Date of Award
Doctor of Philosophy (PhD)
AdS-CFT, anti-de Sitter space, conformal field theory, Holography, hyperbolic space, lattice-holography
Elementary Particles and Fields and String Theory
The Anti-de Sitter/Conformal Field theory (AdS/CFT) correspondence, also known as holography, has been the focus of a great deal of interest and research for the last two decades. It has improved our understanding of general relativity and quantum field theories simultaneously through the interplay between these two different kinds of theories. However, there are still many aspects of holography that we do not understand or demand further analysis. Perturbative quantum field theory and perturbative metric expansion techniques are not equipped to investigate holography in some of the most interesting regimes such as the strongly interacting gravitational theory in anti-de Sitter space (that potentially can have a dual field theory at the boundary with a small number of degrees of freedom, hence away from the large N limit).
In this thesis, I have attempted to contribute to understanding holography in these more difficult regimes. Specifically, I have used lattice techniques in Wick-rotated Anti-de Sitter (AdS) spacetime to investigate holography. After a review of the properties of Anti-de Sitter space and some continuum results, lattice construction of the hyperbolic space and the algorithms used for analysis are discussed. Our initial attempt investigated free scalar fields in hyperbolic space in two and three spacetime dimensions. It has been demonstrated that properties of the bulk lattice geometry information are encoded in the dual CFT through mapping the mass spectrum of the bulk scalar field to the scaling dimension of the boundary CFT operators. Holography has also been investigated for the nearest-neighbor Ising spin model on a hyperbolic space at various temperatures. The principal focus has been measuring boundary correlation functions over a range of temperatures. This yields a temperature-dependent scaling exponent of the boundary operator. Scaling the magnetic susceptibility data at different lattice volumes also allows us to compute the same scaling exponent, demonstrating good agreement with the correlation function measurements. A phase transition temperature in the bulk geometry can be seen to correlate with a minimal boundary scaling dimension. Lattice simulation results are complemented with a discussion of the high-temperature expansion and the duality (in the sense of Kramers and Wannier).
Finally, we show how lattice techniques can be used to probe strong fluctuations of the bulk geometry. A new relation of the scaling dimensions to the bulk mass is derived in this regime. We show that the backreaction of fermionic matter fieldscan dramatically alter the nature of the bulk space. Hyperbolic space configuration dominates the dynamically generated ensemble of simplicial manifolds in the limit of a large number of K\"ahler-Dirac fields.
Apart from the apparent advantage of probing duality in this uncharted territory of strong geometry fluctuation and strongly coupled field theories in hyperbolic space, the tools developed in this thesis should allow investigation of different aspects of quantum information science and tensor network ideas.
Asaduzzaman, Muhammad, "Investigation of Holographic Lattice Theories" (2022). Dissertations - ALL. 1651.