U(N) Gauged N=2 Supergravity and Partial Breaking of Local N=2 Supersymmetry

0.22 MB·English

by  Hiroshi Itoyama

March, 2006
OCU-PHYS 243
hep-th/0603180
U(N) Gauged = 2 Supergravity and
N
Partial Breaking of Local = 2 Supersymmetry
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H. Itoyamaa,b and K. Maruyoshia
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J

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0 a Department of Mathematics and Physics, Graduate School of Science
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1 Osaka City University
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b Osaka City University Advanced Mathematical Institute (OCAMI)
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3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
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r Abstract
a
We study a U(N) gauged = 2 supergravity model with one hypermultiplet
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parametrizing SO(4,1)/SO(4) quaternionic manifold. Local = 2 supersymmetry is
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knowntobespontaneouslybrokento = 1intheHiggsphaseofU(1) U(1).
graviphoton
N ×
Several properties are obtained of this model in the vacuum of unbroken SU(N) gauge
group. In particular, we derive mass spectrum analogous to the rigid counterpart and
put the entire resulting potential on this vacuum in the standard superpotential form
of = 1 supergravity.
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∗e-mail: [email protected]
†e-mail: [email protected]
I. Introduction
For more than a decade, = 2 supersymmetry both in its local and rigid realizations
N
has played an important role in the theoretical developments of quantum field theory and
particlephysics. Ithasledustothesubject ofexactlydeterminedlowenergyeffectiveactions
[1, 2] and has inspired the construction of Lagrangians based on special K¨ahler geometry
[3, 4, 5, 6, 7, 8, 9, 10]. These achievements have proven valuable in order to analyze some of
the phenomena which occur in string theory.
Spontaneous breaking of = 2 supersymmetry to = 1 is an interesting problem
N N
in the light of its implications of string theory to the low energy = 1 supersymmety,
N
which is phenomenologically promising. We give here a partial list of the references of this
subject on the linear realizations [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In
particular, spontaneous partial breaking of rigid = 2 supersymmetry in the U(N) gauge
N
model with or without hypermultiplets has been demonstrated recently [22, 23, 24, 25] under
a generic breaking pattern of the U(N) gauge symmetry. Several other properties of this
model have been obtained. It should be emphasized that the partial breaking of rigid = 2
N
supersymmetry is realized here in the Coulomb phase of overall U(1), the Nambu-Goldstone
fermion being the superpartner of the massless photon and that both interact with the
SU(N) sector thanks to the non-Lie algebraic property of the prepotential.
There are already considerable differences between the rigid special K¨ahler geometry and
its local counterpart and between the hyperk¨ahler geometry and the quaternionic geometry
as have been emphasized in the literatures [7, 8, 9]. This is bound to be reflected in the
comparison of the vacuum analysis of a rigid = 2 effective action with its supergravity
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counterpart. This will be a thrust of the present paper. Spontaneous partial breaking of
local = 2 supersymmetry has been studied in [11, 12, 13, 17, 18, 19, 20, 21]. It was noted
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from the beginning that both the Higgs and the super-Higgs mechanisms must take place
simultaneously and that the vacuum must lie in the Higgs phase of U(1) U(1).
graviphoton
×
The tight structure of the spectrum produced by the mechanisms requires at least one
hypermultiplet with two U(1) translational isometries to be introduced in the models.
In this paper, we study some of the basic and yet unexplored properties of the U(N)
gauged = 2 supergravity model in which local = 2 supersymmetry is partially broken
N N
spontaneously. In particular, the holomorphic section of our model is chosen as a generic
function which leads to the nontrivial scalar coupling terms and the scalar potential. (In
reference [12], a simple form of the section has been adopted.) In the next section, we
briefly review U(N) gauged = 2 supergravity in four dimensions and consider the model
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which contains a U(N) vector multiplet and a hypermultiplet parametrizing SO(4,1)/SO(4)
2
quaternionic manifold. Because of the choice of the section we need careful consideration of
the vacuum, which is done in section three. We consider and solve the vacuum conditions of
the model under the assumption of unbroken SU(N) gauge symmetry. The second vacuum
condition, which is a variation of the potential with respect to the hypermultiplet scalar bu,
was not considered before and the super-Higgs mechanism can not operate without this one.
Partial breaking of local = 2 supersymmetry is exhibited. In section four, we derive the
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mass spectrum of the model and interpret it in terms of = 1 on-shell supermultiplets. In
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section five, we construct the entire Lagrangian on this vacuum and express it in terms of
two superpotentials which are related to each other by a simple relation (5.9). The resulting
form conforms to the standard form of = 1 supergravity.
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II. U(N) Gauged = 2 Supergravity
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The field contents of U(N) Gauged = 2 Supergravity are summarized as follows:
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supergravity multiplet
•
consisting of the vierbein ei (i,µ = 0,1,2,3), two gravitini ψA (A = 1,2) and the
µ µ
graviphoton A0. (The upper and the lower position of the index A represent left and
µ
right chirality respectively.)
vector multiplet
•
consisting of a gauge boson Aa, two gaugini λaA and a complex scalar za. The index a
µ
(a = 1,...,N2) labels the generators of the U(N) gauge group and a = N2 n refers
≡
to the overall U(1). (The notation on the chirality is opposite to that of the gravitini,
namely, the upper and the lower position denote right and left chirality respectively.)
hypermultiplet
•
consisting of two hyperini ζα (α = 1,2) and four real scalars bu (u = 0,1,2,3). (The
upper and the lower position of the index α represent left and right chirality respec-
tively.)
General construction of the Lagrangian of gauged = 2 supergravity has been given in
N
[3, 5, 7]. We exhibit the parts of the Lagrangian and the supersymmetric transformation
laws which are necessary for our analysis of the vacuum.
3
A. Vector Multiplet
The manifold associated with the vector multiplet is special K¨ahler of the local type [3, 4,
5, 6, 7, 8, 9, 10]. It is equipped with a holomorphic section,
XΛ(z)
Ω(z) = , Λ = 0,1,...,n (2.1)
FΛ(z) !
The index 0 refers to the graviphoton part. In terms of this section, the K¨ahler potential is
given by,
= logi Ω Ω¯ = logi(X¯ΛF XΛF¯ ), (2.2)
Λ Λ
K − | − −
where (cid:10) (cid:11)
I
0
i Ω Ω¯ iΩT Ω . (2.3)
| ≡ − I 0 ! ∗
−
(cid:10) (cid:11)
The non-holomorphic section is introduced by
LΛ XΛ
V = e /2Ω = e /2 , (2.4)
K K
MΛ ! ≡ FΛ !
and its covariant derivative is
1 fΛ
U V = (∂ + ∂ )V a . (2.5)
a a a a
≡ ∇ 2 K ≡ hΛa !
|
One characteristic property of = 2 supergravity, which follows from the special K¨ahler
N
geometry of the local type, is the existence of totally symmetric rank-three tensor C such
abc
that
aUb = iCabcgcd∗U¯d∗. (2.6)
∇
(See, for example, [7, 8]). The generalized gauge coupling matrix ¯ is introduced via the
ΛΣ
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following relations:
M¯ = ¯ L¯Σ, h = ¯ fΣ. (2.7)
Λ NΛΣ Λ|a NΛΣ a
The solution is given in terms of two (n+1) (n+1) matrices
×
fΛ h
fIΛ = L¯aΛ !, hΛ|I = M¯ΛΛ|a ! (2.8)
as
¯ = h (f 1)I. (2.9)
NΛΣ Λ|I − Σ
It is well-known that this quantity appears in the kinetic term of the gauge bosons.
4
To specify the model, we need to choose the holomorphic section. Our choice, which is
essentially that of [13], is
1 1 ∂ (z)
X0(z) = , F (z) = 2 (z) za F ,
0
√2 √2 F − ∂za
(cid:18) (cid:19)
1 1 ∂ (z)
Xaˆ(z) = zaˆ, F (z) = F , (2.10)
√2 aˆ √2 ∂zaˆ
1 ∂ (z) 1
Xn(z) = F , F (z) = zn,
√2 ∂zn n −√2
where the index aˆ = 1,...,n 1, labels the generators of SU(N) subgroup. It has been
−
obtained from the derivatives of the holomorphic function F(X0,Xa) = (X0)2 (Xa/X0),
F
that is, ∂F/∂XΛ and performing the symplectic transformation Xn F ,F Xn. The
n n
→ − →
K¨ahler potential and its derivatives are given by
= log , (2.11)
0
K − K
i
∂ = ¯ (zc z¯c) , (2.12)
a 0 a a ac
K 2 F −F − − F
gab∗ = ∂a(cid:0)∂b∗ (cid:1)
K
i
= ∂a ∂b∗ ( ab ¯ab), (2.13)
K K− 2 F −F
0
K
where = ∂ /∂za and
a
F F
1
= i ¯ (za z¯a)( + ¯ ) . (2.14)
0 a a
K F −F − 2 − F F
(cid:18) (cid:19)
Furthermore, the covariant derivative of fΛ is
a
ie /2
∇afb0 ≡ √K2 Cabcgcd∗∂d∗K
1
∂ f0 +Γc f0 + ∂ f0
≡ a b ab c 2 aK b
e /2 1
= K ∂a∂b ∂a ∂b + gcd∗(∂a∂b 0∂d∗ +∂a∂b∂d∗ 0)∂c , (2.15)
√2 K− K K K K K K
(cid:18) K0 (cid:19)
ie /2
∇afbn ≡ √K2 Cabcgcd∗(F¯nd +∂d∗KF¯n)
e /2
K
= ( ∂ ∂ +∂ ∂ )
nab a b n a b n
√2 F − K KF KF
e /2
+ K gcd∗(∂a∂b 0∂d∗ +∂a∂b∂d∗ 0)( nc +∂c n). (2.16)
√2 K K K F KF
0
K
5
The Christoffel symbol is defined as Γcab = −gcd∗∂bgad∗. These equations will be used in the
analysis of the potential term.
In order gauge the vector multiplet, first introduce the Killing vectors which are defined
by
kac∂c = facbzb∂c, kac∗∂¯c∗ = facbz¯b∗∂¯c∗, (2.17)
where fa is the structure constant of the U(N) gauge group satisfying
bc
[t ,t ] = ifc t . (2.18)
a b ab c
We will deal with the case in which the Lie derivative satisfies
Λ
L
0 = LΛK = kΛb∂bK+kΛb∗∂b∗K. (2.19)
The covariant derivative of the scalar fields, for example, takes the standard form:
za = ∂ za +AΛka
∇µ µ µ Λ
= ∂ za +faAbzc. (2.20)
µ bc µ
B. Hypermultiplet
Four real scalar components bu of the hypermultiplet span the quaternionic manifold which
is taken to be SO(4,1)/SO(4). The quaternionic geometry is in general determined by a
triplet of quaternionic potentials,
Ωx = Ωx dbu dbv (2.21)
uv ∧
1
= dωx + ǫxyzωy ωz, x = 1,2,3,
2 ∧
where ωx = ωxdbu are the SU(2) connections. In this paper, we take the same parametriza-
u
tions as that of [12, 13]. The above quantities read
1 1 1
ωx = δx, Ωx = δx, Ωx = ǫxyz. (2.22)
u b0 u 0u −2(b0)2 u yz 2(b0)2
The metric h of this manifold is
uv
1
h = δ , (2.23)
uv 2(b0)2 uv
while the symplectic vielbein is
1
αAdbu,(α,A = 1,2), αA = ǫαβ(db0 iσxdbx) A, (2.24)
Uu U 2b0 − β
6
where σx are the standard Pauli matrices.
Let us introduce the Killing vectors ku and the momentum maps x associated with two
Λ PΛ
U(1) translational isometries of this quaternionic manifold [13]:
ku = g δu3 +g δu2, ku = 0, ku = g δu2,
0 1 2 aˆ n 3
1 1
x = (g δx3 +g δx2), x = 0, x = g δx2. (2.25)
P0 b0 1 2 Paˆ Pn b0 3
Here g ,g ,g , R are coupling constants. These constants play the same role as the super-
1 2 3
∈
potential and the Fayet-Iliopoulos term do in the rigid theory [14, 22, 24, 25].
C. The Lagrangian of = 2 Supergravity
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Let us write the parts of the Lagrangian of the = 2 gauged supergravity which is needed
N
in our analysis:
= √ g( + V(z,z¯,b)+...), (2.26)
kin mass
L − L L −
where
ǫµνλσ
Lkin = R+gab∗∇µza∇µz¯b∗ +huv∇µbu∇µbv + √ g(ψ¯µAγν∇λψAσ −ψ¯Aµγν∇λψσA)
−
1 1
+ (Im ) FΛFΣµν + (Re ) FΛFΣµν
4 N ΛΣ µν 4 N ΛΣ µν
−igab∗λ¯aAγµ∇µλbA∗ −2iζ¯αγµ∇µζα +...,e (2.27)
LYukawa = 2SABψ¯µAγµνψνB +igab∗WaABλ¯bA∗γµψBµ +2iNαAζ¯αγµψAµ
+ αβζ¯ ζ + α ζ¯ λaB + λ¯aλbB +h.c., (2.28)
M α β MaB α MaA|bB A
V(z,z¯,b) = gab∗kΛakΣb∗L¯ΛLΣ +gab∗faΛf¯bΣ∗PΛxPΣx +4 huvkΛukΣvL¯ΛLΣ −3 L¯ΛLΣPΛxPΣx. (2.29)
Here FΛ are the field strengths of the U(N) gauge fields and that of the graviphoton field,
µν
and F˜Λ are their Hodge duals. The supersymmetry transformation laws of the fermions are
µν
δψ = iS γ ǫB +..., (2.30)
Aµ AB µ
δλaA = WaABǫ +..., (2.31)
B
δζ = NAǫ +.... (2.32)
α α A
7
The matrices appearing in the supersymmetry transformation laws and in eq. (2.28) are
composed of the geometric quantities listed in the last two subsections:
i
S = (σ ) xLΛ, (2.33)
AB 2 x ABPΛ
WaAB = ǫABkaL¯Λ +i(σ )AB xgab∗f¯Λ WaAB +WaAB (2.34)
Λ x PΛ b∗ ≡ 1 2
NA = 2 A kuL¯Λ, (2.35)
α Uαu Λ
αβ = Aα Bβǫ [ukv]LΛ, (2.36)
M −Uu Uv AB∇ Λ
α = 4 α kufΛ, (2.37)
MbB − UBu Λ b
1
MaA|bB = 2 ǫABgac∗kΛc∗fbΛ +i(σx)ABPΛx∇bfaΛ (2.38)
(cid:0)1 + 2 . (cid:1) (2.39)
≡ MaA|bB MaA|bB
We obtain explicit forms of these matrices from (2.11)-(2.13), (2.17) and (2.22)-(2.25):
ie /2 i(g +g ) g
K 2 3 n 1
S = F , (2.40)
AB
−2√2b0 g1 i(g2 +g3 n) !
F
0 1
WaAB = ie /2 a , (2.41)
1 − K D 1 0 !
−
WaAB = eK/2 gab∗ g2∂b∗K+g3(F¯nb+∂b∗KF¯n) ig1∂b∗K , (2.42)
2 √2b0 ig1∂b∗ g2∂b∗ +g3(¯nb+∂b∗ ¯n) !
K K F KF
ie /2 g i(g +g ¯ )
NA = K 1 − 2 3Fn , (2.43)
α √2b0 i(g2 +g3 ¯n) g1 !
F −
ie /2 i(g +g ) g
αβ = K − 2 3Fn 1 , (2.44)
M √2b0 g1 i(g2 +g3 n) !
− F
α = √2ieK/2 g1∂aK i(g2∂aK+g3(Fna+∂aKFn)) , (2.45)
MbB − b0 i(g2∂a +g3( nb+∂a n)) g1∂a !
− K F KF K
ie /2 0 1
1;aAbB = K gac∗(∂b +∂b ) c , (2.46)
M | − 2 K D 1 0 !
−
= 1 g2∇bfa0+g3∇bfan −ig1∇bfa0
M2;aA|bB −2b0 −ig1∇bfa0 g2∇bfa0+g3∇bfan !
= ieK/2C gcd∗ g2∂b∗K+g3(F¯nb+∂b∗KF¯n) −ig1∂b∗K . (2.47)
abc
2√2 ig1∂b∗ g2∂b∗ +g3(¯nb+∂b∗ ¯n) !
− K K F KF
8
Here we have introduced
i
a = fa z¯b∗zc. (2.48)
D √2 bc
III. Partial Breaking of = 2 Local Supersymmetry
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By the gaugingof hypermultiplet, the scalar potential takes a nontrivial formandis given
by
e
V(z,z¯,b) = eKgab∗DaDb + (b0K)2gab∗DaxD¯bx∗
e
K ( x + x )( x + x ¯ ), (3.1)
−2(b0)2 E M Fn E M Fn
with
1
Dx = ( x∂ + x( +∂ )), (3.2)
a √2 E aK M Fna aKFn
x = (0, g , g ),
2 1
E
x = (0, g , 0).
3
M
The first term comes from the U(N) gauging of the vector multiplet while the second and
the last terms correspond to gauging of the hypermultiplet.
Let us find the conditions which determine the minimum of the potential. Let us first
consider the variations of V with respect to za. The derivative of the second and the third
terms of V reads
e
K (∂ )gbc∗DxD¯x +(∂ gbc∗)DxD¯x +gbc∗(∂ Dx)D¯x
(b0)2 aK b c∗ a b c∗ a b c∗
(cid:0) (cid:1)
e 1
= (b0K)2gbc∗D¯cx∗ ∂aDbx −(∂bK)Dax + ged∗(∂a∂bK0∂d∗K+∂a∂b∂d∗K0)Dex
(cid:18) K0 (cid:19)
ie
= K C gbd∗D¯x gce∗D¯x , (3.3)
(b0)2 abc d∗ e∗
where we have used (2.15),(2.16) in the last equality. Thus, the first vacuum condition is
e
h∂cVi = h∂c eKgab∗DaDb i+h(b0K)2iCacdgab∗D¯bx∗gde∗D¯ex∗i = 0, (3.4)
(cid:0) (cid:1)
The second vacuum condition is to be with respect to the hypermultiplet scalar bu. As the
potential contains only b0, the condition reads
∂V e
= K 2gab∗DxD¯x ( x + x )( x + x ¯ ) = 0. (3.5)
h∂b0i −(b0)3h a b∗ − E M Fn E M Fn i
9
As we search for the vacua with unbroken SU(N) gauge symmetry in this paper, we will
workontheconditionhzai = δanλ. ThenhDai = h√i2fabcz¯b∗zci = 0holdsandh∂c(eKgab∗DaDb)i =
0. For concreteness, we assume a form of the gauge invariant function (z) as the one which
F
parallels that of [24]:
iC
(z) = (zn)2 + (z), (3.6)
F − 2 G
k
C
(z) = ltr(zat )l, (3.7)
a
G l!
l=0
X
where C R and C are constant. We will see that C must be nonvanishing in order for the
l
∈
inverse of the K¨ahler metric to exist.
Let us compute the expectation value of the derivative of
F
= δ ,
a an n
hF i hF i
= δ ,
na an nn
hF i hF i
= δ iC ,
hFaˆˆbi aˆˆbhFnn − i
= δ , (3.8)
nab ab nnn
hF i hF i
where the explicit form of and that of are respectively
n nn
hF i hF i
C
= l λl 1 +iCλ,
n −
hF i (l 1)!
l −
X
C
= l λl 2. (3.9)
nn −
hF i (l 2)!
l −
X
It is easy to compute ∂ ,
a
K
i e
∂ = h Ki ¯ (λ λ¯)
a a a an
h Ki − 2 hF −F − − F i
= δ ∂ . (3.10)
an n
h Ki
The K¨ahler metric g is
ab
g11∗
h i
g 0
 11 
h ∗i
..
gab∗ =  . , (3.11)
h i  ... 
 
 
 0 gnn 
 h i 
 
10