Higher Rank Conformal Fields in the $Sp(2M)$ Symmetric Generalized Space-Time

TL;DR

This work develops a $Sp(2M)$-invariant conformal framework in the generalized space-time $\mathcal{M}_M$ with matrix coordinates $X^{\alpha\beta}$ and organizes massless higher-spin fields as rank-$r$ tensor products of the singleton representation. Using the unfolded dynamics formalism, it reformulates equations as covariant constancy conditions and analyzes their content via $\sigma_-$ cohomology, identifying dynamical fields and their differential constraints. The central results show that bilinear currents built from rank-1 fields satisfy rank-2 equations in $\mathcal{M}_M$ and that rank-2 fields in $\mathcal{M}_M$ are equivalent to rank-1 fields in $\mathcal{M}_{2M}$, establishing an AdS/CFT‑type duality structure that extends across ranks to a chain of higher-order correspondences. Normalizable rank-2 solutions correspond to purely positive/negative frequency components and map to rank-1 data in the doubled space, while non-normalizable mixed sectors are excluded, reinforcing the proposed brane localization picture and suggesting a field-theoretical realization of brane-like preon states in higher-spin theory.

Abstract

We study various $Sp(2M)$ invariant field equations corresponding to rank $r$ tensor products of the Fock (singleton) representation of $Sp(2M)$. These equations are shown to describe localization on ``branes'' of different dimensions embedded into the generalized space-time $\M_M$ with matrix (i.e., ``central charge'') coordinates. The case of bilinear tensor product is considered in detail. The conserved currents built from bilinears of rank 1 fields in $\M_M$ are shown to satisfy the field equations of the rank 2 fields in $\M_M$. Also, the rank 2 fields in $\M_M$ are shown to be equivalent to the rank 1 fields in $\M_{2M}$.