Matryoshka of Special Democratic Forms

TL;DR

This paper explores special democratic p-forms with symmetric components, revealing a nested structure across dimensions and identifying invariant forms related to exceptional Lie groups and spin groups.

Contribution

It introduces a framework for mapping special democratic forms across dimensions and uncovers a remarkable nested structure involving invariant forms like G_2 and Spin(7).

Findings

Identifies a U(3)-invariant 2-form in six dimensions.

Constructs a G_2-invariant 3-form in seven dimensions.

Describes a Spin(7)-invariant 4-form in eight dimensions.

Abstract

Special p-forms are forms which have components \phi_{\mu_1...\mu_p} equal to +1,-1 or 0 in some orthonormal basis. A p-form \phi\in \Lambda^p R^d is called democratic if the set of nonzero components {\phi_{\mu_1...\mu_p}} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1,...,d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P \geq p and D \geq d. In particular, we display a remarkable nested stucture of special forms including a U(3)-invariant 2-form in six dimensions, a G_2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form \Omega in ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form.