The Minimal Unitary Representation of E_8(8)

TL;DR

This work constructs the minimal unitary representation of $E_{8(8)}$ on $L^2(\mathbb{R}^{29})$ by quantizing its 58-dimensional minimal coadjoint orbit, using a 5-graded decomposition under $E_{7(7)} \times SL(2,\mathbb{R})$ to obtain covariant, compact formulas. It provides both a Schrödinger/coordinate realization and an oscillator/Bargmann-Fock realization with explicit generators and the quartic invariant $I_4$ of $E_{7(7)}$, showing the $E_{8(8)}$ Casimir equals the constant $-120$. Under $E_{7(7)} \times SL(2,\mathbb{R})$ the representation decomposes into an integral of irreps labeled by $I_4$, and a truncation to the $SU(2,1)$ subgroup yields an irreducible minimal $SU(2,1)$ representation with a discrete spectrum of $SU(1,1)$ representations, including singleton cases. The results have potential applications in string and M-theory, notably in superconformal quantum mechanics and black hole spectroscopy, and motivate future work on connections to nonlinear quasi-conformal algebras that could extend this framework.

Abstract

We give a new construction of the minimal unitary representation of the exceptional group E_8(8) on a Hilbert space of complex functions in 29 variables. Due to their manifest covariance with respect to the E_7(7) subgroup of E_8(8) our formulas are simpler than previous realizations, and thus well suited for applications in superstring and M theory.

Figures (1)

• Figure 1: Numbering of simple roots of $E_{8(8)}$