# The effective supergravity of little string theory

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## Abstract

In this work we present the minimal supersymmetric extension of the five-dimensional dilaton-gravity theory that captures the main properties of the holographic dual of little string theory. It is described by a particular gauging of \(\mathcal {N}=2\) 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 coordinate solution of the equations of motion (with flat string frame metric) breaks half of the supersymmetries to \(\mathcal {N}=1\) in four dimensions. Interest in the linear dilaton model has lately been revived in the context of the clockwork mechanism, 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 provides a framework with interesting phenomenological consequences. It offers a way to address the hierarchy when the string scale is at the TeV scale [1, 2, 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, 7, 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 string-frame metric and the dilaton linear in the extra dimension [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 corresponding effective supergravity which in the minimal case is \(\mathcal {N}=2\). In principle, there should be a generalisation with more supersymmetries, or equivalently in higher dimensions. The \(\mathcal {N}=2\) 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 dilaton 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 \(\mathcal {N}=1\) in four dimensions.

The outline of the paper is the following. In Sect. 2, we review the gauged \(\mathcal {N}=2\) supergravity in five dimensions, based on the references [14, 15, 16, 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 LST and identify it with a particular choice of the gauging of the \(\mathcal {N}=2\) supergravity. We also show that the linear dilaton solution preserves half of the supersymmetries, i.e. \(\mathcal {N}=1\) in four dimensions. In Sect. 4, we write the complete Lagrangian, including the fermion terms, depending on three constant parameters. In Sect. 5, we derive the spectrum classified using the 4d Poincaré symmetry and we conclude with some phenomenological remarks. Finally, there are three appendices containing our conventions, the equations of motion with the linear dilaton solution, and some explicit calculations that we use in the study of supersymmetry transformations.

## 2 Gauged \(\mathcal {N}=2\), \(D=5\) supergravity

*SU*(2)-doublet \(\psi _M^i\), where

*i*is the

*SU*(2) index, and the graviphoton, while the \(\mathcal {N}=2\) Maxwell multiplet contains a real scalar \(\phi \), an

*SU*(2) fermion doublet \(\lambda ^i\) and a gauge field. Upon coupling

*n*Maxwell multiplets to pure \(\mathcal {N}=2\), \(D=5\) supergravity, the total field content of the coupled theory can be written as

*n*-dimensional space \(\mathcal {M}\) that has metric \(g_{xy}\,\) that is symmetric for our purposes, while the spinor fields \(\lambda ^{ia}\) transform in the

*n*-dimensional representation of

*SO*(

*n*), which is the tangent space group of \(\mathcal {M}\), so that

*n*-dimensional \(\mathcal {M}\) can be seen as a hypersurface of an \((n+1)\)-dimensional space \(\mathcal {E}\) with coordinates

*U*(1) subgroup of

*SU*(2) generates a scalar potential

*P*, with

*g*is the

*U*(1) coupling constant, \(\Gamma _\mu \) is the \(\Gamma \)-matrix in five spacetime dimensions and the dots stand again for terms that vanish in the vacuum.

*s*. In the following, we use

*t*to denote the additional coordinate on \(\mathcal {E}\), namely \(\xi ^I=\xi ^I(s, t)\,\), \(I=0,1\). The effective supergravity related to the 5-dimensional model for the gravity dual of LST is given by

*a*is a constant parameter. Indeed in the graviton-dilaton system obtained from string compactifications in five dimensions, the first term corresponds to the tree-level contribution (identifying

*t*with the inverse heterotic string coupling) and the second term to the one-loop correction [18].

^{1}

*A*,

*B*are constant parameters. Using (10) we then find the potential to be

*s*take the form

## 3 The 5D dual of LST

^{2}

*C*a constant parameter. The background bulk metric is then

^{3}

*y*, we consider the fifth component of the first of the Eq. (12) in the vacuum

^{4}

## 4 Final Lagrangian

*U*(1), the Lagrangian acquires the additional terms

*I*,

*J*indices. Moreover,

*A*appears only through the combination

*gA*in the additional terms \(\mathcal {L}'\) induced by the gauging, we choose to set \(A=1\). Moreover, at tree-level we may set \(a=0\), as discussed in Sect. 2. The final Lagrangian then takes the formThis Lagrangian has three free parameters:

*g*, \(\upsilon ^0\) and \(\upsilon ^1\).

## 5 Spectrum and concluding remarks

*y*is compactified on an interval [1], allowing to introduce the Standard Model (SM) on one of the boundaries. This spectrum is valid for the graviton, dilaton and their superpartners by supersymmetry. Notice that the 5d graviton zero-mode has five polarisations that correspond to the 4d graviton, a KK vector and the radion. For the rest of the fields, special attention is needed because of the gauging that breaks half of the supersymmetry around the linear dilaton solution.

Indeed, one of the 4d gravitini acquires a mass fixed by *g*, giving rise to a massive spin-3/2 multiplet together with two spin-1 vectors. These are the 5d graviphoton and the additional 5d vector that have non-canonical, dilaton dependent, kinetic terms, as one can see from the Lagrangian (47). Using the background (28), (29), one finds that the *y*-dependence of the vector kinetic terms at the end of the first line of (47) is \(\exp {\{\pm \sqrt{3}C\}}\) with the plus (minus) sign corresponding to the 5d graviphoton \(I=0\) (extra vector \(I=1\)). It follows that they both acquire a mass given by the mass gap.

We conclude with some comments on some possible phenomenological implications of the above lagrangian. One has to dimensionally reduce it from \(D=5\) to \(D=4\), upon compactification of the *y*-coordinate. Moreover, one has to introduce the SM, possibly on one of the boundaries, a radion stabilization mechanism and the breaking of the leftover supersymmetry. An interesting possibility is to combine all of them along the lines of the stabilisation proposal of [3] based on boundary conditions.

There are several possibilities for Dark Matter (DM) candidates in this gravitational sector. There are two gravitini that, upon supersymmetry breaking can recombine to form a Dirac gravitino [19] or remain two different Majorana ones. Depending on the nature of their mass, the exact freeze-out mechanism will be different. There are three possible dark photons \(A^0_\mu \), \(A^1_\mu \) and the KK *U*(1) coming from the 5d metric that could also be DM or their associated gaugini could also play a similar role, again depending on the compactification of the extra coordinate, on how supersymmetry breaking is implemented, as well as on the radion stabilisation mechanism. In general there could be a very rich phenomenology in the gravitational sector.

Regarding LHC or FCC phenomenology it is going to depend on how the SM fields are included in this setup, we will leave that to a forthcoming publication [20]. In general this theory will have KK massive resonances that could be strongly coupled to the SM in a similar fashion as in Randall–Sundrum [13] models.

*Note added* After the completion of this work, we received the paper [21] which contains very similar results.

## Footnotes

- 1.
Note a change of notation between

*s*and*t*compared to Ref. [18]. - 2.
We neglect the remaining spectator five dimensions of the string background which play no role in the properties of the model relevant for our analysis.

- 3.
The details of this calculation are given in the Appendix C.

- 4.
The details of this calculation are given in the Appendix C.

## Notes

### Acknowledgements

The work was partially funded by the Swiss National Science Foundation and by the National Science Foundation under Grant no. PHY-1520966. I.A. would like to thank Amit Giveon for enlightening discussions. The work of S.P. is partially supported by the National Science Centre, Poland, under research Grants DEC-2014/15/B/ST2/02157 and DEC-2015/18/M/ST2/00054.

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