In this thesis, the problem of counting gauge invariant operators in certain
supersymmetric theories is discussed.
These objects have a very important role in supersymmetric gauge theories,
since they can be used to describe the space of zero-energy solutions,
called moduli space, of such theories. In order to approach the counting
problem, a technique is used based on a function known in Algebraic Geometry
as the Hilbert series. For the examined theories, this can be considered
a a partition function counting gauge invariant operators in the field theory
according to their charges under quantum global symmetries.
In the first part of the thesis, particular focus will be given to the application
of the Hilbert series to conformal Chern-Simons theories living on the
world-volume of M2-branes probing different toric Calabi-Yau 4-fold singularities.
It will be shown how the Hilbert series can be combined with the
brane tiling formalism to characterise the mesonic moduli space of vacua of
a given theory through its generators and the relations they satisfy. Then,
toric duality for these theories will be presented, with special attention to
the role played by Hilbert series in making such feature manifest between
two or more theories. Finally, Chern-Simons theories living on M2-branes
probing cones over smooth toric Fano 3-folds and their mesonic Hilbert
series will be presented.
In the second part, it will be shown how the Hilbert series can be applied
to counting gauge invariant operators in supersymmetric generalisations of
Quantum Chromodynamics, known as SQCD theories. The discussion will
hinge on a specific class of theories, with N multiplets transforming in the
fundamental and anti-fundamental and one in the adjoint representation of
the gauge group. For each classical group, the Hilbert series of the moduli
space will be used to determine the dimension on the spaces, their generators
and to argue that they are all Calabi-Yau manifolds.