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JHEP07(2012)011

Published for SISSA by Springer

Received: March 27, 2012

Accepted: June 14, 2012

Published: July 2, 2012

Higher derivative extension of 6D chiral gauged

supergravity

Eric Bergshoeff,aFrederik Coomans,bErgin Sezgincand Antoine Van Proeyenb

aCentre for Theoretical Physics, University of Groningen,

Nijenborgh 4, 9747 AG Groningen, The Netherlands

bInstituut voor Theoretische Fysica, Katholieke Universiteit Leuven,

Celestijnenlaan 200D B-3001 Leuven, Belgium

cGeorge and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,

Texas A&M University,

College Station, TX 77843, U.S.A.

E-mail: E.A.Bergshoeff@rug.nl, Frederik.Coomans@fys.kuleuven.be,

Sezgin@physics.tamu.edu, Antoine.VanProeyen@fys.kuleuven.be

Abstract: Six-dimensional (1,0) supersymmetric gauged Einstein-Maxwell supergravity

is extended by the inclusion of a supersymmetric Riemann tensor squared invariant. Both

the original model as well as the Riemann tensor squared invariant are formulated off-

shell and consequently the total action is off-shell invariant without modification of the

supersymmetry transformation rules. In this formulation, superconformal techniques, in

which the dilaton Weyl multiplet plays a crucial role, are used. It is found that the gaug-

ing of the U(1) R-symmetry in the presence of the higher-order derivative terms does not

modify the positive exponential in the dilaton potential. Moreover, the supersymmetric

Minkowski4× S2compactification of the original model, without the higher-order deriva-

tives, is remarkably left intact. It is shown that the model also admits non-supersymmetric

vacuum solutions that are direct product spaces involving de Sitter spacetimes and negative

curvature internal spaces.

Keywords: Field Theories in Higher Dimensions, Space-Time Symmetries, Supergravity

Models

ArXiv ePrint: 1203.2975

Open Access

doi:10.1007/JHEP07(2012)011

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JHEP07(2012)011

Contents

1 Introduction

1

2 Off-shell gauged (1,0) supergravity

2.1Off-shell Poincar´ e action

2.2 Coupling to an off-shell vector multiplet

2.3 Elimination of auxiliary fields

3

3

6

7

3 An alternative off-shell formulation9

4Inclusion of the RµνabRµνabinvariant

4.1Construction of the RµνabRµνabinvariant

4.2The total gauged R + R2supergravity lagrangian

13

13

16

5 Vacuum solutions

5.1Bosonic field equations

5.2 Vacuum solutions without fluxes

5.3Vacuum solutions with 2-form flux

5.4Vacuum solutions with 3-form flux

5.5 Spectrum in Minkowski spacetime

18

18

21

22

23

24

6 Conclusions25

1 Introduction

Higher-order curvature terms in supergravity theories are of considerable importance for

different reasons. They can be considered as higher-order correction terms (in α′) to an

effective supergravity Lagrangian of a (compactified) string theory (see, e.g., [1]). These

Lagrangians are supersymmetric only order by order in the perturbation parameter α′. On

the other hand off-shell formulations for different curvature squared invariants in 4, 5 and

6 dimensions have been constructed in [2–7]. These invariants, added to a pure off-shell

supergravity theory, are exactly supersymmetric and can be considered in their own right.

The off-shell nature of these theories implies that they contain auxiliary fields. It is well-

known that, when adding higher derivative terms to the Lagrangian, the auxiliary fields

become propagating. Hence, the elimination of these auxiliary fields becomes much harder

since their field equations are not algebraic anymore. Assuming that the dimensionful

parameter in front of the higher derivative part of the Lagrangian is very small, one can

solve the auxiliary field equations perturbatively and eliminate these fields order by order

in the small parameter. It remains an open question if and how the on-shell Lagrangian

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JHEP07(2012)011

obtained in this way is related to the compactified string Lagrangian, which does not

contain any auxiliary fields to begin with.1

Theories containing higher-order curvature terms can provide corrections to black

hole entropies [9–11] and can source higher-order effects in the AdS/CFT correspon-

dence [12, 13]. When considering these theories as toy models on their own they can

be compactified to lower dimensions. A particular case to consider is the compactification

to three dimensions [8]. A particular feature of three dimensions is that D = 3 gravitons

are non-propagating when only considering 2-derivative Lagrangians. Instead, the addition

of higher-derivative terms can turn these non-propagating modes into propagating massive

graviton modes, see, e.g., [14] and references therein. These theories can then be regarded

as simple toy models to study quantum gravity.

In this paper we study higher-order corrections to a six-dimensional (1,0) supersym-

metric U(1)R gauged Einstein-Maxwell supergravity theory, usually referred to as the

Salam-Sezgin model [15], which is a special case of a Sp(n)×Sp(1)Rgauged matter-coupled

supergravity theory that was first obtained in [16]. We shall refer to this more general case

as 6D chiral gauged supergravity as well. An intriguing feature of the Salam-Sezgin model

is that it allows a compactification over S2to a four-dimensional Minkowski spacetime

while retaining half of the supersymmetry [15]. One of the purposes of this work is to

investigate whether this feature survives after the addition of higher-order derivative cor-

rections. To facilitate the addition of such higher-order corrections to the model we will

first construct its off-shell formulation. It turns out that this is only possible for the dual

formulation of the model where the 2-form potential˜B has been replaced by a dual 2-form

potential B [17, 18]. This has the effect that the curvature of the original 2-form potential

no longer contains a Maxwell-Chern-Simons term, but that instead a term of the form

B ∧ F ∧ F, where F is the Maxwell field strength, appears in the Lagrangian.

To construct the off-shell formulation we will make use of the superconformal tensor

calculus. As a first step we will review the construction of off-shell minimal D = 6 super-

gravity [19, 20]. In this construction one makes use of the dilaton Weyl multiplet (obtained

by coupling the regular Weyl multiplet to a tensor multiplet) coupled to a linear multiplet

as compensator. After fixing the conformal symmetries, this theory still has a remaining

U(1) R-symmetry which is gauged by an auxiliary vector Vµ. We will couple this ‘pure’

theory to an Abelian vector multiplet and show that after solving for the auxiliary Vµ, the

gauging proceeds via the vector Wµof the Abelian vector multiplet.

After constructing the off-shell formulation of the gauged (1,0) supergravity theory, we

investigate its deformation by an off-shell curvature squared invariant [2, 3]. To construct

this invariant it is essential to make use of the dilaton Weyl multiplet. We review the

construction of this higher-derivative term and add it to the off-shell (1,0) supergravity

theory. Next, we study the gauging procedure in the presence of the Riemann tensor

squared invariant.

1The elimination of auxiliary fields in higher derivative theories has been discussed in [4]. A conjec-

tured duality between a supergravity Lagrangian with the auxiliary fields eliminated perturbatively and a

compactified string Lagrangian, without auxiliary fields, can be found in section 5 of [8].

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As a first step towards understanding the properties of the higher-derivative extension

of the model we perform a systematic search for vacuum solutions. We construct both su-

persymmetric as well as non-supersymmetric solutions. For one particular supersymmet-

ric solution, namely six-dimensional Minkowski spacetime, we calculate the fluctuations

around this background and show how these fluctuations fit into supermultiplets.

This paper is organized as follows. In section 2 we review the off-shell version of the

(1,0) supergravity model [19, 20] and describe its gauging. In section 3, we introduce

an alternative off-shell formulation of the model in view of the fact that it is best suited

for the addition of the Riemann tensor squared invariant [2]. In section 4 we discuss the

construction of the Riemann tensor squared invariant and arrive at the total Lagrangian

for the higher-derivative extended 6D chiral gauged supergravity theory. In section 5, we

investigate the vacuum solutions of this model. We summarize and comment further on

our results and on some interesting open problems in the Conclusions section. Throughout

the paper we follow the notation given in appendix A of [20].

2 Off-shell gauged (1,0) supergravity

In this section we present an off-shell version of the dual formulation [17, 18] of the Salam-

Sezgin model [15, 16]. In the first subsection we give the off-shell Lagrangian of pure

supergravity plus a tensor multiplet as constructed in [19, 20]. In the next subsection we

couple a vector multiplet to this theory and show that the resulting Einstein-Maxwell model

leads to a non-trivial U(1) gauge symmetry that is not gauged by an auxiliary vector field.

In the last subsection we show that after eliminating the auxiliary fields one ends up with a

Lagrangian in which the U(1) gauge symmetry is effectively gauged by the physical vector

of the vector multiplet. We furthermore show that, after dualizing the 2-form potential

into a dual 2-form potential, this Einstein-Maxwell model is nothing else than the original

Salam-Sezgin model.

2.1Off-shell Poincar´ e action

The off-shell (1,0) supergravity action has been constructed by means of a superconformal

tensor calculus in which the off-shell so-called dilaton Weyl multiplet with independent

fields

{eµa,ψi

and Weyl weights (−1,−1/2,0,0,0,5/2,2), respectively, is coupled to an off-shell linear

multiplet consisting of the fields

µ,Bµν,Vij

µ,bµ,ψi,σ }

(2.1)

{Eµνρσ,Lij,ϕi}, (2.2)

with Weyl weights (0,4,9/2), respectively. The fields (ψi

Weyl spinors labelled by a Sp(1)Rdoublet index, the fields B and E are two- and four-forms

with tensor gauge symmetries, respectively, bµis the dilatation gauge field and Lijare three

real scalars. An appropriate set of gauge choices for obtaining off-shell supergravity with

µ,ψi,ϕi) are symplectic Majorana-

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the Einstein-Hilbert term, namely L = eR + ···, is given by

Lij=

1

√2δij,ϕi= 0,bµ= 0(2.3)

which fixes the dilatations, conformal boost and special supersymmetry transformations.

Moreover, the first of the gauge choices in (2.3) breaks Sp(1)Rdown to U(1)R. This set of

gauge choices leads to an off-shell multiplet containing 48+48 degrees of freedom described

by the fields [19] (see table 5 of [20])

eµa(15), V′

The field Vµ is the gauge field of the surviving U(1)Rgauge symmetry. It arises in the

decomposition

Vij

where the traceless part V′ij

method was employed in [19] where the bosonic action was given, and a procedure for

obtaining the full action was provided. This full action, including the quartic terms, was

constructed in [20]. The Lagrangian up to quartic fermion terms is given by [19, 20]2

??L=1=1

4EµEµ+

−1

2

−1

The indication L = 1 in the left-hand side indicates all the gauge choices (2.3). Here we

have defined the field strength for the 2-form potential and the dual of the field strength

for the 4-form potentials as follows3

µ

ij(12), Vµ(5), Bµν(10), σ (1), Eµνρσ(5); ψµi(40), ψi(8). (2.4)

µ= V′ij

has no gauge symmetry. A superconformal tensor calculus

µ+1

2δijVµ,

V′ij

µδij= 0, (2.5)

µ

e−1LR

2R −1

−1

2σ−2∂µσ∂µσ −1

1

√2EµVµ−

24σ−2Fµνρ(B)Fµνρ(B) + V′

1

4√2Eρ¯ψi

µijV′µij

µγρµνψj

νδij

¯ψµγµνρDν(ω)ψρ− 2σ−2¯ψγµD′

48σ−1Fµνρ(B)

µ(ω)ψ + σ−2¯ψνγµγνψ ∂µσ (2.6)

?

?¯ψλγ[λγµνργτ]ψτ+ 4σ−1¯ψλγµνργλψ − 4σ−2¯ψγµνρψ.

Fµνρ(B) = 3∂[µBνρ], (2.7)

Eµ=

1

24e−1εµν1···ν5∂[ν1Eν2···ν5]. (2.8)

The U(1)Rcovariant derivatives Dµ(ω) and the full SU(2) covariant derivatives D′µ(ω) are

given by

?

D′

4ωµabγab

Dµ(ω)ψi

ν=∂µ+1

?

4ωµabγab

?

?

ψi

ν−1

ψi−1

2Vµδijψνj, (2.9)

µ(ω)ψi=∂µ+1

2Vµδijψj+ Vµ′ijψj, (2.10)

2We use the conventions of [20]. In particular, the spacetime signature is (− + + + ++), γa1···a6=

εa1···a6γ∗, γ∗ǫ = ǫ,¯ψiψj = −¯ψjψi and¯ψiγµψj =¯ψjγµψi. These conventions differ from those in [19] in

using signature (− + ...+) rather than the Pauli convention (+ + ...+), in rescaling Vi

−1/2, and the minus sign in the definition of the Ricci tensor. The signature change merely results in

rescaling εµ1...µ6by a factor of i.

3Note that the definition of Eµhere is purely bosonic, and it differs from the definition used in [19, 20],

where it is a superconformal covariant expression with fermionic bilinear terms.

µj by a factor of

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