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

Non-fermi liquid fixed point in a Wilsonian theory of quantum critical metals

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2015.Recently there has been signi cant interest in new types of metals called non-Fermi liquids, which cannot be described by Landau Fermi liquid theory. Landau Fermi liquid theory is a theoretical model used to describe low energy interacting fermions or quasiparticles. There is a growing interest in constructing an e ective eld theory for these types of metals. One of the paradigms to understand these metals is by the use of Wilsonian renormalization group (RG) to study a theoretical toy model consisting of fermions coupled to a gapless order parameter eld. Here we will study fermions coupled to gapless bosons (order parameter) below the upper critical dimension (d = 3). We will treat both fermions and bosons on equal footing and construct an e ective eld theory which only integrates out high momentum modes. Then we compute the one-loop RG ows for the Yukawa coupling and four-Fermi interaction. We will discuss log2 and log3 subleties associated with the one loop RG ows for the four-Fermi interaction and how they can be circumvented