Physics of conformal field theories

Boundary conformal field theory (BCFT) is simply the study of conformal field theory (CFT) in domains with a boundary. It gains its significance because, in some ways, it is mathematically simpler: the algebraic and geometric structures of CFT appear in a more straightforward manner; and because it has important applications: in string theory in the physics of open strings and D-branes, and in condensed matter physics in boundary critical behavior and quantum impurity models. In this article, however, I describe the basic ideas from the point of view of quantum field theory, without regard to particular applications nor to any deeper mathematical formulations.Boundary conformal field theory (BCFT) is simply the study of conformal field theory (CFT) in domains with a boundary. It gains its significance because, in some ways, it is mathematically simpler: the algebraic and geometric structures of CFT appear in a more straightforward manner; and because it has important applications: in string theory in the physics of open strings and D-branes, and in condensed matter physics in boundary critical behavior and quantum impurity models. In this article, however, I describe the basic ideas from the point of view of quantum field theory, without regard to particular applications nor to any deeper mathematical formulations.

arXiv:hep-th/0411189Nov 21, 200419 citations

Introduction to Conformal Field Theories

Edoardo Lauria

Springer thesesJan 1, 20190 citations

Modern Approach to 2D Conformal Field Theory

Yuya Kusuki

The primary aim of these lecture notes is to introduce the modern approach to two-dimensional conformal field theory (2D CFT). The study of analytical methods in two-dimensional conformal field theory has developed over several decades, starting with BPZ. The development of analytical methods, particularly in rational conformal field theory (RCFT), has been remarkable, with complete classifications achieved for certain model groups. One motivation for studying CFT comes from its ability to describe quantum critical systems. Given that realistic quantum critical systems are fundamentally RCFTs, it is somewhat natural that the analytical methods of RCFT have evolved significantly. CFTs other than RCFTs are called irrational conformal field theories (ICFTs). Compared to RCFTs, the study of ICFTs has not progressed as much. Leaving aside whether there is physical motivation or not, ICFTs inherently possess a difficulty that makes them challenging to approach. However, with the development of quantum gravity, the advancement of analytical methods for ICFTs has become essential. The reason lies in the AdS/CFT correspondence. AdS/CFT refers to the relationship between $d+1$ dimensional quantum gravity and $d$ dimensional CFT. Within this correspondence, the CFT appears as a non-perturbative formulation of quantum gravity. Except in special cases, this CFT belongs to ICFT. Against this backdrop, the methods for ICFTs have developed rapidly in recent years. Many of these ICFT methods are indispensable for modern quantum gravity research. Unfortunately, these cannot be learned from textbooks on 2D CFTs, such as Yellow book. These lecture notes aim to fill this gap. Specifically, we will cover techniques that have already been applied in many studies, such as HHLL block and monodromy method, and significant results that have become proper nouns, such as Hellerman bound and HKS bound.The primary aim of these lecture notes is to introduce the modern approach to two-dimensional conformal field theory (2D CFT). The study of analytical methods in two-dimensional conformal field theory has developed over several decades, starting with BPZ. The development of analytical methods, particularly in rational conformal field theory (RCFT), has been remarkable, with complete classifications achieved for certain model groups. One motivation for studying CFT comes from its ability to describe quantum critical systems. Given that realistic quantum critical systems are fundamentally RCFTs, it is somewhat natural that the analytical methods of RCFT have evolved significantly. CFTs other than RCFTs are called irrational conformal field theories (ICFTs). Compared to RCFTs, the study of ICFTs has not progressed as much. Leaving aside whether there is physical motivation or not, ICFTs inherently possess a difficulty that makes them challenging to approach. However, with the development of quantum gravity, the advancement of analytical methods for ICFTs has become essential. The reason lies in the AdS/CFT correspondence. AdS/CFT refers to the relationship between $d+1$ dimensional quantum gravity and $d$ dimensional CFT. Within this correspondence, the CFT appears as a non-perturbative formulation of quantum gravity. Except in special cases, this CFT belongs to ICFT. Against this backdrop, the methods for ICFTs have developed rapidly in recent years. Many of these ICFT methods are indispensable for modern quantum gravity research. Unfortunately, these cannot be learned from textbooks on 2D CFTs, such as Yellow book. These lecture notes aim to fill this gap. Specifically, we will cover techniques that have already been applied in many studies, such as HHLL block and monodromy method, and significant results that have become proper nouns, such as Hellerman bound and HKS bound.

arXiv:2412.18307Dec 24, 20240 citations

Basics in Conformal Field Theory

Ralph Blumenhagen, Erik Plauschinn

Lecture notes in physicsJan 1, 20097 citations

Introduction to conformal field theory

Springer eBooksDec 4, 20072 citations

Entanglement Entropy and AdS/CFT

Koushik Bar, Supervisors Dr. Sunandan Gangopadhyay, Prasanta Kumar Panigrahi

These features the study of conformal quantum field theories, in dimensions greater than two dimension and very special case of two dimensions. Over the last two decades, the understanding and mathematical formulation of such theories has developed to a very mature state, and conformal field theory (CFT) has in uenced both mathematics and physics. As such, it can be considered a prototype example for a constructive interplay between these two subjects. The main feature of a conformal field theory is the invariance under conformal transformations. We also review the development in the field of entanglement entropy, the holographic aspect of it, and the AdS/CFT conjuncture. Describe the Ruy-Takayangai conjecture of Entanglement Entropy and study the reconstruction of the AdS metric using the same for two different case, which are, 1+1 CFT in Infinite Line, and 1+1 CFT in compact circle.These features the study of conformal quantum field theories, in dimensions greater than two dimension and very special case of two dimensions. Over the last two decades, the understanding and mathematical formulation of such theories has developed to a very mature state, and conformal field theory (CFT) has in uenced both mathematics and physics. As such, it can be considered a prototype example for a constructive interplay between these two subjects. The main feature of a conformal field theory is the invariance under conformal transformations. We also review the development in the field of entanglement entropy, the holographic aspect of it, and the AdS/CFT conjuncture. Describe the Ruy-Takayangai conjecture of Entanglement Entropy and study the reconstruction of the AdS metric using the same for two different case, which are, 1+1 CFT in Infinite Line, and 1+1 CFT in compact circle.

May 1, 20180 citations

d+1

dimensional quantum gravity and

d

dimensional CFT. Within this correspondence, the CFT appears as a non-perturbative formulation of quantum gravity. Except in special cases, this CFT belongs to ICFT. Against this backdrop, the methods for ICFTs have developed rapidly in recent years. Many of these ICFT methods are indispensable for modern quantum gravity research. Unfortunately, these cannot be learned from textbooks on 2D CFTs, such as Yellow book. These lecture notes aim to fill this gap. Specifically, we will cover techniques that have already been applied in many studies, such as HHLL block and monodromy method, and significant results that have become proper nouns, such as Hellerman bound and HKS bound.