| Summary: | Doctor of Philosophy
A thesis submitted to the Faculty of Science, University of The Witwatersrand, in
ful llment of the requirements for the degree of Doctor of Philosophy === The AdS/CFT correspondence relates conformal eld theories in d dimensions
to theories of quantum gravity, on negatively curved spacetimes in d+1 dimensions.
The correspondence holds even for free CFTs which are dual to higher
spin theories. Motivated by this duality, we consider a systematic study of
primary operators in free CFTs.
We devise an algorithm to derive a general counting formula for primary operators
constructed from n copies of a scalar eld in a 4 dimensional free conformal
eld theory (CFT4). This algorithm is extended to derive a counting
formula for fermionic elds (spinors), O(N) vector models and matrix models.
Using a duality between primary operators and multi-variable polynomials,
the problem of constructing primary operators is translated into solving for
multi-variable polynomials that obey a number of algebraic and di erential
constraints. We identify a sector of holomorphic primary operators which
obey extremality conditions. The operators correspond to polynomial functions
on permutation orbifolds. These extremal counting of primary operators
leads to palindromic Hilbert series, which indicates they are isomorphic to the
ring of functions de ned on speci c Calabi-Yau orbifolds. The class of primary
operators counted and constructed here generalize previous studies of primary
operators.
The data determining a CFT is the spectrum of primary operators and the
OPE coe cients. In this thesis we have determined the complete spectrum of
primary operators in free CFT in 4 dimensions. This data may play a role in
attempts to give a derivation of a holographic dual to CFT4. Another possible
application of our results concern recent studies of the epsilon expansion,
which relates explicit data of the combinatorics of primary elds and OPE
coe cients to anomalous dimensions of an interacting xed point === MT 2018
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