Prepared for submission to JHEP

One-modulus Calabi-Yau fourfold
reductions and higher-derivative terms

T.W. Grimm,a K. Mayer,a and M. Weissenbacherba Utrecht University,

Princetonplein 5, 3584 CE Utrecht, The Netherlandsb Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo,

Kashiwa-no-ha 5-1-5, 277-8583, Japan

E-mail: t.w.grimm@uu.nl, k.mayer@uu.nl,
matthias.weissenbacher@ipmu.jp
Abstract: In this note we consider M-theory compactified on a warped Calabi–Yau
fourfold including eight-derivative terms in the eleven-dimensional action known in
the literature. We dimensionally reduce this theory on geometries with one Kähler
modulus and determine the resulting three-dimensional Kähler potential and complex
coordinates. The logarithmic form of the corrections suggests that they might admit
a physical interpretation in terms of one-loop corrections to the effective action.
Including only the known terms the no-scale condition in three dimensions is broken,
but we discuss caveats to this conclusion. In particular, we consider additional
new eight-derivative terms in eleven dimensions and show that they are strongly
constrained by compatibility with the Calabi-Yau threefold reduction. We examine
their impact on the Calabi-Yau fourfold reduction and the restoration of the no-scale
property.

Contents
1 Introduction

1

2 Setting up higher-derivative Calabi-Yau fourfold reductions
2.1 Three-dimensional gauged N = 2 supergravity
2.2 Higher derivative corrections in M-theory
2.3 Calabi–Yau fourfold solution including higher derivatives

3
3
5
6

3 M-theory on Calabi–Yau fourfolds with higher-derivative corrections
7
3.1 Computation of the quantum-corrected Kähler potential
7
3.2 Comments on loop corrections and M/F-theory duality
10
4 Towards a completion of the Ĝ2 R̂3 sector
4.1 Ansatz for a general basis of relevant Ĝ2 R̂3 terms
4.2 Constraints from IIA on Calabi–Yau threefolds
4.3 Constraints from M-theory on Calabi–Yau threefolds
4.4 The Ĝ2 R̂3 -extension on Calabi–Yau fourfolds

12
13
14
15
16

5 Conclusions

17

A Higher-derivative terms

18

B Reduction results

19

C Basis of relevant Ĝ2 R3 terms

20

1

Introduction

Compactifications of eleven-dimensional supergravity, which is the long wavelength
limit of M-theory, has been proven to be a fruitful playground for studying various aspects of both conceptual and phenomenological relevance. Compactifying M-theory
on a Calabi–Yau three- or fourfold leads to a theory with N = 2 supersymmetry infive or three dimensions, respectively. The compactifications of the two-derivative
action in eleven dimensions were worked out in [1, 2]. It is, however, well known
that eleven-dimensional supergravity receives quantum corrections in the form of
higher-derivative operators that are suppressed by six or more powers of the elevendimensional Planck length `M . The most prominent of these corrections are terms–1–quartic in the Riemann tensor that also have counterparts in both Type II string
theories. These terms are known to give rise to quantum corrections to the moduli
space of N = 2 compactifications of Type II string theories on Calabi-Yau threefolds[3–5].
Besides their phenomenological significance, higher-derivative corrections in elevendimensions can play an important role in the physics of black objects. These terms ineleven dimensions can lead to corrections to the macroscopic black hole entropy [6–9]. Phenomenological applications of M-theory compactifications are mostly within
the framework of F-theory [10]. The physics of F-theory is most conveniently studied in the dual formulation given by M-theory on an elliptically fibered Calabi-Yau
manifold, see e.g. [11, 12]. The actual physics is then encoded in the non-trivial
elliptic fibration. Quantum corrections arising in M-theory compactifications mightthen lift to corrections in F-theory using the M-theory to F-theory duality. Amongthese quantum corrections could possibly be a correction to the three-dimensional
Kähler potential stemming from the R̂4 terms in eleven dimensions. Even thoughmuch work was done in the past few years to identify such a correction a definite
answer remained elusive, see [13–19]. Due to the lacking of a fundamental formulation of F-theory we take its dual formulation in terms of M-theory as a definition of
F-theory. It is obvious from this point of view, that before attempting to lift certain
corrections to four dimensions one first has to understand the dual three-dimensional
configuration reasonably well.
In this note we aim to elaborate on the existence of a correction to the threedimensional N = 2 Kähler potential by considering the most simple setup. We
6
corrections and reduce themconsider M-theory including known and conjectured `M
on a Calabi–Yau fourfold with only one modulus, namely the overall volume of thecompact manifold. We use the corrected fourfold solution involving fluxes and warping found in [16, 20, 21] and derive a three dimensional effective action including
the gravity multiplet and one vector multiplet. We show that the resulting action
is compatible with three-dimensional N = 2 supersymmetry. Upon dualizing thevector multiplet into a chiral multiplet we derive the corrected Kähler potential andassociated complex coordinates. We show that the result breaks the no-scale condition in three dimensions leading to a non-vanishing scalar potential for the overall
volume. Our results allow us to make some first observations about the M-theory
to F-theory limit. We point out that the corrections are reminiscent of one-loop
corrections found in three-dimensional effective theories obtained after integratingout massive charged modes arising in a circle compactification from four to three
dimensions [22–27].
One important caveat to point out is the fact that the eight-derivative terms in
the eleven-dimensional action have not been established by a supersymmetrization
procedure or conclusively determined by string amplitudes. Therefore, one might
worry that our result obtained from dimensional reduction of the terms suggest in–2–the literature might change significantly if further new higher-derivative terms areincluded. To check their influence we therefore examine the dimensional reductionof a basis of potentially relevant eight-derivative terms of the form Ĝ2 R̂3 , where Ĝ
is the eleven-dimensional four-form field strength. We constrain the coefficients in a
general ansatz by demanding compatibility with compactification on a Calabi–Yauthreefold. We find that a specific combination can potentially cancel the logarithmiccorrection and restore the no-scale condition. However, due to the suggested physicalinterpretation of the logarithmic correction, we believe that the qualitative featuresof our result will remain unchanged once the actual supersymmetric combination of
the Ĝ2 R̂3 terms has been determined.
This paper is organized as follows. In section 2 we review three-dimensional
gauged N = 2 supergravity with gauged shift symmetries and also state the dictionary for dualizing vector multiplets into chiral multiplets in three dimensions.
6
We then summarize the `M-corrected eleven-dimensional supergravity action and theconsidered fourfold solution with background fluxes and a non-trivial warp-factor.
In section 3 we present our reduction ansatz and perform the dimensional reductionof the higher-derivative terms known in the literature including one Kähler structure modulus. From the results of the dimensional reduction to three dimensions
we then infer the Kähler potential and coordinate describing the N = 2 supersymmetric theory. Section 4 contains a general basis of relevant eight-derivative terms
and constrains their coefficients using five- and four-dimensional supersymmetry arguments. We discuss that such additional terms can have severe consequences forthe one-modulus reduction.

2

Setting up higher-derivative Calabi-Yau fourfold reductions

This section collects the relevant material to perform the one-modulus dimensional
reduction in section 3. We introduce three-dimensional gauged N = 2 supergravity,
the considered eleven-dimensional effective action including higher derivative terms,
and the eleven-dimensional background solution.
2.1

Three-dimensional gauged N = 2 supergravity

In this section we briefly review N = 2 gauged supergravity in three dimensions.
Three-dimensional maximal and non-maximal supergravities and their gaugings were
exhaustively discussed in [28]. The case which relevant for our setting is an N = 2
supergravity theory with a gauged shift symmetry and was studied, for example, in[29, 30]. This shift symmetry corresponds to an isometry of the geometry describing
the scalar field space of the N = 2 theory. We consider three-dimensional N = 2
supergravity coupled to chiral multiplets whose complex scalars are denoted by N a .
The gaugings are realized along certain isometries X̃ ab . We also use a constant–3–

embedding tensor denoted by Θab . The N = 2 action then reads
Z

(3)
a
b̄
a
b
1
1
SN =2 =
R
?
1
−
K
DN
∧
?
D
N̄
−
Θ
A
∧
F
−
V
+
V
? 1,
T
F
ab
a
b̄
2
2

(2.1)

M3

where Kab̄ = ∂N a ∂N̄ b̄ K is a Kähler metric with Kähler potential K. The gauge
covariant derivative DN a is defined by
DN a = dN a + Θbc X̃ ab Ac .

(2.2)

The scalar potential in (2.1) is given by
VT = K ab̄ ∂a T ∂b̄ T − T 2 ,

(2.3)


VF = eK K ab̄ Da W Db W − 4|W |2 ,where T is a real function of the chiral fields N a which will be given more explicitly below. Note that the vectors entering (2.1) via a Chern-Simons term is
non-dynamical. In (2.3) we introduced the hermitian matrix K ab̄ = (K −1 )ab̄ and aholomorphic superpotential W .
Dualizing the action.
We now split the chiral fields as N a = (M I , TΛ ) toobtain a dual description of the action (2.1) by dualizing the chiral multiplets withbosonic component TΛ into vector multiplets. The detailed procedure can be foundin [29, 30], we will therefore just quote the result. Since the dualization is in general
not possible, one has to assume that the action (2.1) is invariant under shifts of
Im TΛ . The relevant gauging is achieved by choosing a constant embedding tensorandX̃ ΛΣ = −2i δ ΛΣ ,¯X̃ IJ = X̃ I J = 0 ,X̃ ΛI = 0 ,ΘIJ = 0 .

(2.4)

The dual action is then given by
Z
¯
(3)
1
SN =2, dual =R ? 1 − K̃M I M̄ J DM I ∧ ?DM̄ J + 14 K̃LΛ LΣ dLΛ ∧ ? dLΣ
2
Z M3


1
+
K̃ Λ Σ F Λ ∧ ? F Σ + 21 ΘΛΣ AΛ ∧ F Σ + F Λ ∧ Im K̃LΛ M I DM I
4 L L
ZM3

−
VT + VF ? 1 .
(2.5)
M3

The physical couplings in (2.5) such as K̃LΛ LΣ = ∂LΛ ∂LΣ K̃ are now derived froma kinetic potential K̃, which is defined in terms of the Kähler potential K by a
Legendre transformation
K(M, T ) = K̃(M, L) − Re TΛ LΛ ,

–4–

(2.6)

where the real coordinates conjugate to Re TΛ are defined by LΛ = −2KTΛ =−2∂TΛ K. The scalars LΛ are now scalars in (propagating) vector multiplets. Using
the Legendre transformation (2.6) one can derive many expressions relating derivatives of K with derivatives of K̃ among which the most important ones areΛΣKTΛ T̄Σ = − 14 K̃ L L ,Re TΛ = K̃LΛ ,∂LΛΛ Σ= 12 K̃ L L .
∂TΣ

(2.7)

These relations are extensively used in the explicit dualization procedure. The scalarpotential VT reads in the vector multiplet variables
I

J¯

Λ

Σ

VT = K̃ M M̄ ∂M I¯T ∂M̄ J¯T − K̃ L L ∂LΛ T ∂LΣ T − T 2 ,T = − 21 LΛ ΘΛΣ LΣ .

(2.8)

The scalar potential VF is in the vector multiplet language given by
i
h

¯
2
K
Σ
Λ
M I M̄ J
VF = e K̃
DM I W DM J W − 4 + L K̃LΛ LΣ L |W | .

(2.9)

The only relevant part to compute the scalar potential in the chiral multiplet formulation (2.3) will be the function T given in (2.8) and the Kähler potential K, as we
will assume a constant superpotential later.
2.2

Higher derivative corrections in M-theory

In the following we will review eight-derivative terms in eleven dimensions available
in the literature [31–39] that will be relevant for the computation of the corrected
three-dimensional, two-derivative effective action arising upon compactification on a
Calabi–Yau fourfold Y4 . These corrections are given by two sectors. The well known,purely gravitational eight-derivative terms R̂4 are supplemented by terms involving
the four-form field strength. The bosonic part of the classical two-derivative N = 1
action in eleven dimensions is given byZ
1
1
2
2κ11 S11 =R̂ ∗ˆ 1 − Ĝ ∧ ∗ˆ Ĝ − Ĉ ∧ Ĝ ∧ Ĝ .
(2.10)
2
6
M11
These terms are supplemented by certain eight-derivative couplings, such as thefamous R̂4 terms
Z2
1
2κ11 SR̂4 =t̂8 t̂8 − 24
11 11 R̂4 ∗ˆ 1 − 32 213 Ĉ ∧ X̂8(2.11)
M11

which are related to the R-symmetry and conformal anomaly of the world-volume
theory of a stack of N M5 branes [35]. In addition, there are eight-derivative terms
containing the four-form field strength. The latter take the schematic form [39]
Z


2
1
ˆ Ĝ 2 R̂2 ∗ˆ 1+256 Ẑ Ĝ∧ˆ∗ Ĝ . (2.12)
2κ11 SĜ =− t̂8 t̂8 + 96
11 11 Ĝ2 R̂3 ∗ˆ 1+ ŝ18 ∇
M11

–5–

The last term in (2.12) was argued to be necessary to ensure Type IIA/M-theory
duality when considering Calabi–Yau threefold compactifications [5]. The detailed
form of the higher-derivative couplings in (2.11)-(2.12) is relegated to appendix A.
ˆ Ĝ 2 R̂2 can be
The detailed index structure of the terms of the schemaric form ∇found e.g. in [17] appendix A.
2.3

Calabi–Yau fourfold solution including higher derivatives

In this section we review the fourfold solutions including eight-derivative terms studied in [16, 20, 21]. The background solution is taken to be an expansion in terms ofthe dimensionful parameter 1
2(4π κ211 ) 3
α =,(2π)4 32 · 213
292κ211 = (2π)5 `M,(2.13)

which reduces to the ordinary direct product solution R1,2 × Y4 without fluxes andwarping to lowest order in α. All terms at and including O(α2 ) are kept, while higher
orders are neglected. At higher order both a warp-factor A(0) and fluxes are induced.
The solution then takes the form


2 (2)
2 (2)
2 (2)
hdŝ2 i = eα Φ e−2α A ηµν dxµ dxν + 2eα A gi̄ dz i dz̄ ̄ ,(2.14)
2 (2) hĜi = α G(1) + dvolR1,2 ∧ d e−3α A .
(2.15)
Using this ansatz one can work out the constraints following from the equations of
motion. It turns out that the metric gi̄ is given by an expansiongi̄ = gi̄(0) + α2 gi̄(2) ,gi̄(2) ∼ ∂i ∂¯̄ ∗(0) J (0) ∧ J (0) ∧ F4 ,(2.16)

where g (0) is the lowest order, Ricci-flat Calabi–Yau metric and J (0) is its associatedKähler form. We furthermore denote with F4 the non-harmonic part of the third
Chern form, which will however be irrelevant for the following discussion, as it drops
out of all expressions in the effective action. The metric solution also includes an(0)∗(0) c(0)and a warp-factor A(2) (z, z̄) satisfying
overall Weyl factor Φ(2) = − 5123 ∧J
3the warp-factor equation2∆(0) e3α A(2)

1 2 (1)dvol(0)∧ G(1) − 32 213 α2 X8(0) = 0 .
Y4 + α G
2(2.17)The background value of the four-form field strength is parametrized by the internal
flux G(1) ∈ H 4 (Y4 ), which is self-dual with respect to the lowest order Calabi–Yaumetric, and a piece proportional to the volume form dvolR1,2 on R1,2 .
1

We follow the conventions of [35].

–6–

3

M-theory on Calabi–Yau fourfolds with higher-derivative
corrections

In this section we perform the dimensional reduction for the case of a Calabi–Yau
fourfold Y4 with h1,1 = 1. This simplified setup allows us to determine the corrected
Kähler potential and infer the broken no-scale property. Intermediate results of thedimensional reduction are deferred to appendix B.
3.1

Computation of the quantum-corrected Kähler potential

We now perform the dimensional reduction of eleven-dimensional supergravity on
the background reviewed in section 2.3. We will do this for the simplified caseh1,1 = dim H 1,1 (Y4 ) = 1. The single Kähler modulus is then given by the volume of
the Calabi–Yau fourfold V. In other words, we expand the Kähler form in a single(1, 1)-form ω asZ
1
1
J = V4ω ,ω4 = 1 ,
(3.1)
4! Y4
where we have normalized ω to avoid cluttering by numerical factors. Note that ω
is harmonic with respect to the zeroth-order background metric gi̄(0) .
The simplified analysis with h1,1 = 1 comes with two main advantages. Firstly,
one can deduce from the warp-factor equation (2.17) the dependence of the warpfactor on the volume. Secondly, the couplings in the three-dimensional effective
action are all topological as opposed to the case for general h1,1 considered in [17, 19].
1
the warp-factor equation should scale
Upon a rescaling of the metric gi̄(0) → V 4 gi(0)j̄homogeneously, mapping a solution of the equation to another solution. This implies
that Ã(z, z̄, V) = V −1 A(2) , where A(2) is a solution of the warp-factor equation withrespect to the metric gi̄(0) . We already noted that the correction to the metric gi̄(2)decouples from the effective action since it only contributes total derivatives. In thefollowing we therefore drop the metric correction from any expression. The reduction
ansatz for the metric and the M-theory four form field strength is thus


1
2
2
2
dŝ2 = eα Φ e−2α Ã gµν dxµ dxν + 2eα Ã V 4 iωi̄ dz i dz̄ ̄ ,
(3.2)
2 
Ĝ = α G(1) + dvolR1,2 ∧ dY4 e−3α A + F ∧ ω ,(3.3)
where F = dA is the field strength of a three-dimensional vector from the expansion
3of Ĉ along the harmonic (1,1)-form ω, Φ = V − 4 Φ(2) and Ã = V −1 A(2) . Before we
continue with the reduction, let us introduce the useful quantities
Z
Z
Z
1
1
3
4
χ̃ = (2π)
c3 ∧ ω ,A =Ã J =
A(2) ω 4 ,
(3.4)
4!
4!
Y4
Y4
Y4
where χ̃ is a constant depending on the topology of Y4 , and A is a constant dependingon the warp-factor profile and background metric of Y4 .

–7–

We will now perform the dimensional reduction of the eleven dimensional action
including eight derivative couplings of interest. The resulting theory is has N = 2
supersymmetry in three dimensions and contains the gravity multiplet, whose bosonicfield is the three-dimensional metric gµν , and a vector multiplet formed by the 3D
vector and the volume V along with their fermionic superpartners. Focusing only on
the bosonic part of the action, we first use the reduction results in appendix B andthen perform a Weyl rescaling to Einstein frame in three dimensions. The resulting
action for the kinetic terms takes the form
Z h
3
2
1
9
κ11 Skin =(3.5)
R ? 1 − 16d log V ∧ ? d log V − V 2 F ∧ ? F
2
M3

3+ 92 α2 V −1 A d log V ∧ ? d log V + 216 α2 V − 4 χ̃ d log V ∧ ? d log V
i
3
1
2
2
2
4
− 6 α V A F ∧ ? F + 384 α V χ̃ F ∧ ? F .
3

To compare this result with the general action (2.5) we first define L = V − 4 −7
3 α2 A V − 4 to rewrite (3.5) as
Z
1
1
1
(3)
2
κ11 Skin =R ? 1 − 2 dL ∧ ? dL − 2 F ∧ ? F
(3.6)
L
L
M3 2
Z
1
1
+
384 α2 χ̃ F ∧ ? F + 384 α2 χ̃ dL ∧ ? dL .
LL
M3
It is now easily seen that (3.6) takes the standard form
Z
1
1
1
(3)
Sstand =R ? 1 + G̃LL (L) dL ∧ ? dL + G̃LL (L) F ∧ ? F ,
4
4
M3 2
with
G̃LL (L) = −


4
4
1
2
1
−
384
α
χ̃
L
= − 2 + 1536 α2 χ̃ .
2
L
L
L

(3.7)

(3.8)

We can integrate the metric G̃LL to obtain the kinetic potential K̃(L) and coordinateK̃ = 4 log L + 1536 α2 χ̃ L log(L) − 1 + 4 ,
L=V

− 43

− 3 α2 A V

− 74

,

(3.9)
(3.10)

where we have chosen the integration constants in a convenient way.
Determining the Kähler potential. We will now dualize the vector multiplet
to a chiral multiplet, whose metric derives from a Kähler potential. This is achievedby a Legendre transformation of the kinetic potential as outlined in section 2.1
K = K̃ − L Re T ,Re T = ∂L K̃ .

–8–

(3.11)

One thus derives the Kähler potential K(T + T̄ ) to be
K = 4 log L − 1536 α2 χ̃ LZ
1 Z
4α2 A 42
3
= −3 loge
J + 512 α (2π)
c3 ∧ J ,4! Y4
Y4(3.12)

with corresponding coordinate
4+ 1536α2 χ̃ log L
L
1
3= 4V 4 + 12α2 V − 4 A − 1152 α2 χ̃ log V .

Re T =(3.13)All quantities in the Kähler potential (3.12) now depend on the modulus V. On
a first sight one might be wonder about that the unusual correction to the Kähler
coordinate ∝ log V. We will comment on the physical interpretation of this correction
in section 3.2.
Let us stress that the analysis of [19] also lead to a Kähler potential of theform (3.12). However, in the analysis performed there, it was not possible to fix
all the coefficients in K unambiguously. Furthermore, the discussion of the Kähler
coordinates was incomplete, due to the presence of many non-topological terms inthe more moduli case. In this one modulus analysis we were able to avoid these
problems. Nevertheless, it is important to point out that the pure warping result of[19], which was inspired by [40], agrees with our findings here.
The no-scale condition and the scalar potential. The essential key point
6of this note is that the no-scale condition in three dimensions is broken once `M
suppressed corrections to the Kähler potential in (3.12) are taken into account. We
can straightforwardly compute
Z
KT2
2
3 −1
T T̄
KT K KT̄ == 4 − 1536 α (2π) V
c3 ∧ J ,(3.14)KT T̄
Y4
which indeed shows that the no-scale condition is broken. We can now also use thisresult to determine the scalar potential. We first evaluate
VT = K T T̄ ∂T T ∂T̄ T − T 2−1 1  ∂Re T −1 ∂T 2
3
− T 2 = 0 + O(α3 ) ,= 16 V 2 + . . .
2 ∂V∂Vwhere we used3
T (V) = − 21 α Θ V − 2 + . . .

1
Θ=
2

,

Z
ω ∧ ω ∧ G(1) .

(3.15)

Y4

This means that the only scalar potential comes from the breaking of the no-scalecondition. It enters the effective action via the F-term scalar potential
2 Z

T T̄
2
3 2 |W0 |K
VF = e K DT W DT W − 4|W | = −1536 (2π) αc3 ∧ J ,(3.16)
V 4 Y4–9–

R
which has a runaway direction for V → ∞ if Y4 c3 ∧ J < 0 2 . We assumed a constant superpotential W0 in (3.16) which may arise from stabilizing complex structure
moduli and inserting their fixed values in the GVW superpotential [41]
Z
1
W = 3
G(1) ∧ Ω ,Ω ∈ H 4,0 (Y4 ) .
(3.17)`M Y4
The runaway behavior of (3.16) for large volume V signals an instability of thesolution for the case of a non-vanishing W0 . This raises doubts about the validityof the reduction for such a non-vanishing W0 as recently stressed in [42]. Let us
emphasize that the vacuum solution around which we expand was supersymmetricand therefore W0 should actually vanish in the vacuum for our analysis to be selfconsistent.
3.2

Comments on loop corrections and M/F-theory duality

One of the main motivations to study M-theory compactifications on Calabi–Yau
fourfolds is its duality to 4D F-theory models with minimal supersymmetry. Uponcompactifying the 4D F-theory action on a circle and taking the F-theory limit ofvanishing torus-fiber volume vol(T2 ) → 0, one can infer F-theory data from thematching with the 3D M-theory compactification. Since F-theory requires an elliptic
fibration with h1,1 > 1 our analysis does not immediately apply to this case. Nevertheless we will try to give some first interpretation of the result (3.12) in the context
of this duality.
Let us assume for a moment that our result is valid beyond the one modulus case.
While this is semingly straightforward for the Kähler potential K it is less obvious
how to generalize the complex coordinates TI . A reasonable assumption appears tobe that the TI contain a correction proportional toZ
3χ̃I = (2π)
c3 ∧ ωI ,
I = 1, . . . , dim H 1,1 (Y4 ) ,(3.18)
Y4

multiplied with the volume of some submanifold of Y4 . The 3D N = 2 theory with thisKähler potential K and coordinates TI can then be thought of as a circle-compactified
4D N = 1 theory with an infinite tower of (massive) Kaluza-Klein states. The 3D
Wilsonian effective action which is valid below some energy scale Λcutoff is then calculated by integrating out the massive fields. These massive fields running in loops canthen in general modify the physical couplings in the effective action up to arbitraryorder in the diagrammatic loop expansion, unless some non-renormalization theorem
comes to the rescue. These loop effects are certainly important when employing
M/F-theory duality. In particular, the duality may mix classical contributions on
An example with this property and h1,1 = 1 is the sextic fourfold. For the sextic one findsc ∧ ω = −420.
Y4 3
2

R

– 10 –

one side of the duality with quantum corrections to the effective action on the otherside of the duality. The inclusion of one-loop corrections turned out to be tracking
the Chern-Simons terms through the M-theory to F-theory limit in compactifications
to three dimensions [43–45] and five dimensions [44, 46, 47].
The simplest 4D setting to start with is a supergravity theory with only the N =1 gravity multiplet, i.e. a pure supergravity theory. Considering this theory on the
background R1,2 ×S 1 the leading perturbative correction to the 3D Kähler coordinate
was determined in [27]. It was inferred from one-loop determinants of fluctuationsof the graviton and the gravitino (plus their ghosts) around the aforementioned
background. The correction was found to be
Re T01-loop = 2π 2 Mpl2 R2 +

7log Mpl2 R2 ,48

(3.19)

where R is the radius of the S 1 and Mpl is the 4D Planck’s mass. This logarithmiccorrection to the lowest order complex structure is reminiscent of the log V correction
to the Kähler coordinate from the M-theory reduction given in (3.12). However, it
is well-known that F-theory setting will not only lead to pure N = 1 supergravity,
but include some moduli fields. These are counted by the Hodge numbers h1,1 , h3,1
and h2,1 of Y4 . Hence, one would need to generalize the analysis of [27] to include
further complex fields in 4D.
We are not aware of a study of such a more general setting. However, perturbative and non-perturbative quantum corrections to N = 2 supersymmetric gauge
theories without gravity were studied intensively from various point of views, forexample, in [22–26]. We review here the case of having a U (1)Nc gauge theory with
Nf flavors labelled by a = 1, . . . , Nf with Coulomb branch masses qja Lj , where qia
are the charges under the i-th U (1) factor and Li is the real scalar in the i-th U(1)vector multiplet. For simplicity we set any further real and complex masses to zero.
The one-loop corrected Kähler coordinates are then determined to be (see e.g. [24])Re Ti1-loop =

X
1
j
C
L
+
qia log qja Lj .
ij
2
e
a

(3.20)

The corresponding kinetric potential was shown to be of the formK̃(Li ) =

X
1
i j
C
L
L
+
qia Li (log qja Lj − 1) .
ij
2e2
a

(3.21)

The corrected coordinates are those of a 3D one-loop Wilsonian effective action withall the massive flavors integrated out. If one now thinks about these coordinates as
coming from a 4D N = 1 supersymmetric F-theory model compactified on a circle
one is led to identify the massive KK-modes on the circle with the massive modes
that have been integrated out in order to obtain (3.20). From the M-theory sideof the duality the massive states responsible for the loop corrections might admit– 11 –

an interpretation in terms of M2-brane states wrapping curves Ca ∈ H2 (Y4 ). Thesestates have a mass qIa LI proportional toZZavol(Ca ) =
J ,
qI =
ωI .
(3.22)Ca

Ca

Remarkably, the L found in (3.10) has the elegant more-moduli generalization
Z
1
vI
2
I
,
VW =
L =
e3α A J 4 ,(3.23)
VW
4!
where v I are the expansion coefficients in J = v I ωI and VW is the standard warped
volume. As suggested by our result (3.10) this LI does not include any higherderivate corrections. Therefore, their contribution via quantum corrections to thethree-dimensional effective action looks exactly as in (3.21).
Note that the results (3.20), (3.21) are obtained in a theory without gravity.
Coupling to gravity will lead to a logarithmic terms such that the coupled result is
expected to be of the form 3
X
qia Li (log qja Lj − 1) ,(3.24)K̃(Li ) = − log(e2 − Cij Li Lj ) +awhich expands to (3.21) for small Cij . Appropriately combining (3.19) with (3.24)
one indeed finds an immediate resemblance with our reduction result (3.9). Even
though this is not a proof that the correction found in (3.12) is a loop correction in
the effective action, this reasoning supports this interpretation.

Towards a completion of the Ĝ2 R̂3 sector

4

In this section we consider the possibility of having additional Ĝ2 R̂3 terms in theeleven-dimensional action. Unfortunately, as of now the supersymmetric completionof the Ĝ2 R̂3 -sector is not known (see, however, [38]). The terms we used in section 3
have been lifted from corresponding terms in the Type IIA effective action, whicharise at the level of the five-point functions in the Type IIA superstring. Partial
indirect conclusions can be drawn at the level of the six-point function [39]. However,full results remain absent at the level of the six-point function, and especially athigher order n-point functions. It is thus desirable to discuss possible extensions of
the Ĝ2 R̂3 beyond the known terms. In this section we study a complete extension of
the eleven-dimensional Ĝ2 R̂3 sector relevant for our Calabi–Yau fourfold reduction.
Instead of computing string amplitudes we take a more indirect approach here.
We construct a complete basis of eight-derivative terms of the schematic form Ĝ2 R̂3 ,Strictly speaking one has to include more vector multiplets L0 , Lα with e2 and Cij dependingon these fields. L0 is the dual variable to T01-loop .
3– 12 –

which can contribute to the kinetic terms of the 3D vectors. To constrain candidate
terms we follow the same strategy as in our previous work [5]. Namely, we derive
restrictions on the higher-dimensional action by looking at constraints arising fromlower-dimensional supersymmetry. In others words determine the possible extensionsof the eleven-dimensional Ĝ2 R̂3 sector, which is compatible with 4D and 5D, N = 2
supersymmetry upon dimensional reduction. It turns out that these arguments arevery restrictive and allow us to parametrize the Ĝ2 R̂3 basis with only five parameters.
Moreover, these Ĝ2 R̂3 terms can be chosen to be consistent with the partially known
six-point function results [39].4
The implications for our current work is the observation that by dimensionallyreducing the general five-parameter extension of Ĝ2 R̂3 -terms generically can modify
the kinetic couplings of the 3D vectors. In section 4.4 we perform the dimensional
reduction of the Ĝ2 R̂3 -extensions to three space-time dimensions on a Calabi–Yau
fourfold with arbitrary h1,1 . We find that this sector naturally gives rise to the same
corrections to the 3D kinetic terms of the vectors depending on Zmn̄rs̄ ∝ 8 8 RY3 4 asfound in previous work [5]. This result will be presented in section 4.4.
4.1

Ansatz for a general basis of relevant Ĝ2 R̂3 terms

Let us next discuss the general form of the relevant terms in the basis of Ĝ2 R̂3 . The
Ĝ2 R̂3 terms contributing to the three-dimensional effective action are those, which do
not contain any Ricci tensors or Ricci scalars as these vanish trivially on a Calabi–
Yau manifold. Also contractions in which the two four-form field strengths shareless than two common indices can not lead to a kinetic term for the 3D vectors.
Taking into account the first Bianchi identity for the Riemann tensor, a minimalˆ Ĝ)2 R̂2basis of these terms is given in appendix C. Note that terms of the form (∇never contribute to the two-derivative kinetic term of the vector in the one modulus
reduction because of the closure of the Kähler form. The general expansion of terms
which may contribute in addition to (2.12) to the 3D action is thenZ
17
X
2
extra, gen
22κ11 S=αβi Bi ∗ˆ1 ,(4.1)
M11 i=1

for some coefficients βi ∈ R.
To restrict the parameters in the ansatz (4.1) we first take a detour to Calabi–Yauthreefold compactifications of M-theory with arbitrary number of Kähler deformations h1,1 (Y3 ) in section 4.3. It is known that the resulting theory will be a 5D
ungauged supergravity theory with eight real supercharges (N = 2). Compatibility with N = 2 supergravity will then lead to constraints on the parameters of the4

It would be interesting to study the constrains on this sector arising from demanding that the
new structures arising in the Type IIA effective supergravity theory should vanish at the order of the
five-point one-loop string scattering amplitude with two NS-NS two-form field and three graviton
vertex operator insertions.

– 13 –

complete basis of eleven-dimensional Ĝ2 R̂3 terms. Furthermore, such novel terms
in eleven dimensions will generically contribute H 2 R3 terms to the 10D Type IIA
effective action which we discuss in section 4.2, where H is the NS-NS three-form
field strength. We compute the imprint of the the general basis of Ĝ2 R̂3 terms by
circular dimensional reduction on R1,9 × S 1 to the ten-dimensional IIA effective action. Analogously to the 5D argument we then perform a dimensional reduction
on a Calabi–Yau threefold and compare the findings to 4D, N = 2 supersymmetry,
analogously to [5].
4.2

Constraints from IIA on Calabi–Yau threefolds

In this section we first reduce the basis of seventeen Ĝ2 R̂3 -terms in the M-theory
effective action to ten-dimensional Type IIA supergravity on R1,9 × S 1 . The only
terms relevant for us are the ones which arise fromφĜ11M N O = e 3 HM N O ,(4.2)

where M, N, O = 1, . . . , 10, and 11 denotes the direction along S 1 . We then proceedby dimensionally reducing the ten-dimensional novel eight-derivative H 2 R3 terms
contributing to the Type IIA supergravity action to four dimensions on a Calabi–
Yau threefold. The zero-mode expansion of the NS-NS two-form field gives rise toh1,1 (Y3 ) scalars bI
B2 = bI ωI .
(4.3)
To be compatible with 4D, N = 2 supersymmetry the novel H 2 R3 terms cannot
modify the kinetic terms of the b-scalars [5]. Note that in four dimensions the bscalars combine into the complexified Kähler moduli ta = ba + iv a , where v a are theh1,1 (Y3 ) Kähler moduli. Furthermore, the kinetic couplings arise from a prepotential
f (ta ) given by1
ζ(3)f (t) = Kabc ta tb tc − iχ(Y3 ) .
(4.4)
3!
2(2π)3and are thus heavily constrained. Note that χ(Y3 ) denotes the Euler–characteristicof the Calabi–Yau threefold. This computation is tedious, we therefore only give theconstraints we find
β1 = c1 ,1β3 = 16c3 ,β5 = c2 ,1
β7 = −12c2 + 12c3 ,1
β9 = 4 c3 ,β11 = − 41 c3 ,β13 = 12 c3 ,1
β15 = − 24
(144c2 − c3 + c4 ) ,1
β17 = 32 (c3 − c4 ) ,β2 = 81 (c3 − c4 ) ,β4 = −2c2 ,β6 = 61 (−2c1 + 24c2 − c3 ) ,β8 = 41 (−2c3 + c4 − 4c5 ) ,β10 = c5 ,β12 = 41 c3 .
β14 = c5 ,β16 = 41 (−2c3 + c4 ) ,– 14 –(4.5)

where the five real cofficients ci parametrize the solution to the constraints. It isremarkable that only five of the original 17 parameters βi remain independent.
One may furthermore wish to check the compatibility of the novel induced H 2 R3terms making use of the Type IIA–Heterotic duality. Compactifying Type IIA on
K3 is dual to the Heterotic string on T4 . For our purpose it is enough to show thatwhen compactifying the novel H 2 R3 -terms on K3 those do not induce any correction
to the 6D action, in particular no four-derivative terms, which results in one further
constraint on the parameters 5
c3 = c4 .

(4.6)

This concludes that by tuning one further parameter the proposed maximal extensionof Ĝ2 R̂3 -terms in the M-theory effective action is fully consistent with the indirect
six-point functions results discussed in [39].
4.3

Constraints from M-theory on Calabi–Yau threefolds

Similarly one can obtain constraints on the coefficients βi by demanding compatibility
with N = 2 supersymmetry in 5D upon compactification on a Calabi–Yau threefold.
This non-renormalization of the vector multiplets in 5D, N = 2 supergravity can
immediately be understood from its description in terms of a real prepotential F(X I )and real special coordinates X I . The physical scalars in the vector multiplets haveto obey the relation
F(X I ) = 3!1 CIJK X I X J X K = 1 ,(4.7)

where CIJK is a totally symmetric, constant tensor. The latter tensor is in turn
determined by the U (1) Chern-Simons terms ∼ CIJK AI F J F K ⊂ L (5D) , which do6
not receive `M–corrections. Therefore both the tensor CIJK and also the physicalI
6).
scalars X remain uncorrected at O(`M
In order to obtain the constraints from N = 2 supergravity in 5D we dimensionally reduce the action (4.1) with general coefficients βi on a Calabi–Yau threefold
Y3 to five dimensions. We are interested in the kinetic terms for the vectors andtherefore expandĜ = F I ∧ ωIY3 ,(4.8)

where F I are the 5D vectors in vector multiplets, ωIY3 ∈ H 1,1 (Y3 ) form a basis of
harmonic (1, 1)-forms on Y3 , and I = 1, . . . , dim H 1,1 (Y3 ). The constraint imposedby supersymmetry is then that upon reducing (4.1) and applying Schouten identities
on Y3 the terms (4.1) do not contribute to the 5D couplings. Since these constraints
are expected to be very similar to the ones in Type IIA, we do not state them here
but instead work with the ones found in the previous section.
5

The constraint arises from imposing the vanishingχ(K3) H (6D) µνρ H (6D) µ ν1 ρ1 R(6D) µµ1 νν1 , with µ, ν = 1, . . . , 6.

– 15 –

ofthefour-derivative

terms

4.4

The Ĝ2 R̂3 -extension on Calabi–Yau fourfolds

Let us next discuss the implications of the maximal Ĝ2 R̂3 -extension to the M-theory
effective action on the kinetic couplings of the vectors arising in Calabi–Yau fourfold
compactifications. We have found in the previous sections 4.3 and 4.2 that the Ĝ2 R̂3 extension can be parametrized by four independent coefficients β1 , β2 , β3 and β5 . To
derive the kinetic terms of the 3D vector one expands the four-form field strength asĜ = F I ∧ ωI ,(4.9)

where F I are the 3D vectors in vector multiplets, ωI ∈ H 1,1 (Y4 ) form a basis of
harmonic (1, 1)-forms on Y4 , and I = 1, . . . , dim H 1,1 (Y4 ). Then the Ĝ2 R̂3 -extension
reduced on a Calabi–Yau fourfold results in
Z
Z h
i

(3)
I
J
2
F ∧?F
2i 144c2 +c3 Zmn̄ ωIn̄m ωJr r +4 c3 Zmn̄rs̄ ωIn̄m ωJs̄r ∗8 1
2κ11 ∆Skin =M3

Y4(4.10)
provided we impose c4 = 0. For the precise definition of the rank four tensor Zmn̄rs̄ =8 8 RY3 4 mn̄rs̄ and Zmn̄ = Zmn̄r r see [17]. The structures (4.10) have been found toarise in the kinetic term from the known Ĝ2 R̂3 -terms in eleven dimensions [17]. It is
intriguing that the 4D, 5D and 6D constraints on the general Ĝ2 R̂3 -extension lead torestrictions which naturally give rise to these familiar structures when compactified
on a fourfold.
Since the focus of this work was on the one-modulus Calabi–Yau fourfold reduction it is interesting to restrict (4.10) to this case. One then finds a shift in thecorrection to the kinetic term of the 3D vector
Z
1
(3)
2
2κ11 ∆Skin = −144 c2 V − 4 χ̃ F ∧ ?F .
(4.11)
M3

Clearly this implies that one can choose c2 to cancel the corrections found in section
3. This would restore the no-scale property and remove the logarithmic corrections
to the coordinates. We do not expect, however, that this is the case, due to thephysical evidence presented in section 3.2.
Let us conclude that the extensive checks we have performed in this section allow
in principle for a Ĝ2 R̂3 -extension of the M-theory effective action, which is fully
consistent with other results known in the literature. However, a string scatteringinterpretation at the level of the six-point function or a Noether coupling procedure
involving the Ĝ2 R̂3 -sector would be desirable to clarify the existence of new terms.
However, let us stress that our checks pose the tightest constraints currently known
on the completeness of the Ĝ2 R̂3 -terms. We will discuss the implications of thepossible Ĝ2 R̂3 -extension to the kinetic terms of the vectors (4.10) in Calabi–Yau
fourfold compactifications with arbitrary h1,1 in a forthcoming work.

– 16 –

5

Conclusions

In this work we extended the study of three-dimensional two-derivative N = 2 effec6
tive actions arising Calabi–Yau fourfold compactifications when including `M–higherderivative terms in eleven-dimensions. Such higher-derivative terms are generally
needed when considering background fluxes, since only then a non-trivial solution
to the eleven-dimensional equations of motion and tadpole constraints exists. While6
at leading order including only a Calabi-Yau metric, the `M–corrected solution does
admit a non-trivial warp-factor as well as other higher-derivative corrections. It isa non-trivial task to perturb around this solution and derive the resulting threedimensional effective action.
To find a fully explicit result, we focused in this work on the special case of a fourfold with one Kähler modulus h1,1 = 1 and frozen complex structure moduli. Hence,
we derived the three-dimensional effective action for the volume modulus and thevector arising from expanding the M-theory three-form along the normalized Kähler
form. Dimensionally reducing the known higher-derivative terms and matching with
N = 2 supergravity in three dimensions made it necessary to introduce a corrected
Kähler potential (3.12) and complex coordinate (3.13). Remarkably, we find for the
first time the explicit form of the Kähler coordinates and show that it contains alogarithmic correction in the fourfold volume. Using this result it was easy to showthat this system of Kähler potential and coordinate breaks the no-scale condition at
6
).
O(`M
The logarithmic correction to the complex coordinate suggests that it might
capture a one-loop correction as we discussed in detail in section 3.2. In particular,
viewing the three-dimensional effective action as arising from a four-dimensional N =1 F-theory effective action on a circle, the logarithmic corrections are expected from
integrating out heavy M2-branes wrapped on curves. In the one-modulus reduction
we have found that the scalar in the vector multiplet is precisely a curve volume
multiplied by the warped volume of the Calabi-Yau fourfold. Furthermore, we haveseen that the kinetric potential has a form that is expected when integrating out
Kaluza-Klein modes arising from the circle. It is an interesting open task to showthat this interpretation is correct by matching the numerical factors of the loop
computation with the result from topology.
It is important to stress again that one can be critical with our findings of section
3 referring to the possibility that not all Ĝ2 R3 -terms are known in the literature.
We therefore critically examined the possibility of having additional unknown terms
in the eleven-dimensional action that might be relevant for our discussion. Thiscan be motivated by the fact that the eleven-dimensional eight-derivative terms arelifted versions of Type IIA terms, which are only tested at the level of the five-point
function. To draw general conclusions we constructed a basis of potentially relevant
eleven-dimensional terms and imposed constraints required by compatibility with– 17 –

four- and five-dimensional N = 2 supergravity. The constraints we found turn out
to be surprisingly severe. Nevertheless we have shown that the remaining terms
can crucially influence the reduction of section 3 and potentially restore the no-scalecondition. While our physical interpretation suggests that the general form of thecomplex coordinates is indeed correct, one can hope that a detailed study of the
more-moduli space allows to gain more insights for the precise numerical factors.

Acknowledgements.
K.M. and M.W. would like to thank the Max-Planck-Intitut für Physik in Munich
for hospitality, where parts of this work were carried out. M.W. was supported by
the WPI program of Japan.

– 18 –

A

Higher-derivative terms

In this appendix we collect the explicit expressions for the higher derivative terms ineleven dimensions relevant in the main part.
Z


1
1
SR 4 = 2t̂8 t̂8 − 11 11 R̂4 ∗ˆ1.
(A.1)
2κ11 M11
24
The two quantities t̂8 t̂8 R̂4 and 11 11 R̂4 in (A.1) have the index representationt̂8 t̂8 R̂4 = t8 M1 ···M8 t8 N1 ···N8 R̂N1 N2 M1 M2 · · · R̂N7 N8 M7 M8 ,(A.2)

11 11 R̂4 = 11 R1 R2 R3 M1 ···M8 11 R1 R2 R3 N1 ···N8 R̂N1 N2 M1 M2 · · · R̂N7 N8 M7 M8 .

(A.3)

The tensor t̂8 is defined ash
1− 2 ĝ N1 N3 ĝ N2 N4 ĝ N5 N7 ĝ N6 N8 + ĝ N1 N5 ĝ N2 N6 ĝ N3 N7 ĝ N4 N8 + ĝ N1 N7 ĝ N2 N8 ĝ N3 N5 ĝ N4 N6t̂8N1 ···N8 = 16
+ 8 ĝ N2 N3 ĝ N4 N5 ĝ N6 N7 ĝ N8 N1 + ĝ N2 N5 ĝ N6 N3 ĝ N4 N7 ĝ N8 N1 + ĝ N2 N5 ĝ N6 A7 ĝ N8 N3 ĝ N4 N1i− (N1 ↔ N2 ) − (N3 ↔ N4 ) − (N5 ↔ N6 ) − (N7 ↔ N8 ) .
(A.4)
These R4 -terms are furthermore supplemented by another term quartic in the Riemann tensor. This term however also comprises a three form Ĉ3 . This piece of thehigher-derivative action then has the form
32 213SX̂8 = − 22κ11ZĈ3 ∧ X̂8

(A.5)

M11

where eight form X̂8 is defined as

2 
1
1
4
2
X̂8 =
Tr R̂ −
Tr R̂
192
4

(A.6)

which is in terms of the (real) curvature two form
1
R̂M N = R̂M N N1 N2 dxN1 ∧ dxN2 .
2

(A.7)

In addition to these quartic Riemann tensor terms it was conjectured in [39] that thecomplete Ĝ4 dependence at O(Ĝ24 ) is captured by introducing
1 ···M8t̂8N1 ···N8 ĜN1 M1 R1 R2 ĜN2 M2 R1 R2 R̂N3 N4 M3 M4 · · · R̂N7 N8 M7 M8t̂8 t̂8 Ĝ2 R̂3 =t̂M
8

11 11 Ĝ2 R3 =11 RM1 ···M10 11 RN1 ···N10 ĜN1 N2 M1 M2 ĜN3 N4 M3 M4 R̂N5 N6 M5 M6 · · · R̂N9 N10 M9 M10 .

– 19 –

B

Reduction results

In this appendix we collect the intermediate results of the reduction, especially ofthe higher-derivative terms 6 .
• Kinetic terms from t̂8 t̂8 R̂4 .
Z
Z
1
41536 χ(Y4 ) ? 1 − 72 V 4 χ̃ d log V ∧ ? d log V(B.1)
t̂8 t̂8 R̂ ∗ˆ 1 =M3

M11• Kinetic terms from 11 11 R̂4 .
Z
Z
1
1
1
4
768 V 4 R ? 1 + 24 V 4 χ̃ d log V ∧ ? d log V−11 11 R̂ ∗ˆ 1 =24 M11Z M3+1536 χ(Y4 ) ? 1
(B.2)
M3

• Kinetic terms from t̂8 t̂8 R̂3 Ĝ24 .
Z
Z
3 2t̂8 t̂8 R̂ Ĝ4 ∗ˆ 1 = 1152
M11

1

V − 4 χ̃ F ∧ ? F

(B.3)

M3

• Kinetic terms from 11 11 R̂3 Ĝ24 .
Z
Z
1
1
3 2
11 11 R̂ Ĝ4 ∗ˆ 1 = −384
V − 4 χ̃ F ∧ ? F
96 M11
M3• Kinetic terms from Ẑ Ĝ24 .
Z
Z
256 Ẑ Ĝ4 ∧ ∗ˆ Ĝ4 = 1024
M11

1

V − 4 χ̃ F ∧ ? F ,

(B.4)

(B.5)

M3

where the scalar Ẑ is
1
Ẑ =R̂M1 M2 M3 M4 R̂M3 M4 M5 M6 R̂M5 M6 M1 M2 −2R̂M1M2 M3M4 R̂M2M5 M4M6 R̂M5M1 M6M2 .
12• Einstein Hilbert and kinetic term for Ĝ4 .
Z

1R̂ ∗ˆ 1 − 14 Ĝ4 ∧ ∗ˆĜ4 =
2

M11

Z
+
ZM3
+
ZM3
+

Z
M3

2
1
V e3α A23
1 − 384 α2 V − 4 χ̃ R ? 1 − 14 G(1) ∧ ∗(0) G(1)73
V d log V ∧ ? d log V − 16
α2 A(2) d log V ∧ ? d log V
161

240 α2 V 4 χ̃ d log V ∧ ? d log V11

1

V 2 F ∧ ? F − 3 α2 V − 2 A(2) F ∧ ? F + 128 α2 V − 2 χ̃ F ∧ ? F

M3

(B.6)
6

The dimensional reduction was performed using the xAct bundle [48–50].

– 20 –

• Chern-Simons term.
2κ211 SCS =Z

1− Ĉ3 ∧ Ĝ4 ∧ Ĝ4 =6
M11Z
αΘA ∧ F ,
M3

1
Θ=
2

Z
ω ∧ ω ∧ G(1) .
Y4

(B.7)

ˆ Ĝ 2 R̂2 .
• Kinetic terms from ∇Zˆ Ĝ 2 R̂2 ∗ˆ 1 = 0ŝ18 ∇

(B.8)

M11

C

Basis of relevant Ĝ2 R3 terms

The basis for the potentially relevant eight-derivative terms involving the four-form
field strength isB1 = GM5 M7 M8 M9 GM6 M7 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M3 M4 M6
B2 = GM4 M6 M8 M9 GM5 M7 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M3 M6 M7
B3 = GM4 M5 M8 M9 GM6 M7 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M3 M6 M7
B4 = GM6 M7 M8 M9 GM6 M7 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M4 M3 M5
B5 = GM6 M7 M8 M9 G4M6 M7 M8 M9 RM M4 M2 M5 RM M1 M2 M3 RM1 M4 M3 M5
B6 = GM5 M7 M8 M9 GM6 M7 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M4 M3 M6
B7 = GM5 M7 M8 M9 GM6 M7 M8 M9 RM M4 M2 M5 RM M1 M2 M3 RM1 M4 M3 M6
B8 = GM3 M6 M8 M9 GM5 M7 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M4 M6 M7
B9 = GM3 M5 M8 M9 GM6 M7 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M4 M6 M7
B10 = GM3 M6 M8 M9 GM5 M7 M8 M9 RM M4 M2 M5 RM M1 M2 M3 RM1 M4 M6 M7
B11 = GM3 M5 M8 M9 GM6 M7 M8 M9 RM M4 M2 M5 RM M1 M2 M3 RM1 M4 M6 M7
B12 = GM4 M7 M8 M9 GM5 M6 M8 M9 RM M4 M2 M5 RM M1 M2 M3 RM1 M6 M3 M7
B13 = GM3 M7 M8 M9 GM5 M6 M8 M9 RM M2 M4 M5 RM M1 M2 M3 RM1 M6 M4 M7
B14 = GM3 M7 M8 M9 GM5 M6 M8 M9 RM M4 M2 M5 RM M1 M2 M3 RM1 M6 M4 M7
B15 = GM5 M7 M8 M9 GM6 M7 M8 M9 RM M1 M2 M4 RM M1 M2 M3 RM3 M5 M4 M6
B16 = GM4 M6 M8 M9 GM5 M7 M8 M9 RM M1 M2 M4 RM M1 M2 M3 RM3 M5 M6 M7
B17 = GM4 M6 M8 M9 GM5 M7 M8 M9 RM M1 M2 M3 RM M1 M2 M3 RM4 M5 M6 M7 .

– 21 –(C.1)

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