Abelian tensor hierarchy in 4D, N = 1 superspace

Abstract

With the goal of constructing the supersymmetric action for all fields, massless and massive, obtained by Kaluza-Klein compactification from type II theory or M-theory in a closed form, we embed the (Abelian) tensor hierarchy of p-forms in four-dimensional, N = 1 superspace and construct its Chern-Simons-like invariants. When specialized to the case in which the tensors arise from a higher-dimensional theory, the invariants may be interpreted as higher-dimensional Chern-Simons forms reduced to four dimensions. As an application of the formalism, we construct the eleven-dimensional Chern-Simons form in terms of four-dimensional, N = 1 superfields.

Chern-Simons actions and their gaugings in 4D, N =1 superspace

“…in section 4 we formulate this system in superspace, thereby gauging the abelian superspace hierarchy of reference [13]. To set up the conventions and quote some results which are useful for the rest of the paper we recall how to formulate non-abelian gauge fields in superspace in section 4.1.…”

“…[13]. For us, a Chern-Simons action will be a sum of terms, each of which is the integral over spacetime of the wedge product of one potential and some number of field strengths.…”

“…Our starting point will be the abelian Chern-Simons actions that we constructed in ref. [13], but where we promote all derivatives to covariant derivatives, and use the field strengths constructed in (4.44). When expanded in components, these actions will contain the bosonic Chern-Simons actions of section 5 (along with more pieces involving other component fields), and are gauge-invariant when we restrict to the abelian case.…”

“…[13], and take its variations, we need to make a couple of definitions, 6) and another application of the Chern-Simons superfield construction, where we are given three arguments, one of which is a chiral spinor superfield ψ and the other two are real scalar superfields U 1 and U 2 . Then we have…”

“…This hierarchy simplifies dramatically if we turn off the g gauging and in reference [13], we embedded this "abelian tensor hierarchy" into 4D, N = 1 superspace. In this paper we gauge this superspace hierarchy to obtain a non-abelian tensor hierarchy in 4D, N = 1 superspace.…”