A hitchhiker's guide to quantum field theoretic aspects of $\mathcal{N}=4$ SYM theory and its deformations

TL;DR

This thesis explores the quantum field theoretic properties of deformed $ ext{N}=4$ SYM theories, revealing how multi-trace interactions affect conformal invariance, integrability, and phase transitions, with detailed calculations and new methodological tools.

Contribution

It provides a detailed analysis of $eta$- and $ ext{N}=4$ SYM deformations, including the effects of multi-trace interactions on conformality, integrability, and thermal properties, along with new computational tools.

Findings

Multi-trace interactions induce running couplings that break conformal invariance.

The deformed theories share the same deconfinement temperature as the undeformed theory at one-loop.

A new Mathematica package, FokkenFeynPackage, automates Feynman diagram calculations.

Abstract

In this thesis, we investigate properties of the one-parameter $\beta$- and the three-parameter $\gamma_i$-deformed descendents of $\mathcal{N}=4$ SYM theory. We find additional multi-trace interactions that must be included in the deformed theories to prevent persistent divergences in observables. For the $\gamma_i$-deformation, we show in an explicit Feynman-diagrammatic calculation that these interactions have running coupling constants which spoil the conformal invariance of the quantised theory, even in the 't Hooft limit. We show that the running coupling constant in principle also affect the anomalous dimensions of composite operators by calculating the $K=L$ loop leading order wrapping corrections to the operators tr$\bigl(\phi_i^L)$. For the $\beta$-deformation, we analyse the impact of multi-trace couplings on the one-loop spectrum and determine the complete one-loop dilatation operator of the conformal $\beta$-deformation in the 't Hooft limit. Based on the field-theoretic data from the $\beta$- and $\gamma_i$-deformation, we propose a test to determine whether supersymmetry and/or exact conformal invariance are necessary prerequisites of the quantum integrability found for $\mathcal{N}=4$ SYM theory. We also generalise the P\'{o}lya-theoretic approach to the thermal one-loop partition function to be also applicable in the setting of the $\beta$- and $\gamma_i$-deformation. We find that the deconfinement phase-transition temperature in the deformed theories is the same as in the undeformed theory at one-loop level and we conjecture that it remains the same even non-perturbatively. Finally, we also provide many technical details concerning Feynman-diagram techniques and present the tool FokkenFeynPackage which implements these rules into Mathematica.