The present work is addressed to defects and boundaries in quantum field theory considering the application to AdS/CFT correspondence. We examine interactions of fermions with spins localised on these boundaries. Therefore, an algebra method is emphasised adding reflection and transmission terms to the canonical quantisation prescription. This method has already been applied to bosons in two space-time dimensions before. We show the possibilities of such reflection-transmission algebras in two, three, and four dimensions. We compare with models of solid state physics as well as with the conformal field theory approach to the Kondo effect.
The present work is addressed to defects and boundaries in quantum field theory considering the application to AdS/CFT correspondence. We examine interactions of fermions with spins localised on these boundaries. Therefore, an algebra method is emphasised adding reflection and transmission terms to the canonical quantisation prescription. This method has already been applied to bosons in two space-time dimensions before. We show the possibilities of such reflection-transmission algebras in two, three, and four dimensions. We compare with models of solid state physics as well as with the conformal field theory approach to the Kondo effect.
We propose a prescription for describing correlation functions in higher-dimensional defect conformal field theories (DCFTs) by those in ancillary conformal field theories (CFTs) without defects, which is a vast generalization of the image method in two-dimensional boundary CFTs. A correlation function of n operators inserted away from a defect in a DCFT is represented by a correlation function of 2n operators in the ancillary CFT, each pair of which is placed symmetrically with respect to the defect. We establish the correspondence by matching the constraints on correlation functions imposed by conformal symmetry on both sides. Our method has the potential to shed light on new aspects of DCFTs from the viewpoint of conventional CFTs.
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.
We present a large and universal class of new boundary states which break part of the chiral symmetry in the underlying bulk theory. Our formulas are based on coset constructions and they can be regarded as a non-abelian generalization of the ideas that were used by Maldacena, Moore and Seiberg to build new boundary states for SU(N). We apply our expressions to construct defect lines joining two conformal field theories with possibly different central charge. Such defects can occur e.g. in the AdS/CFT correspondence when branes extend to the boundary of the AdS-space.
We consider conformal boundaries and defects in Euclidean conformal field theory (CFT) at finite temperature and infinite volume. We distinguish defects that wrap the thermal circle and defects that are localised at a point on it. In both set-ups, and for any dimension and co-dimension, we classify bulk one-point functions of scalar, vector and spin-2 primaries. We relate the low-temperature expansion of bulk scalar one-point functions to zero-temperature defect CFT data and new finite-temperature data of defect primaries. For localised defects, we derive one-point bootstrap sum rules using periodicity and the defect operator expansion. We consider several examples in free bulk CFTs: unitary and non-unitary boundary CFTs, a wrapping monodromy defect, and a localised magnetic line. We determine thermal defect CFT data in all examples. For defects wrapping the thermal circle, we compute thermodynamic observables. For localised defects, we observe a Hurwitz zeta function decomposition of one-point functions suggestive of an underlying dispersion relation.
Quantum field theory (QFT) is a rich and robust part of mathematical and theoretical physics, with a wide range of applications in statistical physics, condensed matter physics, particle physics and string theory. Many (if not most) of the applications in four space-time dimensions require some sort of systematic approximations. In contrast, quantum field theories in lower dimensions often display special properties which open the door, at least in principle, to their exact solution. In many cases, the methods used to obtain exact solutions are peculiar to low-dimensional space-times, but the exact solutions thus obtained can teach us a great deal about more realistic QFTs in higher dimensions. Thus, lower-dimensional QFTs provide excellent models for exploring deep theoretical and computational issues, in addition to their direct applications to physical processes in reduced dimensions, which include polymers, boundary fluctuations, spin chains, and quantum Hall systems. Such models are also vital testing grounds for fundamental theoretical ideas such as integrability, phase transitions, collective and non-perturbative quantum behaviour, and non-equilibrium statistical mechanics. We should also mention the effective dimensional reduction which occurs in the high temperature limit, or is induced by disorder. Low-dimensional quantum field theory thus sits at a unique intersection point of contemporary mathematics and physics, allowing many of these issues to be addressed rigorously and analytically. Surprising inter-relations continue to arise, from the ubiquitous nature of the Bethe ansatz and Calogero models, to the link between integrable spin chains and the anomalous dimensions of operators in gauge theories. Aspects of the latter connection have been known at least since the 1990s; they have recently received tremendous attention in connection with N =4 supersymmetric Yang--Mills theory (in four space-time dimensions) and the AdS/CFT correspondence. One unifying idea is the Bethe ansatz, whose 75th anniversary we celebrate this year, and whose wide-ranging impact in diverse branches of modern mathematical physics is made abundantly clear in this special issue. In addition, new ideas and techniques of significance to both mathematics and physics have emerged: the recent and spectacular progress on geometrical aspects of phase transitions made possible through the ideas of stochastic (Schramm) Loewner evolution, or SLE, is a particularly noteworthy example. All papers in this special issue are invited, refereed contributions from leading experts. Many are pedagogical reviews, capturing the current state of knowledge in a particular subject, putting the subject in context and outlining open challenges. Others are research papers, reflecting the rapid flux of new ideas and results in this rich and exciting field. Together, they represent a broad sample of the entire field. We would like to take this opportunity to thank all of the authors for their efforts in making this special issue possible, and Kazuhiro Hikami for his help in the early stages of this project. We hope it inspires many new developments!
The goal of these notes is to introduce, in a very elementary way, the idea of the anti de-Sitter/Conformal Field Theory (AdS/CFT) correspondence to condensed matter physicists. This theory relates a gravity theory in a (d+1)- dimensional anti-de Sitter spacetime to a strongly-coupled d- dimensional quantum field theory living on its boundary. The AdS/CFT correspondence can be used to study finite temperature real time processes, such as response functions and dynamics far from equilibrium in quantum critical points in condensed matter systems. Computation of these quantities is reduced to solving classical gravitational equations in one higher dimension than the original theory.