Generalised holonomies and K(E$_9$)

TL;DR

The paper probes $K(\mathfrak{e}_9)$, the involutory subalgebra of the affine algebra $\mathfrak{e}_9$, uncovering an infinite family of unfaithful representations that, under $K(\mathfrak{e}_9)$ action, factor through two mutually commuting parabolic algebras with Levi subalgebra $\mathfrak{so}(16)_+ \oplus \mathfrak{so}(16)_-$. It introduces a detailed ideal structure, including chiral ideals and their commuting quotients, and shows how truncations produce finite-dimensional representations that can be pulled back to $K(\mathfrak{e}_9)$ via two embeddings $\rho_{\pm}$. The work demonstrates that the generalized holonomy in these representations decomposes into two decoupled chiral parabolics, with a clear route to uplift to $K(\mathfrak{e}_{10})$ subject to nontrivial consistency conditions; it also provides explicit spinorial realizations up to spin 7/2 and discusses how the results inform the larger symmetry picture in exceptional geometry and M-theory. The extension to $K(\mathfrak{e}_{10})$ hinges on the action of the remaining Berman generator $x_1$, and the paper derives concrete uplift constraints, illustrating both successes (in certain cases) and obstructions (as in the example with $V_0=16$, $V_1=128_c$) that illuminate the rich structure of these infinite-dimensional algebras.

Abstract

The involutory subalgebra K(E$_9$) of the affine Kac-Moody algebra E$_9$ was recently shown to admit an infinite sequence of unfaithful representations of ever increasing dimensions arXiv:2102.00870. We revisit these representations and describe their associated ideals in more detail, with particular emphasis on two chiral versions that can be constructed for each such representation. For every such unfaithful representation we show that the action of K(E$_9$) decomposes into a direct sum of two mutually commuting (`chiral' and `anti-chiral') parabolic algebras with Levi subalgebra $\mathfrak{so}(16)_+\,\oplus\,\mathfrak{so}(16)_-$. We also spell out the consistency conditions for uplifting such representations to unfaithful representations of K(E$_{10}$). From these results it is evident that the holonomy groups so far discussed in the literature are mere shadows (in a Platonic sense) of a much larger structure.

Figures (1)

• Figure 1: Dynkin diagram of $E_{10}$ with nodes labelled.