JHEP04(2017)103
As we will see throughout our presentation, the use of superforms streamlines the con-
struction of geometric invariants and simplifies or obviates many cumbersome and delicate
calculations. We have attempted to make this paper self-contained but have been neces-
sarily brief in our review of superform methods. A pedagogical introduction to superforms
in the context of tensor hierarchies may be found in reference [30].
2 Prepotential formalism
The papers [13, 14] were motivated by the goal of writing a supersymmetric theory in
D > 4 dimensions, particularly eleven-dimensional supergravity, in 4D N = 1 language.
The results obtained were more general, with no assumptions being made about whether
the four-dimensional tensor hierarchy had been obtained from a higher-dimensional theory
or not. In the present work we will not be as careful to maintain this full generality, although
this choice is primarily made to keep the notation simple. Instead we will implicitly assume
that the four-dimensional tensor hierarchy arises from a p-form in D dimensions, where
the D-dimensional theory is being put on a background R
4
× M, with M a (D − 4)-
dimensional internal space. Note that p should be odd in order for us to have a non-trivial
Chern-Simons action.
In this case, the bosonic four-dimensional tensor hierarchy is comprised of axions a,
which are zero-forms in spacetime and p-forms on M, spacetime one-forms A
a
which are
(p − 1)-forms on M, spacetime two-forms B
ab
valued in internal (p − 2)-forms, spacetime
three-forms C
abc
valued in internal (p − 3)-forms, and spacetime four-forms D
abcd
valued in
internal (p−4)-forms. Note that if p = 3 there simply are no D
abcd
fields, and if p = 1 there
are only axions and 1-forms. These forms can be multiplied, using the wedge product for
forms on M, and if D = n(p+1)−1 for some n > 1, then we can construct a D-dimensional
Chern-Simons action by wedging one potential and n − 1 field-strengths and integrating
the resulting D-form over R
4
× M. By integrating just over M, we get a 4D Chern-Simons
action for the tensor hierarchy.
Additionally, if we are reducing a supergravity theory in D dimensions, then we can also
incorporate the 4D gauge fields coming from off-diagonal components of the D-dimensional
metric. These are spacetime one-forms which are tangent vectors on M , and their corre-
sponding non-abelian gauge group is the group of diffeomorphisms on M (whose Lie algebra
can be identified with Γ(T M), the space of vector fields on M with the usual Lie bracket).
In [13], it was explained how to embed these structures into 4D, N = 1 superfields.
The non-abelian gauge vectors A
a
were promoted to T M-valued super-1-forms A
A
(with
the lowest components of the superfield A
a
matching the bosonic fields of the same name)
which were used to build gauge covariant super-derivatives D
A
= {D
a
, D
α
,
¯
D
˙α
} by
D
a
= ∂
a
− (L
A
)
a
, D
α
= D
α
− (L
A
)
α
,
¯
D
˙α
=
¯
D
˙α
− (L
A
)
˙α
. (2.1)
Here L
A
is the Lie derivative along the vector field A which acts on differential forms of
the internal space.
– 5 –
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