$d$-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.

conformal symmetry. We can extract the coefficient by contracting with a (bare) three-point function of shadow operators, This is a tetrahedron: a 6j symbol , = The functional form of a three-point function is fixed by conformal symmetry. We can extract the coefficient by contracting with a (bare) three-point function of shadow operators, This is a tetrahedron: a 6j symbol , = The overlap of two partial waves -a group theoretic quantity-and the planar Feynman diagrams in an SYK correlation function -a dynamical quantity-are just two different ways of splitting a tetrahedron , =

, =
The overlap of two partial waves -a group theoretic quantity-and the planar Feynman diagrams in an SYK correlation function -a dynamical quantity-are just two different ways of splitting a tetrahedron , =

, =
There is a third context in which the 6j symbol appears: loop diagrams in AdS. The preamplitude for the triangle diagram is a 6j symbol

AdS
There is a third context in which the 6j symbol appears: loop diagrams in AdS.    Bosonic d-dimensional model would give same diagrams, but has negative directions. It is only formally defined.

AdS
One can construct a well-defined two-dimensional supersymmetric SYK model.
Perhaps a bosonic higher dimensional model will be found i , with Lagrangian theory will, at large N , be dominated by the same

Higher Dimensions
Bosonic d-dimensional model would give same diagrams, but has negative directions. It is only formally defined.
One can construct a well-defined two-dimensional supersymmetric SYK model.
Perhaps a bosonic higher dimensional model will be found i , with Lagrangian theory will, at large N , be dominated by the same

Higher Dimensions
Bosonic d-dimensional model would give same diagrams, but has negative directions. It is only formally defined.
One can construct a well-defined two-dimensional supersymmetric SYK model.
Perhaps a bosonic higher dimensional model will be found i , with Lagrangian theory will, at large N , be dominated by the same SYK 6-pt function of fundamentals (3-pt of bilinears) The first diagram is what we call the contact diagram, and the second is planar.
Let us look at the planar diagram. As we said before, it corresponds to three 3-pt functions glued together.

=
The contribution of the planar diagrams is given by gluing together three partially amp three-point functions [52], as depicted in Fig. 2(a), where the partially amputated three-point function involves stripping of the propagator on t leg. Since the inverse of the propagator is proportional to the two-point function of the sha 2 d vol(SO(d 1)) two of the three operators are scalars, t 0 was given previously in (2.38). Using refore write, tegral of four three-point structures, glued into a "tetrahedron" graph. Specifihree-point structures shares exactly one coordinate. This is a particular example the conformal group SO(d + 1, 1).
l can also be thought of as an overlap between conformal partial waves as follows.
sider the case J 1 = J 2 = 0. There is then a unique three-point structure for drop the structure label a. Performing the integrals over x a and x 3 in (2.50), We extract the coefficient by contracting with the functional form of a shadow three-point function.
A 6j symbol, as advertised earlier One can derive a simple formula for the diagrams appearing in the higher point functions.

4-pt function fundamentals (sum of ladders)
In 1d, h is dimension • These are simple rules for summing an infinite number of diagrams. It doesn't matter that the four-point function is made up of ladders. These apply to any four-point functions.
• This is not just an OPE expansion. The are the analytically extended OPE coefficients of the single-trace operators. The four-point function is a sum of conformal blocks of single-trace operators and double-trace operators. This emerges upon closing the contour.
• These are simple rules for summing an infinite number of diagrams. It doesn't matter that the four-point function is made up of ladders. These apply to any four-point functions.
• This is not just an OPE expansion. The are the analytically extended OPE coefficients of the single-trace operators. The four-point function is a sum of conformal blocks of single-trace operators and double-trace operators. This emerges upon closing the contour.
Computing the 6j Symbol the manifest tetrahedral symmetry of the 6j symbol is broken by choices of whic orm first. In calculating the 6j symbol, we will be forced to make such choices, an answer will not be manifestly symmetric under S 4 ; the presence of this symmetry nstitutes a strong check of our calculation.
with (3.1), we see that the integrals over x 5 and x 6 are simple to evaluate, eac nformal partial wave, and we recognize the 6j symbol to be the overlap of tw s previously shown in Fig. 1 ,

Need to evaluate integral
The conformal partial wave is a sum of a conformal block and the shadow block In 1d, the conformal block is the hypergeometric function 2 F 1 of a single cross ratio. One can evaluate the integral for the 6j symbol directly, to find a 4 F 3 In higher dimensions, the integral is harder. One would like to somehow make the integral factorize, into a product of onedimensional integrals. It turn out one can do this, by an appropriate analytic continuation of the contour into Lorentzian signature.
The conformal partial wave is a sum of a conformal block and the shadow block In 1d, the conformal block is the hypergeometric function 2 F 1 of a single cross ratio. One can evaluate the integral for the 6j symbol directly, to find a 4 F 3 In higher dimensions, the integral is harder. One would like to somehow make the integral factorize, into a product of onedimensional integrals. It turn out one can do this, by an appropriate analytic continuation of the contour into Lorentzian signature.
The conformal partial wave is a sum of a conformal block and the shadow block In 1d, the conformal block is the hypergeometric function 2 F 1 of a single cross ratio. One can evaluate the integral for the 6j symbol directly, to find a 4 F 3 In higher dimensions, the integral is harder. One would like to somehow make the integral factorize, into a product of onedimensional integrals. It turns out one can do this, by an appropriate analytic continuation of the contour into Lorentzian signature.
al wave, is a Euclidean inversion formula. It may be obtained starting from a four-poi en in contour integral form, applying orthogonality of partial waves. The function I ,J has poles at physic ions with residues encoding the OPE coe cients, In fact, one can apply Caron-Huot's Lorentzian inversion formula to our integral, which is a special case, with a four-point function that is a conformal partial wave.
First, recall that one can expand any four-point function in terms of partial waves Trivially, using the orthogonality of the partial waves, one can invert this e first line we introduced the coe cient function I ,J , dividing by n ,J for conve function I ,J contains all of the theory-specific information in the four poin and it will be the focus of this paper. In the second line we inserted (1.2) an bed the second term by extending the region of integration of the first term ormalizable contributions will be discussed in appendix B.2.
e can now understand how to recover the OPE presentation in (1.1): we defo ur of integration over to the right, picking up poles along the real axis ions of physical operators. The residues are proportional to p ,J . ften, we imagine using (1.1) and (1.4) to determine the four-point function in e we know the OPE data or expansion coe cient I ,J . However, for some applic useful to imagine applying the logic in reverse. Then we assume that the fou ion (or some contribution to it) is given, and we want to evaluate the corresp or coe cient function I ,J . To do this we take the pairing of with the fou ion. Using (1.3) and (1.4), we find an inversion formula 2 is formula, all four points are integrated over d-dimensional Euclidean space. B gauge-fixing the SO(d+1, 1) symmetry, this can be reduced to an integral ove al wave, is a Euclidean inversion formula. It may be obtained starting from a four-poi en in contour integral form, applying orthogonality of partial waves. The function I ,J has poles at physic ions with residues encoding the OPE coe cients, In fact, one can apply Caron-Huot's Lorentzian inversion formula to our integral, which is a special case, with a four-point function that is a conformal partial wave.
First, recall that one can expand any four-point function in terms of partial waves Trivially, using the orthogonality of the partial waves, one can invert this e first line we introduced the coe cient function I ,J , dividing by n ,J for conve function I ,J contains all of the theory-specific information in the four poin and it will be the focus of this paper. In the second line we inserted (1.2) an bed the second term by extending the region of integration of the first term ormalizable contributions will be discussed in appendix B.2.
e can now understand how to recover the OPE presentation in (1.1): we defo ur of integration over to the right, picking up poles along the real axis ions of physical operators. The residues are proportional to p ,J . ften, we imagine using (1.1) and (1.4) to determine the four-point function in e we know the OPE data or expansion coe cient I ,J . However, for some applic useful to imagine applying the logic in reverse. Then we assume that the fou ion (or some contribution to it) is given, and we want to evaluate the corresp or coe cient function I ,J . To do this we take the pairing of with the fou ion. Using (1.3) and (1.4), we find an inversion formula 2 is formula, all four points are integrated over d-dimensional Euclidean space. B gauge-fixing the SO(d+1, 1) symmetry, this can be reduced to an integral ove al wave, is a Euclidean inversion formula. It may be obtained starting from a four-poi en in contour integral form, applying orthogonality of partial waves. The function I ,J has poles at physic ions with residues encoding the OPE coe cients, In fact, one can apply Caron-Huot's Lorentzian inversion formula to our integral, which is a special case, with a four-point function that is a conformal partial wave.
First, recall that one can expand any four-point function in terms of partial waves Trivially, using the orthogonality of the partial waves, one can invert this e first line we introduced the coe cient function I ,J , dividing by n ,J for conve function I ,J contains all of the theory-specific information in the four poin and it will be the focus of this paper. In the second line we inserted (1.2) an bed the second term by extending the region of integration of the first term ormalizable contributions will be discussed in appendix B.2.
e can now understand how to recover the OPE presentation in (1.1): we defo ur of integration over to the right, picking up poles along the real axis ions of physical operators. The residues are proportional to p ,J . ften, we imagine using (1.1) and (1.4) to determine the four-point function in e we know the OPE data or expansion coe cient I ,J . However, for some applic useful to imagine applying the logic in reverse. Then we assume that the fou ion (or some contribution to it) is given, and we want to evaluate the corresp or coe cient function I ,J . To do this we take the pairing of with the fou ion. Using (1.3) and (1.4), we find an inversion formula 2 is formula, all four points are integrated over d-dimensional Euclidean space. B gauge-fixing the SO(d+1, 1) symmetry, this can be reduced to an integral ove formula d dimensions, as a double integral over cross-ratios,  Applying this we find the 6j symbols in 2d and 4d. It is expressed in terms of a product of two 4 F 3 's ese factors cancel the | | in the measure in (3.10), once again leading to a sum of factorized e-dimensional integrals.
The final result for the 6j symbol is (3.40) the notation of (3.4), the ordering in (3.39) is J 4 ( , J; 0 , J 0 | 1 , 2 , 3 , 4 ). This completes our evaluation of the 6j symbol. A simple check is that in the limit that one the operator dimensions goes to zero, the corresponding partial wave degenerates into a product two-point functions. For example, in four dimensions we have, (3.37) as given in (3.11) and k h i 2h ( ) was given in (3.14). The four-dimensional partial wave terms of these blocks in Eq. (2.15). For convenience, we have defined the "spin e Weyl reflection [25], e J ⌘ 2 J.
( 3.38) is straightforward to proceed as in two dimensions. One di↵erence is that in the four ase, both the s and t-channel partial waves contribute factors of 1/( ). However, cancel the | | 2 in the measure in (3.10), once again leading to a sum of factorized nal integrals.
l result for the 6j symbol is nts 1, 2, 3, 4. Thus, dDisc u t vanishes. To obtain the vanishing of a t-channel we can project this statement onto its di↵erent eigenspaces under monodromy of l.
we only need to include the integral over the region R 1 in (3.24), i.e. the integral The final result for the 6j symbol is, singularities. Analogous poles exist for spinning operators, for the appropriate spinning structure and Lorentz represe

6j symbols
In this section we compute 6j symbols for principal ser and four dimensions. We define the 6j symbol for these rep integral of a product of four conformal three-point structu where O i denotes an operator with dimension i 2 d 2 + iR of an operator, e O i , has dimension e i = d i . In tot representations O 1 , . . . , O 6 , together with four indices a, three-point structures. For most of this section, we cons three-point structure for the given representations, so we c It is useful to represent the 6j symbol graphically as a the six edges represent positions which are integrated ove point structure. Notice that every one of the six operato shadow operators appears once. Our notation for the 6j sy and (3.2) contain edges that do not meet at a vertex. For ex in any of the four three-point structures.
In defining the 6j symbol (3.1), we made an arbitrary structures contains the operator versus its shadow. A relat