  Rapid advancements in the experimental capabilities with ultracold
alkaline-earth-like atoms (AEAs) bring to a surprisingly near term the prospect
of performing quantum simulations of spin models and lattice field theories
exhibiting SU($N$) symmetry. Motivated in particular by recent experiments
preparing high density samples of strongly interacting ${}^{87}$Sr atoms in a
three-dimensional optical lattice, we develop a low-energy effective theory of
fermionic AEAs which exhibits emergent multi-body SU($N$)-symmetric
interactions, where $N$ is the number of atomic nuclear spin levels. Our theory
is limited to the experimental regime of (i) a deep lattice, with (ii) at most
one atom occupying each nuclear spin state on any lattice site. The latter
restriction is a consequence of initial ground-state preparation. We fully
characterize the low-lying excitations in our effective theory, and compare
predictions of many-body interaction energies with direct measurements of
many-body excitation spectra in an optical lattice clock. Our work makes the
first step in enabling a controlled, bottom-up experimental investigation of
multi-body SU($N$) physics.
