# All Gauged Curvature Squared Supergravities in Five Dimensions

Gregory Gold<EMAIL_ADDRESS>School of Mathematics and Physics, University
of Queensland, St Lucia, Brisbane, Queensland 4072, Australia  Jessica Hutomo
<EMAIL_ADDRESS>School of Mathematics and Physics, University of
Queensland, St Lucia, Brisbane, Queensland 4072, Australia  Saurish Khandelwal
<EMAIL_ADDRESS>School of Mathematics and Physics, University of
Queensland, St Lucia, Brisbane, Queensland 4072, Australia  Mehmet Ozkan
<EMAIL_ADDRESS>Department of Physics, Istanbul Technical University,
Maslak 34469 Istanbul, Turkey  Yi Pang<EMAIL_ADDRESS>Center for Joint
Quantum Studies and Department of Physics,
School of Science, Tianjin University, Tianjin 300350, China
Gabriele Tartaglino-Mazzucchelli<EMAIL_ADDRESS>School
of Mathematics and Physics, University of Queensland, St Lucia, Brisbane,
Queensland 4072, Australia

###### Abstract

We present a complete basis to study gauged curvature-squared supergravity in
five dimensions. We replace the conventional ungauged Riemann-squared action
with a new Log-invariant, offering a comprehensive framework for all gauged
curvature-squared supergravities. Our findings address long-standing
challenges and have implications for precision tests in the AdS/CFT
correspondence.

Introduction.—Twenty-six years after its discovery, the AdS/CFT correspondence
has entered a new era in which precision tests beyond the leading order have
become increasingly important, owing to developments in both field theory and
gravity. On the one hand, integrability and localization techniques allow one
to compute the observables in superconformal field theories (SCFT) exactly at
finite couplings. On the other hand, the development of superconformal tensor
calculus and superspace techniques – see the reviews Freedman:2012zz ;
Lauria:2020rhc ; Kuzenko:2022skv ; Kuzenko:2022ajd  – in conjunction with the
computational capabilities offered by computer algebra programs, has
significantly advanced the construction of exact, off-shell higher-derivative
supergravity models.

In this letter, we present all gauged curvature-squared supergravity
invariants in five dimensions based on the off-shell dilaton Weyl multiplet.
After going on-shell, our invariants describe the most general four-derivative
corrections to the five-dimensional minimal gauged supergravity, which is a
universal sector to all string compactifications preserving at least eight
supercharges. The gauged aspect is necessary to accommodate a supersymmetric
anti-de Sitter (AdS) solution, and thus is of broad interest in holography. In
particular, due to recent advancements in AdS black hole microstate counting
Benini:2015eyy ; Benini:2016rke ; Cabo-Bizet:2018ehj ; Choi:2018hmj ;
Benini:2018ywd ; Honda:2019cio ; Benini:2020gjh ; Agarwal:2020zwm  using the
dual CFT, a precise matching between the gravity and CFT results at the next
to leading order clearly requires the knowledge of the complete curvature-
squared supergravity actions. Previous works have made attempts to compute
four-derivative corrections based on partial results in the literature, and
certain assumptions were made. Using our full results, it can be shown that,
in fact, some of the assumptions are invalid, thus finally furnishing the
stage for new, next-to-leading order analyses on the gravity side of the
AdS/CFT correspondence.

The construction of gauged curvature-squared invariants is notoriously hard,
as opposed to their ungauged counterparts, which have been fully known for
more than a decade Bergshoeff:2011xn ; Ozkan:2013nwa ; Ozkan:2013uk . The
primary difficulty stems from the absence of a straightforward transition from
ungauged to gauged theories. In fact, the complete basis of invariants must be
constructed from completely different starting points. For instance, certain
ungauged curvature-squared models are attainable through the application of
superconformal tensor calculus, utilizing the dilaton Weyl multiplet. In
contrast, their gauged counterparts necessitate the use of a modified version
of the same multiplet Coomans:2012cf , which has an entirely different field
content and transformation rules. Furthermore, the deformation necessary for
the construction of gauged supergravity models renders certain established
higher-derivative supergravity building techniques impractical, thus further
complicating the task. In fact, it takes an interplay between superconformal
tensor calculus Coomans:2012cf  and superspace techniques Butter:2014xxa ,
together with a series of new, daunting computations finalized only in the
results presented here, to yield the complete set of gauged curvature-squared
invariants.

This letter aims to explicitly show that the past challenges can be overcome
by changing the basis of curvature-squared supergravities, which previously
employed the Weyl tensor squared, Riemann tensor squared, and Ricci scalar
squared as fundamental building blocks. We demonstrate that by replacing the
Riemann-squared action with the Log-invariant, in which the leading term comes
with the Ricci tensor squared, it is possible to explicitly establish all
gauged curvature-squared supergravities in five dimensions. The outcomes
presented in our letter mark a significant advancement, paving the way to a
complete study of physical results beyond the leading supergravity
approximation in five dimensions. This development holds particular promise
for precision tests of the AdS5/CFT4 correspondence. In this context, we
derive the anomaly coefficients in the dual SCFT4, which apparently depend on
all curvature-squared couplings.

Construction of the invariants.—We start by introducing the field content of
the standard Weyl multiplet of conformal supergravity in five dimensions
Bergshoeff:2001hc . Our notation and conventions correspond to that of
Bergshoeff:2001hc . We denote the spacetime indices by $\mu,\nu,\cdots$,
Lorentz indices by $a,b,\cdots$, SU(2) indices by $i,j,\cdots$, and spinor
indices by $\alpha,\beta,\cdots$. The multiplet is described by a set of
independent gauge fields: the vielbein $e_{\mu}{}^{a}$, the gravitino
$\psi_{\mu}{}_{\alpha}^{i}$, the SU(2) gauge fields $V_{\mu}{}^{ij}$, and a
dilatation gauge field $b_{\mu}$. The other gauge fields associated with the
remaining symmetries, including the spin connection $\omega_{\mu}{}^{ab}$, the
$S$-supersymmetry connection $\phi_{\mu}{}_{\alpha}^{i}$, and the special
conformal connection ${f}_{\mu}{}^{a}$, are composite fields, i.e., they are
determined in terms of the other fields by imposing certain curvature
constraints. The standard Weyl multiplet also contains a set of matter fields:
a real antisymmetric tensor $T_{ab}$, a fermion $\chi_{\alpha}^{i}$, and a
real scalar $D$. A more detailed discussion of the superconformal
transformations of the various fields can be found, e.g., in Bergshoeff:2001hc
; Coomans:2012cf .

Below we will make use of a variant multiplet of conformal supergravity, known
as the gauged dilaton Weyl multiplet Coomans:2012cf ; Note2 . For this
multiplet, the independent gauge fields remain the same as the standard Weyl
multiplet, but the matter content is replaced with
$\\{\sigma,C_{\mu},B_{\mu\nu},L_{ij},E_{\mu\nu\rho},N,\psi^{i},\varphi^{i}\\}$.
This is obtained by coupling the standard Weyl multiplet to on-shell vector
and linear multiplets. The vector multiplet consists of a scalar field
$\sigma$, the gaugino $\psi_{\alpha}^{i}$, an abelian gauge vector $C_{\mu}$
with field strength $G_{\mu\nu}=2\partial_{[\mu}C_{\nu]}$, and an $\rm{SU(2)}$
triplet of auxiliary fields $Y^{ij}=Y^{(ij)}$. The linear multiplet contains
an $\rm{SU(2)}$ triplet of scalars $L^{ij}=L^{(ij)}$, a gauge three-form
$E_{\mu\nu\rho}$, a scalar $N$, and an $\rm{SU(2)}$ doublet
$\varphi^{i}_{\alpha}$. The bosonic matter fields of the vector and the
standard Weyl multiplet are then expressed as follows Coomans:2012cf

$\displaystyle Y^{ij}$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle g}{\scriptstyle
2}}}\sigma^{-1}L^{ij}+{\rm f.t.}~{},$ $\displaystyle T_{ab}$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}\sigma^{-1}G_{ab}+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
48}}}\sigma^{-2}\epsilon_{abcde}H^{cde}+{\rm f.t.}~{},$ $\displaystyle D$
$\displaystyle=$ $\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}\sigma^{-1}\nabla^{a}\nabla_{a}\sigma+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
8}}}\sigma^{-2}(\nabla^{a}\sigma)\nabla_{a}\sigma-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 32}}}R$ (1) $\displaystyle-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 16}}}\sigma^{-2}G^{ab}G_{ab}-({\textstyle{\frac{\scriptstyle
26}{\scriptstyle 3}}}T^{ab}-2\sigma^{-1}G^{ab})T_{ab}$
$\displaystyle+{\textstyle{\frac{\scriptstyle g}{\scriptstyle
4}}}\sigma^{-2}N+{\textstyle{\frac{\scriptstyle g^{2}}{\scriptstyle
16}}}\sigma^{-4}L^{2}+{\rm f.t.}~{},$

where “f.t.” stands for omitted fermionic terms and
$H_{abc}=e_{a}{}^{\mu}e_{b}{}^{\nu}e_{c}{}^{\rho}H_{\mu\nu\rho}$ denotes the
three-form field strength
$H_{\mu\nu\rho}:=3\partial_{[\mu}B_{\nu\rho]}+\frac{3}{2}C_{[\mu}G_{\nu\rho]}+\frac{1}{2}gE_{\mu\nu\rho}$.
In the above, the covariant derivative is denoted by

$\displaystyle\nabla_{a}=e_{a}{}^{\mu}\big{(}\partial_{\mu}-\omega_{\mu}{}^{bc}M_{bc}-b_{\mu}\mathbb{D}-V_{\mu}{}^{ij}U_{ij}\big{)}~{},$
(2)

with $M_{ab}$, $\mathbb{D}$, and $U_{ij}$ being the Lorentz, dilatation, and
SU(2) generators, respectively. The dilatation connection $b_{\mu}$ is pure
gauge and will be set to zero throughout. The mapping (1) allows us to easily
convert every invariant involving a coupling to the standard Weyl multiplet to
that written in terms of the gauged dilaton Weyl multiplet. The ungauged map
and the models can simply be obtained by setting $g=0$ in (1). In this case,
the fields of the linear multiplet decouple from the map (1), and the
multiplet reduces to the ungauged dilaton Weyl multiplet with $32+32$ off-
shell degrees of freedom Bergshoeff:2001hc ; Fujita:2001kv .

In the superconformal tensor calculus, the so-called BF action principle plays
a fundamental role in the construction of general supergravity-matter
couplings, see Butter:2014xxa ; Kugo:2000af ; Fujita:2001kv ; Kugo:2002vc ;
Bergshoeff:2001hc ; Bergshoeff:2002qk ; Bergshoeff:2004kh  for the 5D case. It
is based on an appropriate product of a linear multiplet with an Abelian
vector multiplet:

$\displaystyle e^{-1}{\cal L}_{\rm BF}=\,A_{{a}}E^{{a}}+\rho
N+\mathcal{Y}_{ij}L^{ij}+\rm f.t.~{}.$ (3)

Here we use $\\{\rho,A_{\mu},\mathcal{Y}_{ij},\lambda^{i}_{\alpha}\\}$ to
denote the field content in an arbitrary vector multiplet, and the bosonic
part of the constrained vector $E_{a}$ is related to the three-form gauge
field $E_{abc}$ via $E_{a}=-\frac{1}{12}\epsilon_{abcde}\nabla^{b}E^{cde}$. In
any construction that involves composite expressions for the fields of the
linear multiplet in terms of the vector multiplet, the BF-action yields a
vector-coupled action in the (gauged) dilaton Weyl background. Using the off-
shell map given in Ozkan:2013uk , a vector multiplet can be identified with
fields in the gauged dilaton Weyl multiplet as

$\displaystyle{\mathcal{Y}}^{ij}$ $\displaystyle\to\frac{1}{4}{\rm
i}\sigma^{-1}\bar{\psi}^{i}\psi^{j}-\frac{g}{2}\sigma^{-1}L^{ij}\,,$
$\displaystyle\rho$ $\displaystyle\to\sigma\,,$ $\displaystyle A_{\mu}$
$\displaystyle\to C_{\mu}\,,$ $\displaystyle\lambda^{i}$
$\displaystyle\to\psi^{i}\,,$ (4)

which gives rise to off-shell models that are purely expressed in terms of the
fields of the (gauged) dilaton Weyl multiplet. By appropriately choosing
primary composite linear multiplets, eq. (3) becomes the building block for
constructing various curvature-squared invariants.

In the superconformal approach, the off-shell formulation of minimal 5D
supergravity can be achieved by coupling the standard Weyl multiplet to two
off-shell conformal compensators: a vector multiplet and a linear multiplet.
Within this setup, supersymmetric completions of the Weyl tensor squared and
Ricci scalar squared were constructed in Hanaki:2006pj  and Ozkan:2013nwa ,
respectively. The Weyl tensor-squared invariant is based on a composite linear
multiplet comprised solely of standard Weyl multiplet fields. To construct the
Ricci scalar-squared invariant, one starts by defining a composite vector
multiplet in terms of a linear multiplet. This composite vector multiplet is
then substituted into the vector multiplet action obtained using (3).

While the Weyl tensor-squared and Ricci scalar-squared actions based on the
dilaton Weyl multiplet were presented as ungauged models Ozkan:2013nwa , they
can simply be gauged by using the maps (1) and (4). At this point, it is
worthwhile to mention that the on-shell results for these gauged actions
differ from those presented by Liu:2022sew . The reason is that Ref.
Liu:2022sew  assumes that the map between the standard Weyl and the dilaton
Weyl multiplet is not modified in the gauged case. However, as shown in
Coomans:2012cf  and presented as in eq. (1), these expressions are indeed
deformed. For completeness, we present the off-shell gauged results in the
Supplemental Material.

The third invariant necessary to obtain all the curvature-squared models in
five dimensions was constructed as the Riemann-squared invariant in the
ungauged dilaton Weyl basis in Bergshoeff:2011xn . However, this model does
not refer to the standard Weyl multiplet; hence, the prescription to obtain
gauged models cannot be applied. Furthermore, the construction methodology
cannot be extended to the gauged dilaton Weyl multiplet.

Alternatively, a third independent, locally superconformal invariant
containing the Ricci tensor-squared term can be constructed Butter:2014xxa ;
Gold:2023dfe , which provides the correct basis to study gauged curvature-
squared supergravity, as we shall discuss momentarily. In this case, the
lowest component of the composite linear multiplet is given by the field
$L^{ij}_{\rm Log}$. This is obtained by making use of the standard Weyl
multiplet and by acting with six $Q$-supersymmetry transformations on the
field $\log{\rho}$, with $\rho$ being the lowest component of a compensating
vector multiplet Note3 . The rest of the composite “Log multiplet” is then
obtained by acting with up to two more $Q$-supersymmetry transformations on
$L^{ij}_{{\rm Log}}$. Due to the complexity of computing up to eight
supersymmetry transformations, the explicit form of the Log multiplet,
including all fermionic terms, has been obtained only recently with the aid of
the Cadabra software Cadabra-1 ; Cadabra-2 . These lengthy results will be
published elsewhere GJSG-2023 , see also Gold:2023dfe , and GJSGMY-2023  for
the complete analysis of the gauged supergravity case. Inserting the resulting
composite multiplet into (3) yields the explicit form of a new “Log invariant”
which will be presented in GJSG-2023  in the standard Weyl basis. Then, the
Log invariant in the gauged dilaton Weyl background can be obtained by
employing the map (1) and (4). For the purpose of this letter, it suffices to
present its bosonic sector in the gauge

$\displaystyle\sigma=1~{},\qquad b_{\mu}=0~{},\qquad\psi^{i}=0~{}.$ (5)

The gauged Log invariant in the dilaton Weyl background, which includes a
Ricci-squared term, reads

$\displaystyle e^{-1}{\cal L}_{{\rm Log}}$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
6}}}R_{{a}{b}}R^{{a}{b}}+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
24}}}R^{2}+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
6}}}R^{{a}{b}}G^{2}_{{a}{b}}+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
3}}}RH_{{a}{b}}G^{{a}{b}}-{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}R_{{a}{b}}H^{{a}{c}}G^{{b}}{}_{{c}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 3}}}RH^{2}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
12}}}\epsilon^{{a}{b}{c}{d}{e}}C_{{a}}V_{{b}{c}}\,^{ij}V_{{d}{e}ij}$ (6)
$\displaystyle+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
6}}}V^{{a}{b}ij}V_{{a}{b}ij}-2({{H}}^{2})^{2}+{\textstyle{\frac{\scriptstyle
16}{\scriptstyle
3}}}H^{2}_{{a}{b}}H^{{a}{c}}G^{b}{}_{c}-{\textstyle{\frac{\scriptstyle
4}{\scriptstyle 3}}}H^{2}H_{{a}{b}}G^{{a}{b}}+{\textstyle{\frac{\scriptstyle
2}{\scriptstyle
3}}}H_{{a}{b}}H_{{c}{d}}\big{(}G^{{a}{b}}G^{{c}{d}}-2G^{{a}{c}}G^{{b}{d}}\big{)}$
$\displaystyle+{\textstyle{\frac{\scriptstyle 2}{\scriptstyle
3}}}H^{2}G^{2}-{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}H^{{2}{a}{b}}G^{2}_{{a}{b}}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
3}}}H_{{a}{b}}G^{{a}{b}}G^{2}+G^{2}_{{a}{b}}H^{ac}G^{b}{}_{c}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 48}}}(G^{2})^{2}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 24}}}G^{4}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
6}}}\nabla_{{c}}G^{{a}{c}}\nabla^{{b}}G_{{a}{b}}$
$\displaystyle+2\nabla_{{a}}H_{{b}{c}}\nabla^{[{a}}H^{{b}{c}]}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
48}}}\epsilon^{{a}{b}{c}{d}{e}}\nabla^{{f}}G_{{e}f}(4H_{{a}{b}}-G_{{a}{b}})(4H_{{c}{d}}-G_{{c}{d}})$
$\displaystyle+{\textstyle{\frac{\scriptstyle g}{\scriptstyle
6}}}\Big{(}RN-4NH_{{a}{b}}G^{{a}{b}}-2NG^{2}+V_{{a}{b}}\,^{ij}L_{ij}(G^{{a}{b}}+4H^{{a}{b}})+12NH^{2}-6\nabla^{{a}}\nabla_{{a}}N\Big{)}$
$\displaystyle-{\textstyle{\frac{\scriptstyle g^{2}}{\scriptstyle
24}}}\Big{(}2RL^{2}-L^{2}(G^{2}-4G^{{a}{b}}H_{{a}{b}}-24H^{2})+4N^{2}+6\nabla^{{a}}L^{ij}\nabla_{{a}}L_{ij}\Big{)}+{\textstyle{\frac{\scriptstyle
2}{\scriptstyle 3}}}N{L}^{2}{g}^{3}+{\textstyle{\frac{\scriptstyle
5}{\scriptstyle 24}}}{L}^{4}{g}^{4}~{},$

where
$V_{ab}{}^{ij}=2\partial_{[a}V_{b]}{}^{ij}-2V_{[a}{}^{k(i}V_{b]k}{}^{j)}$.
Furthermore, we have used the following notations:
$H^{ab}=-\frac{1}{12}\epsilon^{abcde}H_{cde}$, $H^{2}=H^{ab}H_{ab}$,
$G^{2}=G^{ab}G_{ab}$, $H^{2}_{ab}:=H_{a}{}^{c}H_{bc}$,
$G^{2}_{ab}:=G_{a}{}^{c}G_{bc}$, $G^{4}=G^{2ab}G^{2}_{ab}$, and
$H^{4}=H^{2ab}H^{2}_{ab}$.

Note that the supersymmetric Riemann-squared invariant can be obtained by
taking the following linear combination of the ungauged Weyl-squared invariant
presented in Bergshoeff:2011xn  and setting $g=0$ in the Log invariant (6)

$\displaystyle{\cal L}_{{\rm Riem}^{2}}={\cal L}_{\rm Weyl^{2}}+2{\cal L}_{\rm
Log}|_{g=0}~{}.$ (7)

The resulting action is identical to the one presented in Bergshoeff:2011xn
up to total derivatives.

Going on shell and dual CFT.—Now let us study a certain linear combination of
the Einstein-Hilbert and all three curvature-squared invariants

$(16\pi G){\cal L}_{2\partial+4\partial}={\cal L}_{\rm EH}+\lambda_{1}{\cal
L}_{\rm Weyl^{2}}+\lambda_{2}{\cal L}_{\rm Log}+\lambda_{3}{\cal L}_{R^{2}}\
.$ (8)

where $G$ is Newton’s constant and all the invariants are given in the gauged
dilaton Weyl multiplet background. ${\cal L}_{\rm Weyl^{2}}$ and ${\cal
L}_{R^{2}}$ respectively denote the Weyl tensor squared and Ricci scalar
squared actions which are obtained by employing the maps (1) and (4) in the
standard Weyl multiplet results of Ozkan:2013nwa . Their explicit form is not
crucial here, but they are given in the Supplemental Material for the reader’s
convenience. The two-derivative Lagrangian ${\cal L}_{\rm EH}$ is obtained by
using the linear multiplet action in the standard Weyl multiplet basis
Coomans:2012cf ; Ozkan:2013uk  and the sequential use of the maps (1) and (4).
Note that, in this section, we rescaled the Lagrangians such that the
coefficient of their leading curvature-squared term is normalized to unity. To
go on-shell, we fix the gauge according to (5) and break SU(2) down to U(1) by
choosing

$\displaystyle L_{ij}$ $\displaystyle={\textstyle{\frac{\scriptstyle
1}{\scriptstyle\sqrt{2}}}}\delta_{ij}L\,,$ $\displaystyle V_{a}^{ij}$
$\displaystyle=V_{a}^{{}^{\prime}ij}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}\delta^{ij}V_{a}\ .$ (9)

Consequently, the two-derivative Lagrangian becomes

$\displaystyle e^{-1}{\cal L}_{\rm EH}=L(R-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
2}}}G_{ab}G^{ab}+4H_{ab}H^{ab}+2V_{a}^{{}^{\prime}ij}V^{{}^{\prime}a}_{ij})$
$\displaystyle+L^{-1}\partial_{a}L\partial^{a}L-2L^{-1}E_{a}E^{a}-2\sqrt{2}E_{a}V^{a}-2N^{2}L^{-1}$
$\displaystyle-4gC_{a}E^{a}-2gNL-4gN-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}g^{2}L^{3}+2g^{2}L^{2}\ .$ (10)

From the total Lagrangian (8), several auxiliary fields can be solved from
their field equations up to ${\cal O}(\lambda_{i})$

$\displaystyle N$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}gL(2+L)+{\cal O}(\lambda_{i})\ ,$ $\displaystyle\quad E_{a}$
$\displaystyle=$ $\displaystyle{\cal O}(\lambda_{i})\,,\quad
V^{{}^{\prime}ij}_{a}={\cal O}(\lambda_{i})\ .$ (11)

To arrive at the five-dimensional gauged minimal supergravity, we first
dualize $B_{\mu\nu}$ to a new 1-form gauge field $\widetilde{C}_{\mu}$
following the procedure in Coomans:2012cf ; Ozkan:2013uk . We then truncate
the model consistently by imposing

$L=1+{\cal O}(\lambda_{i})\,,\quad\widetilde{C}_{a}=C_{a}+{\cal
O}(\lambda_{i})\ .$ (12)

Following (12), the field equation of $E_{abc}$ now implies

$V_{a}=-{\textstyle{\frac{\scriptstyle 3}{\scriptstyle\sqrt{2}}}}gC_{a}+{\cal
O}(\lambda_{i})\ .$ (13)

Plugging (11)–(13) back to the total Lagrangian (8), one obtains the on-shell
theory up to first order in $\lambda_{i}$. It is important to note that in the
procedure outlined above, the ${\cal O}(\lambda_{i})$ terms arising from
substituting (11)–(13) to the two-derivative action either vanish
(proportional to the leading order equations of motion of auxiliary fields) or
can be removed by field redefinitions Argyres:2003tg . To recover the standard
convention of minimal supergravity, we rescale the graviphoton and the U(1)
coupling according to $C_{a}\rightarrow{\textstyle{\frac{\scriptstyle
1}{\scriptstyle\sqrt{3}}}}C_{a},\,g\rightarrow\sqrt{2}g$. To conclude,
following Myers:2009ij , the resulting Lagrangian can be further simplified by
redefining the metric and the U(1) gauge field. Eventually, the on-shell model
is recast in the form below

$\displaystyle(16\pi G)e^{-1}{\cal
L}_{2\partial+4\partial}=c_{0}R+12c_{1}g^{2}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}c_{2}G_{ab}G^{ab}$
$\displaystyle+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
12\sqrt{3}}}}c_{3}\epsilon^{abcde}C_{a}G_{bc}G_{de}+\lambda_{1}{\cal L}_{\rm
GB}|_{\rm onshell}\ ,$ (14)

where the various coefficients are

$\displaystyle c_{0}$ $\displaystyle=$ $\displaystyle
1+({\textstyle{\frac{\scriptstyle 28}{\scriptstyle
3}}}\lambda_{1}-20\lambda_{2}-4\lambda_{3})g^{2}\,,$ $\displaystyle c_{1}$
$\displaystyle=$ $\displaystyle 1+({\textstyle{\frac{\scriptstyle
50}{\scriptstyle 9}}}\lambda_{1}-{\textstyle{\frac{\scriptstyle
28}{\scriptstyle 3}}}\lambda_{2}+{\textstyle{\frac{\scriptstyle
52}{\scriptstyle 3}}}\lambda_{3})g^{2}\ ,$ $\displaystyle c_{2}$
$\displaystyle=$ $\displaystyle 1+({\textstyle{\frac{\scriptstyle
64}{\scriptstyle 9}}}\lambda_{1}-{\textstyle{\frac{\scriptstyle
92}{\scriptstyle 3}}}\lambda_{2}-{\textstyle{\frac{\scriptstyle
76}{\scriptstyle 3}}}\lambda_{3})g^{2}\ ,$ $\displaystyle c_{3}$
$\displaystyle=$ $\displaystyle
1-12(\lambda_{1}+3\lambda_{2}+3\lambda_{3})g^{2}\ ,$ (15)

and the on-shell Gauss-Bonnet invariant is given by

$\displaystyle{\cal L}_{\rm GB}|_{\rm
onshell}=R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 8}}}G^{4}$ $\displaystyle-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}W_{abcd}G^{ab}G^{cd}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2\sqrt{3}}}}\epsilon^{abcde}C_{a}R_{bc}{}^{fg}R_{defg}~{},$
(16)

where $W_{abcd}$ is the Weyl tensor. This on-shell action is consistent with
the generic result presented in Cassani:2022lrk  for the proper choice of
parameters. Based on the on-shell model (14) we find that the AdS5 radius
receives corrections from the higher-derivative terms and is given by

$\ell=g^{-1}(1+{\textstyle{\frac{\scriptstyle 8}{\scriptstyle
9}}}g^{2}\lambda_{1}-{\textstyle{\frac{\scriptstyle 16}{\scriptstyle
3}}}g^{2}\lambda_{2}-{\textstyle{\frac{\scriptstyle 32}{\scriptstyle
3}}}g^{2}\lambda_{3})\ .$ (17)

The effective Newton’s constant from Eq. (14) is then

$G_{\rm eff}=G+G(-{\textstyle{\frac{\scriptstyle 28}{\scriptstyle
3}}}\lambda_{1}+20\lambda_{2}+4\lambda_{3})g^{2}\ .$ (18)

The AdS5 vacuum preserves maximal eight supercharges Gutowski:2004ez ;
Gutowski:2004yv  and the dual field theory should be a $D=4,\,\mathcal{N}=1$
CFT. Utilizing (17) and (18) the $a$ and $c$ Weyl anomaly coefficients of the
dual CFT can be obtained via the standard holographic renormalization
procedure Henningson:1998gx ; Fukuma:2001uf

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{\pi}{8g^{3}G}-\frac{9\pi(\lambda_{2}+\lambda_{3})}{2gG}~{},$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{\pi}{8g^{3}G}+\frac{\pi(2\lambda_{1}-9\lambda_{2}-9\lambda_{3})}{2gG}~{},$
(19)

using which one finds the results above are consistent with $R$-symmetry
anomaly whose coefficients are related to those of the Weyl anomaly via
Anselmi:1997am ; Cassani:2013dba

$5a-3c=\frac{\pi c_{3}}{4g^{3}G}~{},\quad a-c=-\frac{\pi\lambda_{1}}{gG}\ .$
(20)

Conclusions and outlook.—In this letter, we provide the correct and complete
basis to study curvature-squared gauged supergravity in five dimensions. Based
on the new results, we successfully computed the anomaly coefficients
governing dual four-dimensional SCFTs. As four-dimensional SCFTs are
characterized by two anomaly coefficients, one would naturally anticipate the
emergence of only two independent linear combinations among the three four-
derivative couplings. Indeed, our analysis confirms this expectation, with
$\lambda_{2}$ and $\lambda_{3}$ consistently appearing together in the anomaly
coefficients as the combination $\lambda_{2}+\lambda_{3}$. However, when
examining the on-shell action (14), it becomes evident that $\lambda_{2}$ and
$\lambda_{3}$ do not share this combination. This suggests that while two
curvature-squared invariants may suffice for calculating BPS quantities
Baggio:2014hua ; Bobev:2021qxx ; Bobev:2022bjm ; Cassani:2022lrk , the
computation of generic physical parameters may require the incorporation of
all three invariants.

We can also generalize the results by coupling multiple vector multiplets
which also enjoy an off-shell formulation. Given the simple form the 6D
ungauged Gauss-Bonnet invariant Liu:2013dna ; Novak:2017wqc ; Butter:2018wss
and the relation between dilaton Weyl multiplets in these two dimensions
Bergshoeff:2011xn , it may be feasible to reformulate the 5D Gauss-Bonnet
invariant into a more elegant expression that facilitates the construction of
intriguing solutions. For instance, the non-existence of supersymmetric AdS5
black ring solutions in the two-derivative theory Grover:2013hja  raises the
intriguing question of whether this situation changes in the presence of
higher-derivative interactions. Our new invariants enable the computation of
corrections to the entropy of ${\textstyle{\frac{\scriptstyle 1}{\scriptstyle
16}}}$-BPS black holes Gutowski:2004ez ; Gutowski:2004yv ; Cvetic:2004ny ;
Chong:2005hr ; Kunduri:2006ek , thereby extending the precision test of black
hole microstate counting to the next to leading order. It is also interesting
to extend the recently proposed equivariant localization
BenettiGenolini:2023ndb ; BenettiGenolini:2023kxp  beyond the leading two-
derivative cases.

Acknowledgements.—The work of G.G., J.H., S.K., and G.T.-M. was supported by
the Australian Research Council (ARC) Future Fellowship FT180100353, and by
the Capacity Building Package of the University of Queensland. G.G. and S.K.
are supported by postgraduate scholarships at the University of Queensland.
M.O. acknowledges the support by the Outstanding Young Scientist Award of the
Turkish Academy of Sciences (TUBA-GEBIP). The work of Y.P. is supported by
National Natural Science Foundation of China (NSFC) under grant No. 12175164
and the National Key Research and Development Program under grant No.
2022YFE0134300.

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SUPPLEMENTAL MATERIAL
(APPENDICES)

In the Supplemental Material to our letter, we present the bosonic sectors of
the off-shell Weyl-squared and scalar curvature-squared invariants in the
gauged dilaton Weyl background. These results can be obtained by starting from
the invariants written in a standard Weyl multiplet background, then applying
the map of fields (1) which defines the gauged dilaton Weyl multiplet.

##  Appendix A A. The gauged Weyl-squared invariant

The bosonic sector of the gauged Weyl-squared invariant in the gauged dilaton
Weyl background, eq. (1), and in the gauge (5), is given by:

$\displaystyle e^{-1}{\cal L}_{{\rm Weyl}^{2}}$ $\displaystyle=$
$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
8}}}\epsilon^{{a}{b}{c}{d}{e}}C_{{a}}R_{{b}{c}fg}R_{{d}{e}}{}^{fg}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
6}}}\epsilon^{{a}{b}{c}{d}{e}}C_{{a}}V_{{b}{c}}\,^{ij}V_{{d}{e}ij}+{\textstyle{\frac{\scriptstyle
2}{\scriptstyle 3}}}V^{{a}{b}ij}V_{{a}{b}ij}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
4}}}R_{{a}{b}{c}{d}}R^{{a}{b}{c}{d}}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 3}}}R_{{a}{b}}R^{{a}{b}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 12}}}{R}^{2}$ (22)
$\displaystyle+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
3}}}R_{{a}{b}{c}{d}}({{G}}^{{a}{b}}G^{{c}{d}}-2H^{{a}{b}}H^{{c}{d}}-3{{H}}^{{a}{b}}{{G}}^{{c}{d}})-{\textstyle{\frac{\scriptstyle
4}{\scriptstyle
3}}}R_{{a}{b}}{{H}}^{{a}{c}}G^{{b}}{}_{{c}}+{\textstyle{\frac{\scriptstyle
16}{\scriptstyle
3}}}R^{{a}{b}}{{H}}^{2}_{{a}{b}}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 3}}}R{{H}}_{{a}{b}}G^{{a}{b}}-{\textstyle{\frac{\scriptstyle
4}{\scriptstyle 3}}}RH^{2}$
$\displaystyle-4(H^{2})^{2}-8H^{4}-{\textstyle{\frac{\scriptstyle
16}{\scriptstyle 3}}}H^{2}H_{{c}{d}}G^{{c}{d}}-{\textstyle{\frac{\scriptstyle
40}{\scriptstyle
3}}}H^{2}_{{a}{d}}H^{a}{}_{{c}}G^{{c}{d}}+{\textstyle{\frac{\scriptstyle
8}{\scriptstyle 3}}}H^{2}G^{2}+{\textstyle{\frac{\scriptstyle 2}{\scriptstyle
3}}}H_{{a}{b}}H_{{c}{d}}G^{{a}{b}}G^{{c}{d}}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 12}}}(G^{2})^{2}$
$\displaystyle-{\textstyle{\frac{\scriptstyle 16}{\scriptstyle
3}}}H^{2}_{{a}{b}}G^{2{a}{b}}-{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}H_{{a}{b}}H_{{c}{d}}G^{{a}{c}}G^{{b}{d}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
3}}}H_{{a}{b}}G^{{a}{b}}G^{2}+2G^{2{a}{b}}G^{{c}}{}_{{b}}H_{{c}{a}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 3}}}(\nabla^{{a}}{G_{{b}{c}}})\nabla_{{a}}{G^{{b}{c}}}$
$\displaystyle+{\textstyle{\frac{\scriptstyle 8}{\scriptstyle
3}}}(\nabla^{{a}}H_{{b}{c}})\nabla_{{a}}H^{{b}{c}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 2}}}G^{4}+{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}\epsilon^{{a}{b}{c}{d}{e}}H_{{a}{b}}H_{{c}{d}}\nabla^{{f}}{G_{{e}{f}}}-2\epsilon^{{a}{b}{c}{d}{e}}H_{{b}{f}}(\nabla_{{a}}H_{{c}}{}^{{f}})G_{{d}{e}}$
$\displaystyle-{\textstyle{\frac{\scriptstyle 2}{\scriptstyle
3}}}\epsilon^{{a}{b}{c}{d}{e}}H_{{a}{b}}(\nabla^{{f}}{G_{{c}{f}}})G_{{d}{e}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
24}}}\epsilon^{{a}{b}{c}{d}{e}}(\nabla^{{f}}{G_{{a}{f}}})G_{{b}{c}}G_{{d}{e}}$
$\displaystyle-g\left(-{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}NH_{ab}G^{ab}+{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}NG^{2}-8NH^{2}+{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}V_{ab}{}^{ij}L_{ij}H^{ab}-{\textstyle{\frac{\scriptstyle 2}{\scriptstyle
3}}}V_{ab}{}^{ij}L_{ij}G^{ab}-{\textstyle{\frac{\scriptstyle 2}{\scriptstyle
3}}}RN\right)$ $\displaystyle+g^{2}\left({\textstyle{\frac{\scriptstyle
1}{\scriptstyle 3}}}L^{2}H^{ab}G_{ab}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 3}}}L^{2}G^{2}+2L^{2}H^{2}-{\textstyle{\frac{\scriptstyle
8}{\scriptstyle 3}}}N^{2}+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
6}}}RL^{2}\right)-{\textstyle{\frac{\scriptstyle 4}{\scriptstyle
3}}}g^{3}NL^{2}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
6}}}g^{4}L^{4}~{},$

where $H^{4}:=H^{2}{}^{ab}H^{2}_{ab}$ and other combinations of product of
fields are defined after eq. (6).

##  Appendix B B. The gauged scalar curvature-squared invariant

The bosonic sector of the gauged scalar curvature-squared invariant in the
gauged dilaton Weyl background, eq. (1), and in the gauge (5), is given by:

$\displaystyle e^{-1}{\cal L}_{R^{2}}$ $\displaystyle=$
$\displaystyle\mathbf{Y}^{ij}\mathbf{Y}_{ij}-2{\nabla}^{{a}}(NL^{-1}){\nabla}_{{a}}(NL^{-1})-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
8}}}\epsilon_{{a}{b}{c}{d}{e}}C^{{a}}\mathbf{G}^{{b}{c}}\mathbf{G}^{{d}{e}}+NL^{-1}G^{{a}{b}}\mathbf{G}_{{a}{b}}-N^{2}L^{-2}G^{ab}G_{ab}$
(23a) $\displaystyle+4N^{2}L^{-2}H^{ab}G_{ab}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle
4}}}\mathbf{G}^{{a}{b}}\mathbf{G}_{{a}{b}}-4NL^{-1}H_{ab}\mathbf{G}^{{a}{b}}-4gN^{3}L^{-2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 16}}}g^{4}L^{4}$
$\displaystyle+g^{2}\Big{(}{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}L^{ij}{\nabla}^{a}{\nabla}_{a}{L_{ij}}-{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 4}}}RL^{2}-H^{2}L^{2}+{\textstyle{\frac{\scriptstyle
1}{\scriptstyle 8}}}G^{2}L^{2}-{\textstyle{\frac{\scriptstyle 5}{\scriptstyle
2}}}N^{2}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}E^{a}E_{a}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2}}}{\nabla}^{a}{L}{\nabla}_{a}{L}\Big{)}~{},$ where,
$\displaystyle\mathbf{G}_{{a}{b}}$ $\displaystyle=$ $\displaystyle
4\nabla_{[a}(L^{-1}E_{b]})+2L^{-1}L_{ij}(V_{ab}{}^{ij})-2L^{-3}L_{ij}\big{(}\nabla_{[a}L^{ik}\big{)}\nabla_{b]}L_{k}{}^{j}~{},$
$\displaystyle\mathbf{Y}^{ij}{}$ $\displaystyle=$
$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle
4}}}L^{-1}\left(4\nabla^{a}\nabla_{a}{L^{ij}}-2R{L^{ij}}-8H^{2}{L^{ij}}+G^{2}{L^{ij}}\right)$
(23b)
$\displaystyle+L^{-3}\left(-N^{2}{L^{ij}}-E^{a}E_{a}{L^{ij}}-2E^{a}L^{k(i}\nabla_{a}L_{k}{}^{j)}-{L_{kl}}\nabla^{a}{L^{k(i}}\nabla_{a}{L^{j)l}}\right)\
.$