Irregular conformal blocks and connection formulae for Painlevé V functions

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.

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

Journal of High Energy Physics ◽

10.1007/jhep12(2020)019 ◽

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.

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Dispersion formulas in QFTs, CFTs and holography

Journal of High Energy Physics ◽

10.1007/jhep05(2021)098 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

David Meltzer

Keyword(s):

Fourier Transform ◽

Correlation Functions ◽

Momentum Space ◽

Conformal Blocks ◽

Dispersion Formula ◽

Space Dispersion

Abstract We study momentum space dispersion formulas in general QFTs and their applications for CFT correlation functions. We show, using two independent methods, that QFT dispersion formulas can be written in terms of causal commutators. The first derivation uses analyticity properties of retarded correlators in momentum space. The second derivation uses the largest time equation and the defining properties of the time-ordered product. At four points we show that the momentum space QFT dispersion formula depends on the same causal double-commutators as the CFT dispersion formula. At n-points, the QFT dispersion formula depends on a sum of nested advanced commutators. For CFT four-point functions, we show that the momentum space dispersion formula is equivalent to the CFT dispersion formula, up to possible semi-local terms. We also show that the Polyakov-Regge expansions associated to the momentum space and CFT dispersion formulas are related by a Fourier transform. In the process, we prove that the momentum space conformal blocks of the causal double-commutator are equal to cut Witten diagrams. Finally, by combining the momentum space dispersion formulas with the AdS Cutkosky rules, we find a complete, bulk unitarity method for AdS/CFT correlators in momentum space.

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Combinatorial expansions of conformal blocks

Theoretical and Mathematical Physics ◽

10.1007/s11232-010-0067-6 ◽

2010 ◽

Vol 164 (1) ◽

pp. 831-852 ◽

Cited By ~ 36

Author(s):

A. V. Marshakov ◽

A. D. Mironov ◽

A. Yu. Morozov

Keyword(s):

Conformal Blocks

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Connection formulae for differential representations in Stokes flow

Journal of Computational and Applied Mathematics ◽

10.1016/s0377-0427(00)00651-8 ◽

2001 ◽

Vol 133 (1-2) ◽

pp. 283-294 ◽

Cited By ~ 3

Author(s):

G. Dassios ◽

P. Vafeas

Keyword(s):

Stokes Flow ◽

Connection Formulae

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U(1) Chern-Simons theory and c=1 conformal blocks

Physics Letters B ◽

10.1016/0370-2693(89)90920-9 ◽

1989 ◽

Vol 223 (1) ◽

pp. 61-66 ◽

Cited By ~ 98

Author(s):

Michiel Bos ◽

V.P. Nair

Keyword(s):

Simons Theory ◽

Conformal Blocks ◽

Chern Simons ◽

Chern Simons Theory

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Parabolic Connection Formulae in Quantum Mechanics

Annals of Physics ◽

10.1006/aphy.1996.5663 ◽

1997 ◽

Vol 255 (2) ◽

pp. 305-332 ◽

Cited By ~ 5

Author(s):

L.V. Chebotarev

Keyword(s):

Quantum Mechanics ◽

Connection Formulae

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TETRAHEDRA AND POLYNOMIAL EQUATIONS IN TOPOLOGICAL FIELD THEORY

International Journal of Modern Physics A ◽

10.1142/s0217751x91001738 ◽

1991 ◽

Vol 06 (20) ◽

pp. 3571-3598 ◽

Cited By ~ 1

Author(s):

NOUREDDINE CHAIR ◽

CHUAN-JIE ZHU

Keyword(s):

Field Theory ◽

Topological Field Theory ◽

Fundamental Representation ◽

Conformal Blocks ◽

Conformal Field ◽

Witten Theory ◽

Topological Field ◽

General Program ◽

The One ◽

Chern Simons

Some tetrahedra in SUk(2) Chern-Simons-Witten theory are computed. The results can be used to compute an arbitrary tetrahedron inductively by fusing with the fundamental representation. The results obtained are in agreement with those of quantum groups. By associating a (finite) topological field theory (FTFT) to every rational conformal field theory (RCFT), we show that the pentagon and hexagon equations in RCFT follow directly from some skein relations in FTFT. By generalizing the operation of surgery on links in FTFT, we also derive an explicit expression for the modular transformation matrix S(k) of the one-point conformal blocks on a torus in RCFT and the equations satisfied by S(k), in agreement with those required in RCFT. The implication of our results on the general program of classifying RCFT is also discussed.

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