What's the importance of all four fundamental forces being "curvature"?

Question

I've heard about how, in a gauge theory, the gauge covariant derivative of the field around a closed curve is generally not zero, and this is how you can quantify force or field strength. And that this is the same basic idea as curvature, with the gauge field being equivalent to the connection.

So since gravity is already known to be curvature, we can say that all the forces of nature are curvature in their own way. So what's the significance of that? Is there some deeper reason that we should expect that to be the case? And are the current unification programs based on that similarity?

Answer 1

When we study non-gravitational fundamental interactions, we distinguish internal symmetries associated with only such interactions from the external symmetries of spacetime. For all fundamental interactions, there is a finite-dimensional Lie group characterizing that interaction's symmetries. In the case of gravity, the Lie derivative of Killing vector fields on the spacetime manifold defines the associated Lie algebra's structure constants; for the other interactions, there is a "space" that plays a role analogous to this manifold, but it's not spacetime itself. Instead, it's a space of legal values for a field over spacetime.

For example, electromagnetism's $U(1)$ symmetry (let's put electroweak unification aside for the moment) is the rotational invariance of $|\phi|$ for $\phi\in\Bbb C$ with $|\phi|:=\sqrt{\phi^\ast\phi}$, or equivalently for $\phi\in\Bbb R^2$ with $|\phi|:=\sqrt{\phi\cdot\phi}$. (I'm denoting the set of values $\phi$ can have at each point in spacetime, say $\Bbb R^4$, so as a function $\phi\in X^{\Bbb R^4}$ for $X=\Bbb C$ or, somewhat less helpfully in QFT, $X=\Bbb R^2$.) So if there is a space which is "curved" in this context by electromagnetism, it is not spacetime per se.

As for unification implications, Wikipedia notes

For ordinary Lie algebras, the gauge covariant derivative on the space symmetries... cannot be intertwined with the internal gauge symmetries... this is the content of the Coleman–Mandula theorem. However, a premise of this theorem is violated by the Lie superalgebras (which are not${}^\dagger$ Lie algebras!) thus offering hope that a single unified symmetry can describe both spatial and internal symmetries: this is the foundation of supersymmetry

${}^\dagger$ emphasis in the original.

Answer 2

In the 1920s–1940s, people developed a unified classical theory of gravity and electromagnetism using just this sort of approach. It's called Kaluza-Klein theory. Some aspects of it even generalize to classical non-abelian Yang-Mills theories (R. Montgomery: Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations). I think I've heard there's some subtle problem with its quantum version that prevents it from describing quantum non-abelian Yang-Mills theories well. That might be one reason it's not talked about much these days.