Higher Spin Representations of K(E₁₀)

Abstract

We review the recently constructed non-trivial fermionic representations of the infinite-dimensional subalgebra K(e 10 ) of the hyperbolic Kac-Moody algebra e 10 . These representations are all unfaithful (and more specifically, of finite dimension). In addition we present their decompositions under the various finite-dimensional subgroups associated with some maximal supergravities in dimensions D ≤ 11, and the projectors for 'spin-7 2 ' which have not been given before. Those representations that have not been derived from supergravity still have to find a role and a proper physical interpretation in the conjectured correspondence between E 10 and M-theory. Nevertheless, they provide novel mathematical structures that could shed some light on fundamental questions in supergravity and on the possible role of K(E 10 ) as an 'R-symmetry' of Mtheory, and perhaps also on the algebra e 10 itself.

On Spinorial Representations of Involutory Subalgebras of Kac–Moody Algebras

“…This is related to the analogue of the spin-5 2 representation studied in [KN5]. The spin-3/2 representation of supergravity [NS, KNP] can also be obtained from this construction by taking a further quotient.…”

“…These actions extend the so-called 'generalised holonomy' groups in the physics literature [DS, DL, H] to an infinite-dimensional context. Furthermore, and most relevant for the present work, some new finite-dimensional representations beyond supergravity were identified in [KN3,KN5], and their structure was further analysed in [KNV], where surprising features were discovered, such as the generic non-compactness of the quotient algebras (and quotient groups) obtained by dividing the original algebra by the annihilating ideal of the given representation. These representations were further studied in [HKL, LK] where in particular the action of the corresponding (covering) group representation ofK(G) [GHKW] was clarified; moreover, in complete analogy to the finite-dimensional situation it turns out that in the simple-laced case this two-fold covering groupK(G) is simply connected with respect to its natural topology [HK] (induced from the Kac-Peterson topology on the corresponding Kac-Moody group).…”

Generalised holonomies and K(E9)

“…Since the finite-dimensional spin- 1 2 and spin-3 2 representations furnish only infinitesimal glimpses of the huge Kac-Moody-type symmetry, it is important to understand the structure of its representations better. First steps were taken for K(e 10 ) by the construction of two new 'higher spin' representations in [25,26] (see also [21] for a review), where the 'spin' actually refers to DeWitt superspace. In contrast to the spin- 1 2 and spin-3 2 representations these can no longer be deduced from supergravity.…”

“…Unlike the quotient to W N , removing further so(16) representations within one of the constituent V i may require a careful check that the resulting space JHEP09(2021)054 is a representation of K(e 9 ). Nevertheless, the known examples of K(e 10 ) representations suggest that such extra truncations may be necessary for the existence of the over-extended Berman generator x 1 and for a consistent uplift to K(E 10 ) [26].…”

U-Dualities in Type II and M-Theory: A Covariant Approach

“…For larger U-duality groups the problem is even more complicated as first one encounters E 10 , which is an extension of the E 8 by two imaginary roots, and then E 11 , which must encode timelike direction as well. Some progress in the construction of such infinitely dimensional extended spaces and the corresponding theories has been made in [22][23][24][25].…”