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

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

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

Automatic Clustering with Single Optimal Solution

Determining optimal number of clusters in a dataset is a challenging task. Though some methods are available, there is no algorithm that produces unique clustering solution. The paper proposes an Automatic Merging for Single Optimal Solution (AMSOS) which aims to generate unique and nearly optimal clusters for the given datasets automatically. The AMSOS is iteratively merges the closest clusters automatically by validating with cluster validity measure to find single and nearly optimal clusters for the given data set. Experiments on both synthetic and real data have proved that the proposed algorithm finds single and nearly optimal clustering structure in terms of number of clusters, compactness and separation.Comment: 13 pages,4 Tables, 3 figure

(Anti-)Symmetrizing Wave Functions

The construction of fully (anti-)symmetric states with many particles, when the single particle state carries multiple quantum numbers, is a problem that seems to have not been systematically addressed in the literature. A quintessential example is the construction of ground state baryon wave functions where the color singlet condition reduces the problem to just two (flavor and spin) quantum numbers. In this paper, we address the general problem by noting that it can be re-interpreted as an eigenvalue equation, and provide a formalism that applies to generic number of particles and generic number of quantum numbers. As an immediate result, we find a complete solution to the two quantum number case, from which the baryon wave function problem with arbitrary number of flavors follows. As a more elaborate illustration that reveals complications not visible in the two quantum number case, we present the complete class of states possible for a system of five fermionic particles with three quantum numbers each. Our formalism makes systematic use of properties of the symmetric group and Young tableaux. Even though our motivations to consider this question have their roots in SYK-like tensor models and holography, the problem and its solution should have broader applications.Comment: v3: journal version, contains slightly expanded discussions and example