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 Conformal blocks from celestial gluon amplitudes. Part II. Single-valued correlators

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Wei Fan ◽  
Angelos Fotopoulos ◽  
Stephan Stieberger ◽  
Tomasz R. Taylor ◽  
Bin Zhu
Keyword(s):  
Field Operator ◽  
Conformal Block ◽  
Primary Field ◽  
Branch Points ◽  
Minimal Models ◽  
Group Representations ◽  
Conformal Blocks ◽  
Crossing Symmetry ◽  
Entire Complex ◽  
Complex Spin

Abstract In a recent paper, here referred to as part I, we considered the celestial four-gluon amplitude with one gluon represented by the shadow transform of the corresponding primary field operator. This correlator is ill-defined because it contains branch points related to the presence of conformal blocks with complex spin. In this work, we adopt a procedure similar to minimal models and construct a single-valued completion of the shadow correlator, in the limit when the shadow is “soft.” By following the approach of Dotsenko and Fateev, we obtain an integral representation of such a single-valued correlator. This allows inverting the shadow transform and constructing a single-valued celestial four-gluon amplitude. This amplitude is drastically different from the original Mellin amplitude. It is defined over the entire complex plane and has correct crossing symmetry, OPE and bootstrap properties. It agrees with all known OPEs of celestial gluon operators. The conformal block spectrum consists of primary fields with dimensions ∆ = m + iλ, with integer m ≥ 1 and various, but always integer spin, in all group representations contained in the product of two adjoint representations.

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 fR Gravity Effects on Charged Accelerating AdS Black Holes Using Holographic Tools

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Adil Belhaj ◽  
Hasan El Moumni ◽  
Karima Masmar
Keyword(s):  
Black Holes ◽  
Correlation Functions ◽  
Entanglement Entropy ◽  
De Sitter ◽  
Four Dimensions ◽  
Holographic Entanglement Entropy ◽  
Ads Black Holes ◽  
Gravity Effects ◽  
Behavioral Situation ◽  
Point Correlation

We investigate numerically fR gravity effects on certain AdS/CFT tools including holographic entanglement entropy and two-point correlation functions for a charged single accelerated Anti-de Sitter black hole in four dimensions. We find that both holographic entanglement entropy and two-point correlation functions decrease by increasing the acceleration parameter A, matching perfectly with literature. Taking into account the fR gravity parameter η, the decreasing scheme of the holographic quantities persist. However, we observe a transition-like point where the behavior of the holographic tools changes. Two regions meeting at such a transit-like point are shown up. In such a nomination, the first one is associated with slow accelerating black holes while the second one corresponds to a fast accelerating solution. In the first region, the holographic entanglement entropy and two-point correlation functions decrease by increasing the η parameter. However, the behavioral situation is reversed in the second one. Moreover, a cross-comparison between the entropy and the holographic entanglement entropy is presented, providing another counterexample showing that such two quantities do not exhibit similar behaviors.

BRAID GROUP REPRESENTATIONS ASSOCIATED WITH ${\mathfrak{sl}}_m$

1996 ◽  
Vol 05 (05) ◽  
pp. 637-660 ◽  
Author(s):  
RUTH J. LAWRENCE
Keyword(s):  
Braid Group ◽  
Correlation Functions ◽  
Jones Polynomial ◽  
Hecke Algebra ◽  
Young Diagrams ◽  
Group Representations ◽  
Point Correlation ◽  
The One ◽  
Elementary Topology

It has been seen elsewhere how elementary topology may be used to construct representations of the Iwahori-Hecke algebra associated with two-row Young diagrams, and how these constructions are related to the production of the same representations from the monodromy of n-point correlation functions in the work of Tsuchiya & Kanie and to the construction of the one-variable Jones polynomial. This paper investigates the extension of these results to representations associated with arbitrary multi-row Young diagrams and a functorial description of the two-variable Jones polynomial of links in S3.

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 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.

scholarly journals Further Results on a Function Relevant for Conformal Blocks

2020 ◽  
Author(s):  
Vincent Comeau ◽  
◽  
Jean-François Fortin ◽  
Witold Skiba ◽  
◽  
...  
Keyword(s):  
Differential Operator ◽  
Recurrence Relation ◽  
Correlation Functions ◽  
Previous Article ◽  
Proper Action ◽  
Conformal Blocks ◽  
Point Correlation

We present further mathematical results on a function appearing in the conformal blocks of four-point correlation functions with arbitrary primary operators. The H-function was introduced in a previous article and it has several interesting properties. We prove explicitly the recurrence relation as well as the D<sub>6</sub>-invariance presented previously. We also demonstrate the proper action of the differential operator used to construct the H-function.

scholarly journals Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

2021 ◽  
Vol 11 (5) ◽  
Author(s):  
Nikita Nemkov ◽  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Theta Functions ◽  
Conformal Dimension ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Crossing Symmetry ◽  
Point Correlation ◽  
Structure Constants ◽  
Twist Fields

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

scholarly journals CORRELATION FUNCTIONS OF THE TRI-CRITICAL 3-STATE POTTS MODEL

2004 ◽  
Vol 19 (28) ◽  
pp. 2135-2145
Author(s):  
S. BALASKA ◽  
K. DEMMOUCHE
Keyword(s):  
Integral Representation ◽  
Potts Model ◽  
Operator Algebra ◽  
Correlation Functions ◽  
The Other ◽  
Critical Fields ◽  
Conformal Blocks ◽  
Bootstrap Approach ◽  
Point Correlation ◽  
Structure Constants

We build the Z3 invariants fusion rules associated to the (D4,A6) conformal algebra. This algebra is known to describe the tri-critical Potts model. The 4-point correlation functions of critical fields are developed in the bootstrap approach, and on the other hand, they are written in terms of integral representation of the conformal blocks. By comparing both expressions, one can determine the structure constants of the operator algebra.

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