  In this thesis we calculate the dimension of the conformal manifold (DCM) for
a class of 3d $\mathcal{N}=4$ supersymmetric theories called the
star-shaped-quiver (SSQ) theories. These theories are the mirror dual theories
of class $\mathfrak{s}$ theories, constructed as a compactification of the 4d
class $\mathcal{S}$ theories on a circle $\mathbb{S}^{1}$. The 4d class
$\mathcal{S}$ theories themselves are constructed as a compactification of the
6d $\mathcal{N}=\left(2,0\right)$ superconformal field theories on a Riemann
surface $C_{g,s}$ (with genus $g$ and $s$ punctures). The IR fixed point of
these theories can be strongly coupled, and an interesting probe of the fixed
point is the conformal manifold, the space of all exactly marginal
deformations. The supersymmetric index is a tool, developed in recent years,
that allows to calculate the DCM for supersymmetric theories. The index encodes
within it the exactly marginal operators, but to deduce the DCM involves a few
more tools that transform the problem into a group-theoretic one. We employ
these tools for the SSQ theories, first calculate the supersymmetric index and
then calculate the DCM. Our results are that the DCM of the 3d SSQ (or class
$\mathfrak{s}$) theories scales as $\sim g^{4}$ and $\sim s^{2}$, which is
significanly larger that the DCM of the related 4d class $\mathcal{S}$ theories
which scales linearly with $g$ and $s$.
