Conformal Field Theories (CFTs) are very interesting object: their enhanced
symmetry offers the possibility to study and solve them only using first principles,
without relying on perturbative approaches. Recently, the conformal bootstrap approach has proven very effective in this direction and has offered a new way to explore
the space of CFTs. A key ingredients of this method are the conformal blocks, whose
starting point is the study of three-point functions of conformal primaries.
The already powerful constraints imposed by conformal symmetry become stronger
when the theory under exam is supersymmetric. In this context it is necessary to
compute superconformal blocks, namely linear combination of conformal ones.
This thesis aims to analyze 4d N = 1 Superconformal Field Theories (SCFTs) which
enjoy a flavor symmetry. A prototypical example are the infrared fixed points of
superQCD in the conformal window. In that case the flavor symmetry is SU(N) ×
SU(N). We want to study the conserved current, to lay the groundwork for an indepth study of theories with any global symmetry.
In 4d N = 1 SCFTs, flavor currents jµ are part of a short supermultiplets, whose superprimary is a scalar operator J. In this work we start from the most general three-point
function involving two flavor supermultiplets and a third generic supermultiplet.
We begin by showing that in superspace there are only two independent structures
for which such correlator is non-zero and this constrains the third supermultiplet to
have vanishing R-charge and to transform in a suitable Lorentz representations (j, ¯j)
such that |j − ¯j| = 0, 2.
Once we have obtained the more generic form of the three-point function, in order
to find the OPE coefficients of the current, it is necessary to expand the correlator
in its components.
In our work we focus on extracting all three-point functions involving a flavor current,
the scalar superprimary and a third conformal primary.
To this aim we introduce differential operators acting in superspace, projecting the
supermultiplets onto its components. The striking consequence of this formalism is
that we easily obtain linear relations among operators in the same supermultiplet.
Finally, we obtain all OPE coefficients for the non supersymmetric correlation function ⟨OjµJ⟩. This allows to write the superconfromal blocks as a linear combination
of non-supersymmetric blocks, which are known. The next logical step is to expand
the same superspace correlator further and extract all three-point functions involving
two conserved current and a third operator. These correlators, combined with our
results, will allow a numerical bootstrap study of the four-point function ⟨jµjνjρjσ⟩,
opening a window on all 4d N = 1 SCFTs enjoying a global SU(N) symmetry.