scholarly journals Conformal four-point correlation functions from the operator product expansion

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Jean-François Fortin ◽  
Valentina Prilepina ◽  
Witold Skiba
Keyword(s):  
Lorentz Group ◽  
Correlation Functions ◽  
Operator Product Expansion ◽  
Gegenbauer Polynomials ◽  
Operator Product ◽  
Symmetric Tensors ◽  
Conformal Blocks ◽  
Relevant Group ◽  
Point Correlation

Abstract We show how to compute conformal blocks of operators in arbitrary Lorentz representations using the formalism described in [1, 2] and present several explicit examples of blocks derived via this method. The procedure for obtaining the blocks has been reduced to (1) determining the relevant group theoretic structures and (2) applying appropriate predetermined substitution rules. The most transparent expressions for the blocks we find are expressed in terms of specific substitutions on the Gegenbauer polynomials. In our examples, we study operators which transform as scalars, symmetric tensors, two-index antisymmetric tensors, as well as mixed representations of the Lorentz group.

scholarly journals Logarithmic Operators in Conformal Field Theory and the ${\mathcal W}_\infty$-Algebra

1997 ◽  
Vol 12 (21) ◽  
pp. 3723-3738 ◽  
Author(s):  
A. Shafiekhani ◽  
M. R. Rahimi Tabar
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Operator Product Expansion ◽  
Field Theories ◽  
Operator Product ◽  
Tensor Operator ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Point Correlation

It is shown explicitly that the correlation functions of conformal field theories (CFT) with the logarithmic operators are invariant under the differential realization of Borel subalgebra of [Formula: see text]-algebra. This algebra is constructed by tensor-operator algebra of differential representation of ordinary [Formula: see text]. This method allows us to write differential equations which can be used to find general expression for three- and four-point correlation functions possessing logarithmic operators. The operator product expansion (OPE) coefficients of general logarithmic CFT are given up to third level.

scholarly journals Conformal two-point correlation functions from the operator product expansion

2020 ◽  
Vol 2020 (4) ◽  
Author(s):  
Jean-François Fortin ◽  
Valentina Prilepina ◽  
Witold Skiba
Keyword(s):  
Correlation Functions ◽  
Operator Product Expansion ◽  
Operator Product ◽  
Point Correlation

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

scholarly journals Efficient rules for all conformal blocks

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Jean-François Fortin ◽  
Wen-Jie Ma ◽  
Valentina Prilepina ◽  
Witold Skiba
Keyword(s):  
Operator Product Expansion ◽  
General Procedure ◽  
Conformal Block ◽  
Irreducible Representations ◽  
Operator Product ◽  
Projection Operators ◽  
Space Operator ◽  
Conformal Blocks ◽  
General Rules

Abstract We formulate a set of general rules for computing d-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism [1]. With these rules, the procedure for determining any conformal block of interest is reduced to (1) identifying the relevant projection operators and tensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate the bookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We present several concrete examples to illustrate the general procedure as well as to demonstrate and test the explicit application of the rules. In particular, we consider four-point functions involving scalars S and some specific irreducible representations R, namely 〈SSSS〉, 〈SSSR〉, 〈SRSR〉 and 〈SSRR〉 (where, when allowed, R is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations.

scholarly journals Isgur–Wise functions, Bjorken–Uraltsev sum rules and their Lorentz group interpretation

2015 ◽  
Vol 30 (10) ◽  
pp. 1543008
Author(s):  
L. Oliver ◽  
J.-C. Raynal
Keyword(s):  
Lorentz Group ◽  
Operator Product Expansion ◽  
Positive Measure ◽  
Sum Rules ◽  
Integral Formula ◽  
Operator Product ◽  
Lorentz Transformations ◽  
Sum Rule ◽  
Heavy Quark Limit ◽  
Higher Derivatives

In the heavy quark limit of QCD, using the Operator Product Expansion and the non-forward amplitude, as proposed by Nikolai Uraltsev, we formulate sum rules that generalize Bjorken and Uraltsev sum rules. We recover the Uraltsev lower bound for the slope of the Isgur–Wise (IW) function, that we generalize to higher derivatives. We show that these results have a clear interpretation in terms of the Lorentz group, since the IW function is given by an overlap between the initial and final light clouds, related by Lorentz transformations. Both the Lorentz group and the Sum Rules approaches are equivalent. Moreover, we formulate an integral representation of the IW function with a positive measure. Inverting this integral formula, we obtain the measure in terms of the IW function, allowing one to formulate criteria to decide if a given ansatz for the IW function is compatible or not with the sum rule constraints. We compare these theoretical constraints to some forms proposed in the literature.

scholarly journals USE OF NILPOTENT WEIGHTS IN LOGARITHMIC CONFORMAL FIELD THEORIES

2003 ◽  
Vol 18 (25) ◽  
pp. 4747-4770 ◽  
Author(s):  
S. MOGHIMI-ARAGHI ◽  
S. ROUHANI ◽  
M. SAADAT
Keyword(s):  
Correlation Functions ◽  
Brst Symmetry ◽  
Momentum Tensor ◽  
Field Theories ◽  
Scale Transformation ◽  
Operator Product ◽  
Conformal Weight ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Point Correlation

We show that logarithmic conformal field theories may be derived using nilpotent scale transformation. Using such nilpotent weights we derive properties of LCFT's, such as two and three point correlation functions solely from symmetry arguments. Singular vectors and the Kac determinant may also be obtained using these nilpotent variables, hence the structure of the four point functions can also be derived. This leads to non homogeneous hypergeometric functions. Also we consider LCFT's near a boundary. Constructing "superfields" using a nilpotent variable, we show that the superfield of conformal weight zero, composed of the identity and the pseudo identity is related to a superfield of conformal dimension two, which comprises of energy momentum tensor and its logarithmic partner. This device also allows us to derive the operator product expansion for logarithmic operators. Finally we discuss the AdS/LCFT correspondence and derive some correlation functions and a BRST symmetry.

CORRELATION FUNCTIONS AND OPE IN TWO-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (25) ◽  
pp. 2271-2279 ◽  
Author(s):  
YOSHIAKI TANII ◽  
SHUN-ICHI YAMAGUCHI
Keyword(s):  
Correlation Functions ◽  
Field Theories ◽  
Operator Product ◽  
Theory Approach ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Dimensional Gravity ◽  
Point Correlation ◽  
Liouville Field Theory ◽  
The Continuum

We compute a class of four-point correlation functions of physical operators on a sphere in the unitary minimal conformal field theories coupled to 2-dimensional gravity. We use the continuum Liouville field theory approach and they are obtained as integrals over the moduli (positions of the operators). We examine the integrands near the boundaries of the moduli space and compare their singular behaviors with the operator product expansion.

scholarly journals Conformal blocks from celestial gluon amplitudes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Wei Fan ◽  
Angelos Fotopoulos ◽  
Stephan Stieberger ◽  
Tomasz R. Taylor ◽  
Bin Zhu
Keyword(s):  
Gauge Group ◽  
Correlation Functions ◽  
Delta Function ◽  
Adjoint Representation ◽  
Conformal Block ◽  
Group Representations ◽  
Conformal Blocks ◽  
Four Dimensions ◽  
Point Correlation ◽  
Complex Spin

Abstract In celestial conformal field theory, gluons are represented by primary fields with dimensions ∆ = 1 + iλ, λ ∈ ℝ and spin J = ±1, in the adjoint representation of the gauge group. All two- and three-point correlation functions of these fields are zero as a consequence of four-dimensional kinematic constraints. Four-point correlation functions contain delta-function singularities enforcing planarity of four-particle scattering events. We relax these constraints by taking a shadow transform of one field and perform conformal block decomposition of the corresponding correlators. We compute the conformal block coefficients. When decomposed in channels that are “compatible” in two and four dimensions, such four-point correlators contain conformal blocks of primary fields with dimensions ∆ = 2 + M + iλ, where M ≥ 0 is an integer, with integer spin J = −M, −M + 2, …, M − 2, M. They appear in all gauge group representations obtained from a tensor product of two adjoint representations. When decomposed in incompatible channels, they also contain primary fields with continuous complex spin, but with positive integer dimensions.

scholarly journals A Bäcklund Transformation for Elliptic Four-Point Conformal Blocks

2018 ◽  
Vol 30 (07) ◽  
pp. 1840012
Author(s):  
André Neveu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Correlation Functions ◽  
Integral Transformation ◽  
Bäcklund Transformation ◽  
Integral Representations ◽  
Liouville Theory ◽  
Partial Differential ◽  
Conformal Blocks ◽  
Point Correlation

We apply an integral transformation to solutions of a partial differential equation for the five-point correlation functions in Liouville theory on a sphere with one degenerate field [Formula: see text]. By repeating this transformation, we can reach a whole lattice of values for the conformal dimensions of the four other operators. Factorizing out the degenerate field leads to integral representations of the corresponding four-point conformal blocks. We illustrate this procedure on the elliptic conformal blocks discovered in a previous publication.

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