Conformal field theory physics notes by jakob schwichtenberg. Exactly solvable models of conformalinvariant quantum field. Minimal models of conformal field theory with such additional symmetries are considered. The reason is that in two dimensions conformal symmetry is a most interesting in. An introduction to conformal field theory iopscience.
Infinitedimensional lie algebras in 4d conformal quantum. A useful summary article introduction to conformal field theory and infinite dimensional algebras by david olive may be found in the book physics, geometry and topology, h. They enjoy an infinite symmetry algebra the conformal algebra. Various attempts at a theory of quantum gravity led to the development of string theory. A conformal field theory cft is a quantum field theory that is invariant under conformal transformations. In string theory where a cft is coupled to 2d gravity, the conformal symmetry is an infinite dimensional yet very lowdimensional, relatively to the starting group residual symmetry from the diff x weyl group which is a fullfledged local symmetry generators depend on both world sheet coordinates arbitrarily. Nuclear physics b241 1984 333380 northholland publishing company infinite conformal symmetry in two dimensional quantum field theory a a belavin, a m polyakov and a b zamolodchikov l d landau institute for theoretical physics, academy of sciences, kosygina 2. His paper infinite conformal symmetry in two dimensional quantum field theory, with alexander belavin, and alexander zamolodchikov, founded two dimensional conformal field theory.
By twisting the commutation relations between creation and annihilation operators, we show that quantum conformal invariance can be implemented in the 2d moyal plane. Nature these books still belong on the essential reading list for anyone wanting to gain a deep understanding of the subject. Infinite conformal symmetry in two dimensional quantum field theory a. Part ii surveys more advanced topics of conformal field theory such as the representation theory of the virasoro algebra, conformal symmetry within string theory, an axiomatic approach to euclidean conformally covariant quantum field theory and a mathematical interpretation of the verlinde formula in the context of moduli spaces of holomorphic. I do know quantum mechanicsclassical mechanics, but im not really an. Twisted conformal symmetry in noncommutative two dimensional quantum field theory. Both volumes of superstring theory are likely to remain standard reference works for years to come. In this sense a two dimensional field theory can exhibit a goldstone boson, although it must be free. In this paper we consider a conformal field theory with an infinite symmetry generated by nonlocal currents with fractional spins parafermions. I started reading about conformal field theory a few weeks ago. Pdf download conformal field theory free unquote books. Aascientific council for cybernetics of the ussr academy of sciences, ussr. Many two dimensional conformal theories are related to this model through the addition of different gauge symmetries. Conformal field theory is often studied in two dimensions where there is an infinite dimensional group of local conformal transformations, described by the holomorphic functions.
Chapter 9 foundations of twodimensional conformal quantum. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations. This is the situation that we are used to when studying quantum. Conformal field theory last updated january 24, 2020. Conformal boundary conditions and threedimensional. A two dimensional conformal field theory is a quantum field theory on a euclidean two dimensional space, that is invariant under local conformal transformations in contrast to other types of conformal field theories, two dimensional conformal field theories have infinite dimensional symmetry algebras. Entanglement entropy and conformal field theory iopscience. Quantum field theory stanford encyclopedia of philosophy.
Conformal invariance and surface critical behaviour j cardy mathematical background. Infinite conformal symmetry in twodimensional quantum field theory. Conformal field theory cft is a recent example of a physical theory undergoing the geometrization process. Nevertheless, there is a sense in which nearly a d s 2 gravity is well defined. The dual conformal field theory has one null dimension and can in certain cases be described as the discrete light cone quantization of a two dimensional deformed symmetricproduct conformal field. I tried to include references to most relevant papers, but the emphasis was on papers. We present an investigation of the massless, twodimensional, interacting field theories. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries and the case of several disjoint intervals. In a quantum field theory the conformal symmetry gives rise to ward identities that interrelate various correlation functions. It is shown that the local fields forming the operator algebra can be. Functional representations of conformal symmetry in. Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. Symmetries in conformal field theory chris elliott these are elementary notes on virasoro and a ne lie algebra symmetries in 2d conformal eld theory, prepared for a seminar talk at northwestern. Infinite additional symmetries in twodimensional conformal.
In two dimensions, there is an infinite dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Just as the algebra of conformal transformations is infinite dimensional, conformal supersymmetry is likewise an infinite dimensional algebra. Nuclear physics b241 1984 333380 northholland publishing company infinite conformal symmetry in twodimensional quantum field theory a a belavin, a m polyakov and a b zamolodchikov l d landau institute for theoretical physics, academy of sciences, kosygina 2. Consider a two dimensional theory with minkowski signature. The dual conformal field theory has one null dimension and can in certain cases be described as the discrete light cone quantization of a twodimensional deformed symmetricproduct conformal field. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of virasoro algebra, and that the. Their basic property is their invariance under an infinitedimensional group. He joined rutgers university 1990 where he cofounded rutgers new high energy theory center, and was named board of governors professor 2005.
The infinite dimensional lie algebras and groups have been suggested as candidates for a unified symmetry of superstring theory. Their basic property is their invariance under an infinite dimensional group of conformal analytic transformations. Due to this invariance, correlation functions must obey linear equations called conformal ward identities. This paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians, and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations. The infinitedimensional lie algebras and groups have been suggested as candidates for a unified symmetry of superstring theory. Infinite conformal symmetry in two dimensional quantum field theory. This is an explicit realization of an infinite dimensional symmetry as a quantum algebra. The full exact solution are known in terms of representation theory and therefore no pertubation theory is needed. Quantum field theory qft is the mathematical and conceptual framework for contemporary elementary particle physics. In order to understand the meaning of these equations, and therefore of conformal field theory, we define the classical limit of a conformal field theory as a limit in which the conformal weights of all primary fields vanish. Fundamental concepts of phase transitions, such as order parameters, spontaneous symmetry breaking, scaling transformations, conformal symmetry and anomalous dimensions, have deeply changed the modern vision of many areas of physics, leading to remarkable developments in statistical mechanics, elementary particle theory, condensed matter physics and string theory. We present a systematic way to construct theories for d 2 conformal supergravity with n 0, 1, 2 or 4 as gauge theories corresponding to infinite dimensio. Conformal symmetry ranks second in this totally unscienti. Zamolodchikov, infinite conformal symmetry in twodimensional quantum field theory, nucl.
In certain special theories socalled minimal models these relations take the form of differential equations whose solutions provide an explicit solution of the theory 29. It has already revealed new relations between different branches of mathematics, to mention only representation theory of infinite dimensional groups, theory of finite. The large n limit of the nonlinear sigma model at the nontrivial fixed point is shown to be an example of a conformal field theory, using zeta function regularization. If the address matches an existing account you will receive an email with instructions to reset your password. Conformal field theory wikimili, the best wikipedia reader. Conformal blocks are not just solutions of the conformal ward identities, but actually elements of a particular basis of solutions. Palchik institute of automation and electxometry 630090, novosibirsk, ussr dedicated to lm gelfand on his 75th birthday abstract the method for exact solution of a certain class of models of conformal quantum. It is also intriguing to notice that conformal field theory of critical percolation possesses a field. Beginners questions concerning conformal field theory. Hidden symmetries in two dimensional field theory arxiv vanity.
Infinite additional symmetries in the two dimensional conformal quantum field theory. A conformal field theory is a quantum field theory which is invariant under conformal transformations. We present an mvestlgauon of the massless, two dimensional, interacting field theories their basic property is their invanance under an lnfimtedlmenslonal group of conformal analytic transformations it is shown that the local fields forlmng the operator algebra can be classified according to the irreducible representations of vtrasoro algebra, and that the correlation functions. In some respects a d s 2 is a bit harder to understand than its higherdimensional siblings. We study the massless quantum field theories describing the critical points in two dimensional statistical systems. While it is possible for a quantum field theory to be scale invariant but not conformallyinvariant, examples are rare.
The wzw model is one of the few quantum field theories that can be solved exactly. A conformal field theory cft is a quantum field theory or statistical mechanics model at the critical point that is invariant under conformal transformations. Infinite conformal symmetry in twodimensional quantum field. The symmetry i will discuss with more detail is two dimensional conformal symmetry there is probably no need to motivate the choice of this symmetry. We present an investigation of the massless, two dimentional, interacting field theories. Belavin a a, polyakov a m and zamolodchikov a b 1984 infinite conformal symmetry in two dimensional quantum field theory nucl. His path integral formulation of string theory had profound and lasting impacts on the conceptual and mathematical understanding of the theory. We present an mvestlgauon of the massless, twodimensional, interacting field theories their basic property is their invanance under an lnfimtedlmenslonal group of conformal analytic transformations it is shown that the local fields forlmng the operator algebra can be classified according to the irreducible representations of vtrasoro algebra, and that the correlation. Joel scherk and john schwarz first proposed in 1974 that string theory could be the quantum theory of gravity. Infinite conformal symmetry in two dimensional quantum field theory a a belavin, a m polyakov and a b zamolodchikov l d landau institute for theoretical physics, academy of sciences, kosygma 2, i17334 moscow, ussr received 22 november 1983 we present an mvestlgauon of the massless, two dimensional, interacting field theories their. The spectrum is in representations of the conformal algebra similar to spectrum of hydrogen atom in representations of o3, or even o4,1. Foundations of twodimensional conformal quantum field theory in this chapter we study twodimensional conformally invariant quantum. He coauthored the famous bpz paper infinite conformal symmetry in twodimensional quantum field theory, with alexander polyakov and alexander belavin.
Zamolodchikov theoretical and mathematical physics volume 65, pages 1205 12 1985 cite this article. Two dimensional conformal field theory these are scale invariant 2d field theories. Infinite conformal symmetry of critical fluctuations in two dimensions. We present an investigation of the massless, twodimentional, interacting field theories. The main reason is that pure gravity in a d s 2 is inconsistent with the existence of finite energy excitations above the a d s 2 vacuum. Infinite additional symmetries in the twodimensional. Quantum field theory, grassmannians, and algebraic curves. Four dimensional conformal field theory models withrational correlation functions n m nikolov, ya s stanev and i t todorovstring partition functions, hilbert schemes and affine lie algebra representations on homology groups loriano bonora, andrey bytsenko and emilio elizaldeconstraint quantization of a worldline. Recently conformal eld theory appeared in yet another context, namely the \adscftcorrespondence, where also higher dimensional super conformal invariance is relevant. Twisted conformal symmetry in noncommutative twodimensional. Twisted conformal symmetry in noncommutative twodimensional quantum field theory. From the representation theory point of view, a primary is the lowest dimension operator in a given representation of the conformal algebra. Analyzing the two dimensional ising model with conformal eld.
His paper infinite conformal symmetry in twodimensional quantum field theory, with alexander belavin, and alexander zamolodchikov, founded twodimensional conformal field theory. Nonlocal parafermion currents in twodimensional conformal. Classical and quantum conformal field theory springerlink. In a quantum field theory the conformal symmetry gives rise to ward identities that interrelate. Infinite conformal symmetry in twodimensional quantum. Additional symmetries in twodimensional conformal field theory generated by spin currents are investigated.
What does infinite conformal symmetry in two dimensional. A realization of the infinite dimensional symmetries of. Their basic property is their invariance under an infinitedimensional group of conformal analytic. In a rather informal sense qft is the extension of quantum mechanics qm, dealing with particles, over to fields, i. On sigma model as a threedimensional conformal field theory. Pdf twisted conformal symmetry in noncommutative two. Motivated by the simplest case of the strongly coupled two flavor schwinger model, i will show that the trivial field theory of a free massless boson in one space dimension has a hidden s u 2. In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. Bpz belavin, polyakov and zamolodchikov, infinite conformal symmetry in twodimensional quantum field theory, nuclear phys. On the representation of operator algebras of the quantum. Conformal symmetry and its breaking in twodimensional. Northholland pubhslung company infinite conformal symmetry. Additional symmetries in twodimensional conformal field theory generated by spin s 89 1. For spins s 52 and s 3, the generators of the symmetry form associative algebras with quadratic determining relations.
Conserved parafermion currents were discovered by one of us v. Infinite additional symmetries in twodimensional conformal quantum field theory a. Zamolodchikov theoretical and mathematical physics volume 65. Infinite conformal symmetry in twodimensional quantum field theory a a belavin, a m polyakov and a b zamolodchikov l d landau institute for theoretical physics, academy of sciences, kosygma 2, i17334 moscow, ussr received 22 november 1983 we present an mvestlgauon of the massless, twodimensional, interacting field theories their. Conformal invariance and applications to statistical mechanics. In preparing these notes i referred to notes of tong ton and ginsparg gin89, as well as the book fs10 of frischman and sonnenschein. Functional representations of conformal symmetry in quantum field theory hugh osborn department of applied mathematics and theoretical physics, wilberforce road, cambridge cb3 0wa, england abstract representations of the conformal algebra are constructed in terms of functional di erential operators depending a scalar eld. Belavin a a, polyakov a m and zamolodchikov a b 1984 infinite conformal symmetry in twodimensional quantum field theory nucl.