In a previous Quill, we saw that one has to do compactifications when dealing with theories in higher dimensions, examples include supergravity (or M-theory) in $D=11$ or superstrings in $D=10$, or the very classic Kaluza-Klein compactification from $D=5$ to $D=4$. We do these compactifications on a very special kind of manifolds for reasonings that are more apt for physics than mathematics.

Let's say we take $10$-dimensional theory and compactify on K like $M_4 \times K$ where $K$ is really a $(3,1)$ manifold of space-time and $K$ is a six-dimensional (compact) Riemannian manifold. The dimensions of $K$ are very macroscopic and $K$ will serve us for three things: 1) Supersymmetry, 2) effective low-energy limit physics, and 3) zero-curvature (Ricci flat). It is thus that $K$ has to be very special, especially from a physicist's point of view. We want to preserve the $\mathcal{N}=1$ (the bare and the simplest number of SUSY) in $M_4$ and we want these $K$ to also provide massless modes of interest. In particular, the holonomy group of the metric of $K$ must be contained in $u(n)$, and in particular, it is of $SU(3)$ holonomy. (It is important for many reasons, to name one, it is crucial for the unbroken supersymmetry in $M_4$.)

Now, there are some good mathematical consequences of these, which were realized by physicists in an amazing period of the 80s following works in algebraic geometry and differential geometry.

This one is rather on a more mathematical side.
Quantum mechanics is just one-dimensional quantum field theory.  In quantum mechanics, one does a lot with Hamiltonian $H$ and the Hilbert space $\mathcal{H}$. $H$ is contained in $\mathcal{H}$, or more appropriately in the algebra. We can now find the time evolution for a state in $\mathcal{H}$ by $$e^{iHt}$$ and something is time invariant if it commutes with $H$. In topological quantum mechanics, we want to, in a sense, kill the time. This can be done, trivially, by setting $H=0$ which limits the eigenspace to a zero eigenspace (which contains only the ground state). It is, however, not interesting in the sense that we do not have much of a theory.

A good alternative is to kill time in a derived sense. This would employ the Hodge theory (for Riemannian manifolds) and cohomology. We can, instead of taking an infinite-dimensional $\mathcal{H} = L^2(M)$ Hilbert space of $L^2$ functions on a manifold $M$, take a space of differential forms $$L^2(p)$$ where $p$ are differential forms. Or more excitingly, attach a complex chain $$(\Omega^\circ, X)$$ where the differential forms are $U(1)$ graded. Doing this adds extra fields to the theory. We can identify, now this has become a SUSY theory, the supercharges $Q,Q^*$ (for $\mathcal{N}=1$ SUSY). These will be our de Rham differential operator $Q=d$ and $Q^*=d^*$ (if you can guess now then we want to pass into cohomology eventually). As in a SUSY algebra, $$[Q,Q] = [Q^*,Q^*] = 0$$ $$[Q, H] = [Q^*,H]=0$$ but $$[Q,Q^*]= H.$$

The Hamiltonian acts on the differential forms $p$ as a Laplace operator, $H=\Delta$. Then one writes $H$ in the form of these supercharges. Also, given $[Q,Q] =0$, one has $d^2=0$ (which is quite a hint for our pass to homology, since $d$ becomes a boundary operator.). If one takes the de Rham cohomology (or Q-cohomology) $H^*(M)$, then Hamiltonian $H$ acts as $H=0$ (and thus the zero eigenvalues), and for the complex chain $(\Omega^\circ, M)$, one has $H$ homotopic to zero.

In this sense, we killed the time, but in a derived sense. In the process, we have gained a bigger algebra in the form of $\mathcal{N}=1$ SUSY algebra (some super Lie-group). Similarly, one can find theories with a larger number of SUSY supercharges. For a good reference on this, see Witten's paper and the notes on E-M and Langlands by Ben-Zvi.

Similarly, if we have a map $\Pi: A \to B$ where $A, B$ are two schemes, there is a pullback from the sheaves on $B$ to $A$

$$\mathcal{O}_B \to \Pi_* \mathcal{O}_A$$

and this also works well with the restrictions. Now, there are morphisms between the ringed spaces $(A,\mathcal{O}_A)$ and also works with restrictions. For example, $X \subset A$ be an open affine cover, then there is a morphism of ringed spaces

$$(X, \mathcal{O}_{A|X}) \to (A,\mathcal{O}_{A})$$

One also knows that the morphisms of ringed spaces admit compositions. Then one sees that the category of ringed spaces is formed (with objects being ringed spaces and the morphisms of ringed spaces as maps between them).

If taken an open covering over $A$ like $\cup_i\  \mathcal{U}_i = A$, then the morphisms agree on the overlap and there exists a unique map for each $\mathcal{U}_i \to B$ which will be $\Pi_i$. Now, a morphism of ringed spaces does not mean always a morphism of rings. 

For reference, you know what to look for (the rising sea). 

Before I end this post; the inaugural issue of Anveshanā (a bi-yearly magazine) is up now and contains, among many things, interviews and perspectives on mathematics, physics, and arts. It can be accessed at the following:

https://anveshanamagazine.github.io/ or link to full pdf.

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.