Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge C≤1

Dotsenko, Victor; Fateev, V.A.

doi:10.1016/s0550-3213(85)80004-3

Cited by 715 publications

(504 citation statements)

References 12 publications

Supporting

Mentioning

481

Contrasting

Unclassified

Order By: Relevance

“…We see that in this normalization the special structure constants C (E) ± (α) andC (E) ± (α) are given simply by the Dotsenko-Fateev "screening" integrals [13], i.e., are trivial if no "screening" is required The correlation functions here are taken over the free scalar field ϕ(x) and the "screening operators" e −2iβϕ and e 2iβ −1 ϕ are weighted with the coupling constants −M and −M respectively, whileM is determined by the following relation πM γ(−β −2 ) = πM γ(−β 2 ) −β −2 (C.5)…”

Section: "Exponential" Normalization In Gmmmentioning

confidence: 89%

Three-point function in the minimal Liouville gravity

2005

View full text Add to dashboard Cite

The problem of the structure constants of the operator product expansions in the minimal models of conformal field theory is revisited. We rederive these previously known constants and present them in the form particularly useful in the Liouville gravity applications. Analytic relation between our expression and the structure constant in Liouville field theory is discussed. Finally we present in general form the three-and two-point correlation numbers on the sphere in the minimal Liouville gravity. Liouville gravityBy gravity usually we imply a dynamic theory of the metric structure on certain manifold. In the two dimensional case the latter is supposed to be a two-dimensional surface Σ (compact or non-compact) of certain topology and equipped with a Riemann metric g ab (x). To avoid problems with moduli in this paper I always imply that Σ is a sphere. Also I consider the euclidean version of the gravity, i.e., g ab is always non-degenerate and has positive signature. In the path integral approach the problem is reduced to the evaluation of a functional integral over all Riemann metric D[g] modulo the diffeomorphism equivalent g ab (x). E.g., the gravitational partition function of a sphere is formally written as(1.1)1 On leave of absence from Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117259 Moscow, Russia.2 Laboratoire Associé au CNRS URA-768Here A eff [g] is supposed to be an effective action induced by some generally covariant "matter" field theory living on the surface. General covariance ensures that A eff [g] is invariant under diffeomorphisms.In general case of massive matter A eff [g] is a non-local and quite complicated functional of the metric. The problem (1.1) appears very complicated. There is a drastic simplification however, if all the matter inducing A eff [g] is "critical", i.e., described by a conformal field theory (CFT). In this case the form of A eff [g] is very universal and simple, being called the Liouville action. This fact was first discovered by A.Polyakov in 1981 [1] by direct computations with free fields. In general CFT this statement simply follows from the form of conformal anomaly. Moreover, conceptually this form of effective action can be taken (with few additional assumptions) as the very definition of CFT.Due to the diffeomorphism invariance of A eff [g] there is a gauge fixing problem in (1.1). One of the most convenient gauge choices is the conformal gauge where the coordinates on Σ are (locally) chosen in the way that (this is always possible in two dimensions) g ab (x) = e 2bφ(x) δ ab (1.2)The scale factor here is described by a quantum field φ(x) called the Liouville field (see below for the definition of the parameter b). One can also fix the gauge in a covariant way choosing an arbitrary metric g (0) ab (x) as the reference one and requiring g ab (x) = e 2bφ(x) g (0) ab (x) (1.3)In the latter approach the Liouville field φ is a usual scalar under coordinate transformations. Gauge (1.2) then implies a particular choice of coord...

show abstract

Section: "Exponential" Normalization In Gmmmentioning

confidence: 89%

Three-point function in the minimal Liouville gravity

2005

View full text Add to dashboard Cite

show abstract

“…The induced gravity action I[g] being a non-local functional, it is natural to introduce one, or several fields in addition to the metric that render the action local. The minimal way to achieve this is by means of a single scalar field, B(x), as in Feigin-Fuks theory [68][69][70], which has a nonminimal coupling to the metric. Consider the following local action, invariant under general coordinate transformations applied to g µν and B:…”

Section: Jhep02(2016)167mentioning

confidence: 99%

The unitary conformal field theory behind 2D Asymptotic Safety

Nink

Reuter

2016

J. High Energ. Phys.

View full text Add to dashboard Cite

Being interested in the compatibility of Asymptotic Safety with Hilbert space positivity (unitarity), we consider a local truncation of the functional RG flow which describes quantum gravity in d > 2 dimensions and construct its limit of exactly two dimensions. We find that in this limit the flow displays a nontrivial fixed point whose effective average action is a non-local functional of the metric. Its pure gravity sector is shown to correspond to a unitary conformal field theory with positive central charge c = 25. Representing the fixed point CFT by a Liouville theory in the conformal gauge, we investigate its general properties and their implications for the Asymptotic Safety program. In particular, we discuss its field parametrization dependence and argue that there might exist more than one universality class of metric gravity theories in two dimensions. Furthermore, studying the gravitational dressing in 2D asymptotically safe gravity coupled to conformal matter we uncover a mechanism which leads to a complete quenching of the a priori expected Knizhnik-Polyakov-Zamolodchikov (KPZ) scaling. A possible connection of this prediction to Monte Carlo results obtained in the discrete approach to 2D quantum gravity based upon causal dynamical triangulations is mentioned. Similarities of the fixed point theory to, and differences from, non-critical string theory are also described. On the technical side, we provide a detailed analysis of an intriguing connection between the Einstein-Hilbert action in d > 2 dimensions and Polyakov's induced gravity action in two dimensions.

show abstract

“…The matrix model expression of the conformal block on a sphere is given by reinterpreting the Dotsenko-Fateev integral representation of it as the (beta-deformed) matrix integral [34][35][36][37] with a logarithmic potential. When the central charge c = 1 + 6Q 2 (Q = b + 1/b) equals one, or equivalently, b = i, the conformal block is represented by the usual matrix model.…”

Section: Introductionmentioning

confidence: 99%

Seiberg-Witten curve via generalized matrix model

Maruyoshi

Yagi

2011

J. High Energ. Phys.

View full text Add to dashboard Cite

Abstract:We study the generalized matrix model which corresponds to the n-point toric Virasoro conformal block. This describes four-dimensional N = 2 SU(2) n gauge theory with circular quiver diagram by the AGT relation. We first verify that the generalized matrix model is obtained from the perturbative calculation of the Liouville correlation function. We then derive the Seiberg-Witten curve for N = 2 gauge theory as a spectral curve of the generalized matrix model.

show abstract

Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge C≤1

Cited by 715 publications

References 12 publications

Three-point function in the minimal Liouville gravity

Three-point function in the minimal Liouville gravity

The unitary conformal field theory behind 2D Asymptotic Safety

Seiberg-Witten curve via generalized matrix model

Contact Info

Product

Resources

About