This post will contain only one ACT.

The kind of duality that exists between the electric charge and magnetic charge is very advanced and that extends to some of the wild mathematics. Let's explore some of the mathematics here. 

What mathematicians study as the Fourier transform between the lattice $H^2(M,\mathbb{Z})$ and the dual (complexified) torus $T$ is the same as electric-magnetic duality in $\mathcal{N}=4$ super Yang-Mills theory with gauge group $U(1)$. There is a lot to unpack there.

In a past Quill, we saw a twist that produces $\mathcal{N}=1$ quantum mechanics, and that killed time (so the Hamiltonian is zero in cohomology) in a derived sense. We will now see the $U(1)$ gauge theory. For mathematicians, it is a connection on the $U(1)$ bundle, and for physicists, it is the theory of electromagnetism in three dimensions. Now, we will do a twist of a similar kind here to get a bigger Lie algebra (super Lie algebra). There are two kinds of twists (see Ben-Zvi's lectures on Langlands). But the idea is to include some more operators than the Hamiltonian only and study the cohomology. We already have seen the standard SUSY algebra in this post. 

A-type Twist: We include the operators (like the Q-charge with nilpotency $Q^2=0$) and 16 supercharges. Now, instead of asking for the cohomology on the ordinary Hilbert space of $L^2$ functions, we want to look at the cohomology on the space of connections modulo gauge equivalence. The obtained cohomology is again a de-Rham cohomology with differential forms.

B-type Twist: Now, we include a complex connection corresponding to Higgs fields, to the total connection is $d+A+i\sigma$ where $\sigma$ is one-form on the three-dimensional manifold $M$ (see this). Now, the bundle is not ordinary $U(1)$ but a complex bundle which we will call $\mathbb{C}$, which will also complexify the structure group of the bundle. Anyhow, the cohomology is now studied for the connection on the $\mathbb{C}$ bundle. In this case, the vector space contains the holomorphic functions on $M$. The cohomology that will be of interest, in this vector space, is the Dolbeault cohomology with operators like $\bar{\partial}^2=0$ and so on. 

Now, the A-twist can be associated with the locally constant functions in the zeroth de Rham cohomology, i.e., $H^2(M,\mathbb{Z})$. This $\mathbb{C}(H^2(M,\mathbb{Z}))$ corresponds to the dual lattice of the complexified torus (that is locally how the holomorphic functions would look like in the B-twist). More precisely, $\mathbb{C}(H^2(M,\mathbb{Z}))$ and $\mathbb{C}(T_\mathbb{C}$) are related by the Fourier series. And the lattice $H^2(M,\mathbb{Z})$ and the dual torus $T_\mathbb{C}$ are related by the Fourier transform.

Coming to physics, in A-twist, the 't Hooft operators create magnetic charges and the Wilson operators in the B-twist create electric charges. From the above duality, these operators are also related. So A-twist and B-twist are the same theory with different elementary charges. This is a manifestation of electric-magnetic duality. A-twist is gauge group $G$ and the B-twist is with dual group $^VG$.

A moduli space in algebraic geometry, differential geometry, and algebraic topology is about the classifications of objects which are equivalent and those which are not equivalent. (In topology, it is usually called a classifying space.) In essence, it is a geometric interpretation of the solutions. For example, a moduli space of Riemman surface, a moduli space of Elliptic curves, and so on. 

Coming back to the compactifications, here, the moduli space is of Ricci flat metrics. The moduli of the compactification consist of the scalar fields (the massless modes) which are the solutions of the equations that are written for motion. These scalar fields correspond to an infinite tower of modes, also called Kaluza-Klein modes. One can take a family of Ricci flat metric $\mathcal{G}$ and a perturbation around it (as we discussed above). We can then show that some perturbations $\delta \mathcal{G}$ will correspond to these varying of $\mathcal{G}$ (which is equivalent to saying that the perturbations would not hurt the conditions of $K$ that we started with). These will still solve the equations of motion. And each variation (in $\delta \mathcal{G}$) will give us a massless scalar field in our compactified $4-$manifold. As is remarked in this paper, there is no need for symmetry for these variations (like one needs for the description of Goldstone bosons).

One then is left with different values, which only differ by the varying scalar fields. We can then write a family of the vacuas and call it the moduli space $\mathcal{M}$. Now, this is not very exciting because it predicts a large number (a very big number) of vacua for describing our four-dimensional world. This is why moduli stabilization is considered. Moduli is lifted to make the massless modest pick up some mass. (There are non-supersymmetric and supersymmetric realizations of this though.)