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arXiv:2411.03381 · Journal of Physics A Mathematical and Theoretical

Young Researchers School 2024 Maynooth: lectures on CFT, BCFT and DCFT

Christian Northe

Published Dec 19, 2024• 3 citations
arXivHTMLPDF

Abstract

Abstract These notes were presented at the Young Researchers School (YRS) in Maynooth in April 2024 and provide an introduction to conformal field theory (CFT), boundary CFT and Defect CFT. This class is mostly self-contained and includes exercises with solutions. The first part of these notes is concerned with the basics of CFT, and was taught by the author during the pre-school for the YRS 2024. Here the aim is to convey the notion of conformal families, their fusion and the construction of partition functions. The second part of these notes is dedicated to boundaries and defects in CFT and was presented by the author at the main school. As far as boundaries are concerned, emphasis is placed on boundary operators and their state spaces, as well as the boundary state formalism with the Cardy constraint. Topological defects are discussed in analogy, i.e. defect state spaces and the relevant consistency constraint are derived. Verlinde lines are constructed as their simplest solution and their properties are inspected.

DOI
10.1088/1751-8121/ada1b4

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Abstract These notes were presented at the Young Researchers School (YRS) in Maynooth in April 2024 and provide an introduction to conformal field theory (CFT), boundary CFT and Defect CFT. This class is mostly self-contained and includes exercises with solutions. The first part of these notes is concerned with the basics of CFT, and was taught by the author during the pre-school for the YRS 2024. Here the aim is to convey the notion of conformal families, their fusion and the construction of partition functions. The second part of these notes is dedicated to boundaries and defects in CFT and was presented by the author at the main school. As far as boundaries are concerned, emphasis is placed on boundary operators and their state spaces, as well as the boundary state formalism with the Cardy constraint. Topological defects are discussed in analogy, i.e. defect state spaces and the relevant consistency constraint are derived. Verlinde lines are constructed as their simplest solution and their properties are inspected.
Abstract Motivated by questions about quantum information and classification of quantum field theories, we consider Conformal Field Theories (CFTs) in spacetime dimension d ≥ 5 with a conformally-invariant spatial boundary (BCFTs) or 4-dimensional conformal defect (DCFTs). We determine the boundary or defect contribution to the Weyl anomaly using the standard algorithm, which includes imposing Wess-Zumino consistency and fixing finite counterterms. These boundary/defect contributions are built from the intrinsic and extrinsic curvatures, as well as the pullback of the ambient CFT’s Weyl tensor. For a co-dimension one boundary or defect (i.e. d = 5), we reproduce the 9 parity-even terms found by Astaneh and Solodukhin, and we discover 3 parity-odd terms. For larger co-dimension, we find 23 parity-even terms and 6 parity-odd terms. The coefficient of each term defines a “central charge” that characterizes the BCFT or DCFT. We show how several of the parity-even central charges enter physical observables, namely the displacement operator two-point function, the stress-tensor one-point function, and the universal part of the entanglement entropy. We compute several parity-even central charges in tractable examples: monodromy and conical defects of free, massless scalars and Dirac fermions in d = 6; probe branes in Anti-de Sitter (AdS) space dual to defects in CFTs with d ≥ 6; and Takayanagi’s AdS/BCFT with d = 5. We demonstrate that several of our examples obey the boundary/defect a-theorem, as expected.
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This lecture note covers topics on boundary conformal field theory, modular transformations and the Verlinde formula, and boundary logarithmic CFT. An introductory review on CFT with boundary and a discussion of its applications to logarithmic cases are given. LCFT at c=−2c=-2c=−2 is mainly discussed.
These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions, with applications related to topics of particular interest: topics include the conformal bootstrap program, boundary conformal field theory, and applications related to the AdS/CFT correspondence. We assume the reader to be familiar with quantum mechanics at the graduate level and to have some basic knowledge of quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help.
These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions, with applications related to topics of particular interest: topics include the conformal bootstrap program, boundary conformal field theory, and applications related to the AdS/CFT correspondence. We assume the reader to be familiar with quantum mechanics at the graduate level and to have some basic knowledge of quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help.
These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran. These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories. The two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic c=-2 theory.