World-volume Effective Action of Exotic Five-brane in M-theory

0.24 MB·

by  Tetsuji Kimura

TIT/HEP-650
World-volume Effective Action of
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1 Exotic Five-brane in M-theory
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Tetsuji Kimuraa,b, Shin Sasakic and Masaya Yatad
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a Research and Education Center for Natural Sciences, Keio University
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Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, JAPAN
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p tetsuji.kimura at keio.jp
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h b Department of Physics, Tokyo Institute of Technology
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Tokyo 152-8551, JAPAN

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c Department of Physics, Kitasato University
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8 Sagamihara 252-0373, JAPAN
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shin-s at kitasato-u.ac.jp
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. d Department of Physics, National University of Singapore
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0 2, Science Drive 3, Singapore 117542, Singapore
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phymasa at nus.edu.sg
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a Abstract
We study the world-volume effective action of an exotic five-brane, known as the M-theory
53-brane (M53-brane) in eleven dimensions. The supermultiplet of the world-volume theory
is the = (2,0) tensor multiplet in six dimensions. The world-volume action contains three
N
Killing vectors kˆM (Iˆ = 1,2,3) associated with the U(1)3 isometry. We find the effective
Iˆ
T-duality rule for the eleven-dimensional backgrounds that transforms the M5-brane effective
action to that of the M53-brane. We also show that our action provides the source term for
the M53-brane geometry in eleven-dimensional supergravity.
1 Introduction
It is known that there are various kinds of extended objects in string theories. Among other things,
D-branes, the most familiar extended objects in superstring theories, have been studied in various
contexts. There are other extended objects such as Kaluza-Klein monopoles (KK5-branes) and
NS5-branes. They are BPS solutions in supergravity theories whose background geometries are
governed by single valued functions of space-time.
A remarkable fact about these branes are that they are related by U-duality in string theories.
U-duality isakeypropertytounderstandthenon-perturbativedefinitionofstringtheories, namely,
M-theory. BPS states in d-dimensional supermultiplets obtained from the M-theory compactified
onthe(11 d)-dimensional torus T11−d areinterpretedas BPSbraneswrappingcycles inthetorus.
−
TheBPSbranesconsistofD-branes,NS5-branes,KK5-branesandwaves. Theyarestandardbranes
which have been studied extensively. In addition to these branes, in d 8 dimensions, there are
≤
other extended objects known as exotic branes [1, 2]. The background geometries of the exotic
branes are described by multi-valued functions of space-time and exhibit non-trivial monodromies
given by the U-duality group. This kind of geometry is known as the non-geometric background
and called the U-fold [3].
We have studied exotic five-branes in string theories. In particular, we have studied the exotic
52-brane1 from the viewpoint of the worldsheet theories [6, 7, 8, 9, 10], the background geometries
2
as supergravity solutions [11, 12], the supersymmetry conditions [13] and the world-volume theory
[14]. Remarkably, the world-volume theory is the most direct approach to study the dynamics
of extended objects. Applying the T-duality transformation to the NS5-branes twice, we obtain
the world-volume effective action of the 52-brane in type IIB string theory [15] whose dynamics
2
is governed by the = (1,1) vector multiplet in six dimensions. On the other hand, the type
N
IIA 52-brane contains the 2-form field in the = (2,0) tensor multiplet. We derived the effective
2 N
action of the type IIA 52-brane [14] from the type IIA NS5-brane whose M-theory origin is the
2
M5-brane in eleven-dimensions [16].
We knowthattheBPSextendedobjects intypeIIAstringtheoryhave theireleven-dimensional
M-theory origin. TheD0-brane is obtained from the M-wave. The F-string and the D2-brane come
from the M2-brane while the NS5-brane and the D4-brane come from the M5-brane. The type
IIA KK5-brane and the D6-brane are obtained by the double/direct dimensional reductions of
the KK6-brane, respectively. The D8-brane and also the KK8-brane is expected to come from
a nine-dimensional object in eleven dimensions [17, 18]. It is also important to study M-theory
originsofexoticbranesinstringtheories. Indeed,thereisanexoticfive-braneknownasthe53-brane
[19, 4,5]in eleven-dimensional M-theory. We call this theM53-branein this paper. TheM53-brane
has isometries along three transverse directions to its world-volume. The tension of the M53-brane
is proportional to the volume of the torus T3 realized in the isometry directions. This is just a
generalized KK monopole in eleven dimensions [20]. The supergravity solution associated with
the M53-brane exhibits non-geometric nature whose monodromy is given by the U-duality group
1The522-branehastwoisometriesalongtransversedirectionsanditstensionisproportionaltogs−2. Heregs isthe
string coupling constant. See [2, 4, 5] for the notation of 52.
2
2
SL(3,Z) SL(2,Z) in eight dimensions. The direct/double dimensional reductions of the M53-
×
brane to ten dimensions provide the exotic 52-brane/43-brane in type IIA string theory. Therefore
2 3
the M53-brane is the higher dimensional origin of the type IIA exotic branes. In this paper we
study the world-volume effective action of the exotic M53-brane in eleven dimensions.
The organization of this paper is as follows. In the next section, we introduce the world-volume
effective action of the exotic 52-brane in type IIA string theory. In section 3, we propose the world-
2
volume effective action of the M53-brane. We will show that the action correctly reproduces the
action of the type IIA 52-brane in ten dimensions. In section 4, we will show that the actions of the
2
52-brane and M53-brane give the source terms of these five-brane geometries. Section 5 is devoted
2
to the conclusion and discussions.
2 Effective action of type IIA 52-brane
2
In this section we introduce the world-volume effective action of the exotic 52-brane in type IIA
2
string theory. The background geometry of the 52-brane is a 1/2-BPS solution to the type IIA
2
supergravity and the world-volume theory has 16 conserved supercharges. The geometry of the
52-brane exhibits the monodromy given by the SO(2,2) T-duality group and this is a T-fold.
2
The world-volume action of the 52-brane is obtained from that of the type IIA NS5-brane by
2
repeated T-duality transformations. Thetype IIANS5-brane is obtained by the direct dimensional
reduction of the M5-brane in eleven dimensions. We start from theeffective action of theM5-brane
which has been established in [16]. The supermultiplet of the six-dimensional world-volume theory
on the M5-brane is the = (2,0) tensor multiplet. This consists of a 2-form field A , five scalar
ab
N
fields XMˆ (Mˆ = 1,...,5) and their superpartners. The field strength F = ∂ A ∂ A +
abc a bc b ac
−
∂ A (a,b,c = 0,...,5) satisfies the self-duality condition in six dimensions. The effective action
c ab
of the M5-brane2, the so-called Pasti-Sorokin-Tonin (PST) action, is
√ gˆ
S = T d6ξ det(P[gˆ] +iHˆ∗ )+ − (∂ a∂aa)−1Hˆ∗abcHˆ ∂ a∂da +S , (2.1)
M5 − M5 − ab ab 4 a bcd a WZ
Z (cid:20)q (cid:21)
where T = 1 M6 is the tension of the M5-brane and M is the Planck mass in eleven
M5 (2π)5 11 11
dimensions. The symbol P stands for the pull-back of the background fields:
P[gˆ] = gˆ ∂ XM∂ XN, P[Cˆ(3)] = Cˆ(3) ∂ XM∂ XN∂ XP, (2.2)
ab MN a b abc MNP a b c
where gˆ (M,N = 0,1,...,10) and Cˆ(3) are the metric and the 3-form in eleven-dimensional
MN
supergravity. XM and ξa (a = 0,...,5) are the space-time and world-volume coordinates, respec-
tively. gˆ = P[gˆ] is the induced metric on the M5-brane world-volume and gˆ is the determinant
ab ab
of gˆ . The world-volume indices a,b,c,... = 0,...,5 are raised and lowered by the induced metric
ab
gˆ and its inverse, gˆab. We note that the action (2.1) contains a non-dynamical auxiliary field a
ab
which is needed to write down the action for the self-dual field in a Lorentz covariant way. The
world-volume 2-form field A enters into the action (2.1) with the following combinations:
ab
Hˆ = F P[Cˆ(3)] ,
abc abc abc
−
2We always consider theactions for thebosonic fields in this paper.
3
1 1
Hˆ∗abc = εabcdefHˆ ,
def
3!√ gˆ
−
1 1 1 1
Hˆ∗ = Hˆ∗ ∂ca = ε cdefHˆ ∂ a. (2.3)
ab ∂ a∂ga abc 3!√ gˆ ∂ a∂ga ab def c
g g
−
Here ε is the Levi-Cpivita symbol. The five scalpars XMˆ, which appear in the pull-back of
abcdef
the background fields in the static gauge, represent the transverse fluctuation modes (named the
geometric zero-modes) of the M5-brane in eleven dimensions. The M5-brane action (2.1) has
symmetries of the world-volume U(1) gauge transformations, two field dependent world-volume
gauge transformations and the space-time gauge transformations of the background fields [16, 21,
22]. In the following, we do not consider the Wess-Zumino part S .
WZ
WenowperformthedirectdimensionalreductionoftheM5-braneaction(2.1)totendimensions.
We decompose the eleven-dimensional index M = (µ,♯) where µ = 0,...,9 and ♯ = 10. The former
represents the ten-dimensional space-time, and the latter is the compactified M-circle direction.
The KK ansatz of the eleven-dimensional metric gˆ and the 3-form Cˆ(3) is
MN MNP
gˆ = e−32φ(gµν +e2φCµ(1)Cν(1)) e43φCµ(1) , Cˆ(3) = C(3) , Cˆ(3) = B . (2.4)
MN  e34φCν(1) e34φ  µνρ µνρ µν♯ − µν
 
(1) (3)
Here g is the ten-dimensional metric, φ is the dilaton, C ,C are the R-R 1- and 3-forms.
µν µ µνρ
Applying this dimensional reduction to (2.1), we obtain the action of the type IIA NS5-brane
[23, 14]. The NS5-brane action has symmetries that are inherited from those of the M5-brane. The
world-volume theory of theNS5-braneconsists of thefourscalar fields XI′ (I′ = 1,...,4) which are
the geometric zero-modes, an extra scalar field Y which originates from the geometric zero-mode
along the M-circle direction, a 2-form field whose field strength satisfies the self-duality condition
in six dimensions and an auxiliary field. They comprises the six-dimensional = (2,0) tensor
N
multiplet as expected.
Now we introduce the Killing vector kµ associated with the U(1) isometry of the background
1
fields g ,B ,φ, C(1),C(3). This U(1) isometry is defined along the transverse direction to the
µν µν
NS5-brane world-volume. Applyingthe T-duality transformation to the type IIANS5-brane action
along this direction, we obtain the effective action of the type IIB KK5-brane [24]. In addition to
the couplings to the background fields in type IIB supergravity, the effective action of the KK5-
brane contains couplings to the Killing vector kµ. Geometrically, this corresponds to the isometry
1
in the Taub-NUT space and there is no geometric zero-mode associated with this direction. The
world-volume theory has three geometric zero-modes together with a scalar mode Y. Since the
world-volume theory of the type IIB KK5-brane is governed by the = (2,0) tensor multiplet,
N
one needs one extra scalar field ϕ . This naturally appears in the T-duality transformation as
1
the dual coordinate of the isometry direction. This is nothing but the Lagrange multiplier in the
dualized process in the string world-sheet theory (see Appendix in [14]). We call ϕ the winding
1
zero-mode.
We next introduce the second Killing vector kµ associated with the U(1) isometry along an-
2
other transverse direction to the KK5-brane. By further T-duality transformation, the KK5-brane
4
becomes the type IIA exotic 52-brane. The effective action of the type IIA 52-brane was derived in
2 2
[14]. This is given by
SIIA = T d6ξ (deth[2])e−2φ det(gˇ +λ2(deth[2])−1e2φFˇ(1)Fˇ(1))
522 − 5 − ab a b
Z q
ieφ
det δ b+ Z b
×s a 3! ˇ (deth[2])(gˇcd∂ca∂da) a
(cid:16) N (cid:17)
λ2 1 p 1 λ2e2φgˇgrgˇhsFˇ(1)Fˇ(1)
T d6ξ εabcdefHˇ Hˇ ∂ a∂ a gˇgh r s ,
− 4 5Z 3!Nˇ2gˇpq∂pa∂qa def abg c h " − (deth[2])+λ2e2φgˇuvFˇu(1)Fˇv(1)#
(2.5)
where λ = 2πα′ is the string slope parameter, and T = 1 g−2α′−3 is the tension of the NS5-
5 (2π)5 s
brane. Here gˇ is the ten-dimensional metric on which the T-duality transformations with the two
µν
Killing vectors kµ,kµ have been applied. The explicit form of gˇ is
1 2 µν
1
gˇ = g h[2]IJ i g (i B λdϕ ) i g+(i B λdϕ ) +(µ ν) , (2.6)
µν µν − 2 { kI − kI − I }µ{ kJ kJ − J }ν ↔
h i
where we have defined the “Killing matrix” h[2] = (g +B )kµkν and its inverse h[2]IJ (where
IJ µν µν I J
I,J = 1,2). The inner product of an n-form X(n) with a Killing vector kµ is defined by
(ikX(n))µ1···µn−1 = kνXν(nµ)1···µn−1. (2.7)
The effective induced metric is defined by gˇ = P[gˇ] , and gˇab is the inverse of gˇ . Note that
ab ab ab
in the pull-back, we define P[dϕ ] = ∂ ϕ . In the action (2.5), we have defined the following
I a a I
quantities:
εgbcdef(gˇ +λ2(deth[2])−1e2φFˇ(1)Fˇ(1))Hˇ ∂ a
Z b = ac a c def g ,
a
det(gˇ +λ2(deth[2])−1e2φFˇ(1)Fˇ(1))
pq p q
−
Fˇ(1) = ∂ Y +λ−1P[Cˇ(1)] ,
a a a
λ2e2φ(gˇefFˇ(1)∂ a)2
ˇ2 = 1 e f ,
N − (gˇab∂ a∂ a)((deth[2])+λ2e2φgˇcdFˇ(1)Fˇ(1))
a b c d
Hˇ = F P[Cˇ(3)] λ(P[Bˇ] Fˇ(1)) . (2.8)
abc abc abc abc
− − ∧
HereBˇ,Cˇ(1),Cˇ(3) aretheNS-NSB-field,R-R1-and3-formsonwhichtheT-dualitytransformations
have been applied. Their explicit forms are
1
Bˇ = B h[2]IJ i g (i B λdϕ ) i g+(i B λdϕ ) ,
− 2 { kI − kI − I }∧{ kJ kJ − J }
Cˇ(1) = i i C(3) (i i B)C(1)+ǫIJ(i C(1))(i B λdϕ ), (2.9)
− k1 k2 − k1 k2 kI kJ − J
Cˇ(3) = i i C(5) (i i B)C(3)+ǫIJ(i C(3)) (i B λdϕ )
− k1 k2 − k1 k2 kI ∧ kJ − J
1
ǫIJC(1) (i B λdϕ ) (i B λdϕ )
− 2 ∧ kI − I ∧ kJ − J
1
i i C(3)+(i i B)C(1) ǫKL(i C(1))(i B λdϕ )
− 2 k1 k2 k1 k2 − kK kL − L
h i
5
h[2]IJ i g (i B λdϕ ) i g+(i B λdϕ ) . (2.10)
∧ { kI − kI − I }∧{ kJ kJ − J }
Here we introduced an antisymmetric symbol ǫIJ = ǫJI. Since the 52-brane is a defect brane
− 2
of co-dimension two [25], there are only two geometric zero-modes in the world-volume action.
Indeed, the effective theory contains two scalar fields X1,X2 in the pull-back of the type IIA
supergravity backgrounds [14]. There are also a scalar field Y which comes from the M-circle
and a 2-form field A . In addition, there are two winding zero-modes ϕ ,ϕ introduced in the
ab 1 2
T-duality process. Since the factor deth[2] contains the volume of the 2-torus T2, the tension of
the 52-brane is proportional to g−2(vol(T2)) which is consistent with the definition of the 52-brane.
2 s 2
In our previous paper [14], we have discussed the gauge symmetries of the action (2.5) and its
Wess-Zumino coupling.
The action (2.5) should be obtained from the direct dimensional reduction of the action of the
M53-brane. In the next section, we work out the M53-brane action which is consistent with the
action (2.5).
3 Effective action of M53-brane
In this section, we study the M53-brane effective action in the eleven-dimensional supergravity
background. We concentrate on the purely geometric background where the 3-form field Cˆ(3) is
absent. The effective theory has the following properties. (i) The world-volume theory is governed
by the = (2,0) tensor multiplet in six dimensions. The bosonic fields are a 2-form A whose
ab
N
field strength satisfies the self-duality condition, two scalar fields X1,X2 that correspond to the
geometric zero-modes, three scalar fields ϕˆIˆ (Iˆ = 1,2,3) that correspond to the dual winding
coordinates and an auxiliary field a. (ii) The background is the eleven-dimensional metric gˆ
MN
with the U(1)3 isometry. (iii) The M53-brane couples to three Killing vectors kˆM associated with
Iˆ
the U(1)3 isometry. The effective tension of the M53-brane is proportional to the volume of T3
wheretheU(1)3 isometry is realized. (iv) After thedirect dimensionalreduction to ten dimensions,
the action should reproduce the 52-brane action where the NS-NS B-field and the R-R 3-form C(3)
2
are turned off.
In the following, we explore the structure of the M53-brane action in each world-volume field
sector. In particular, we verify that the proposed structureis consistent with the type IIA52-brane
2
action.
Geometric zero-mode sector
We begin with the following terms in the first line of the type IIA 52-brane effective action (2.5):
2
gˇ +λ2(deth[2])−1e2φFˇ(1)Fˇ(1). (3.1)
ab a b
When C(3) = B = 0, (3.1) becomes
λ2
Π[2](k ,k )∂ Xµ∂ Xν + g (kµ∂ ϕ kµ∂ ϕ )(kν∂ ϕ kν∂ ϕ )
µν 1 2 a b deth[2] µν 1 a 2 − 2 a 1 1 b 2− 2 b 1
6
λ2e2φ
+ ∂ Y∂ Y (i C(1))∂ ϕ (i C(1))∂ ϕ ∂ Y (i C(1))∂ ϕ (i C(1))∂ ϕ ∂ Y
deth[2] a b − k1 a 2− k2 a 1 b − k1 b 2 − k2 b 1 a
h n o n o
+ (i C(1))∂ ϕ (i C(1))∂ ϕ (i C(1))∂ ϕ (i C(1))∂ ϕ , (3.2)
k1 a 2− k2 a 1 k1 b 2− k2 b 1
n on o i
where Π[2](k ,k ) is the projector constructed by kµ,kµ:
µν 1 2 1 2
Π[2](k ,k ) = g h[2]IJ kρkσg g . (3.3)
µν 1 2 µν − I J µρ νσ
Here h[2]IJ is the inverse of h[2] =kµkνg . We first look for the M53-brane origin of the first term
IJ I J µν
in (3.2), i.e., the geometric zero-mode sector. Since the M53-brane is a defect brane and has the
U(1)3 isometry in the transverse directions, there are no geometric zero-modes in these directions.
Only two geometric zero-modes which correspond to the two transverse directions appear in the
pull-back. We expect that the two scalar fields associated with the geometric zero-modes are
described by the gauged sigma model [20]. We start from the Nambu-Goto action as the building
block of the M53-brane world-volume action:
S = T d6ξ det(gˆ ∂ XM∂ XN). (3.4)
M5 MN a b
− −
Z q
In order to obtain the gauged sigma model, we introduce an auxiliary world-volume gauge fields
CIˆ (Iˆ= 1,2,3) and gauge covariantize the pull-back in the action (3.4):
a
D XM = ∂ XM +CIˆkˆM. (3.5)
a a a Iˆ
With this covariantization, theisometry of thebackground metric gˆ becomes a gauge symmetry
MN
in the world-volume. After integrating out the auxiliary gauge field CIˆ, the metric gˆ in (3.4)
a MN
becomes
Πˆ[3] (k) = gˆ ˆh[3]IˆJˆkˆPkˆQgˆ gˆ . (3.6)
MN MN − Iˆ Jˆ PM QN
Thisisaprojectorwhichprojectsoutthethreegeometriczero-modesassociatedwiththeisometries.
Here we have defined the 3 3 Killing matrix hˆ[3] and its inverse hˆ[3]IˆJˆ in eleven dimensions:
× IˆJˆ
hˆ[3] = gˆ kˆMkˆN, hˆ[3]IˆJˆhˆ[3] = δIˆ . (3.7)
IˆJˆ MN Iˆ Jˆ JˆKˆ Kˆ
The action (??) contains two geometric zero-modes. As a generalization of the effective action of
the KK6-brane [20], the action of the M53-brane has the overall factor (dethˆ[3]) which guarantees
IˆJˆ
the correct tension. Indeed, this factor gives the volume of the torus T3 when an appropriate
coordinate system for kˆM is employed. Then the would-be action becomes
Iˆ
S = T d6ξ (dethˆ[3]) det Πˆ[3] (k)∂ XM∂ XN . (3.8)
− M5 − MN a b
Z r (cid:16) (cid:17)
In order to confirm plausibility of the action (3.8), we perform the direct dimensional reduction of
(3.8) to ten dimensions. We decompose the index M = (µ,♯) and define the M-circle direction as
X♯ = Y and the corresponding Killing vector kˆM = δM. The other Killing vectors kˆM become
3 ♯ 1,2
those in the ten-dimensional kµ :
1,2
kˆµ = kµ, kˆ♯ = 0, (I,J = 1,2). (3.9)
I I I
7
Then the dimensional reduction of the Killing matrix is given by
hˆ[3] = e−23φ(gµν +e2φCµ(1)Cν(1))kIµkJν e43φCµ(1)kIµ . (3.10)
IˆJˆ  e43φCµ(1)kJµ e43φ 
 
We can show that the dimensional reduction of the overall volume factor of T3 is evaluated as
dethˆ[3] = deth[2]. (3.11)
We note that the R-R 1-form C(1) and the dilaton dependence cancel out in the dimensional
reduction of dethˆ[3].
We next calculate the dimensional reduction of the projector Πˆ[3] . Using the fact that Πˆ[3]
MN MN
is the projector along X♯ = Y direction and the reduction rule for the inverse of the matrix (3.10),
we find
Πˆ[M3]N(k)∂aXM∂bXN = e−23φΠ[µ2ν](k1,k2)∂aXµ∂bXν. (3.12)
Again, except for the overall dilaton factor e−23φ, the R-R 1-form C(1) and the dilaton dependence
cancel out in the reduction. Then we find that the direct dimensional reduction of (3.8) is given by
T d6ξe−2φ(deth[2]) det Π[2](k ,k )∂ Xµ∂ Xν , (3.13)
5 µν 1 2 a b
− −
Z r (cid:16) (cid:17)
where we have used the relation T = T . The action (3.13) is nothing but the geometric zero-
5 M5
modes sector of the first line in (2.5) where the B-field and the R-R 3-form C(3) are turned off.
We note that the correct volume factor of the torus T2 and the dilaton factor e−2φ emerges in the
action.
Winding zero-modes sector
Wenextconsiderthewindingzero-modesector. SinceΠˆ[3] (k)∂ XM∂ XN containsonlytwoscalar
MN a b
fields, we need threeextra scalar fields ϕˆ (Iˆ= 1,2,3) for the = (2,0) tensor multiplet. They are
Iˆ N
interpreted as winding zero-modes of membranes wrapped on the three torus T3 [26, 27]. In order
to see how these scalar fields appear in the M53-brane action, we examine the terms that contain
two windingzero-modes ϕ ,ϕ and the M-circle scalar Y in (3.2). Sincethe ten-dimensional metric
1 2
g and the R-R 1-form C(1) come from the eleven-dimensional metric gˆ , we expect that the
µν MN
higher dimensional origin of the terms that contain (ϕ ,ϕ ,Y) in (3.2) is given by
1 2
1
I = gˆ (kˆM∂ ϕˆIˆ)(kˆN∂ ϕˆJˆ). (3.14)
ab dethˆ[3] MN Iˆ a Jˆ b
In order to confirm this proposal, we perform the direct dimensional reduction of the term (3.14).
Using the KK ansatz (2.4) of the metric, we find
λ2
Iab = deth[2]e−32φgµν(k1µ∂aϕ2−k2µ∂aϕ1)(k1ν∂bϕ2−k2ν∂bϕ1)
8
λ2
+ deth[2]e34φ (ik1C(1))∂aϕ2−(ik2C(1))∂aϕ1 (ik1C(1))∂bϕ2−(ik2C(1))∂bϕ1
λ2 n on o
− deth[2]e34φ (ik1C(1))∂aϕ2−(ik2C(1))∂aϕ1 ∂bY
λ2 n o
− deth[2]e34φ (ik1C(1))∂bϕ2 −(ik2C(1))∂bϕ1 ∂aY
λ2 n o
+ deth[2]e34φ∂aY∂bY, (3.15)
where we have identified the three scalars and the winding zero-modes as ϕˆIˆ = λ(ϕ , ϕ , Y).
2 1
− −
This precisely reproduces the (ϕ ,ϕ ,Y) terms in (3.2) except for the overall dilaton factor. In the
1 2
determinant of the 6 6 matrix Iab, we extract the dilaton factor e−23φ and leave it outside of the
×
square root. Then the overall factor of the action becomes e−2φ which gives the correct tension of
the 52-brane. Therefore we have confirmed that the (ϕ ,ϕ ,Y) terms in (3.2) emerge from (3.14).
2 1 2
Then the geometric and the winding zero-mode sectors of the M53-brane action are given by
1
S = T d6ξ (dethˆ[3]) det Πˆ[3] (k)∂ XM∂ XN + gˆ (kˆM∂ ϕˆIˆ)(kˆN∂ ϕˆJˆ) .
− M5 − MN a b dethˆ[3] MN Iˆ a Jˆ b
Z r (cid:16) (cid:17)
(3.16)
Gauge and auxiliary fields sector
Next, westudythegaugefieldsector intheM53-braneaction. Thisis theeleven-dimensional origin
of the second line in (2.5). In the type IIA 52-brane action, the 2-form gauge field A contributes
2 ab
to the following terms
ieφ
δa + Z b, (3.17)
b 3! ˇ (deth[2])(gˇcd∂ a∂ a) a
c d
N
where Z b and ˇ are defined in (2.8). Wepfirst look for the higher dimensional origin of Z b. From
a a
N
the discussion in the geometric and the winding zero-mode sectors, we know that the following
quantity in eleven-dimensions
1
Gˆ = Πˆ[3] (k)∂ XM∂ XN + gˆ (kˆM∂ ϕˆIˆ)(kˆN∂ ϕˆJˆ) (3.18)
ab MN a b detˆh[3] MN Iˆ a Jˆ b
is dimensionally reduced to
Gˆab = e−23φ gˇab+λ2(deth[2])−1e2φFˇa(1)Fˇb(1) . (3.19)
(cid:16) (cid:17)
We therefore expect that the M53-brane counterpart of Z b is given by
a
εgbcdefGˆ ˆ ∂ a
Zˆ b = acHdef g . (3.20)
a
detGˆ
−
Here ˆ should be dimensionally reduced to pHˇ in the type IIA 52-brane action. We find this
Habc abc 2
is given by
ˆ = F +3! ∂ ϕˆ1∂ ϕˆ2+3! hˆ[3]1I ∂ ϕˆ2∂ ϕˆ3 3! hˆ[3]2I ∂ ϕˆ3∂ ϕˆ1, (3.21)
abc abc 3[a b c] I[a b c] I[a b c]
H G G − G
I=1,2 I=1,2
X X
9
where we have defined
hˆ[3]
= (i gˆ) I3(i gˆ) , (I = 1,2),
GIa kI a− kˆ2 k3 a
3
1
= ∂ ϕˆ3+ (i gˆ) . (3.22)
G3a a kˆ2 k3 a
3
Here kˆ2 = gˆ kˆMkˆN. Now, it is straightforward to confirm that Zˆ b is dimensionally reduced to
3 MN 3 3 a
Z b in the following way,
a
Zˆab = e34φZab. (3.23)
We next examine the factor ˇ (deth[2])(gˇcd∂ a∂ a) in (3.17). The matrix gˇab is the inverse of the
c d
N
following effective induced metric:
p
λ2
gˇ = Π[2](k ,k )∂ Xµ∂ Xν + g (kµ∂ ϕ kµ∂ ϕ )(kν∂ ϕ kν∂ ϕ ). (3.24)
ab µν 1 2 a b deth[2] µν 1 a 2− 2 a 1 1 b 2− 2 b 1
It is convenient to consider the following quantity rather than the induced metric itself:
λ2e2φ
gˇ + Fˇ(1)Fˇ(1). (3.25)
ab deth[2] a b
The inverse of the above matrix is evaluated as
λ2e2φFˇ(1)Fˇ(1)gˇacgˇbd
gˇab c d . (3.26)
− (deth[2])+λ2e2φFˇ(1)Fˇ(1)gˇef
e f
We have already observed that the eleven-dimensional origin of (3.25) is (3.18) except for the
dilaton factor. Therefore the eleven-dimensional origin of (3.26) is the inverse of Gˆ , namely, we
ab
seethat gˇab comes fromGˆab. With theseobservations, weexpectthat theeleven-dimensional origin
of gˇab∂ a∂ a and ˇ in (3.17) is given by Gˆab∂ a∂ a. Indeed, the dimensional reduction of this term
a b a b
N
gives
λ2e2φ(gˇcd∂ aFˇ(1))2
Gˆab∂aa∂ba = e23φ(gˇab∂aa∂ba)"1− (gˇef∂ a∂ a)(deth[2]+cλ2ed2φgˇghFˇ(1)Fˇ(1))#
e f g h
= e23φ(gˇab∂aa∂ba) ˇ2. (3.27)
N
Note that the dilaton factor e32φ which appears in (3.27), together with the factor e34φ in (3.23),
reproduces the correct factor eφ in (3.17). Then we conclude that the eleven-dimensional origin of
(3.17) is
i 1 εgbcdefGˆ ˆ ∂ a
δ c+ abHdef g . (3.28)
a
3! dethˆ[2] Gˆgh∂ a∂ a detGˆ
g h −
p q p
Finally, we look for the eleven-dimensional origin of the third line in (2.5). Using the induced
metric Gˆ , it is straightforward to show that the following term
ab
1 1 1
T d6ξ εabcjkl ˆ ˆ ∂ a∂ aGˆde (3.29)
−4 M5 3!Gˇgh∂ a∂ a HjklHbcd a e
Z g h
reproduces the third line in (2.5).
10