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 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 On 2d CFTs that interpolate between minimal models

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Exactly Solvable ◽  
Structure Constants

We investigate exactly solvable two-dimensional conformal field theories that exist at generic values of the central charge, and that interpolate between A-series or D-series minimal models. When the central charge becomes rational, correlation functions of these CFTs may tend to correlation functions of minimal models, or diverge, or have finite limits which can be logarithmic. These results are based on analytic relations between four-point structure constants and residues of conformal blocks.

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.

scholarly journals YUKAWA COUPLINGS FOR BOSONIC ZN ORBIFOLDS: THEIR MODULI AND TWISTED SECTOR DEPENDENCE

1992 ◽  
Vol 07 (33) ◽  
pp. 3059-3070 ◽  
Author(s):  
S. STIEBERGER ◽  
D. JUNGNICKEL ◽  
J. LAUER ◽  
M. SPALIŃSKI
Keyword(s):  
Correlation Functions ◽  
Target Space ◽  
Twisted Sector ◽  
Yukawa Couplings ◽  
Point Correlation ◽  
Field Correlation ◽  
Twist Fields

The three-point correlation functions with twist fields are determined for bosonic ZN orbifolds. Both the choice of the modular background (compatible with the twist) and of the (higher) twisted sectors involved are fully general. We point out a necessary restriction on the set of instantons contributing to twist field correlation functions not obtained in previous calculations. Our results show that the theory is target space duality invariant.

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 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.

ON THE CLASSIFICATION OF W-ALGEBRAS

1995 ◽  
Vol 10 (10) ◽  
pp. 1413-1448
Author(s):  
DIRK VERSTEGEN
Keyword(s):  
Central Charge ◽  
Virasoro Algebra ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Bootstrap Approach ◽  
Finite Set

We review and extend the conformal bootstrap approach to the classification of quantum W-algebras. These are extensions of the Virasoro algebra by a finite set of primary fields. Explicit forms are given for the most general crossing-symmetric four-point functions. Together with a large c expansion of the conformal blocks, this gives a powerful tool for finding all W-algebras that are associative for generic values of the central charge c.

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.

Some three-point correlation functions in the η-deformed AdS5 × S5

2016 ◽  
Vol 31 (01) ◽  
pp. 1550224 ◽  
Author(s):  
Plamen Bozhilov
Keyword(s):  
Field Theory ◽  
Angular Momentum ◽  
String Theory ◽  
Correlation Functions ◽  
Finite Size ◽  
Semiclassical Approach ◽  
Point Correlation ◽  
Structure Constants

We compute some normalized structure constants in the [Formula: see text]-deformed [Formula: see text] in the framework of the semiclassical approach. This is done for the cases when the “heavy” string states are finite-size giant magnons carrying one angular momentum and for three different choices of the “light” state: primary scalar operators, dilaton operator with nonzero momentum, singlet scalar operators on higher string levels. Since the dual field theory is still unknown, the results obtained here must be considered as conjectures or as predictions from the string theory side.

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