We study the 6j symbol for the conformal group, and its appearance in three seemingly unrelated contexts: the SYK model, conformal representation theory, and perturbative amplitudes in AdS. The contribution of the planar Feynman diagrams to the three-point function of the bilinear singlets in SYK is shown to be a 6j symbol. We generalize the computation of these and other Feynman diagrams to d dimensions. The 6j symbol can be viewed as the crossing kernel for conformal partial waves, which may be computed using the Lorentzian inversion formula. We provide closed-form expressions for 6j symbols in d = 1, 2, 4. In AdS, we show that the 6j symbol is the Lorentzian inversion of a crossing-symmetric tree-level exchange amplitude, thus efficiently packaging the doubletrace OPE data. Finally, we consider one-loop diagrams in AdS with internal scalars and external spinning operators, and show that the triangle diagram is a 6j symbol, while one-loop n-gon diagrams are built out of 6j symbols.

/pdf/d-dimensional-syk-ads-loops-and-6j-symbols-2be7dlr0nm.pdf

d-dimensional SYK, AdS loops, and 6j symbols

We study the $6j$ symbol for the conformal group, and its appearance in three seemingly unrelated contexts: the SYK model, conformal representation theory, and perturbative amplitudes in AdS. The contribution of the planar Feynman diagrams to the three-point function of the bilinear singlets in SYK is shown to be a $6j$ symbol. We generalize the computation of these and other Feynman diagrams to $d$ dimensions. The $6j$ symbol can be viewed as the crossing kernel for conformal partial waves, which may be computed using the Lorentzian inversion formula. We provide closed-form expressions for $6j$ symbols in $d=1,2,4$. In AdS, we show that the $6j$ symbol is the Lorentzian inversion of a crossing-symmetric tree-level exchange amplitude, thus efficiently packaging the double-trace OPE data. Finally, we consider one-loop diagrams in AdS with internal scalars and external spinning operators, and show that the triangle diagram is a $6j$ symbol, while one-loop $n$-gon diagrams are built out of $6j$ symbols.

/pdf/d-dimensional-syk-ads-loops-and-6j-symbols-4zuttuww6n.pdf

$d$-dimensional SYK, AdS Loops, and $6j$ Symbols

We compute exactly various 4−point correlation functions of shortest scalar operators in bi-scalar planar four-dimensional “fishnet” CFT. We apply the OPE to extract from these functions the exact expressions for the scaling dimensions and the structure constants of all exchanged operators with an arbitrary Lorentz spin. In particular, we determine the conformal data of the simplest unprotected two-magnon operator analogous to the Konishi operator, as well as of the one-magnon operator. We show that at weak coupling 4−point correlation functions can be systematically expanded in terms of harmonic polylogarithm functions and verify our results by explicit calculation of Feynman graphs at a few orders in the coupling. At strong coupling we obtain that the correlation functions exhibit the scaling behaviour typical for semiclassical description hinting at the existence of the holographic dual.

/pdf/exact-correlation-functions-in-conformal-fishnet-theory-ol1998ai6s.pdf

Exact Correlation Functions in Conformal Fishnet Theory

We present the first-principles derivation of a weak-strong duality between the fishnet theory in four dimensions and a discretized stringlike model living in five dimensions. At strong coupling, the dual description becomes classical and we demonstrate explicitly the classical integrability of the model. We test our results by reproducing the strong coupling limit of the four-point correlator computed before nonperturbatively from the conformal partial wave expansion. Because of the extreme simplicity of our model, it could provide an ideal playground for holography with no supersymmetry. Furthermore, since the fishnet model and N=4 super Yang-Mills theory are continuously linked, our consideration could shed light on the derivation of AdS/CFT for the latter. For simplicity, in this Letter we restrict our considerations to a large subset of all states.

Derivation of the Holographic Dual of a Planar Conformal Field Theory in 4D.

We provide the eigenfunctions for a quantum chain of N conformal spins with nearest-neighbor interaction and open boundary conditions in the irreducible representation of SO(1,5) of scaling dimension Δ=2-iλ and spin numbers l=l[over ˙]=0. The spectrum of the model is separated into N equal contributions, each dependent on a quantum number Y_{a}=[ν_{a},n_{a}] which labels a representation of the principal series. The eigenfunctions are orthogonal and we computed the spectral measure by means of a new star-triangle identity. Any portion of a conformal Feynmann diagram with square lattice topology can be represented in terms of separated variables, and we reproduce the all-loop "fishnet" integrals computed by B. Basso and L. Dixon via bootstrap techniques. We conjecture that the proposed eigenfunctions form a complete set and provide a tool for the direct computation of conformal data in the fishnet limit of the supersymmetric N=4 Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.

Exactly Solvable Magnet of Conformal Spins in Four Dimensions.

We propose a D-dimensional generalization of 4D biscalar conformal quantum field theory recently introduced by Gurdogan and one of the authors as a particular strong-twist limit of γ-deformed N=4 supersymmetric Yang-Mills theory Similar to the 4D case, the planar correlators of this D-dimensional theory are conformal and dominated by "fishnet" Feynman graphs The dynamics of these graphs is described by the integrable conformal SO(1,D+1) spin chain In 2D, it is the analogue of Lipatov's SL(2,C) spin chain for the Regge limit of QCD but with the spins s=1/4 instead of s=0 Generalizing recent 4D results of Grabner, Gromov, Korchemsky, and one of the authors to any D, we compute exactly at any coupling a four-point correlation function dominated by the simplest fishnet graphs of cylindric topology and extract from it exact dimensions of operators with chiral charge 2 and any spin together with some of their operator product expansion structure constants

Biscalar Integrable Conformal Field Theories in Any Dimension.

We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar 4-point correlation functions given by conformal “fishnet” Feynman graphs. These diagrams are represented by a fragment of a regular square lattice of power-like propagators, arising in the recently proposed integrable bi-scalar fishnet CFT. The formula is derived from first principles, using the formalism of separated variables in integrable SL(2, ℂ) spin chain. It is generalized to anisotropic fishnet, with different powers for propagators in two directions of the lattice.

/pdf/basso-dixon-correlators-in-two-dimensional-fishnet-cft-22j0ij6l8j.pdf

Basso-Dixon correlators in two-dimensional fishnet CFT

We study the Feynman graph structure and compute certain exact four-point correlation functions in chiral CFT4 proposed by O. Gurdŏgan and one of the authors as a double scaling limit of γ-deformed $$ \mathcal{N}\mathrm{Unknown}\ \mathrm{character}\ \left(0\mathrm{x}\mathrm{F}700\right)\ \mathrm{from}\ "\mathrm{Euclid}\ \mathrm{Math}\ \mathrm{One}"\ \left(0\mathrm{x}3\mathrm{D}\right) = 4 $$
 SYM theory. We give full description of bulk behavior of large Feynman graphs: it shows a generalized “dynamical fishnet” structure, with a dynamical exchange of bosonic and Yukawa couplings. We compute certain four-point correlators in the full chiral CFT4, generalizing recent results for a particular one-coupling version of this theory — the bi-scalar “fishnet” CFT. We sum up exactly thecorresponding Feynman diagrams, including both bosonic and fermionic loops, by Bethe-Salpeter method. This provides explicit OPE data for various twist-2 operators with spin, showing a rich analytic structure, both in coordinate and coupling spaces.

/pdf/generalized-fishnets-and-exact-four-point-correlators-in-1vzhplp2o5.pdf

Generalized fishnets and exact four-point correlators in chiral CFT 4

We study the Feynman graph structure and compute certain exact four-point correlation functions in chiral CFT$_4$ proposed by O.Gurdogan and one of the authors as a double scaling limit of $\gamma$-deformed $\mathcal{N}=4$ SYM theory. We give full description of bulk behavior of large Feynman graphs: it shows a generalized "dynamical fishnet" structure, with a dynamical exchange of bosonic and Yukawa couplings. We compute certain four-point correlators in the full chiral CFT$_4$, generalizing recent results for a particular one-coupling version of this theory -- the bi-scalar "fishnet" CFT. We sum up exactly the corresponding Feynman diagrams, including both bosonic and fermionic loops, by Bethe-Salpeter method. This provides explicit OPE data for various twist-2 operators with spin, showing a rich analytic structure, both in coordinate and coupling spaces.

/pdf/generalized-fishnets-and-exact-four-point-correlators-in-2qkqudizxh.pdf

Enrico Olivucci

Papers

Biscalar Integrable Conformal Field Theories in Any Dimension.

Exactly Solvable Magnet of Conformal Spins in Four Dimensions.

Basso-Dixon correlators in two-dimensional fishnet CFT

Generalized fishnets and exact four-point correlators in chiral CFT 4

Generalized Fishnets and Exact Four-Point Correlators in Chiral CFT$_4$