Eur. Phys. J. C (2018) 78:146
https://doi.org/10.1140/epjc/s10052-018-5632-4
Regular Article - Theoretical Physics
The effective supergravity of little string theory
Ignatios Antoniadis
1,2,a
, Antonio Delgado
3,b
, Chrysoula Markou
1,c
, Stefan Pokorski
4,d
1
Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE, 75005 Paris, France
2
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
3
Department of Physics, University of Notre Dame, 225 Nieuwland Hall, Notre Dame, IN 46556, USA
4
Faculty of Physics, Institute of Theoretical Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
Received: 17 November 2017 / Accepted: 10 February 2018
© The Author(s) 2018. This article is an open access publication
Abstract In this work we present the minimal supersym-
metric extension of the five-dimensional dilaton-gravity the-
ory that captures the main properties of the holographic dual
of little string theory. It is described by a particular gauging
of N = 2 supergravity coupled with one vector multiplet
associated with the string dilaton, along the U (1) subgroup
of SU(2) R-symmetry. The linear dilaton in the fifth coor-
dinate solution of the equations of motion (with flat string
frame metric) breaks half of the supersymmetries to N = 1
in four dimensions. Interest in the linear dilaton model has
lately been revived in the context of the clockwork mecha-
nism, which has recently been proposed as a new source of
exponential scale separation in field theory.
1 Introduction
Besides its own theoretical interest, little string theory pro-
vides a framework with interesting phenomenological con-
sequences. It offers a way to address the hierarchy when the
string scale is at the TeV scale [1–3], without postulating large
extra dimensions (in string units) but instead an ultra-weak
string coupling [4,5]. Recently, interest in the holographic
dual to LST (the linear dilaton model) has been revived in
the context of the so-called clockwork models [6–8] which
address the exponential scale separation in field theory in a
new way [9,10].
Little string theory (LST) corresponds to a non-trivial
weak coupling limit of string theory in six dimensions with
gravity decoupled and is generated by stacks of (Neveu–
Schwarz) NS5-branes [11]. Its holographic dual corresponds
to a seven-dimensional gravitational background with flat
a
e-mail: antoniad@lpthe.jussieu.fr
b
e-mail: antonio.delgado@nd.edu
c
e-mail: chrysoula@lpthe.jussieu.fr
d
e-mail: Stefan.Pokorski@fuw.edu.pl
string-frame metric and the dilaton linear in the extra dimen-
sion [12]. Its properties can be studied in a simpler toy model
by reducing the theory in five dimensions. Introducing back
gravity weakly coupled, one has to compactify the extra
dimension on an interval and place the Standard Model on
one of the boundaries, in analogy with the Randall–Sundrum
model [13] on a slice of a five-dimensional (5d) anti-de Sitter
bulk [1].
Since we know that the bulk LST geometry preserves
space-time supersymmetry, in this work we study the corre-
sponding effective supergravity which in the minimal case
is N = 2. In principle, there should be a generalisation
with more supersymmetries, or equivalently in higher dimen-
sions. The N = 2 gravity multiplet contains the graviton,
a graviphoton and the gravitino (8 bosonic and 8 fermionic
degrees of freedom), while the heterotic (or type I) string dila-
ton is in a vector multiplet containing a vector, a real scalar
and a fermion. The corresponding supergravity action [15]
admits a gauging of the U(1) subgroup of the SU(2) R-
symmetry, that generates a potential for the single scalar
field [15,16]. This potential depends on two parameters
allowing a multiple of possibilities with critical or non critical
points, or even flat potential with supersymmetry breaking.
Here, we observe that the vanishing of one of the parameters
generates the runaway dilaton potential of the non-critical
string. This potential has no critical point with 5d maximal
symmetry but it leads to the linear dilaton solution in the fifth
coordinate that preserves 4d Poincaré symmetry. We show
that this solution breaks one of the two supersymmetries,
leading to N = 1 in four dimensions.
The outline of the paper is the following. In Sect. 2,we
review the gauged N = 2 supergravity in five dimensions,
based on the references [14–17], and specialize in the case of
one vector multiplet using the results of the string effective
action of Ref. [18]. In Sect. 3, we present the 5d graviton-
dilaton toy model that describes the holographic dual of
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