultimately be a set of equations on a fixed background spacetime but rather a set of equations from which spacetime itself should somehow emerge. Very different, therefore, from quantum field theory which needs a fixed background and describes all the other forces in the standard model.
This was my attempt at addressing the "not like the other forces". For the quantum dynamics see the sibling comment https://news.ycombinator.com/item?id=41179419 . And thank you for teaching me about the "permanent" and "immanant" of a matrix, but I still have no idea what that comment is supposed to mean - sorry.