Compact vs non-compact scalars in dimensional reduction Starting from the vector multiplet in (5+1) dimensions with 8 supercharges (whose field content is a gauge field $A_\mu$ and a spinor field), if we successively dimensionally reduce to (2+1) dimensions with 8 supercharges, the vector multiplet in (2+1) dimensions has the following bosonic content:


*

*a gauge field $a_\mu$, which is dual to a scalar field $\phi$ [more precisely, the 3d Hodge dual of $f_{(2)} = da_{(1)}$ is a 1-form, which is the field strength of a scalar $\phi$] 

*three scalars $\varphi_i$ ($i = 1, 2, 3$)


Why is the scalar field $\phi$ referred to as a "compact scalar field" and the three scalars $\varphi_i$ referred to as "non-compact scalars"? The reference I have in mind is page 2 of this dissertation by Giulia Ferlito entitled "Mirror Symmetry in 3d supersymmetric gauge theories".
Isn't this a $T^3$ reduction of the 6d vector multiplet to a 3d vector multiplet? In that case, the three circles in $T^3 = S^1 \times S^1 \times S^1$ ought to denote compact directions. So I would have thought that three scalars $\varphi_i$ (which are essentially components of the 6d gauge field $A_\mu$ along the three compact directions, i.e. 3 $S^1$'s) are defined on compact $S^1$'s.
EDIT: For anyone who may stumble upon this post in the future, there is a footnote on page 1 of arXiv:hep-th/1606.01989 which answers part of this question, in addition to the replies by ACuriousMind, Prahar and the answer by user810003.
 A: A scalar field is called compact if it is circle-valued ($\phi(x) \sim \phi(x) + 2\pi$), and non-compact if it is $\mathbb{R}$-valued. For an abelian gauge theory with the identification $A \sim A + \mathrm{d}\alpha$, we say that the gauge group is U(1) if $\alpha$ is compact, or $\mathbb{R}$ if $\alpha$ is non-compact. The two theories differ in their spectra of gauge-invariant operators. For example, consider the Wilson loop:
$$W_{qC}(A) = e^{iq\oint_C A}.$$
Under a gauge transformation, 
$$W_{qC}(A) \mapsto W_{qC}(A) e^{i q \oint_C \mathrm{d}\alpha}.$$
If the closed loop $C$ is homologically trivial ($C = \partial D$ for some 2-disk $D$), then the Wilson loop is manifestly gauge invariant for any $q \in \mathbb{R}$. (Indeed it may be written purely in terms of the gauge-invariant field strength in this case). However, if $C$ is a non-trivial 1-cycle, we must be more careful. In a U(1) gauge theory (with $\alpha$ circle-valued), $\oint_C \mathrm{d}\alpha$ computes the winding number of $\alpha$ (times $2\pi$). Then $q$ must be an integer for the Wilson loop to be gauge-invariant. On the other hand, if $\alpha$ is $\mathbb{R}$-valued, it cannot wind, and $\oint_C \mathrm{d}\alpha = 0$ even if $C$ is non-trivial. Thus, any $q\in \mathbb{R}$ is allowed in this case. 
Similarly, one can ask if 't Hooft operators are gauge-invariant. An 't Hooft operator is the electric-magnetic dual of a Wilson operator. While a Wilson loop is defined by the holonomy of the "electric" gauge field $A$, an 't Hooft loop is defined by the holonomy of the dual "magnetic" gauge field $\hat A$. You can think of a Wilson line as inserting the worldline of a probe electric charge, and an 't Hooft line as inserting the worldline of a magnetic charge. 
In 3d, the dual of $A$ is a scalar $\phi$, and an 't Hooft "line" would take the form 
$$H_p(\phi(x))=e^{i p \phi(x)}.$$
As for the Wilson operators, we would like to determine what values of $p$ lead to gauge-invariant 't Hooft operators for a U(1) or $\mathbb{R}$ gauge theory.
In terms of the original electric degrees of freedom, the 't Hooft operator is defined by the prescription: delete the point $x$ from the 3-manifold and demand that the first Chern class of the gauge field on a 2-sphere surrounding $x$ is $p$:
$$\oint_{S^2} \frac{F}{2\pi} = p.$$
As I already mentioned, this is just the condition that a magnetic charge $p$ resides inside the sphere. 
Let us compute this magnetic flux on a 2-sphere with a given gauge field configuration.  We divide the sphere into northern and southern hemispheres $H_{N,S}$ with local gauge fields $A_{N,S}$. The two patches overlap on the equator, $S^1 = H_N \cap H_S$, where the gauge fields are related by a gauge transformation, $A_N = A_S + \mathrm{d}\alpha$. The magnetic charge is then
$$\oint_{S^2} F = \int_{H_N} \mathrm{d} A_N + \int_{H_S} \mathrm{d} A_S
= \oint_{S^1} (A_N-A_S) = \oint_{S^1} \mathrm{d}\alpha.$$
So $\oint F$ just computes the winding number of $\alpha$. If the gauge group is $\mathbb{R}$, the winding number must vanish, $p=0$, and there are no 't Hooft operators. If the gauge group is U(1), the winding number can be a non-zero integer, and there are gauge-invariant 't Hooft operators labeled by an integer $p$. Returning to the dual description $e^{i p \phi}$, we see that the dual scalar must be compact, $\phi \sim \phi + 2\pi$. A U(1) gauge field is therefore dual to a compact scalar in 3-dimensions. 
