Towards a Finite-NN Hologram

We suggest that holographic tensor models related to SYK are viable candidates for exactly (ie., non-perturbatively in $N$) solvable holographic theories. The reason is that in these theories, the Hilbert space is a spinor representation, and the Hamiltonian (at least in some classes) can be arranged to commute with the Clifford level. This makes the theory solvable level by level. We demonstrate this for the specific case of the uncolored $O(n)^3$ tensor model with arbitrary even $n$, and reduce the question of determining the spectrum and eigenstates to an algebraic equation relating Young tableaux. Solving this reduced problem is conceptually trivial and amounts to matching the representations on either side, as we demonstrate explicitly at low levels. At high levels, representations become bigger, but should still be tractable. None of our arguments require any supersymmetry.Comment: 16 page

Contrasting SYK-like Models

We contrast some aspects of various SYK-like models with large-$N$ melonic behavior. First, we note that ungauged tensor models can exhibit symmetry breaking, even though these are 0+1 dimensional theories. Related to this, we show that when gauged, some of them admit no singlets, and are anomalous. The uncolored Majorana tensor model with even $N$ is a simple case where gauge singlets can exist in the spectrum. We outline a strategy for solving for the singlet spectrum, taking advantage of the results in arXiv:1706.05364, and reproduce the singlet states expected in $N=2$. In the second part of the paper, we contrast the random matrix aspects of some ungauged tensor models, the original SYK model, and a model due to Gross and Rosenhaus. The latter, even though disorder averaged, shows parallels with the Gurau-Witten model. In particular, the two models fall into identical Andreev ensembles as a function of $N$. In an appendix, we contrast the (expected) spectra of AdS$_2$ quantum gravity, SYK and SYK-like tensor models, and the zeros of the Riemann Zeta function.Comment: 45 pages, 17 figures; v2: minor improvements and rearrangements, refs adde

Massive Scattering Amplitudes in Six Dimensions

We show that a natural spinor-helicity formalism that can describe massive scattering amplitudes exists in $D=6$ dimensions. This is arranged by having helicity spinors carry an index in the Dirac spinor {\bf 4} of the massive little group, $SO(5) \sim Sp(4)$. In the high energy limit, two separate kinds of massless helicity spinors emerge as required for consistency with arXiv:0902.0981, with indices in the two $SU(2)$'s of the massless little group $SO(4)$. The tensors of ${\bf 4}$ lead to particles with arbitrary spin, and using these and demanding consistent factorization, we can fix $3-$ and $4-$point tree amplitudes of arbitrary masses and spins: we provide examples. We discuss the high energy limit of scattering amplitudes and the Higgs mechanism in this language, and make some preliminary observations about massive BCFW recursion.Comment: 37 pages; v2: minor improvements, JHEP versio

Characterization of the first hexacoordinate phosphorus compound with S→P←S bonds

The first example of a hexacoordinate phosphorus compound [S{6-t-Bu-4-Me-C6H2O}2]2P+(Cl-· C3H4N2) with two S→P bonds is reported. This compound can be construed as an oxophosphonium salt with double intramolecular coordination by sulfur atoms. X-ray structure reveals a facial arrangement of the ligands with two coordinating sulfur atoms cis to each other. The S→P distance of 2.334 (1) Å is one among very short coordinate bond distances between sulfur and phosphorus

The first structural study on a cyclic tricoordinate phosphorochloridite and a pentacoordinate phosphorane based on 1,2,3,5-protected myo-inositol-a new conformation of 1,3,2-dioxaphosphorinane ring

Treatment of the phosphoramidite {myo-C6H6-2-[OC(O)Ph]-1,3,5-(O3CH)-4,6-(O2P-NH-i-Pr)} with o-chloranil affords the first example of inositol-based pentacoordinate phosphorane {myo-C6H6-2-[OC(O)Ph]-1,3,5-(O3CH)-4,6-(O2P-NH-i-Pr)(1,2-O2C6Cl4)} (9) (X-ray structure) with a trigonal bipyramidal geometry at phosphorus. The six-membered 1,3,2-dioxaphosphorinane ring with the inositol residue has an unusual boat conformation in 9 which is quite different from that found in unrestrained rings investigated before, but is similar to that of its PIII chloro precursor {myo-C6H6-2-[OC(O)Ph]-1,3,5-(O3CH)-4,6-(O2PCl)} (X-ray structure). Also, a convenient and chromatography-free procedure for the protected myo-inositol derivative {myo-C6H6-2-[OC(O)Ph]-1,3,5-(O3CH)-4,6-(OH)2} is reported