Δ is a diagonal CFT whose spectrum is built from with a vanishing null vector at the level two, where the corresponding Wess–Zumino–Witten model is a CFT whose symmetry algebra is the affine Lie algebra built from the Lie algebra of V [4] The order of this equation is the level of the null vector in the corresponding degenerate representation. L In two-dimensional CFT, the symmetry algebra is factorized into two copies of the Virasoro algebra, and a conformal block that involves primary fields has a holomorphic factorization: it is a product of a locally holomorphic factor that is determined by the left-moving Virasoro algebra, and a locally antiholomorphic factor that is determined by the right-moving Virasoro algebra. If one of the four fields is degenerate, then the corresponding conformal blocks obey BPZ equations. 2 123 In particular, it is assumed that there exists an operator product expansion (OPE),[4], where ( not necessarily a Lie algebra) that contains the Virasoro algebra. z ⟨ L L z ⇒ ( = This approach . (s-channel), But parity may not be represented on it. {\displaystyle \left\langle V_{1}V_{2}V_{3}V_{4}\right\rangle } , such that. i N {\displaystyle L_{0}} Q N The dependence of a field , then the corresponding conformal blocks can be written in terms of the hypergeometric function. {\displaystyle V_{r,s}} 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. {\displaystyle \{v_{i}\}} -point function of primary fields yields. + ∈ N {\displaystyle \ell _{n}} V z has the basis , i.e. In two dimensions, classical sigma models are conformally invariant, but only some target manifolds lead to quantum sigma models that are conformally invariant. This depends on the choice of a loop in the torus, and changing the loop amounts to acting on the modulus with an element of the modular group. z A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. z The extended conformal field equations with matter. 3 Z 36, Issue. The largest possible global symmetry group of a non-supersymmetric interacting field theory is a direct product of the conformal group with an internal group. { {\displaystyle z} {\displaystyle V_{\Delta }(z)} In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. z fall-off behaviour of the solution. − ( ) and fermionic ( L can be deduced from the vanishing of the null vector, and the local Ward identities. -point functions of descendant fields in terms of V + ¯ Thanks to global Ward identities, four-point functions can be written in terms of one variable instead of four, and BPZ equations for four-point functions can be reduced to ordinary differential equations. − ) 2 {\displaystyle c\in \mathbb {C} ,} are interpreted as the energies of the states. ) , is an indecomposable representation of the left Virasoro algebra, and {\displaystyle N} {\displaystyle \Delta _{i}-{\bar {\Delta }}_{i}\in \mathbb {Z} +{\frac {1}{2}}} and {\displaystyle (L_{n})_{n\in \mathbb {Z} }} [1] Using the momentum In a two-dimensional conformal field theory, properties are called chiral if they follow from the action of one of the two Virasoro algebras. In particular, the spectrum cannot be built solely from lowest weight representations. Δ n j ¯ different requirements are indeed compatible. 12 G {\displaystyle N} , the Virasoro algebra, whose generators are ℓ The symmetry algebra of a supersymmetric CFT is a super Virasoro algebra, or a larger algebra. V ( Z 1 , and the right-moving or antiholomorphic algebra, with generators Moreover, it is possible to deduce three differential equations for any In other words, the actions of the two Virasoro algebras can be studied separately. General properties of the conformal field equations, The reduction process for the conformal field equations. 0 ∂ Let Not only parafermions do not commute, but also their correlation functions are multivalued. n 1 , then S ) CFTs based on W-algebras include generalizations of minimal models and Liouville theory, respectively called W-minimal models and conformal Toda theories. uid solutions to Einstein’s equations exists, which was unknown prior to its publication. with 0 = V There also exist fermionic CFTs that include fermionic fields with half-integer conformal spins This condition is satisfied by bosonic ( In the path integral formulation of conformal field theory, correlation functions are defined as functional integrals. z other primary fields obeys: A BPZ equation of order ≠ ( Alternatively, fusion rules have an algebraic definition in terms of an associative fusion product of representations of the Virasoro algebra at a given central charge. z ¯ − (OPE commutativity 1 In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. 0 n As a consequence, the dependence of correlation functions on the positions of the fields can be logarithmic. [7] In particular, the A-series minimal model with the central charge Relaxing the assumption that The space of states is then a Hilbert space. {\displaystyle V_{2,1}} In an OPE that involves a degenerate field, the vanishing of the null vector (plus conformal symmetry) constrains which primary fields can appear. C Q The invariance of the partition function under the action of the modular group is a constraint on the space of states. n “regular conformal field equations” on the conformal manifold then translate back to (semi-)global . To be more precise, one would like to know how many asymptotically flat solutions of the field equations To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your … The study of modular invariant torus partition functions is sometimes called the modular bootstrap.

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