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M-theory potential from the G 2 Hitchin functional in superspace
Received: November
M-theory potential from the Hitchin functional in
Katrin Becker 0 2
Melanie Becker 0 2
Sunny Guha 0 2
William D. Linch III 0 2
Daniel Robbins 0 1
College Station 0
TX 0
U.S.A. 0
Texas A&M University,
0 Open Access , c The Authors
1 Department of Physics, University at Albany
2 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy
We embed the component fields of eleven-dimensional supergravity into a superspace of the form X × Y where X is the standard 4D, N = 1 superspace and Y is a smooth 7-manifold. The eleven-dimensional 3-form gives rise to a tensor hierarchy of superfields gauged by the diffeomorphisms of Y . It contains a natural candidate for a G2 structure on Y , and being a complex of superforms, defines a superspace Chern-Simons invariant. Adding to this a natural generalization of the Riemannian volume on X × Y and freezing the (superspin- 32 and 1) supergravity fields on X, we obtain an approximation to the eleven-dimensional supergravity action that suffices to compute the scalar potential. In this approximation the action is the sum of the superspace Chern-Simons term and a superspace generalization of the Hitchin functional for Y as a G2-structure manifold. Integrating out auxiliary fields, we obtain the conditions for unbroken supersymmetry and the scalar potential. The latter reproduces the Einstein-Hilbert term on Y in a form due to Bryant. ArXiv ePrint: 1611.03098
Chern-Simons Theories; M-Theory; Superspaces; Differential and Algebraic
1 Introduction 2 3 4
Superfields and components
Chern-Simons action
Scalar potential
G2 toolbox
In the zoo of supergravity theories, eleven-dimensional supergravity is unique in that it has
the largest possible (manifest) spacetime symmetry group. Despite being, in this sense,
the most fundamental of supergravity theories, it has various quite mysterious properties.
For example, in contrast to its ten-dimensional relatives, there is no theory of critical
superstrings that has it as a low-energy limit. To find a home even somewhat analogous,
one must go to M-theory (which is even more mysterious) and take a massless limit of that.
Another presumably related property is the emergence of an exceptional symmetry of its
(gauged) compactifications on tori.
For applications to the study of physics in lower dimensions, this theory may be
compactified on eleven-dimensional manifolds of the form X × Y and expanded in Kaluza-Klein
modes by integrating over Y . This results in an effective theory on X in which the
contribution of the internal part is organized in a tower of ever-more-massive fields.
An alternative to this approach is to split the eleven-dimensional spacetime as X × Y
and to reorganize the fields into representations of the reduced structure group but
without averaging over the “internal” space. Such backgrounds were precisely the subject of
reference [1], wherein this is referred to as “keeping locality in Y ”. There, the action for
the bosonic part of eleven-dimensional supergravity was decomposed on X × Y explicitly.
Of course it is always possible to keep the full diffeomorphism invariance of the
elevendimensional theory recast in terms of covariant, interacting X and Y parts. What is
somewhat surprising, however, is that this can be organized in a very manageable form [1]. We
would like to construct a superspace action that reproduces the bosonic eleven-dimensional
supergravity action in this form.
As this is presumably impossible (in the na¨ıve sense) for more than 8 real supercharges,
auxiliary fields [2] and non-chiral matter. This complicates the use of such a superspace
description both technically and phenomenologically. Instead, we propose to embed the
Y -dependence. This gives a description of eleven-dimensional supergravity on X × Y with
X a curved superspace modeled on R4|4 and Y a Riemannian 7-manifold. Projecting such
a theory to component fields results in a component supergravity description on the bosonic
submanifold X × Y .
Although the resulting physics is eleven-dimensionally super-diffeomorphism invariant,
with the 7D (bosonic) Riemannian part). Note that precisely this amount of local
superPoincar´e invariance is what one would retain were one to compactify on a manifold Y
admitting a Riemannian metric of G2 holonomy. Although we do not insist on such a
cations in which we refer to X or X as “spacetime” and Y as the “internal space”.
There are various partial realizations of this superspace supergravity program less
ambitious than the construction of the full 11D theory in arbitrary X × Y . For example,
one could attempt to build the linearized action by working out the linearized
superdiffeomorphisms and building an invariant action order-by-order following a superspace Noether
Alternatively, one may attempt to define the theory in a gravitino superfield
fields holding the components of the 3-form but all non-linearly. In such an approach, we
expect the action to ta (...truncated)