Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions

Abstract

We disclose the mathematical structure underlying the gauge field sector of the recently constructed non-abelian superconformal models in six space–time dimensions. This is a coupled system of 1-form, 2-form, and 3-form gauge fields. We show that the algebraic consistency constraints governing this system permit to define a Lie 3-algebra, generalizing the structural Lie algebra of a standard Yang–Mills theory to the setting of a higher bundle. Reformulating the Lie 3-algebra in terms of a nilpotent degree 1 BRST-type operator Q, this higher bundle can be compactly described by means of a Q-bundle; its fiber is the shifted tangent of the Q-manifold corresponding to the Lie 3-algebra and its base the odd tangent bundle of space–time equipped with the de Rham differential. The generalized Bianchi identities can then be retrieved concisely from Q2=0, which encode all the essence of the structural identities. Gauge transformations are identified as vertical inner automorphisms of such a bundle, their algebra being determined from a Q-derived bracket.