A 6-dimensional theory defined on $M_6$ does not easily admit a quantum field theory action (since $\int_{M_6} H \wedge \star H =0$). On $M_6$, one identifies a 3-form self-dual $H$ with a Bianchi identity
$$dH=0$$
which is similar to $dF=0$ for a 2-form $F$ in a four-dimensional gauge theory. Basically, $F$ is just a curvature of a $U(1)$ connection, and $H$ is a curvature of a $U(1)$ gerbe connection. We can nonetheless study $M_4$ by compactifying $M_6$ on 2-torus
$$M_6 = M_4 \times T^2 $$
Or alternatively, $M_5$ Cauchy hypersurface in $M_6$ has a symplectic form that can be quantized to give a Hilbert space. This will serve the data on $M_6$. But instead, we look at $T^2$ and take
$$ T^2 = {\mathbb{C}^2}/{\Lambda} $$
where $\mathbb{C}^2$ is a $u-v$ plane and $\Lambda$ is a lattice parametrized by $1, \tau$. Here $\tau$ is a point in the complex plane. Let us now endow $M_6$ with a metric $g$, and we can decompose this metric
$$g(M_6) = g(M_4) + g(T^2)$$
and we will now fix the metric on $T^2$. Now we will add a real scaling factor for $g(M_4)$, so
$$g(M_6) = t^2\ g(M_4) + g(T^2).$$ On $T^2$, the conformal strcuture is determined by a point $\tau$ in the upper-half of $\mathbb{C}^2$, modulo the action of $SL(2,\mathbb{Z})$. If one takes $t \rightarrow \infty$, one reduces $M_6$ self-dual gauge theory to a four-dimensional gauge theory; since it is conformal invariant, $t^2$ can be dropped from now. But the whole reduction depends on the canonical structure of $T^2$. The reduced gauge theory will have a symmetry $SL(2,\mathbb{Z})$. We now pull back $F$ to $M_6 = M_4 \times T^2$ with only non-trivial $SL(2,\mathbb{Z})$ terms
$$H = F \wedge dx + \star F \wedge dy$$
Since $dH=0$, we have $dF =0 $ and $d \star F=0$, which are the equation of motion for a $U(1)$ gauge theory in four dimensions. So a self-dual theory of 3-form $H$ in six dimensions is related to a 2-form $F$ defining gauge theory in four dimensions. One can generalize it by relating a self-dual theory in $4k+2$ dimensions with $2k$ form curvature in $4k$ dimensions. We now find that $F$ has a coupling parameter which is determined by $T^2$, namely $\tau$, modulo the action of $SL(2,\mathbb{Z})$
$$\tau = \frac{\theta}{2\pi} + \frac{4 \pi i}{e^2}$$
Even though six-dimensional gauge theory does not have an action, 4-dimensional gauge theory has an action which, in this case, is just the usual action for $U(1)$ gauge theory. But the presence of $SL(2,\mathbb{Z})$ symmetry implies the electric-magnetic duality, which we find inevitably in the reduction to 4-dimensional gauge theory. In this heuristic argument, we prove that reducing a gauge theory on $M_6$ to $M_4$ would admit a hidden symmetry $SL(2,\mathbb{Z})$, which is determined by the conformal structure of $T^2$.
