Having Fun with Theoretical Physics

# Higher-Dimensional Supergravity

One of the longest unsolved problems in the study of supersymmetry is an irreducible off-shell formulation containing a finite number of component fields for the ten-dimensional (along with the extended and eleven-dimensional ones) supergravity multiplet has not been presented. An even simpler and unachieved task is to give a reducible off-shell formulation explicitly showing a finite number of component fields, which is the main goal of our series of work on this topic. In the forty years since 11D on-shell supergravity theory was constructed in 1978, a lot of efforts have been made to understand supergravity in superspace.

Inspired by the history of how Einstein constructed General Relativity, we study the linearized Nordstrom supergravity in ten- and eleven-dimensional superspaces in [JHEP07(2019)063]. We found no obstacles to applying the lessons we learned in 4D to higher dimensions. We also derive infinitesimal 10D superspace Weyl transformation laws. The identification of all off-shell ten-dimensional supergeometrical Weyl field strength tensors, constructed from respective torsions, is presented in [JHEP03(2021)074].

## Polytopic SUSY Representation Theory

Since the traditional approach requires a high computational price to be paid to elucidate the $\theta$-expansion for component fields residing within the superfields, we establish a novel approach founded by the polytopic SUSY representation theory. We realize that Lie Algebra techniques, in particular branching rules, Plethysm, and tensor product, provide the key to deciphering the complete list of independent fields that describe a supersymmetric multiplet in arbitrary spacetime dimensions efficiently

11D: [JHEP09(2020)089]

10D: [JHEP02(2020)176]

9D - 5D: [JHEP09(2021)202]

We show the explicit one-to-one correspondence between Lorentz irreps and field variables, leading to an ``adynkrafield'' formalism in which the traditional $\theta$-monomials are replaced by Young Tableaux in [ATMP25(2021)6].

Further elaborations on the ``scan'' machinery and the consequent prepotential candidates for 10D, N = I, IIA, and IIB superconformal supergravity theories are presented as well in [JHEP03(2021)074].

## N=4 Supersymmetric SYK Models

The study of the Sachdev-Ye-Kitaev (SYK) model has received a boost owing to its fantastic features as a solvable large N model. One of the most important properties is its quantum chaotic behavior which makes it a potential candidate for a holographic dual of AdS2 quantum gravity. In the past few years, various generalizations of these models have been considered, in particular, N=1 and 2 supersymmetric extensions of the SYK model have been constructed. For N=4 extension, one quiver model built entirely from 1D chiral supermultiplet has been presented in [Anninos, Anous, Denef, 2016].

In [JHEP06(2021)158] we established a neat algorithm for constructing N=4 models with SYK-like interactions by compactifying from 4D, N=1 supersymmetric Lagrangians involving chiral, vector, and tensor supermultiplets. Quartic fermionic vertices are generated via integrals over the whole superspace, while 2(q-1)-point fermionic vertices are generated via superpotentials.

We contributed to the discussion by including the complex linear supermultiplet in [JHEP03(2022)148]. The models that we constructed have random bosonic interactions with derivative couplings, indicating that N=4 systems with SYK-type interactions must include dynamical bosons.

## SUSY Holography Conjecture

The SUSY Holography conjecture was first proposed in [Gates, Linch, Phillips, 2002] as reducing higher-dimensional supersymmetric models to one dimension, and one-dimensional models encode the structure of their higher-dimensional counterparts. In this program, the key object to study is the one-dimensional models. The problem of classifying off-shell representations of the N-extended 1D super Poincare algebra is closely related to the classification of its graphical representations -- adinkras.

We define the adinkra ``height yielding matrix number'' (HYMN) and study HYMN equivalence classes for all GR(4,4) valise adinkras in [IJMPA2019]. More examples and discussions on GR(4,8), GR(5,8), and GR(6,8) adinkras can be found in [IJMPA2021].

In [JHEP05(2021)077], a conjecture is made that the weight space for 4D, N-extended supersymmetrical representations is embedded within the permutohedra associated with permutation groups S_d. The fact that Klein's Vierergruppe of S_4 plays the role of Hopping operators provides strong evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four-color problem on the truncated octahedron. This observation suggests an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of S_d.