Quantum Field Theory at the Boundary

Abstract

This thesis explores various aspects of boundary conformal field theories. They provide acontinuum description of lattice systems with boundaries when both the bulk and the boundary are tuned to criticality. A given CFT may admit multiple conformally invariant boundary conditions which describe different boundary critical points of the same bulk system. In other words, there is plenty of room at the boundary. We explore this rich boundary behavior of several well-known bulk conformal field theories. After giving a brief introduction to the subject, we start in Chapter 2 by studying conformal boundary conditions in one of the simplest conformal field theories, namely a system of ? free scalars interacting only through the boundary with an ?(?) invariant interaction. By a combination of large ? and epsilon expansions, we provide evidence for the existence of non-trivial ?(?) BCFTs in 1 < ? < 4 where ? is the dimension of the boundary. We then point out that these models are closely related to long range ?(?) models which describe lattice systems with long range interactions. We continue the study of long range ?(?) models in Chapter 3, where we study the spectrum of heavy operators. To be specific, we consider operators with a charge ? under ?(?) symmetry and study their conformal dimensions in the limit of large ? and ? with ?/? held fixed. Chapter 4 is devoted to boundary critical behavior in the interacting ?(?) vector model. To study it, we use the idea that a boundary conformal field theory is Weyl equivalent to a CFT in anti-de Sitter (AdS) space. We recover the known boundary fixed points for ?(?) vector models and study these fixed points in a large ? expansion in general bulk dimension ? as well as in an epsilon expansion near ? = 2, 4 and 6. Then in Chapter 5, we use similar techniques to identify new boundary fixed points in the Gross-Neveu (GN) model. We verify the conjectured boundary F-theorem and compute several pieces of BCFT data at these boundary fixed points. This thesis is based on [1,2] with Simone Giombi and [3,4] with Simone Giombi and Elizabeth Helfenberger.