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Rational Conformal Field Theory
In theoretical physics, a rational conformal field theory is a special type of two-dimensional conformal field theory with a finite number of conformal primaries. In these theories, all dimensions (and the central charge) are rational numbers that can be computed from the consistency conditions of conformal field theory. The most famous examples are the so-called minimal models. More generally, ''rational conformal field theory'' can refer to any CFT with a finite number of primary operators with respect to the action of its chiral algebra. Chiral algebras can be much larger than the Virasoro algebra. Well-known examples include (the enveloping algebra of) affine Lie algebras, relevant to the Wess–Zumino–Witten model In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edwa ..., and W-al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Conformal Field Theory
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. In some cases, this allows them to be solved exactly, using the conformal bootstrap method. Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models. Basic structures Geometry Two-dimensional conformal field theories (CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions. While a CFT might conceivably exist only on a given Riemann surface, its existence on any surface other than the sphere implies its existence on all surfaces. Given a CFT, it is indeed possible to glue two Riemann surfaces where it exists, and obtain t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Field
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. From the representation theory point of view, a primary is the lowest dimension operator in a given representation of the conformal algebra. All other operators in a representation are called ''descendants''; they can be obtained by acting on the primary with the raising generators. History of the concept Primary fields in a ''D''-dimensional conformal field theory were introduced in 1969 by Mack and Salam where they were called ''interpolating fields''. They were then studied by Ferrara, Gatto, and Grillo who called them ''irreducible conformal tensors'', and by Mack who called them ''lowest weights''. Polyakov used an equivalent definition as fields which cannot be represented as derivatives of other fields. The modern terms ''primary fields ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Model (physics)
In theoretical physics, a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified and solved, and found to obey an ADE classification. The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra. Relevant representations of the Virasoro algebra Representations In minimal models, the central charge of the Virasoro algebra takes values of the type : c_ = 1 - 6 \ . where p, q are coprime integers such that p,q \geq 2. Then the conformal dimensions of degenerate representations are : h_ = \frac\ , \quad \text\ r,s\in\mathbb^*\ , and they obey the identities : h_ = h_ = h_\ . The spectrums of minimal models are made of irreducible, degenerate lowest-weight representations of the Virasoro algebra, whose conformal dimensions are of the type h_ with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Lie Algebra
In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra \mathfrak, one considers the loop algebra, L\mathfrak, formed by the \mathfrak-valued functions on a circle (interpreted as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wess–Zumino–Witten Model
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten. A WZW model is associated to a Lie group (or supergroup), and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra). By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra. Action Definition For \Sigma a Riemann surface, G a Lie group, and k a (generally complex) number, let us define the G-WZW model on \Sigma at the level k. The model is a nonlinear sigma model whose action is a functional of a field \gamma:\Sigma \to G: :S_k(\gamma)= -\frac \int_ d^2x\, \mathcal \left (\gamma^ \partial^\mu \gamma, \gamma^ \partial_\mu \gamma \right ) + 2\pi k S^(\gamma). Here, \Sigma is equipped with a flat Eu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
W-algebra
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples. Definition A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields W^(z), including the energy-momentum tensor T(z)=W^(z). For h\neq 2, W^(z) is a primary field of conformal dimension h\in\frac12\mathbb^*. The generators (W^_n)_ of the algebra are related to the meromorphic fields by the mode expansions : W^(z) = \sum_ W^_n z^ The commutation relations of L_n=W^_n are given by the Virasoro algebra, which is parameterized by a central charge c\in \mathbb. This number is also called the central charge of the W-algebra. The commutation relations : _m, W^_n= ((h-1)m-n)W^_ are equivalent to the assumption that W^(z) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
