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 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 Towards Feynman rules for conformal blocks

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Hoback ◽  
Sarthak Parikh
Keyword(s):  
Field Theory ◽  
Effective Field Theory ◽  
Hypergeometric Functions ◽  
Power Series Expansion ◽  
Effective Field ◽  
Conformal Block ◽  
Tree Level ◽  
Conformal Blocks ◽  
Simple Set ◽  
Feynman Rules

Abstract We conjecture a simple set of “Feynman rules” for constructing n-point global conformal blocks in any channel in d spacetime dimensions, for external and exchanged scalar operators for arbitrary n and d. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the n-point comb channel blocks. We prove these rules for all previously known cases, as well as two new ones: the seven-point block in a new topology, and all even-point blocks in the “OPE channel.” The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block beyond those considered in this paper.

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 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 SEIBERG–WITTEN PREPOTENTIAL FROM WZNW CONFORMAL BLOCK: LANGLANDS DUALITY AND SELBERG TRACE FORMULA

2012 ◽  
Vol 27 (22) ◽  
pp. 1250129
Author(s):  
TA-SHENG TAI
Keyword(s):  
Trace Formula ◽  
Conformal Block ◽  
Selberg Trace Formula ◽  
Conformal Blocks ◽  
Langlands Duality

We show how SU(2) Nf = 4 Seiberg–Witten prepotentials are derived from [Formula: see text] four-point conformal blocks via considering Langlands duality.

scholarly journals On Fusion Rules in Logarithmic Conformal Field Theories

1997 ◽  
Vol 12 (10) ◽  
pp. 1943-1958 ◽  
Author(s):  
Michael A. I. Flohr
Keyword(s):  
Quantum Group ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Group Representations ◽  
Fusion Rules ◽  
Conformal Field ◽  
Almost All

We find the fusion rules for the cp,1 series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via c = c3p,3. This leads to the conjecture that (almost) all minimal models with c = cp,q, gcd (p,q) > 1, belong to the class of rational logarithmic conformal field theories.

scholarly journals Dessins d’enfants, Seiberg-Witten curves and conformal blocks

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jiakang Bao ◽  
Omar Foda ◽  
Yang-Hui He ◽  
Edward Hirst ◽  
James Read ◽  
...  
Keyword(s):  
Gauge Theories ◽  
Algebraic Curves ◽  
Partition Functions ◽  
Minimal Models ◽  
Conformal Blocks ◽  
Dessins D’Enfants ◽  
Mirror Map ◽  
Dessins D'enfants

Abstract We show how to map Grothendieck’s dessins d’enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d $$ \mathcal{N} $$ N = 2 supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.

DETERMINATION OF THE SINGULARITIES OF THE DYSON-SCHWINGER EQUATION FOR THE QUARK PROPAGATOR

1992 ◽  
Vol 07 (21) ◽  
pp. 5369-5386 ◽  
Author(s):  
P. MARIS ◽  
H.A. HOLTIES
Keyword(s):  
Branch Point ◽  
Quark Propagator ◽  
Ladder Approximation ◽  
Complex Conjugate ◽  
Branch Points ◽  
Complex Singularities ◽  
Running Coupling ◽  
Entire Complex ◽  
Timelike Region

In this paper we study the analytic continuation of the Dyson-Schwinger equation into the entire complex p2 plane. It is shown that in the usual ladder approximation with a running coupling, there are two complex-conjugate branch points and one real branch point in the timelike region. The real branch point corresponds presumably to a physical mass for a quark; but it is removed by the addition of a so-called confinement term to the bare gluon propagator: with this extra term, there are at least three pairs of complex-conjugate branch points. The origin of these complex singularities is not yet clear.

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.

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