![]() ![]() 16 would permit left-right paths that cross fewer than two double-lines. The green blob should consist of connected diagrams only, or else the diagrams in Fig. A blob labeled “A” represents a sum over connected amputated planar diagrams (for the blob with four external double-lines, this is equivalent to summing over 1PI diagrams). This equation is used to solve for the green blob with six external double-lines. Note that diagrams B and C both arise from the same double-trace term that appears in the action for H after integrating out O. 3), so we will no longer draw them in subsequent figures. The contractible diagrams (namely, the O bubble diagrams) will be cancelled by counterterms (just as in Fig. Diagrams A and B wrap the double-trumpet. Each of the three connected diagrams above is labeled by a letter. Ignoring the H double-lines, a general diagram can be separated into connected diagrams. There can be many additional black double-lines which we have not explicitly drawn (these correspond to the H matrix). Each red double-line corresponds to the O matrix. Each of the two traces in ( 9.3) is represented by a black line. The top and bottom ends of the diagram are identified to obtain the cylinder topology. Reuse & PermissionsĪ ‘t Hooft diagram that contributes to the connected double-trace correlator ( 9.3) at leading order in the genus expansion. A black double-line represents the propagator of the H matrix. Bottom row: we provide an example of one of the infinitely many ways the diagrams in the top row can be filled in with planar ‘t Hooft diagrams involving the H matrix. The counterterm is a single-trace term in the potential and hence is represented by a single loop. In our terminology, the red double-line loop in the center is an example of an “ O bubble diagram.” The right disk represents a correction from the single-trace counterterm potential that is designed to cancel the contribution from the O bubble diagram. One should imagine filling in the regions inside and outside the red double-line loop with the ‘t Hooft diagrams of H in the SSS model of disk and cylinder topology, respectively. The middle disk represents a correction from a single insertion of the double-trace term in V ˜ ( H ) (in general, there could be arbitrarily many insertions, which are all summed over). One should imagine filling in the disk with all possible planar ‘t Hooft diagrams of H. Top row: The left disk represents the disk computation of ⟨ Tr e − β H ⟩ in the SSS model. A nonperturbative treatment of the matrix models discussed in this work is left for future investigations. The UV divergence indicates that the matrix integral saddle of interest is perturbatively unstable. To the extent we have checked, it is possible to reproduce the gravitational double-trumpet, which is UV divergent, from a systematic classification of matrix model ‘t Hooft diagrams. Applied to the models of interest, we find that these cylinder observables depend on the details of the double-scaling limit. We show that in any single-trace, two-matrix model, the genus zero two-boundary expectation value, with up to one O insertion on each boundary, can be computed directly from all of the genus zero one-boundary correlators. Separately, we design a model that reproduces certain double-scaled Sachdev-Ye-Kitaev correlators that may be scaled once more to obtain the disk correlators. One method involves imposing an operator equation obeyed by H and O as a constraint on the two matrices. We describe multiple ways to construct double-scaled matrix models that reproduce the gravitational disk correlators. The non-Gaussian statistics of the matrix elements of O correspond to a generalization of the eigenstate thermalization hypothesis ansatz. We study the matching of the genus zero-, one- and two-boundary expectation values in the matrix model to the disk and cylinder Euclidean path integrals. The single-boundary observables of interest are thermal correlation functions of O. ![]() One matrix is the Hamiltonian H of a holographic disorder-averaged quantum mechanics, while the other matrix is the light operator O dual to the bulk scalar field. We present evidence for a duality between Jackiw-Teitelboim gravity minimally coupled to a free massive scalar field and a single-trace two-matrix model.
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