Exceptional Reductions

TL;DR

This paper systematically derives new identities for exceptional Lie groups E7 and E6 from E8, with applications to supergravity theories and black hole solutions.

Contribution

It introduces a novel reduction method from E8 to E7 and E6, revealing new invariant identities and connecting them to supergravity and black hole physics.

Findings

Derived new identities for E7 and E6 from E8.

Connected invariant tensor structures to supergravity charge quantization.

Highlighted applications in black hole solution analysis.

Abstract

Starting from basic identities of the group E8, we perform progressive reductions, namely decompositions with respect to the maximal and symmetric embeddings of E7xSU(2) and then of E6xU(1). This procedure provides a systematic approach to the basic identities involving invariant primitive tensor structures of various irreprs. of finite-dimensional exceptional Lie groups. We derive novel identities for E7 and E6, highlighting the E8 origin of some well known ones. In order to elucidate the connections of this formalism to four-dimensional Maxwell-Einstein supergravity theories based on symmetric scalar manifolds (and related to irreducible Euclidean Jordan algebras, the unique exception being the triality-symmetric N = 2 stu model), we then derive a fundamental identity involving the unique rank-4 symmetric invariant tensor of the 0-brane charge symplectic irrepr. of U-duality groups, with potential applications in the quantization of the charge orbits of supergravity theories, as well as in the study of multi-center black hole solutions therein.