Effective action for nongeometric fluxes duality covariant actions
JHE
Effective action for nongeometric fluxes duality
Kanghoon Lee 0 1 4
SooJong Rey 0 1 2 4
Yuho Sakatani 0 1 3 4
0 Seoul National University , Seoul 08826 , Korea
1 Institute for Basic Sciences , Daejeon 34047 , Korea
2 School of Physics & Astronomy and Center for Theoretical Physics
3 Department of Physics, Kyoto Prefectural University of Medicine
4 Fields, Gravity & Strings , CTPU
The (heterotic) double field theories and the exceptional field theories are manifestly duality covariant formulations, describing lowenergy limit of various superstring and Mtheory compactifications. These field theories are known to be reduced to the standard descriptions by introducing appropriately parameterized generalized metric and by applying suitably chosen section conditions. In this paper, we apply these formulations to nongeometric backgrounds. We introduce different parameterizations for the generalized metric in terms of the dual fields which are pertinent to nongeometric fluxes. Under certain simplifying assumptions, we construct new effective action for nongeometric backgrounds. We then study the nongeometric backgrounds sourced by exotic branes and find their U duality monodromy matrices. The charge of exotic branes obtained from these monodromy matrices agrees with the charge obtained from the nongeometric flux integral.
Flux compactifications; String Duality

HJEP07(21)5
1 Introduction 2
General framework
2.1
2.2
2.3
2.4
2.5
Parameterization of Lie algebra
The generalized metric
Example: Double Field Theory
Example: Einstein gravity
Effective action for nongeometric fluxes
3
Nongeometric fluxes in EFT: Mtheory
3.1
Parameterization of the generalized vielbein 3.2 3.3
Elevendimensional effective action
Reduction to the type IIA theory
4
Nongeometric fluxes in EFT: type IIB section
4.1
Parameterizations of the generalized vielbein
3.1.1
3.1.2
3.1.3
3.1.4
4.1.1
4.1.2
4.1.3
4.1.4
6.1.1
6.1.2
6.1.3
6.2.1
6.2.2
6.3.1
Exotic branes in the Mtheory 6.3 Exotic branes in the type IIB theory – i –
p7−pbrane
164brane
7
Discussion
A Notations
A.1 Ed(d) algebras: Mtheory section
A.2 Ed(d) algebras: type IIB section
B Calculation of the EFT action
B.1 Redefinitions of coordinates
B.2 External part
B.3 Internal (potential) part
B.4 Summary
C
Doublevielbein formalism for gauged DFT
C.1 Parameterization from defining properties of doublevielbein
C.2 Connection and curvature
C.3 Nongeometric fluxes and action
D Exotic branes
HJEP07(21)5
I do not wish, at this stage, to examine the logical justification
of this form of argumentation; for the present, I am considering
it as a practice, which we can observe in the habits of men
and animals.
Bertrand Russell, ‘Philosophy’.
1
Introduction
Recently, a significant progress has been achieved for novel formulations of supergravity
in which duality symmetries in string and Mtheory compactification are manifest. They
include the double field theory (DFT) [1–7], the exceptional field theory (EFT) [8–26]
(see also [27–34] for closely related attempts) as well as the generalized geometry [35–
40]. One important advantage of these formulations is that they can treat wide variety of
spacetimes, such as nongeometric backgrounds [41–44], that are not globally describable
backgrounds arise quite naturally in superstring theories. Backgrounds sourced by exotic
branes [47–53] are concrete examples. As an application of DFT and related formulations
such as the βsupergravity [54–61], a background of a particular exotic brane, socalled
a 522brane, was studied in [45, 46, 62–72] and the exotic 522brane was identified with a
magnetic source of the nongeometric Qflux [64, 70, 72].
– 1 –
One reason why the exotic 522brane received special attention is that the nongeometric
Qflux, which is intrinsic to the 522brane background, is related to a T duality monodromy,
and the much developed DFTs efficiently describe such background. It is known that
backgrounds of other exotic branes possess other nongeometric fluxes that are related to
the Qflux via U duality transformations [51, 73]. In order to describe such nongeometric
backgrounds, variants of the βsupergravity, which can describe the background of an exotic
pbrane (called a p7−pbrane) or a 164brane, was proposed in [74]. There, each of these
3
exotic branes was identified as the magnetic sources of a nongeometric P flux [75–77] or a
nongeometric Qflux associated with a 6vector, βm1···m6 [74]. However, the reformulation
of [74] is applicable only to a limited situation; coexistence of different nongeometric fluxes
U duality covariant formulation of the supergravity, is a more suitable formulation, and
indeed, backgrounds of the exotic 53brane, 522brane, and the 523brane were studied in
SL(5) EFT [78, 79]. One of the main purposes of this paper is to systematically identify
the nongeometric fluxes in Ed(d) EFT for the cases of 4 ≤ d ≤ 7.
The goal of this paper is to develop effective actions for a certain class of nongeometric
flux backgrounds in Type II string and Mtheories.
Our starting point is the duality
covariant action in an extended field theory, such as the manifestly U duality covariant
EFT. Since the U duality orbit is of infinite order, there are in practice infinitely many
possible parameterization of the U duality group. The key idea is to identify the most
effective parameterization for a given set of nongeometric flux background. Note that
our nongeometric parameterization is efficient for backgrounds with only nongeometric
fluxes. For backgrounds with both geometric and nongeometric fluxes, such as the truly
nongeometric backgrounds of [80], a more general treatment will be required.1
Our construction can be extended to nongeometric flux backgrounds in heterotic string
theories. Heterotic string exhibits O(D, D + 16) or O(D, D + dim G) duality group, where
G is the heterotic YangMills group, E8 × E8 or SO(32), and the corresponding heterotic
DFT [1, 2, 81] provides a duality manifest description of the effective field theory. Again,
the key idea is to identify the most effective parameterization. Through the nongeometric
parameterization of heterotic generalized vielbein, we construct heterotic Qflux which
includes ChernSimons like term and an additional nongeometric bivector flux associated
with the heterotic YangMills field strength. The corresponding nongeometric effective
action can be constructed from O(D, D + dim G) gauged DFT [82–84]. If we take the
maximal Abelian reduction of heterotic YangMills gauge symmetry, G = U(1)16, the
nongeometric gauged DFT reduces to the nongeometric parameterization by Blumenhagen
and Sun [85].
This paper is organized as follows. In section 2, after reviewing some elements of
Lie algebra, we explain the general construction of the generalized metric or vielbein. In
1Note that the section condition or the strong constraint in DFT/EFT can be relaxed through the
generalized ScherkSchwarz reduction [82], which provides all the fluxes in the maximal and halfmaximal
izations of the generalized vielbein; the conventional geometric parameterization and the
dual nongeometric parameterization. Using the two different parameterizations, we write
down two different elevendimensional effective actions. We also consider the dimensional
reduction to the type IIA theory, and obtain the nongeometric fluxes in the type IIA
theory. EFT in terms of the type IIB theory is discussed in section 4 and tendimensional
action for the nongeometric fluxes in the type IIB theory is obtained. In section 5, we
find a parameterization of heterotic DFT relevant for nongeometric fluxes. In section 6,
the relation between the nongeometric fluxes and exotic branes are discussed. Discussions
and future directions are given in section 7. We relegated much of technical details to the
Aij (i, j = 1, . . . , rank g) that has the structure
Aii = 2 ,
Aij ∈ Z≤0 (i 6= j) ,
Aij = 0
⇔
Aji = 0 ,
det Aij > 0 ,
(2.1)
where Z≤0 denotes nonpositive integers. In g, consider the Chevalley basis generators
{Hi, Ei, Fi}, which obey the properties
[Hi, Hi] = 0 ,
[Hi, Ej ] = Aji Ej ,
[Hi, Fj ] = −Aji Fj ,
[Ei, Fj ] = δij Hi ,

1−{Azji
}
[Ei, [· · · , [Ei, Ej ] · · · ]] = 0 ,

1−{Azji
}
[Fi, [· · · , [Fi, Fj ] · · · ]] = 0 .
It is known that the generators {Hi, Ei, Fi}, together with the commutators of Ei or Fi ,
[Ei1 , [· · · , [Eik−1 , Eik ] · · · ]]
and
[Fi1 , [· · · , [Fik−1 , Fik ] · · · ]] ,
2Here, we suppose g is a split real Lie algebra, considering the applications to DFT, g = o(d, d), and
EFT, g = ed(d). Application to a nonsplit case is considered in section 5.
(2.2)
(2.3)
– 3 –
form a complete set of basis of g . In the Chevalley basis, the generators Hi for i =
1, . . . , rank g form the Cartan subalgebra h, the generator Ei is associated with the positive
simple root αi ∈ h∗ with αi(Hj ) = Aij , and the generator Fi is associated with the negative
simple root −αi . We denote the space of positive root by Δ+ and the space of negative
root by Δ−, respectively. For an arbitrary positive root α ∈ Δ+,
we can construct the associated generator as ktuple leftcommutator
HJEP07(21)5
Eα ≡ [Ei1 , [· · · , [Eik−1 , Eik ] · · · ]] .
For the corresponding negative root −α ∈ Δ−, we also construct the associated generator
as ktuple rightcommutator
Fα ≡ [[· · · [Fik , Fik−1 ], · · · ], Fi1 ] .
Denote the space spanned by Eα and Fα (α ∈ Δ+) as n+ and n−, respectively. Then, we
obtain the triangular decomposition by decomposing the Lie algebra g as
α =
k
X αin ,
n=1
g = n
− ⊕ h ⊕ n+ .
The second method is known as the Cartan decomposition. Define the Cartan
involution θ by
θ(Hi) = −Hi ,
θ(Ei) = −Fi ,
θ(Fi) = −Ei .
From the distributive property that θ([s, t]) = [θ(s), θ(t)] for s, t ∈ g, it follows that
θ(Eα) = −Fα ,
θ(Fα) = −Eα
for every
α ∈ Δ+ .
Redefining the generators as we can diagonalize the Cartan involution as
Sα ≡ Eα + Fα
and
Jα ≡ Eα − Fα
for every
α ∈ Δ+ ,
θ(Hi) = −Hi ,
θ(Sα) = −Sα ,
θ(Jα) = +Jα ,
and classify the generators according to the parity under the involution θ:
k = {s ∈ g  θ(s) = +s} = span(Jα)
and
p = {s ∈ g  θ(s) = −s} = span(Hi, Sα) .
We are thus decomposing the Lie algebra g as
obtaining the Cartan decomposition. Since the number of the positive roots is (dim g −
rank g)/2, we have
dim k =
dim g − rank g
2
dim g + rank g
2
.
g = k ⊕ p ,
– 4 –
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
G = Ed(d)
D = dim l1
SL(5)
10
SO(5, 5)
E6(6)
27
E7(7)
56
Although the commutator in p is not closed (since it has the odd parity under θ), the Lie
commutators in k yields a subalgebra, sometimes called the Cartaninvolutioninvariant
subalgebra, which coincides with the maximal compact subalgebra of g .
The third method is known as the Iwasawa decomposition, the decomposition we shall
be using in the present paper. There are two possible types of Iwasawa decomposition.
The positive decomposition is defined by
[h, ZM ] = −
X(ρh)M N ZN
N
(h ∈ g) .
θ(h), Z
M
= −
X(ρh)M N Z
N
(h ∈ g) .
N
≡ −θ ZM , we obtain from (2.17) the following commutator:
Here, ρh is the matrix realization for the element h ∈ g in the l1representation. Defining
dual matrix realization (ρ¯h)M N ≡ δMK (ρh)K L δLN . We then obtain
To render the position of indices consistent, we introduce the fundamental forms, δMN and
δMN , whose components are equal to δMN (and are not generalized tensors), and define the
h, Z
M
= −(ρ¯θ(h))M N Z
N
(h ∈ g) .
g = k ⊕ h ⊕ n+ ,
g = n
− ⊕ h ⊕ k ,
where b+ ≡ h ⊕ n+ is referred to as the positive Borel subalgebra. The negative
decomposition is defined by
where b
− ≡ n
− ⊕ h is referred to as the negative Borel subalgebra.
Associated to the Lie algebra g, we construct the corresponding Lie group G as the
exponential map. We can realize group element g ∈ G in any of the above decomposition
of g. In particular, we can straightforwardly extend the definition of the Cartan involution
θ to an arbitrary group element g ∈ G, and then define an antiinvolution ♯ by
g♯ ≡ θ(g−1) , (ab)♯ = b♯a♯
where
g, a, b ∈ G .
In section 3 and 4, we take the Lie algebra g = ed(d) and its Lie group G = Ed(d) as
the duality symmetry (summarized in table 1). Suppose that the (generalized) momenta
ZM (M = 1, . . . , D ≡ dim l1), which generate abelian translations ([ZM , ZN ] = 0) in the
extended space X of the U duality action, are in the fundamental representation l1 of the
Lie group G [28],
– 5 –
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
([h, v], w¯) + (v, [h, w¯]) = 0
(h ∈ g) ,
(Adg · v, Adg · w¯) = (v, w¯) ,
(2.20)
(2.21)
where Adg · v ≡ g v g−1 (g ∈ G) . We will normalize the abelian generators ZM such that
the scalar product becomes the identity matrix,
ZM , Z
N
= δMN ,
We also introduce a natural Ginvariant scalar product (v, w¯) ≡ vM w¯M for an element v
of the representation l1 spanned by AM and w¯ of the dual representation ¯l1 spanned by
∈ G, v = vM ZM , and w¯ = w¯M Z , we have
M
(eρh )M K (eρ¯θ(h) )N K = δMN .
(2.22)
(2.24)
(2.25)
(2.26)
(2.27)
(eρh )M N = (e−ρ¯θ(h) )N M = eρ¯h♯ N
M = δNL eρh♯
L
K δKM ,
where we defined h
♯ ≡ −θ(h) for h ∈ g , and used the dual representation (ρ¯h)N M =
δNL (ρh)LK δKM in the last equality. This relation shows that the antiinvolution ♯ defined
in (2.16), sometimes called the generalized transpose, acts as the matrix transpose in the
matrix realization of Lie algebra g.
2.2
The generalized metric
hv, wi = hw, vi = vM wN δMN .
– 6 –
identities, ZM , θ(ZN ) = − ZM , Z
symmetric and positivedefinite,
We next study the geometry of extended space X associated with the duality
transformation group G. We shall define the generalized metric MMN of X and explain how to
parameterize MMN in terms of appropriate physical fields (see [11, 86]). We first define a
bilinear form
hv, wi ≡ − v, θ(w) = −vM wN ZM , θ(ZN ) ,
for generalized vectors, v = vM ZM and w = wM ZM , in the l1representation. From the
N
= −δMN = −δMN , we see that the metric (2.26) is
However, as δMN is not a generalized tensor, this metric is not Ginvariant. Indeed, for
general element h ∈ g, we find that the adjoint action
[h, v], w + v, [h, w] = − [h, v], θ(w) − v, [θ(h), θ(w)] = − [h − θ(h), v], θ(w)
is nonzero. However, it is invariant under the maximal compact subgroup, K, of G, since
h = θ(h) for h ∈ k.
Starting from this (constant) positivedefinite metric and a group element g ∈ G, we
now define the generalized metric from the generalized bilinear form
M(v, w) ≡ MMN vM wN
≡ hAdg−1 · v, Adg−1 · wi .
(2.28)
and w¯M ≡ δMN wN , we have
The generalized bilinear form is positivedefinite by construction, and it is defined to be
Ginvariant. We assume that the generalized metric MMN varies over the spacetime, so
the group element g ∈ G should be spacetimedependent as well. Denoting g = e−h (h ∈ g)
θ Adg−1 · w = e[θ(h), · ] ZM w¯M = (eρ¯h )M N Z
N w¯M ,
and the inner product in (2.28) becomes
M(v, w) = vM (eρh )M K (eρ¯θ(h) )N L w¯N (ZK , ZL)
= vM (eρh )M K (eρ¯h )N K w¯N = vM (eρh )M K (eρh♯ )K L δLN wN .
Introducing the generalized vielbein as EM
metric in the conventional form,
A
≡ (eρh )M A, we can express the generalized
MMN = EM
A
EN
B δAB .
Here, the indices A, B run over 1, . . . , D = dim l1, which play the same role as the original
indices M, N but are interpreted as “flat indices.” As h · , · i is Kinvariant, two generalized
metrics constructed from g ∈ G and g · k (k ∈ K), respectively, have the same structure.
Thus, the generalized metric can be parameterized by a coset representative of G/K, and
so the number of the independent parameters is given by dim(G/K) = dim G − dim K.
For an explicit construction of the generalized metric, we find it convenient to use the
Iwasawa decomposition (2.14) and parameterize the representative g ∈ G/H, where H is
the Cartan subgroup, in terms of functions, hi(x) and Aα(x), associated with generators
of the positive Borel subalgebra b+, and the K equivalence class:
g(x) = ePi hi(x) Hi ePα∈Δ+ Aα(x) Eα k(x)
∼
ePi hi(x) Hi ePα∈Δ+ Aα(x) Eα ,
(2.32)
Here, k(x) denotes an element in the compact subgroup K and x refers to the coordinate
system adopted. We can then obtain the generalized metric from the following
generalized vielbein:
EM A(x) = ehi(x) ρHi ePα∈Δ+ Aα(x) ρEα
M
Note that the generalized metric MMN is invariant under the antiinvolution g → g
(i.e. symmetric), while the generalized vielbein is not. Using the above decomposition,
♯
we have
g♯(x) = k♯(x) ePα∈Δ+ Aα(x) Fα ehi(x) Hi
with certain functions ehi(x), Aeα(x) and ek(x) ∈ K, whose relation to hi(x), Aα(x), and k(x)
is in general complicated. This expression for g♯ corresponds to the alternative Iwasawa
decomposition (2.15), so we can obtain the generalized vielbein in terms of the functions
associated with the generators of negative Borel subalgebra b :
−
HJEP07(21)5
(2.34)
(2.35)
(2.36)
(2.37)
(2.38)
The key idea of this paper is that the above replacement g → g♯, which does not change
the generalized metric, generally corresponds to the replacement from the conventional
geometric parameterization of the generalized metric MMN to the dual “nongeometric”
parameterization of it. A transformation between the conventional and the dual
parameterization is sometimes referred to as the exotic duality transformation [62, 74, 87]. In
this paper, we will show that the exotic duality transformation is identifiable with the
generalized transpose.
It remains to confirm the tensorial property of the generalized metric. As we mentioned
above, the flat bilinear form hv, wi = vM wN δMN was not Ginvariant. However, the
generalized bilinear form
M(v, w) = hAdg−1 · v, Adg−1 · wi is invariant under G. This
constrains the transformation rule for the group element g (i.e. the generalized vielbein).
It then follows that, as the Kinvariance of δMN , the transformation rule of the generalized
vielbein generally has the following form:
EM
A
→ gM
N
EN
B kBA
for
g ∈ G, k ∈ K .
2.3
Example: Double Field Theory
Before presenting our new results, we first illustrate the above general consideration for the
DFT. In this case, the Tduality group is G = O(d, d). We can decompose the generators
of g = o(d, d) into representations of the GL(d), the gl(d)generators Kab, Rab = R[ab],
Rab = R[ab] (a, b = 1, . . . , d), which obey the following commutation relations:
[Kab, Kcd] = δbc Kad − δda Kcb ,
[Kab, Rcd] = δbc Rad + δbd Rca ,
[Rab, Rcd] = 4 δ[[ca Kb]d] ,
[Kab, Rcd] = −δca Rbd − δda Rcb .
The Cartan subalgebra h is generated by the diagonal components of Kab: Ha ≡ Kaa (no
summation). The Cartan involution is given by
EM A(x) = ePi ehi(x) ρHi ePα∈Δ+ Aeα(x) ρFα
M
θ(Kab) = −Kba
θ(Rab) = −Rab ,
and
– 8 –
In particular, the (anti)chiral combinations, Ma±b ≡ (Jab ± Tab)/2 , satisfy the algebra for
o(d) × o(d):
[Ma±b, Mc±d] = 2 δ[da Ma±]c − 2 δ[ca Mb±]d ,
[Ma+b, Mc−d] = 0 .
b), Rab} and {Ha, Kab (a > b), Rab}, respectively.
The positive and negative Borel subalgebras, b+ and b−, are spanned by {Ha, Kab (a <
In DFT, we take the fundamental (i.e. vector) representation, whose matrix realization
is given by the matrices,
δac δdb
0
0
−δda δbc
!
0 2 δacdb!
0
0
(ρKcd )AB =
, (ρRcd )AB =
, (ρRcd )AB =
0
−2 δcadb 0
0
!
,
(2.41)
where δacdb ≡ δ
[[ac δbd]] (see appendix A for our conventions). The commutators with the
generalized momenta ZM = (Pm, Pem) are given by
(2.39)
(2.40)
(2.42)
(2.43)
N
(2.44)
(2.45)
(2.46)
and the Cartaninvolutioninvariant subgroup is generated by
and
Note that the variable ZM defined by
[Kab, Pc] = −δca Pb ,
[Rab, Pc] = −2 δc[a Peb] ,
[Rab, Pc] = 0 ,
[Kab, Pec] = δbc Pea ,
[Rab, Pec] = 0 ,
[Rab, Pec] = 2 δ[ca Pb] .
ZM
≡ ηMN ZN
and
ηMN
≡
0 δm!
n
δmn 0
,
M
is in the same representation as Z . We thus see that the O(d, d)invariance of (ZM , Z ) =
δMN is equivalent to the O(d, d)invariance of another metric, ((ZM , ZN )) ≡ ηMN , which is
commonly used in DFT. We also have the K = O(d) × O(d)invariant metric δAB.
We define the generalized vielbein in the gauge of positive Borel subalgebra by
EM A(x) = ePa ha(x) ρKaa ePa<b hab(x) ρKab e 2
1 Pa,b Bab(x) ρRab
M
in ddimensions, and (e−T) is the inverse of the transpose of the vielbein. This generalized
vielbein yields the conventional generalized metric in DFT:
Upon the antiinvolution, g → g♯, the generalized vielbein takes the lowertriangular
HJEP07(21)5
and
β(
2
)(x) =
0
−βab(x) 0
0 !
,
(2.49)
where ema(x) is a lowertriangular matrix. In this case, the generalized metric becomes
where Gmn ≡ ema enb δab .
form, parameterized by
where
,
MMN
Gmn − Bmk Gkl Bln Bmk Gkn!
δmk Bmk
0
δm
k
−Gmk Bkn
!
Gmn
Gkl 0 !
0 Gkl
δ
l
n
−Bln δnl
0
!
,
EM A(x) = E(x) eβ(
2
)(x)
M
A ,
=
=
=
=
e
ema(x)
0
0
e
(e−T)ma(x)
MMN
!
Gemn
(2.47)
(2.48)
(2.50)
(2.51)
studied in [88–92].
where Gemn ≡ eema eenb δab . These dual variables were first introduced and extensively
2.4
Example: Einstein gravity
It is illuminating to compare the above results for DFT with the case of pure Einstein
gravity. In Einstein gravity, the generators Rab and Rab are absent, the
Cartaninvolutioninvariant subgroup is simply generated by the local Lorentz O(d) rotations, and there is
no important difference between the gauges of positive and negative Borel subalgebras.
Indeed, as it is wellknown, when we consider decomposing the spacetime into space and
time, there are two natural parameterizations into upper or lower triangular decomposition:
ArnowittDeserMisner [93]: (gmn) =
LandauLifschitz [94]:
(gmn) =
1 N k!
k
0 δi
1
−gi δik
0
!
−N 2 0 !
0
hkl
These two parameterizations are related simply by a usual local Lorentz transformation.
In comparison, the situation is different in the DFT case. In order to relate two
parameterizations (2.47) and (2.50), we need to use a nontrivial O(d) × O(d) subgroup of the
T duality group. In general, the parameterization (2.47) is suited for the conventional
geometric backgrounds, while (2.50) is suited for nongeometric backgrounds, such as T
folds. As such, we will refer to the latter, negative Borel subalgebra parameterization as
the nongeometric parameterization.
By definition, the actions of the extended field theories are independent of the explicit
parameterization of the generalized metric. However, once we parameterize the
generalized metric in terms of appropriate physical fields, we can straightforwardly construct the
effective actions appropriate for describing dynamics of these field excitations.
As is well known in DFT [7] or EFT [11], parameterizing the generalized metric in
terms of the conventional supergravity fields, we can derive the conventional supergravity
action from DFT or EFT action. For example, if we choose the conventional, geometric
parameterization and impose the section constraint ∂em = 0, we find that the DFT action
is reduced to
HJEP07(21)5
(2.52)
(2.53)
(2.54)
L = e−2φ
1
R(G) + 4 dφ2 − 2 H(3)2
,
where φ is the conventional string dilaton field defined by the Tduality invariant dilaton
≡  
of DFT, e−2d
G 1/2 e−2φ, and the threeform H(3)
strength for the KalbRamond twoform potential B(
2
).
≡ dB(
2
), called the Hflux, is the field
On the other hand, if we choose the dual, nongeometric parameterization (2.50), we
reduce the DFT action to the socalled βsupergravity [54–56, 60, 61]. Although the full
expression is complicated, with the simplifying assumption that indices of βmn contracted
with ∂m always vanishes and the constraint ∂em = 0, the DFT action is reduced to the form
Le = e−2φe
R(Ge) + 4 dφe − 2 
2
1
Q(
1,2
) 2

1/2 e−2φe . Further, we defined
Here, the tilde signifies the nongeometric parameterization, and φe is the dual dilaton field

Q(
1,2
) 2
 ≡ 2 Gem1n1 Gem2n2 Gem3n3 Qm1 m2m3 Qn1 n2n3 ,
Qkmn
≡ ∂kβmn .
The mixedsymmetry tensor,3 Qkmn, is called the nongeometric Qflux. In this paper, we
further generalize the βsupergravity starting from the (heterotic) DFT or EFT.
3
Nongeometric fluxes in EFT: Mtheory
In this section, we consider the elevendimensional supergravity of Mtheory compactified
on a dtorus, Td, equivalently, the tendimensional type IIA supergravity compactified on
a (d − 1)torus, Td−1. This theory possesses the U duality transformation symmetry, and
3This behaves as a tensor only under the simplifying assumption [55].
Ed(d)
Kd
D
αn
SL(5)
SO(5)
10
3
E6(6)
Sp(
4
)
27
6
E7(7)
SU(8)
56
12
various noncompact dimensions, 4 ≤ n ≤ 7.
the EFT provides the manifestly U duality covariant formulation. To construct the EFT,
we consider an exceptional spacetime with the following generalized coordinates:
(XI ) = (xµ , Y M ) (µ, ν
= 0, . . . , n − 1, M = 1, . . . , D) ,
(3.1)
where n ≡ (11 − d) is the dimension of the uncompactified, external spacetime and D is the
dimension of a fundamental representation of the exceptional group Ed(d) whose value for
each n is shown in table 2. In this paper, we consider the cases of noncompact dimensions
n = 4, 5, 6, 7, equivalently, cases of compact dimensions d = 7, 6, 5, 4.
The EFT actions for n = 4, 5, 6, 7 are presented in [14, 15, 22, 23] (see also [24] for
n = 9, [21] for n = 8, and [18] for n = 3). For simplicity, we focus on the following parts
of the action, which are the relevant parts for our purposes:
dnx dDY LEFT
where
LEFT = LEH + Lscalar + Lpot ,
Z
SEFT =
LEH = eR ,
1/2, R is the Ricci scalar of the external metric gµν , and αn is
Here, e abbreviates det gµν 
into account by Lpot.
the integer shown in table 2. Note that the potential part in the EFT action is fully taken
In the EFT, to render the gauge algebra closed, we will impose the section condition of
the form, Y MN
P Q ∂M (· · · ) ∂N (· · · ) = 0, where Y MN
P Q for each EFT is given in [12, 13].4
As is wellknown, there are two natural routes to solve for the section conditions: the
Mtheory section or the type IIB section [14, 98], where all background fields and gauge
parameters depend only on d coordinates xi or d − 1 coordinates xm, respectively. In this
section, we study the Mtheory section and parameterize the generalized metric in terms
4The section condition of DFT can be relaxed in the flux formulation [82, 95, 96] or in the approach
of [97], and the section condition of EFT may be also relaxed in these approaches.
of the conventional/dual fields in eleven dimensions. We relegate the parameterization in
the type IIB section to section 4.
In the Mtheory section, we decompose the internal Ddimensional coordinates Y M
into some representations of SL(d). Explicitly, for each n, we introduce the following
coordinates [11, 28]:
n = 7 : (Y M ) = (xi, yij )
n = 6 : (Y M ) = (xi, yij , yi1···i5 )
n = 5 : (Y M ) = (xi, yij , yi1···i5 )
n = 4 : (Y M ) = (xi, yij , yi1···i5 , zi)
(i, j = 7, 8, 9, M) ,
(i, j = 6, . . . , 9, M) ,
(i, j = 5, . . . , 9, M) ,
(i, j = 4, . . . , 9, M) ,
where the conventional Mtheory circle direction, denoted by xM, is one of the internal
coordinates xi. The section condition is satisfied when all fields are functions only of xi,
the physical coordinates on the dtorus. So, ∂/∂yij = ∂/∂yi1···i5 = ∂/∂zi = 0.
Parameterization of the generalized vielbein
We now examine parameterization of the generalized metric (or vielbein) in the Mtheory
section of the EFT. The generalized metric in the SL(5) EFT was first obtained in [8]
(which in turn is based on the earlier work [99]) as
Subsequently, the same generalized metric (up to an overall factor) was presented in [11]
in the context of E11 program [27, 28], and its extensions to Ed(d) EFT with 5 ≤ d ≤ 7
were also presented (see also [100, 101] for d = 4, 5). The parameterization given in [11]
was obtained by choosing the positive (or uppertriangular) Borel gauge. If we instead
choose the negative (or lower triangular) Borel gauge, we can parameterize the generalized
metric using the socalled dual Ωfields (the explicit form of Ωfields for SL(5) EFT is given
in [78, 79], which we repeat below). As the Ωfields are related to the nongeometric fluxes,
we refer to the latter as nongeometric parameterization.
In the rest of this subsection, we present two parameterizations of the generalized
vielbein, i.e., the conventional parameterization and the nongeometric parameterization,
for 4 ≤ d ≤ 7 (or 4 ≤ n ≤ 7). Using these parameterizations, we define the nongeometric
fluxes in Mtheory and construct the elevendimensional effective actions that are useful
for describing these nongeometric fluxes.
3.1.1
For the g = sl(5) Lie algebra, we decompose the 24 generators to5 [11]
5We relegate their commutators in appendix A.1.
(3.3)
HJEP07(21)5
(3.4)
where Kab are the gl(
4
) generators and Ra1a2a3 and Ra1a2a3 are the generators that
transform as totally antisymmetric under gl(
4
). So, we are decomposing 24 generators into
16 + 4 + 4 generators. Using this decomposition, a group element g of G = SL(5) can be
parameterized as
This element can always be rewritten in the form of positive Borel gauge:
where
k ∈ H = SO(5) .
(3.7)
It turns out that the SO(5) element k does not contribute to the generalized metric.
Disregarding it, the number of independent parameters are 10 + 4, which is equal to the
dimension of the coset space G/H = SL(5)/SO(5).
We can identify the parameters,
eib ≡ (eh)ib ∈ GL(
4
)/SO(
4
) and Aa1a2a3 , as the vielbein and the 3form potential on
the 4torus, respectively. Note that the left index of the matrix (eh) is changed from a to
i in order to interpret it as the curved index.
From the formulas (2.31) and (2.33) and the matrix representations (A.12)–(A.16), the
generalized vielbein and the metric become [8]
MMN ≡ G 5 MMN ,
1
EM
A
1
≡ G 10 EM A ,
G ≡ det Gij ,
Gij ≡ eia ej b δab ,
(EM A) ≡ Eb eA(3) =
e a
i − √12 Aia1a2
0
eia1i12a2
,
(MMN ) = EM
A EN B δAB
=
where
(EbM A) ≡
A(3)
1
≡ 3!
eia
0
0 eia1i12a2
!
,
Aabc ρRabc =
Gij + 12 Aikl Aklj
,
0 − √12 Aab1b2
0
0
!
,
eia1i12a2 ≡ (e−T)i1 [a1 (e−T)i2 a2] ,
δAB ≡
δab
0
0 δa1a2, b1b2
!
Gi1···in, j1···jn ≡ δki11······iknn Gk1j1 · · · Gknjn ,
δa1a2, b1b2 ≡ δca11ca22 δc1b1 δc2b2 ,
and the indices are changed using the vielbein (e.g. Aia1a2 ≡ eic Aca1a2 ) and raised or
lowered using the metric Gij and its inverse.
See appendix A for further details of our conventions. – 14 – (3.8)
(3.9)
If we do not choose the Borel gauge, we can generally parameterize the SL(5)
generalized metric as [78, 79]
(EM A) ≡ Eb eA(3) eΩ(3) =
Choosing Ωijk = 0 or Aijk = 0, we obtain two alternative parameterizations for the
Ωc1c2c3 ρRc1c2c3 =
0
where we defined the Ωmatrix:
generalized metric,
(3.10)
,
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
The first expression is the conventional, geometric parameterization, while the second
expression is the nongeometric parameterization. From these two parameterizations, we
obtain the following relation between the standard fields and the dual fields:
Geij =  
G 1/9
 
E 1/9 Eij ,
Ωij1j2 = (E−1)ik Gj1k1 Gj2k2 Akk1k2 ,
where
Further, associated to the two parameterizations, the external metric is also expressed in
two alternative ways:
Eij ≡ Gij +
Aikl Aklj .
1
2
gµν = G n−2 gµν = Ge n−2 gµν .
e
1
1
We confirm that gµν
and Gij are components of the conventional metric in the
elevendimensional supergravity, denoted by Gµˆνˆ (µ,ˆ νˆ = 0, . . . , 9, M).
3.1.2
n = 6: G
= SO(5, 5)
The generalized metric or vielbein generally has the overall factor,
MM N ≡  
G n−2 MM N ,
1
that comes from the second term in the righthandside of (A.12). In the following, we
focus on the parameterizations of MM N and EM
A
In the present case of G = SO(5, 5), we can similarly parameterize the generalized
vielbein as [8]
0
0
,
0
0
0
.
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
or as
where we defined
(EM
A
) ≡ Eb e
Ω(3)
=
E
b ≡ 0
eia1i12a2
0
0
5
√
5!
0
0
eia1·1····i·5a5
1
− √
2
Ωi1i2a
Ω[i1i2i3 Ωi4i5]a
A(3)
Ω
(3)
e a
i
0
1
3!
1
3!
e a
i
0
e a
i
e
,
0
0
0
1
− √
2
Aia1a2
5
√
5!
Ai[a1a2
,
We can again redundantly parameterize the generalized vielbein as
EM
A
≡ Eb e
A(3) Ω(3)
e
In the case G = E6(6), we can parameterize the generalized vielbein as [8]
or as
EM
Eb ≡ 0
0
,
≡ − 6!
Ac1c2c3 ρRc1c2c3 = 0
Ac1···c6 ρRc1···c6 = 0
Ωc1c2c3 ρRc1c2c3 = − √12 Ωa1a2b
Ωc1···c6 ρRc1···c6 =
0
0
,
0
0
0
0
0
0
0
− √15! Ωa1···a5b
0
0
0
0
0
0 ,
0
0
0
,
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
We remark that the normalization of the 6form is different from that used in [8] by a factor
2. Note also that, in the middle expression of the last line, the minus sign is introduced in
order to make the exotic duality, Ac1···c6 ↔ Ωc1···c6 , coincides with the matrix transpose.
Stated differently, the negative sign comes from the fact that the Cartan involution (A.17)
for Rc1···c6 appears with the positive sign, θ(Rc1···c6 ) = +Rc1···c6 .
3.1.4
In the E7(7) case, we can parameterize the generalized vielbein as [11]
(EM A) ≡ Eb eA(6) eA(3)
or
(EM A) ≡ Eb eΩ(6) eΩ(3) ,
eia
0
Eb ≡ 0
0
eia1i12a2
0
0
0
0
0
0
e a1···a5
i1···i5
0
0
0
e−1 eia
,
− √15! Ωa1···a5b
0
0
0
0
0
0
0
0
0
0
0
0
.
,
(3.34)
(3.35)
0
0
0
,
0
(3.36)
(3.37)
≡ − 6!
Ωc1···c6 ρRc1···c6 =
1
Ac1c2c3 ρRc1c2c3 = 00
0
0
0
0
0
0
0
0
1
4
1
4
35
2
6!√22 δ[ab1 ǫb2]c1···c6 Ωc1···c6
R(G) ≡ R(g) + gµν
1
4 ∂µ Gij ∂ν Gij +
1
4 ∂µ ln G ∂ν ln G
+ R(G) + Gij
∂igµν ∂j gµν
+
∂i ln g ∂j ln g
Fi1···i7 ≡ 7 ∂[i1 Ai2···i7] +
A[i1i2i3 Fi4i5i6i7] ,
Fµ, k 1···k6 ≡ ∂µ Ak1···k6 − 10 A[k1k2k3 ∂µ Ak4k5k6] .
We remark that the parameterizations for Ed(d) with 4 ≤ d ≤ 6 are obtainable by a
truncation of those for E7(7).
3.2
Elevendimensional effective action
The elevendimensional effective action is obtained by solving the section condition such
that the elevendimensional coordinates are given by (xµˆ) ≡ (xµ , xi); see appendix B for
the detailed derivation. For instance, consider the E7(7) EFT in the geometric
parameterization. The action becomes
L = G 2
1
where
R(G)
− g
µν
1
− 2 · 4!
(3.38)
1
1
0
0
0
1
βeα, m1m2m3n ≡ 2 ǫm1m2m3np1p2 βαp1p2 ,
For the external part, we focus on the following twoderivative terms:
LEH = e R(g)
and
e
4αn
Recalling the relation, gµν = G n−12 gµν , the first term is given by
LEH = g 12 G1/2 R(g) + 2 (n − 1) ∂µ e gµν ∂ν ln G− n−12
+ g 21 G n−2
n/2 n − 1
1
= G 2 R(g) +
4(n − 2)
n − 1
4(n − 2)
g
µν ∂µ ln G ∂ν ln G
g
µν ∂µ ln G ∂ν ln G ,
where we defined G 2 = g 12 G 2 and neglected the total derivative term at the second
1 1
equality. For the scalar part, Lscalar, noting that the matrix V has a blockwise upper/lower
triangular form with constant diagonal elements, we obtain
γ
Gµ,ˆ m1···m4 ≡ ∂µˆDm1···m4 − 3 ǫγδ B[m1m2 ∂µˆBδm3m4] ,
Gµ,ˆ m1···m6 ≡ ∂µˆBm1···m6 − 15 B[βm1m2 ∂µˆDm3···m6] + 15 ǫγδ B[m1m2 Bm3m4 ∂µˆBδm5m6] .
β β β γ
e
4αn
g
µν ∂µ McMN ∂νMc
MN
g
µν
Mc
MN
McP Q ωµM P ωµN Q ,
(B.85)
e
− 2αn
where the first term simply becomes
4αn
g
=
µ McM N ∂ν Mc
M N
 
g
4
4
1 µν
g
4
∂µ Gij ∂ν Gij − 4(n − 2)
g
µν
∂µ Gmn ∂ν Gmn − 4(n − 2)
g
µν
1
1
∂µ mαβ ∂ν mαβ
∂µ ln G ∂ν ln G
.
∂µ ln G ∂ν ln G +Lscalar
(mat)
(Mtheory)
∂µ ln G ∂ν ln G
 
where we used, e gµν
• SL(5), SO(5, 5) (geometric):
• E6, E7 (geometric):
L(smcaalat)r =
−  
G 21 gµν
 
L(smcaalat)r = − 2 · 3!
1
G 2
 
R(g) +
+
1 µν
g
4
1 µν
g
4
1 µν
g
4
∂µ Gij ∂ν Gij +
∂µ Gmn ∂ν Gmn +
1 µν
g
4
1 µν
g
4
∂µ mαβ ∂ν mαβ
+ Lscalar
(mat)
g
µν
M M N McP Q ωµM
c
P ωνN
Q ,
We can calculate the explicit form of Lscalar as follows:
= G 21 gµν
and
Mc
M N
(mat)
McP Q = M M N McP Q .
c
g
µν
Gi1i2i3, j1j2j3 ∂µ Ai1i2i3 ∂ν Aj1j2j3 ,
(B.89)
Gei1i2i3, j1j2j3 Sµ i1i2i3 Sν j1j2j3 ,
L(smcaalat)r = −Ge 2 egµν
1
µν
mαβ Gm1m2, n1n2 ∂µ Bmα1m2 ∂ν Bnβ1n2 ,
(B.93)
 
G 2 µν
2
mαβ Gm1m2, n1n2
2!
∂µ Bmα1m2 ∂ν Bnβ1n2
+
Gm1···m4, n1···n4
4!
Gµ, m1···m4 Gν,n1···n4
,
(B.94)
mαβ Gm1m2, n1n2
∂µ Bmα1m2 ∂ν Bnβ1n2
+
+
2!
Gm1···m4, n1···n4
4!
mαβ Gm1···m6, n1···n6
6!
Gµ, m1···m4 Gν,n1···n4
α
β
Gµ, m1···m6 Gν,n1···n6
,
(B.95)
• E7 (geometric):
L(smcaalat)r = −
1
 
G 2 µν
g
2
• SL(5) (nongeometric):
• E7 (nongeometric):
L(smcaalat)r = −
1
µν
e g
L(smcaalat)r = − 2 · 2! e
µν
1
mαβ Gem1m2, n1n2 Qα,µ
m1m2 Qβ,ν n1n2 ,
(B.96)
• SO(5, 5), E6 (nongeometric):
L(smcaalat)r = −
1
µν
mαβ Gem1m2, n1n2 Qα,µ
e
2!
m1m2 Qβ,ν n1n2
4!
+
Gem1···m4, n1···n4 P m1···m4 Pν n1···n4 ,
µ
(B.97)
mαβ Gem1m2, n1n2 Qα,µ
e
2!
m1m2 Qβ,ν n1n2
4!
+
Gem1···m4, n1···n4 P m1···m4 Pν n1···n4
µ
+ e
mαβ Gem1···m6, n1···n6 Qα,µ
6!
m1···m6 Qβ,ν n1···n6 .
(B.98)
Internal (potential) part
The internal part, or the potential part, consists of three terms
Here, we choose the canonical section, (∂M ) = (∂i , 0, . . . , 0), where the index i represents i
in the Mtheory section or m in the type IIB section. In this case, the first and the third
terms can be obtained as follows:
Lpot ≡ L(p1o)t + L(p2o)t + L(p3o)t ,
(
2
)
(3)
(1)
Lpot ≡ e
Lpot ≡ −e 2 M
1
4αn
1
M
.
(B.99)
(B.100)
(B.101)
(B.102)
4(n − 2)2 Gij ∂i ln G ∂j ln G − 2(n − 2)
n
Gij Gkl ∂i ln G ∂lGjk
+
+
4
2
1 Gij ∂i ln g ∂j ln g +
1 Gij ∂i ln G ∂j ln g +
1
On the other hand, as we show later, the second term L(p2o)t can be written as
L(p2o)t = −e 2 Mc
1
MN ∂N Mc
KL ∂LMcMK + ΔL(p2o)t ,
1
2αn
1
− 2αn
1
4
Gmn Gpq ∂m ln G ∂pGnq +
Gmn ∂mmαβ ∂nmαβ
.
(B.104)
(B.105)
(Mtheory)
(type IIB)
=
1
1
G 2
1
1
n − 2
1
1
n − 2
Gij ∂kGil ∂j Gkl +
Gmn ∂pGmq ∂nGpq +
where ΔL(p2o)t does not include derivatives of metric.
Gij Gkl ∂i ln G ∂kGjl + ΔL(p2o)t
1
2(n − 2)2 Gij ∂i ln G ∂j ln G
1
2(n − 2)2 Gmn ∂m ln G ∂n ln G
where we used the formula
R(G) =
1 Gij ∂iGkl ∂jGkl
1 Gij ∂kGil∂jGkl +
∂mmαβ ∂nmαβ
M MN McP Q ωmM
c
P ωnN
Q
1 Gij Gkl ∂i ln G ∂kGjl −
1 ∂i G 21 Gij Gkl ∂jGkl − ∂kGlj
1 Gij Gkl ∂i ln G ∂kGjl +
1 Gij ∂i ln g ∂j ln G +
1
1 ∂i G 21 Gij Gkl ∂jGkl − ∂kGlj ,
,
− 2αn
+Gmn
4
4
− 2
−
1
G 2
We thus obtain the potential as
Thus, comparing this with (B.105), we obtain
G 2 R(G) +
Gij ∂igµν ∂j gµν +
Gij ∂i ln g ∂j ln g
− 2αn
1
G 2
2
e
1
and dropped the boundary term.
Calculation of L(p2o)t.
of ΔL(p2o)t.
case, noticing VM i = δi
M = (V −1)M i and (V T)iM = δi
M = (V −T)iM , we obtain
First, let us calculate L(p2o)t in the case of the conventional parameterization. In this
Here, we show equation (B.105) and determine the explicit form
L(p2o)t = − 2
e
e
e
McP Q ωijP ωklQ = G 2
Gil Gjk McP Q ωijP ωklQ .
We next calculate L(p2o)t in the nongeometric parameterization. In this case, we use
the simplifying assumption [54] that requires any derivatives contracted with the dual
potentials vanishes (e.g. βmn ∂m = 0). In our notation, it can be expressed as
· · · VM i ∂i = · · · δM i ∂i ,
∂i · · · VM i = ∂i · · · δM i
V = V or V −1 ,
(B.110)
where the ellipsis represent arbitrary tensors or derivatives. Using the simplifying
assumption, we obtain
L(p2o)t = − 2 M
Mi ∂iM
Kj ∂jMMK
e
e
= − 2
MN ∂N Mc
KL ∂LMcMK .
(B.111)
where, in the third equality, we used the simplifying assumption and McP i = δj
ji
P Mc ,
and in the fourth equality, we used (V −T)M j = δjM and Mkl = Mckl which are generally
satisfied in the nongeometric parameterization.
Comparing (B.111) with (B.105), we
obtain ΔL(p2o)t = 0 in the nongeometric parameterization.
Summary of the potential Lpot. To summarize, we obtained
1
G 2 R(G) +
Gmn ∂m ln g ∂n ln g
1
4
4
1
4
1
4
+ Gmn ∂mmαβ ∂nmαβ + Lpot
(mat)
Lpot
1
(mat) = −G 2
1
Gi1···i4, j1···j4 Fi1···i4 Fj1···j4 + 2 · 7!
1
Gi1···i7, j1···j7 Fi1···i7 Fj1···j7 ,
• geometric parameterization:
Lpot
(mat) = − 2
1
G 2
1
αn
• nongeometric parameterization:
Lpot
Geij McMN McP Q ωiM P ωjN Q .
More explicit form of L(pmotat) in each case is given as follows:
• SL(5), SO(5, 5), E6 (geometric):
Lpot
(mat) = − 2 · 4!
G 2
1
Gi1···i4, j1···j4 Fi1···i4 Fj1···j4 ,
Gij McMN ωiM P ωjN Q
− Gil Gjk ωijP ωklQ
McP Q ,
(B.113)
.
(mat)
Lpot
1
= − 2 · 3! e
Gij Gei1i2i3, j1j2j3 Sii1i2i3 Sj j1j2j3 ,
(B.117)
• E6, E7 (nongeometric):
(mat)
mαβ Gm1m2m3, n1n2n3 Hmα1m2m3 Hnβ1n2n3 ,
(B.119)
(mat)
Lpot
mαβ Gm1m2m3, n1n2n3
3!
Hmα1m2m3 Hnβ1n2n3
+
Gm1···m5, n1···n5
5!
Gm1···m5 Gn1···n5
,
(B.120)
• E6(6), E7(7) (geometric):
• SL(5) (nongeometric):
(mat)
Lpot
1
= − 2 · 2! e
• SO(5, 5), E6(6) (nongeometric):
(mat)
1
Gmn
• E7(7) (nongeometric):
(mat)
= −
e
Gmn
1
m1···m6 Qβ, nn1···n6 .
(B.123)
mαβ Gmn Gem1m2, n1n2 Qα, m
e
m1m2 Qβ, nn1n2 ,
(B.121)
−
Gem1···m4, n1···n4 Pm
4!
m1···m4 Pnn1···n4 ,
(B.122)
mαβ Gem1m2, n1n2 Qα, m
e
m1m2 Qβ, nn1n2
mαβ Gem1m2, n1n2 Qα, m
e
m1m2 Qβ, nn1n2
−
Gem1···m4, n1···n4 Pm
4!
+ e
mαβ Gem1···m6, n1···n6 Qα, m
6!
m1···m4 Pnn1···n4
2!
2!
Fi1···i4 ≡ 4 ∂[i1Ai2i3i4] ,
Hmα1m2m3 ≡ 3 ∂[m1Bmα2m3] ,
B.4
Summary
δ
Gm1···m5 ≡ 5 ∂[m1Dm2···m5] − 15 ǫγδ B[m1m2 ∂m3Bm4m5]
γ
= 5 ∂[m1Cm2···m5] + 30 H[1m1m2m3 Cm4m5] .
35
2
Fi1···i7 ≡ 7 ∂[i1Ai2···i7] +
A[i1i2i3 Fi4i5i6i7] , (B.124)
In this appendix section, we evaluated several external terms in the EFT action,
4αn
and the potential part, Lpot. Combining these, we obtain
1
L =G 2 R(g) +
4
1 gµν ∂µ Gij ∂νGij +
4
1 gµν ∂µ ln G ∂ν ln G
+ R(G) +
4
1 Gij ∂i ln g ∂j ln g + L(smcaalat)r + Lpot
(mat)
1
≡ G 2 R(G) + L(smcaalat)r + L(pmotat) .
For example, for the E7(7) EFT in the geometric parameterization, this becomes
Gi1···i4, j1···j4 Fi1···i4 Fj1···j4 +
Gi1···i7, j1···j7 Fi1···i7 Fj1···j7 . (B.129)
C
C.1
Doublevielbein formalism for gauged DFT
Parameterization from defining properties of doublevielbein
The previous result from the Iwasawa decomposition provides the upper or lower triangular
parameterization of the generalized vielbein. However, the triangulation breaks the full
local structure group into the diagonal subgroup. If we decompose O(1, D−1 + dim G) as
O(D − 1, 1) × O(dim G), then we choose the diagonal gaugefixing by identifying the two
local Lorentz groups,
O(D − 1, 1) × O(1, D − 1)
O(D − 1, 1)D .
(C.1)
Here, we shall construct the geometric parameterization and the nongeometric
parameterization directly from the defining conditions of doublevielbein. This approach does not
require any gaugefixing condition and ensures manifest O(1, D − 1) × O(1, D−1 + dim G)
covariance. Analogous to the ordinary O(D, D) case, doublevielbein for O(D, D + dim G)
gauged DFT satisfies the following defining properties [127],
VMcpV Mcq = ηpq ,
VMcpV¯ Mc
q¯ˆ = 0 ,
V¯Mcpˆ¯
V¯ Mc
q¯ = η¯ˆpˆ¯q¯ˆ ,
VMcpVNb p + V¯Mcp¯ˆ
V
Nb
p¯ˆ = JˆMcNb ,
(C.2)
(B.125)
(B.126)
(B.127)
(B.128)
where ηmn and η¯ˆpˆ¯q¯ˆ are O(1, D − 1) and O(D − 1, 1 + dim G) metric, respectively. The
doublevielbein is then decomposed as
Mc
V m =
VM m!
Vαm
and
V¯ m¯ˆ =
Mc
V¯M m¯ V¯M a¯!
V¯αm¯
V¯αa¯
.
Note that the usual geometric parameterization is obtained by assuming that the
upperhalf blocks of VM
m and V¯M
m¯ are nondegenerate and by identifying them as a pair of
conventional vielbeins [127]. However, the nondegeneracy assumption can be relaxed in a
consistent manner.
Suppose that the upperhalf blocks of V m and V¯ m¯ are given by
V µm = (e−1)µm + β′µν eνm
and
V¯ µ m¯ = (e¯−1)µm + β′µν e¯νm ,
where eµ m and e¯µ m¯ are two copies of the Ddimensional vielbein corresponding to the same
metric gµν
eµmeνnηmn = −e¯µ m¯ e¯ν n¯η¯m¯ n¯ = gµν ,
and β′ is an arbitrary tensor. Then, V µm and V¯ µ m¯ are not guaranteed to be nondegenerate.
Substituting the previous decomposition ansatz (C.3) and (C.4) into the defining
properties (C.2), we find the most general parameterization that satisfy all the algebraic
constraints (C.2) for VMˆ
Vαm = √
m
1
1
2
2
V¯M m¯ = √
V¯αm¯ = √
1
1
2
2
eµ m + Bµν′ (e−1)νm
− β′νρ eρm !
(e−1)µm
− β′µν eνm
καβ(AT)βµ (e−1)µm
− β′µν eνm
− καβ(A˜T)βµ eµ m ,
e¯µ m¯ + Bµν′ (e¯−1)ν m¯
− β′νρ e¯ρ m¯ !
− β′µν e¯νm¯
(AT)αµ (e¯−1)µ m¯
− β′µν e¯νm¯
− (A˜T)αµ e¯µ m¯ ,
,
,
−Aµ a¯ + Bµν′ A˜νa¯!
A˜µα (φT)αa¯
,
V¯αa¯ = φa¯α + φa¯β(A˜T)βµ Aµα .
and for V¯ m¯ˆ
Mc
Here, Bµν′ and β′µν are defined as
in which Bµν and βµν are antisymmetric tensors.
Bµν′ = Bµν + 1 α′Aµ α(AT)αν ,
β′µν = βµν
1 α′A˜µα (A˜T)αν ,
2
− 2
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
However, if we assume that each blocks of VM m and V¯M m¯ are nondegenerate, this
solution is overparameterized. The physical degrees of freedom are determined by the coset
O(D, D + dim G)
O(D−1, 1) × O(1, D−1 + dim G)
,
and the associated number of degrees of freedom is given by
1
2
1
1
(2D + G)(2D + G − 1) − 2
D(D − 1) − 2 (D + G)(D + G − 1) = D2 + DG ,
where G denotes dim G. The D2 components arise from the gµν , Bµν
or g˜µν , βµν , and
to make up the parameterization.
The geometric parameterization, which is for the conventional heterotic
supergravity [128], is obtained by turning off βµν and A˜µ a¯,
eµ m + Bµν′ (e−1)νm!
e¯µ m¯ + Bµν′ (e¯−1)ν m¯!
(e¯−1)µ m¯
Aµ α(φT)αa¯!
0
,
,
Vαm = √ (AT)αµ (e−1)µm ,
, V¯αm¯ = √ (AT)αµ (e¯−1)µ m¯ ,
1
2
1
2
1
V¯αa¯ = √α′
(φa¯)α .
Under the nondegeneracy assumption, one can show through a field redefinition that
the geometric parameterization is essentially the same as the most general solution (C.6)
and (C.7). On the other hand, if we assume that some of components of V µm
or V¯ µ m¯ are
vanishing, we can define an another class of nongeometric background, which cannot be
related by field redefinition from geometric parameterization [129].
Using the relation the projection operators and doublevielbein:
PMcNb = VMcmηmn(V T)nNˆ
and
PMcNb = V¯ m¯η¯m¯ n¯(V¯ T n¯
¯
Mc
) Nb +V¯Mca¯
κa¯¯b(V¯ T ¯b , (C.13)
) Nb
we construct a geometric parameterization for the projection operators as
and
and
P =
1
2
g + α′A κ At + B′g−1(B′)t
Aκ + B′g−1Aκ 1 + B′g−1
κAt + κ Atg−1(B′)t
1 + g−1(B′)t
κ Atg−1A κ
g−1A κ
κ Atg−1 ,
g−1
P¯ =
1
2
−g − α′A κ At − B′g−1(B′)t
−κAt − κAtg−1(B′)t
1 − g−1(B′)t
−Aκ − B′g−1Aκ
−κAtg−1Aκ − α2′ κ
−g−1Aκ
1 − B′g−1
−κAtg−1 .
−g−1
(C.9)
(C.10)
(C.11)
(C.12)
(C.14)
(C.15)
0
1
eµ m
2 (e−1)µm
− β′µν eν m
1
Vαm = − √2 καβ(A˜T)βµ eµ m ,
¯ m¯ = √
VM
1
− β′µν e¯ν m¯
√α′A˜µα (φT)αa¯
e¯µ m¯
,
1
, V¯α m¯ = − √2 καβ(A˜T)βµ e¯µ m¯ ,
V¯αa¯ = √α′
1
φa¯α .
The corresponding projection operators are constructed as
P =
−A˜κ + β′g˜A˜κ
1 − g˜β′T
−κA˜T + κA˜Tg˜β′T
g˜−1 + β′g˜β′T + α′A˜κA˜T
P¯ =
−κA˜Tg˜A˜κ − α2′ κ
A˜κ − β′g˜A˜κ
1 + g˜β′T
κA˜T
− κA˜Tg˜β′T
−g˜−1 − β′g˜β′T − α′A˜κA˜T
by H = P − P¯ takes the form:
Here, we used Kαβ = −(ta¯)αTκa¯¯bt¯bβ. In this parameterization, it follows that the projection
operators satisfy the complete relation, J = P + P¯ and that the generalized metric defined
H =
g + B′g−1(B′)t + AκAt
κAt + κAtg−1(B′)t
Aκ + B′g−1Aκ
κAtg−1Aκ + α1′ κ
g−1Aκ
B′g−1
κAtg−1 .
g−1
Consider next the nongeometric parameterization. As for the geometric
parameterization, it is simply given by turning off Bµν and Aµ a¯ while keeping β and A˜ in (C.6)
and (C.7):
HJEP07(21)5
,
.
.
(C.16)
(C.17)
(C.18)
(C.19)
(C.20)
(C.21)
and
and
satisfied and that the generalized metric H = P − P¯ is expressed by
Once again, in this parameterization, it follows that the complete relation J = P + P¯ is
H = −κA˜Tg˜
κA˜Tg˜A˜κ + α1′ κ
−A˜κ + β′g˜A˜κ
−g˜β′T
−κA˜T + κA˜Tg˜β′T
g˜−1 + β′g˜β′T + α′A˜κA˜T
One notes that this result is consistent with the parameterization in terms of the Iwasawa
decomposition given in (5.34).
We should remark that, ultimately, the doublevielbein formalism is imperative. For
the bosonic case, the geometric parameterization and the nongeometric parameterization
of doublevielbein, (C.11) and (C.18), respectively, are equivalent to the previous result
constructed by coset representative, as they should. Even though these two approaches
are consistent for the bosonic case, for introducing supersymmetry, the doublevielbein
formalism is the most adequate approach [128, 130, 131].
Connection and curvature
The gauge symmetry for gauged DFT is given by a twisted generalized Lie derivative which
is defined by
(LˆX V )McNb = (Lˆ0X V )McNb − f McPbQbXPbV QbNb − fNbPb
QbXPbV Mc ,
Q
b
The Lˆ0X is the ordinary generalized Lie derivative defined in the ungauged DFT by
(Lˆ0X V )McNb = XPb∂ V Mc
Pb
Nb + (∂McX
Pb − ∂PbXMc)V PbNb + (∂Nb XPb
− ∂PbXNb )V Mc ,
Pb
Lˆ0X d = XMc∂Mcd − 2 ∂McXMc ,
1
where fMcNbPb are the structure constants for YangMills gauge group. The gauge parameter
XMc consists of ordinary generalized Lie derivative part and a YangMills gauge symmetry
part in an O(D, D + dim G) covariant way.
As for the covariant differential operator of the gauge transformations (C.22), we
present a covariant derivative which can be applied to any arbitrary O(D, D + dim G),
Spin(D − 1, 1) and Spin(1, D − 1 + dim G) representations as follows
DˆMc := ∂
Mc + Γ
Mc + Φ
Mc + Φ¯ Mc .
where Φ
Mcmn and Φ¯
are constructed in gauged DFT [83]
Mcm¯ˆ n¯ˆ are spinconnections and ΓMcNbPb is semicovariant connection which
(C.22)
(C.23)
(C.24)
QbRbSb .
(C.25)
(C.26)
(C.27)
(C.28)
ΓPbMcNb = Γ0PbMcNb + δP QbP
McRbP Sb + δPbQbP¯
Nb
Mc
2
RbP¯ Sb fQbRbSb − 3 P + P¯ PbMcNb
Nb
QbRbSbf
where Γ0P MN is the connection for ordinary DFT [127],
Γ0PbMcNb = 2(P ∂PbP P¯)[McNb] + 2(P¯[Mc
QbP¯
R
Nb] b − P[Mc
QbPNb]Rb)∂QbPRbPb
4
− D − 1
PP [McP
Nb]Qb + PPb[McPNb]
Q
b
∂Qbd + (P ∂RbP P¯ [RbQb] ,
and PPbMcNb
QbRbSb and P¯PbMcNb
QbRbSb are ranksix projection operators
PPbMcNb
SbQbRb :=PPbSbP[Mc
PPbMcNb
SbQbRb :=P¯ SbP¯[Mc
Pb
Rb] +
Rb] +
2
2
D − 1 PPb[McPNb]
¯
D − 1 PPb[McP
¯
Nb]
[QbP Rb]Sb ,
which are symmetric and traceless,
PPbMcNbQbRbSb = PQbRbSbPbMcNb = PPb[McNb]Qb[RbSb] ,
PPbMcNbQbRbSb = P¯QbRbSbPbMcNb = P¯Pb[McNb]Qb[RbSb] ,
¯
Here the superscript ‘0’ indicates a quantity defined in the ungauged DFT.
The spinconnections are defined by using the semicovariant derivative
Although these are not gauge covariant, we can project out to the tensor part
Φ
Mcmn = V Nb m∂McVNbn + ΓMcNbPbV Nb mV Pbn ,
Φ¯ Mcm¯ˆ n¯ˆ = V¯ Nb mˆ¯ ∂McV¯Nb n¯ˆ + ΓMcNbPb
V¯ Nb
m¯ˆ V Pbn¯ˆ .
(C.29)
Φp¯mn ,
Φ¯ pm¯ n¯ ,
Φ¯ [a¯¯bc¯] ,
Φa¯mn ,
Φ¯pm¯ a¯ ,
Φ¯ p¯ˆp¯ˆm¯ ,
Φ[pmn] ,
Φpa¯¯b ,
Φ¯ p¯ˆp¯ˆa¯ .
Φppm ,
Φ¯ [p¯m¯ n¯] ,
Φ¯ [p¯m¯ a¯] ,
Φ¯ [p¯a¯¯b] ,
(C.30)
These will be the building block that the formalism uses. Various covariant quantities can
be generated by using these spinconnections and their derivatives [83].
The heterotic DFT action is given by the generalized curvature tensor from
semicovariant curvature tensor S
McNbPbQb as
1
2
SMcNbPbQb =
R
McNbPbQb + RPbQbMcNb − ΓRbMcNb ΓRbPbQb ,
(C.31)
where R
McNbPbQb is defined from the standard commutator of the covariant derivatives
R
McNbPbQb = ∂McΓNbPbQb − ∂Nb ΓMcPbQb + ΓMcPbRbΓ
NbRbQb − ΓNbPbRbΓ
McRbQb + fRbMcNb
R
Γ bPbQb . (C.32)
jection operators
Then, the generalized curvature scalar is defined by contraction of SMcNbPbQb with the
proS := 2P McNb P PbQbS
McPbNbQb
= 2 2∂mΦnmn − ΦmmpΦnnp − 2
3 Φ[mnp]Φmnp − 2
1 Φp¯mnΦp¯mn − 2
1 Φa¯mnΦa¯mn
(C.33)
− fpmnΦpmn
− fp¯mnΦp¯mn
− fa¯mnΦa¯mn .
C.3
Nongeometric fluxes and action
There are several approaches for constructing differential geometry of the gauged DFT [65,
83]. Here, we follow the so called semicovariant formalism [83] which is wellsuited for
To define nongeometric fluxes, we adopt the nongeometric parameterizations of
doublevielbein obtained in (C.17) and (C.18), and substitute them to the definition of
generalized spin connection (C.29). Not all of the components of generalized spin
connection are involved for defining heterotic DFT action. The relevant components of generalized
spinconnection should be invariant under the generalized diffeomorphism for gauged DFT
10See appendix C for the concise review of doublevielbein formalism for gauged DFTs.
Φm¯ np = + √
Φa¯mn = − 2
Φ[mnp] = + √
1
1
2
2
1
2
Φmmn = + √
(e−1)µm ωµmn − 2∂nφ ,
In the above expression, the components of generalized spin connection comprise three
kinds of fluxes that were introduced in (5.40).
Consider now the nongeometric action of heterotic DFT in terms of the nongeometric
fluxes. The action is given by the generalized curvature scalar S, which is defined in (C.33)
in terms of the generalized spinconnections:
Shet =
Z
e−2d 2S ,
1 F˜µν a¯eµm eνn ,
1
1
ωm¯ np − 2 Qρµν e¯ρm¯ eµn eνp + Qρµν eρneµp e¯ν m¯ ,
ω[mnp] − 2 Qρµν eρ[meµ neνp] .
or twisted generalized Lie derivative. They define the nongeometric fluxes
where
D
Exotic branes
them by
S = 2∂mΦnmn − ΦmmpΦnnp − 2
3 Φ[mnp]Φmnp − 2
1 Φp¯mnΦp¯mn − 2
1 Φa¯mnΦa¯mn .
By substituting (C.34) into this action, one can show that (C.36) is equivalent to the
previous nongeometric heterotic action (5.40).
A defect brane refers to a codimensiontwo configuration in type II string theory. Denote
b(nd, c)(n1 · · · nb, m1 · · · , mc, ℓ1 · · · ℓd) ,
for the configuration wrapped or smeared over the 7torus [45, 46, 51] and thus has the mass:
Mb(nd, c) =
1
gsn ls
Rn1 · · · Rnb
l
b
s
Rm1 · · · Rmc
l
c
s
2
constant and bcn ≡ b(nd=0, c) and bn ≡ b(nd=0, c=0).
Here, Ri is the compactification radius in the xidirection and gs is the string coupling
In this paper, we consider compactification on shrinking tori. As an example, consider a
522(34567, 89)brane (6.49) in the E7(7) EFT. In this case, xm (m = 4, . . . , 9) are compactified
on a sixtorus and x3 direction is a noncompact direction. In this case, the “522(34567,
89)brane” is a onedimensional extended object with the tension,
T =
1
2πgs2 ls2
R4 · · · R9
l
5
s
R8R9
l
2
s
2
(C.35)
(C.36)
(D.1)
(D.2)
(D.3)
type IIA theory
01 = D0 (MP(n) = Rn−1)
10(n) = F1(n)
21(n1n2) = D2(n1n2)
41(n1 · · · n4) = D4
52(n1 · · · n5) = NS5
61(n1 · · · n6) = D6
512(n1 · · · n5, n6) = KKM
613(n1 · · · n5, n7)
522(n1 · · · n4)
433(n1 · · · n4, m1m2m3)
253(n1n2, m1 · · · m5)
164(n1, m1 · · · m6)
073(, 3 · · · 9)
0(1, 6)(, n1 · · · n6, m1)
4
P
Mtheory
P(M)
P(n1)
M2 = 23
M5 = 56
KKM = 619
(MP(n) = Rn−1)
M2(nM)
M2(n1n2)
M5(n1 · · · n4M)
M5(n1 · · · n5)
512
215
3 53(n1 · · · n5, m1m2M)
53(n1 · · · n4M, m1m2m3)
6 26(n1n2, m1 · · · m5M)
26(n1M, m1 · · · m6)
0(118, 7) 0(1, 7)(, 3 · · · 9, M)
0(1, 7)(, n1 · · · n6M, m1)
KKM(n1 · · · n6, M)
KKM(n1 · · · n5M, n6)
KKM(n1 · · · n6, n7)
We will still call it a pointlike 522(34567, 89)brane as its mass becomes that of the usual
522(34567, 89)brane after further compactifying the x3direction.
A list of defect branes in the type IIA theory compactified on a seventorus is collected
in table 3. As shown in the table, each defect brane of the type IIA theory can be regarded
as a reduction of a defect brane of the Mtheory compactified on an eighttorus. By using
the relation
ls = RM−1/2 l131/2
and
gs = RM3/2 l1−13/2 ,
where RM is the radius in the Mtheory direction and l11 is the Planck length in eleven
dimensions, a b(nd, c)brane in the type IIA theory can be identified with a defect b(˜d, c)brane
n
in the Mtheory with the mass,
Mb(nd, c) =
˜
l11
˜
n
Rn1 · · · Rnb
RMb
˜
n
l11
2 Rℓ1 · · · Rℓd
3
,
2
b + 2c + 3d − n + 1
2
RMd
3
(D.4)
(D.5)
(D.6)
HJEP07(21)5
Here, the indices ni, mi, ℓi run over 3, . . . , 9, M, where M represents the Mtheory direction.
Note that the subscript n˜ of b(n˜d, c) is usually suppressed.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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