Algebraic structures in the counting and construction of primary operators in free conformal field theory

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 PhilosophyThe 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 pointMT 201