Compactifications have a really long history in physics, the most famous inception was that of Kaluza-Klein theory which was a five-dimensional theory of gravitation and electromagnetism. Generally, one starts with finding the vacua in $M^4 \times S^1$ instead of $M^5$. We assume that the circle's radius is quite small, somewhat of the Planck scale. What are the symmetries of a ground state in $M^4 \times S^1$? These would be the Poincare symmetries of $M^4$ and the gauge $U(1)$ symmetries of $S^1$. And these symmetries would be available for us to observe as (local) gauge symmetries in our $M^4$ (non-compact real world) since Einstein's equations are covariant. After imposing the general covariance and taking a metric on five independent dimensions $g_{\mu \nu}$ and this contains additional fields. These fields are in $g_{\mu 5}$ which is a vector field and satisfies Maxwell's equations. In this way, one has a solution of Minkowski's space in five dimensions and as well as the Electromagnetism. If one does some oscillations around this ground state, one gets a number of massive modes and some finite massless modes which are the spin-two gravitons and spin-one photons (also includes the Brans-Dicke scalar).

Now, similarly, we hope to compactify the solutions of Supergravity and Superstrings (and M-theory, F-theory) to obtain a vacua similar to ours. I do not wish to discuss the realistic interpretation of these works as in phenomenology, yet. The classic example of compactification is of kind $M^4 \times K$ where $K$ is a Calabi-Yau manifold with some desired holonomy. For example, for the unbroken SUSY, we generally want K with $SU(3)$ holonomy,, and the Calabi-Yau manifold is perfect for this. Now, as discussed in the Kaluza-Klein case, the local symmetry group would be coming from $K$. In the case of the standard model, the symmetries are $SU(3) \times SU(2) \times U(1)$, thus $K$ must admit these when talking about finding the standard model upon compactifications. (One could argue that there is a spontaneous compactification, as Scherk and Cremmer did, of the vacuum from the $M^4 \times K$ to our four-dimensional world, and the extra dimensions, which are the $dim K$, are very microscopic.)

The famous superstring compactification was done by Candelas, et.al, of the heterotic string theory with gauge group $E_8 \times E_8$ and $SO(32)$ as they were shown to have the anomaly cancellation. Similarly, one could talk about the supergravity with same gauge groups. For supergravity, one can have a maximum $11$ dimensions to hold the supersymmetry and one more dimension could predict massless modes with spin more than two, which is not possible. In fact for $K$ to have the symmetry group $SU(3) \times SU(2) \times U(1)$, $K$ should be of minimum seven dimensions.

Upon compactification, we generally obtain a moduli of solutions, since there is no unique solution. We will hopefully discuss these moduli later.

Now, I come to the other part of this post which is merely a small shout-out. Physicists are mainly interested in the geometric Langlands program, as it was shown that S-duality (or electric-magnetic duality) is a manifestation of Langlands duality by Olive, Witten, Kapustin, Goddard, and so on in different works. Generally, there is a duality between the $\mathcal{A}$ side and $\mathcal{B}$ side, as in mirror symmetry. Generally, theoretical physics does not have much progress with the Real Langlands as they are generally number-theoretical advancements. The same is true with the local (which is about p-adic fields) and global (which is about number fields) Langlands correspondence. (For some more developments of Langlands in physics, see https://arxiv.org/abs/2409.04677)

However, Scholze (and Fargues) worked on a proposal to refine the local Langlands program, to make it more geometric, to make it like more geometric Langlands. Scholze's new paper appeared this week about the same subject and now the key word is motives.