Infinite-Dimensional Algebraic $\mathfrak{Spin}$($N$) Structure in Extended/Higher Dimensional SUSY Holoraumy for Valise and On-Shell Supermultiplet Representations

TL;DR

The paper investigates how holoraumy, a higher-dimensional analogue of the SUSY commutator structure, interacts with Hodge duality beyond four dimensions. It shows that a holoraumy–duality link persists in 6D and 4D contexts but disappears in 10D, while 4D $\mathcal{N}=4$ vector–tensor multiplets can exhibit such links; through dimensional reduction to 1D, holoraumy yields an infinite-dimensional extension of $\mathfrak{Spin}(N)$, revealing a deep holographic connection between higher-dimensional SUSY and one-dimensional representations. The authors provide explicit 10D, 6D, and 4D Lagrangians and transformation laws, derive on-shell holoraumy for vector and tensor multiplets, and develop a 2D $(4,0)$ truncation to demonstrate the emergence of an infinite generator algebra via holoraumy. These results illuminate the algebraic structure underlying extended SUSY and suggest a powerful framework for embedding higher-dimensional kinematics into lower-dimensional, adinkra-like representations. The work highlights the subtle interplay between dualities, holoraumy, and dimensional reduction, with implications for SUSY holography and the classification of off-shell and on-shell multiplets.

Abstract

We explore the relationship between holoraumy and Hodge duality beyond four dimensions. We find this relationship to be ephemeral beyond six dimensions: it is not demanded by the structure of such supersymmetrical theories. In four dimensions for the case of the vector-tensor $\cal N$ = 4 multiplet, however, we show that such a linkage is present. Reduction to 1D theories presents evidence for a linkage from higher-dimensional supersymmetry to an infinite-dimensional algebra extending $\mathfrak{Spin}(N)$.