Table Of ContentEquivariant Reduction of U(4) Gauge Theory over S2 S2 and
F × F
and the Emergent Vortices
2
1 Sec¸kin Ku¨rk¸cu¨oˇglu
0
2
n
Middle East Technical University,
a
J Department of Physics,
7 Dumlupinar Boulevard, 06800, Ankara, Turkey
2
e-mail: [email protected]
]
h
t
-
p
e
h
[
2
Abstract
v
8
2
7
0 We consider a U(4) Yang-Mills theory on M×SF2 ×SF2 where M is an arbitrary
. Riemannian manifold and S2 S2 is the product of two fuzzy spheres spon-
1 F × F
0 taneously generated from a SU( ) Yang-Mills theory on which is suitably
2 N M
coupled to six scalars in the adjoint of U( ). We determine the SU(2) SU(2)-
1
N ×
: equivariantU(4)gaugefieldsandperformthedimensionalreductionofthetheory
v
i over S2 S2. The emergent model is a U(1)4 gauge theory coupled to four com-
X F × F
plex and eight real scalar fields. We study this theory on R2 and find that, in
r
a certain limits, it admits vortex type solutions with U(1)3 gauge symmetry and
discuss some of their properties.
1 Introduction
Recently, therehasbeensignificantadvancesinunderstandingthestructureofgaugetheories
possessing fuzzy extra dimensions [1, 2] (for a review on fuzzy spaces see [3]). It is known
that in certain SU( ) Yang-Mills theories on a manifold , which are suitably coupled to
N M
a set of scalar fields, fuzzy spheres may be generated as extra dimensions by spontaneous
symmetry breaking. The vacuum expectation values (VEVs) of the scalar fields form the
fuzzy sphere(s), while the fluctuations around the vacuum are interpreted as gauge fields
over S2 or S2 S2 [2, 4]. The resulting theories can therefore be viewed as gauge theories
F F × F
over M S2 and M S2 S2 with smaller gauge groups; which is further corroborated
× F × F × F
by the expansion of a tower of Kaluza-Klein modes of the gauge fields. Inclusion of fermions
into this theory was considered in [4, 5]. For instance, in [5] an appropriate set of fermions
in 6D allowed for an effective description of Dirac fermions on M4 S2, which was further
× F
affirmed by a Kaluza-Klein modes expansion over S2. It was also found that a chirality
F
constraint on the fermions leads to a description in terms of ”mirror fermions” in which each
chiral fermion comes with a partner with opposite chirality and quantum numbers.
It appears well motivated to investigate equivariant parametrization of gauge fields and
perform dimensional reduction over the fuzzy extra dimensions to shed some further light
into the structure of these theories. Essentially, it is possible to use the well known coset
space dimensional reduction (CSDR) techniques to achive this task. To briefly recall the
latter consider a Yang-Mills theory with a gauge group S over the product space
M ×
G/H. G has a natural action on its coset, and requiring the Yang-Mills gauge fields to
be invariant under the G action up to S gauge transformations leads to a G-equivariant
parametrization of the gauge fields and subsequently to the dimensional reduction of the
theory after integrating over the coset space G/H [6, 7]. CSDR techniques have been widely
used as a method in attempts to obtain the standard model on the Minkowski space M4
starting from a Yang-Mills-Dirac theory on the higher dimensional space M4 G/H (for
×
a review on this topic reader can consult [7]). The widely known, prototype example of
CSDR is the SU(2)-equivariant reduction of the Yang-Mills theory over R4 to an abelian
Higgs model on the two-dimensional hyperbolic space H2, which was formulated by Witten
[8] prior to the development of the formal approach of [6], and it led to the construction of
instanton solutions with charge greater than 1.
Another approach, parallel to the CSDR scheme, using the language of vector bundles
and quivers is also known in the literature [9]. In recent times, this approach has been
employed in a wide variety of problems, including the formulation of quiver gauge theory of
non-Abelianvortices over R2d correspondingtoinstantonsonR2d S2, R2d S2 S2 [10,11],
θ θ × θ × ×
to the construction of vortex solutions over Riemann surfaces which become integrable for
appropriate choice of the parameters [12] and to the construction of non-Abelian monopoles
over R1,1 S2 in [13]. In [14], reduction of the Yang-Mills-Dirac theory on M S2 is
× ×
considered with a particular emphasis on the effects of the non-trivial monopole background
onthephysicalparticlespectrumofthereducedtheory. Dimensionalreductionoverquantum
sphereis recently studied and led to the formulation of q-deformed quiver gauge theories and
non-Abelian q-vortices [15].
Both of these techniques have also been applied to Yang-Mills theories over R2d S2 [16],
θ ×
1
where R2d is the 2d dimensional Groenewald-Moyal space; a prime example of a noncom-
θ
mutative space. In this framework, Donaldson-Uhlenbeck-Yau (DUY) equations of a U(2k)
Yang-Mills theory have been reduced to a set of equations on R2d whose solutions are given
θ
by BPS vortices on R2d and the properties of the latter have been elaborated.
θ
Startingwiththearticle[17],wehaveinitiatedinvestigations ontheequivariantreduction
of gauge theories over fuzzy extra dimensions. In [17] the most general SU(2)-equivariant
U(2)gaugefieldover S2 havebeenfound,anditwasutilizedtoperformthedimensional
M× F
reduction over S2. It was shown that for = R2 the emergent theory is an Abelian
F M
Higgs type model which has non-BPS vortex solutions corresponding to the instantons in
the original theory. There it was also found that these non-BPS vortices attract or repel
dependingon theparametersinthemodel. Thisarticlehasbeenfollowed upbyinvestigating
the situation in which is also a noncommutative space [18]. Performing the SU(2)-
M
equivariant dimensional reduction of this theory led to a noncommutative U(1) theory which
couples adjointly to a set of scalar fields. On the Groenewald-Moyal plane = R2 the
M θ
emergent models admit noncommutative vortex as well as fluxon solutions, which are non-
BPS and devoid of a smooth commutative limit as θ 0.
→
As we have noted earlier, gauge theory on M4 S2 S2 has been recently investigated
× F × F
in [4]. For this purpose authors of [4] have considered a SU( ) gauge theory on M4, which
N
is suitably coupled to six scalar fields in the adjoint of U( ). The model has the same field
N
contentasthatofthebosonicpartoftheN = 4SUSYYang-Mills theory, butcomestogether
with a potential breaking the N = 4 supersymmetry and the R-symmetry which is a global
SU(4). The deformed potential makes possible (after spontaneous symmetry breaking) the
identification of the VEV’s of the scalars with S2 S2 and the fluctuations around this
F × F
vacuum as gauge fields on S2 S2. Structure of fermions in this theory is elaborated in [4].
F × F
In a related article, it was shown that twisted fuzzy spheres can be dynamically generated as
extra dimensions starting from a certain orbifold projection of a N = 4 SYM theory whose
consequences have been discussed in [19]. For a review on these results [20] can beconsulted.
In the present article, we investigate the SU(2) SU(2) equivariant formulation of a
×
U(4) gauge theory over S2 S2. Starting from the SU( ) gauge theory model described
F × F N
above, but now put on some Riemannian Manifold , we focus on a U(4) gauge theory
M
on S2 S2 after spontaneous symmetry breaking. We determine the SU(2) SU(2)-
M× F × F ×
equivariant U(4) gauge fields and perform the dimensional reduction of the theory over
S2 S2. The emergent model is a U(1)4 gauge theory coupled to four complex and eight
F × F
realscalar fields. Westudythistheory on R2 andfindthat, incertain limits, itadmits vortex
type solutions with U(1)3 gauge symmetry and discuss some of their properties.
Our work in the rest of the paper is organized as follows. In section 2, we give the basics
of the SU( ) gauge theory over and indicate how the gauge theory over dynamically
N M M
develops S2 S2 as extra dimensions. This is followed by a systematic construction of the
F × F
SU(2) SU(2)-equivariant U(4) gauge field using essentially the SO(4) SU(2) SU(2)
× ≈ ×
representation theory. In section 3, we present the results of the equivariant reduction over
S2 S2 and give the reduced action in full, and find that the emergent model is a
M× F × F
U(1)4 gauge theory coupled to four complex and eight real scalar fields. This is ensued by a
discussionofthestructureofthereducedaction. Insection4,wepresentnon-trivialsolutions
ofthereducedactiononR2 fortwodifferentlimitingcasesoftheparametersa anda inthe
L R
2
theoryanddemonstratethat, theseparticularmodelshavevortex solutions withU(1)3 gauge
symmetry which tend to attract or repel at the critical point of the parameter space gg˜= 1.
For completeness, brief definitions of S2 and S2 S2 are given in appendix A and basics of
F F × F
theU( )gaugetheoryover S2 andtheU(2)-equivariantgaugefieldparametrizationare
N M× F
discussed in appendix B. In appendix C, we collect the explicit expressions after dimensional
which is presented in section 3.
2 U(4) Gauge Theory over S2 S2
M× F × F
i. Gauge theory on S2 S2:
M× F × F
We start with an SU( ) gauge theory coupled adjointly to six scalar fields Φ ,(i =
i
N
1, ,6). The relevant action is given in the form [4]
···
1
S = Tr F F +(D Φ ) (D Φ ) +V(Φ). (2.1)
N 4g2 µ†ν µν µ i † µ i
ZM (cid:16) (cid:17)
In this expression, A are su( ) valued anti-Hermitian gauge fields, Φ (i = 1, 6) are six
µ i
N ···
anti-Hermitian scalars transformingin the adjoint of SU( ) and D Φ = ∂ Φ +[A ,Φ ] are
µ i µ i µ i
N
the covariant derivatives.
It is assumed further that Φ ,(i = 1, ,6) transform in the vector representation of a
i
···
global SU(4) = SO(6) group.
∼
When considered on the four dimensional Minkowski spacetime M4, depending on the
formofthepotentialtermV(Φ),theaction(2.1)correspondstothebosonicpartoftheN = 4
super Yang-Mills theory with the global SU(4) being its R-symmetry, or a modification of it
thereof. The potential may have the form
V(Φ) = V (Φ)+V (Φ), (2.2)
N=4 break
where the first term corresponds to the potential of the N = 4 super Yang-Mills theory
6
1
V (Φ) = g2 [Φ ,Φ ]2, (2.3)
N=4 4 4 i j
i,j
X
while the second term breaks both the N = 4 supersymmetry and the R-symmetry. It
also worths to mention that the above action (2.1) descends from a ten-dimensional N = 1
super Yang-Mills theory by dimensional reduction. We will not review this here as it is not
necessary for our purposes, however a quick discussion can be found in [4].
We would like to see now how the product of two fuzzy spheres emerges as extra di-
mensions from this theory as a consequence of spontaneous breaking of the original gauge
symmetry. Following the discussion in [4], we consider a potential of the form
1 1 1
V(Φ)= V (ΦL)+ V (ΦR)+ V (ΦL,R)+a2VL(Φ )+a2VR(Φ ), (2.4)
g2 1 g2 1 g2 1 L 2 L R 2 R
L R LR
where
ΦL = Φ , ΦR = Φ , (a = 1,2,3), (2.5)
a a a a+3
3
and
V1(ΦL)= TrNFaLb†FaLb, FaLb = [ΦLa ,ΦLb]−εabcΦLc
V1(ΦR) = TrNFaRb†FaRb, FaRb = [ΦRa ,ΦRb ]−εabcΦRc
V (ΦL) = Tr (ΦLΦL+˜b )2, V (ΦR) = Tr (ΦRΦR+˜b )2
2 a a L 2 a a R
N N
V1(ΦL,R) = Tr Fa(bL,R)†Fa(bL,R), Fa(bL,R) = [ΦLa ,ΦRb ]. (2.6)
N
We observe that the potential V(Φ) is positive definite, and it is possible to pick ˜b and
L
˜b as the quadratic Casimirs of respectively SU(2) and SU(2) with IRR’s labeled by ℓ
R L R L
and ℓ
R
˜b = ℓ (ℓ +1), ˜b = ℓ (ℓ +1), 2ℓ , 2ℓ Z. (2.7)
L L L R R R L R
∈
If it is further assumed that = (2ℓ +1)(2ℓ +1)n, (n Z), then the configuration
L R
N ∈
ΦL = X(2ℓL+1) 1 1 ,
a a ⊗ (2ℓR+1)⊗ n
ΦR = 1 X(2ℓR+1) 1 , (2.8)
a (2ℓL+1)⊗ a ⊗ n
[ΦL,ΦR]= 0, (2.9)
a b
is a global minimum of the potential V(Φ) where X(2ℓL+1) and X(2ℓR+1) are the anti-
a a
Hermitian generators of SU(2) and SU(2) respectively in the IRR’s ℓ and ℓ , with the
L R L R
commutation relations
[X(2ℓL+1),X(2ℓL+1)]= ε X(2ℓL+1), [X(2ℓR+1),X(2ℓR+1)]= ε X(2ℓR+1). (2.10)
a b abc c a b abc c
This vacuum configuration spontaneously breaks the SU( ) down to U(n) which is the
N
commutant of ΦL,ΦR in (2.8).
a a
Defining
i i
xˆL = X(2ℓL+1) 1 , xˆR = 1 X(2ℓR+1), (2.11)
a ℓ (ℓ +1) a ⊗ (2ℓR+1) a (2ℓL+1)⊗ ℓ (ℓ +1) a
L L R R
p xˆLaxˆLa = 1, xˆRaxˆRa = 1. p (2.12)
thevacuumisaproductoftwofuzzyspheresS2 S2 generatedbyxˆL andxˆR. (seeappendix
F× F a a
A for a description of S2 S2).
F × F
Fluctuations about this vacuum give a U(n) gauge theory over S2 S2. We can write
F × F
ΦL = XL +AL, ΦR = XR +AR (2.13)
a a a a a a
where AL,AR u(2ℓ +1) u(2ℓ +1) u(n) with the short-hand notation X(2ℓL+1)
a a L R a
∈ ⊗ ⊗ ⊗
1 1 =:XL and 1 X(2ℓR+1) 1 =:XR.
(2ℓR+1)⊗ n a (2ℓL+1)⊗ a ⊗ n a
Thus, ΦL,ΦR are the “covariant coordinates” on S2 S2, and the associated curvatures
a a F × F
FL, FR, FL,R take their familiar form after expanding according to (2.13)
ab ab ab
FL = [XL,AL] [XL,AL]+[AL,AL] ε AL,
ab a b b a a b abc c
− −
FR = [XR,AR] [XR,AR]+[AR,AR] ε AR,
ab a b b a a b abc c
− −
FL,R = [XL,AR] [XR,AL]+[AL,AR]. (2.14)
ab a b − b a a b
4
Therefore, we can interpret the spontaneously broken theory as a U(n) gauge theory on
S2 S2 withA := (A ,AL,AR)asthegaugefieldsandF asthecorrespondingfield
M× F× F M µ a a MN
strength. The VL and the VR serve as constraint terms to suppress the normal components
2 2
of the gauge fields on each of the fuzzy spheres, in similar manner as discussed for the case
of a single fuzzy sphere in [2, 17].
It is important to point out that, this gauge theory can be called the “standard” Yang-
Mills theory on S2 S2 if we take g = g = √2g := g˜, scale the scalar fields as
M× F × F L R L,R
Φ˜ = √2g˜Φ and take g˜g = 1, since only then it takes the form of the L2 norm of F .
i MN
We also note for future use that, with the developments above
1
Tr = Tr Tr Tr (2.15)
N n(2ℓL +1)(2ℓR +1) Mat(2ℓL+1)⊗ Mat(2ℓR+1)⊗ Mat(n)
where Mat(k) denotes the algebra of k k matrices.
×
Finally, it is also usefulto remark that there are other possibilities for the vacuum config-
uration as discussed in [4] which for instance lead to S2 S2 carrying magnetic fluxes under
F × F
the U(1) component of the unbroken gauge group SU(n) SU(m) U(1) after spontaneous
× ×
symmetry breaking.
ii. The SU(2) SU(2)-Equivariant Gauge Field
×
We will now formulate the SU(2) SU(2) = SO(4)-equivariant, U(4) gauge theory on
L× R ∼
S2 S2. The gauge fields carry the fundamental representation of U(4). We introduce
M× F × F
SO(4) symmetry generators under which A is a scalar up to a U(4) gauge transformation,
µ
that is carrying the SO(4) IRR (0,0) and AL and AR are SO(4) tensors carrying the IRRs
a a
(1,0) and (0,1), respectively. In other words, AL is a vector under the SU(2) and a scalar
a L
under the SU(2) , whereas AR is an SU(2) vector and an SU(2) scalar.
R a R L
On S2 S2 the SU(2) SU(2) = SO(4) rotational symmetry is implemented by the
F × F × ∼
adjoint actions adXL and adXR (see appendix A):
a a
adXL = [XL, ], adXR = [XR, ], [adXL,adXR] = 0. (2.16)
a a a a a a
· · · ·
Let’s introduce the anti-Hermitian symmetry generators
LL
ωL = X(2ℓL+1) 1 1 1 1 i a ,
a a ⊗ (2ℓR+1)⊗ 4− (2ℓL+1)⊗ (2ℓR+1)⊗ 2
LR
ωR = 1 X(2ℓR+1) 1 1 1 i a . (2.17)
a (2ℓL+1)⊗ a ⊗ 4− (2ℓL+1)⊗ (2ℓR+1)⊗ 2
Here LL and LR are 4 4 matrices whose structure will be given shortly. They are chosen
a a
×
so that ωL and ωR fulfill the consistency conditions
a a
[ωL,ωL] = iε ωL,
a b abc c
[ωR,ωR] = iε ωR, (2.18)
a b abc c
[ωL,ωR] = 0. (2.19)
a b
In order to write down the matrices LL and LR consider first the 4 4 matrices denoted
a a
×
as e (m,n = 1,2,3,4) whose all entries are zero except the entry on the mth row and the
mn
5
nth column which is 1. We let
J = iε e , K = i(e e ), (2.20)
a abc bc a a4 4a
− − −
and define
LL = J +K , LR = J K . (2.21)
a a a a a a
−
These matrices fulfill
[LL,LL] = 2iε LL,
a b abc c
[LR,LR] = 2iε LR,
a b abc c
[LL,LR] = 0. (2.22)
a b
Therefore we have altogether six anti-symmetric SU(4) matrices generating the two sub-
groups SU(2) and SU(2) . Remaining nine symmetric generators of SU(4) may be taken
L R
as LLLR. Together with the 4 4 identity 1 , LL, LR and LLLR span U(4) and furnish a
a b × 4 a a a b
basis for the fundamental representation of U(4).
LL and LR form a 4 4 basis of the Lie algebra so(4) = su(2) su(2). In addition, these
a a
× ⊕
matrices satisfy the relations
LLLL = iε LL+δ 1 ,
a b abc c ab 4
LRLR = iε LR+δ 1 , (2.23)
a b abc c ab 4
which permits to view them as two sets of 4 4‘Pauli Matrices”.
×
From the point of view of the SO(4) representation theory LL, LR carry the reducible
a a
representations of SU(2) and SU(2) . LL carries two copies of the IRR (1,0), whereas
L R a 2
LR carries two copies of the IRR (0, 1), which can be clearly observed from their Casimir
a 2
operators with the eigenvalues 3.
As the gauge fields AL and AR on S2 S2 are u(4) valued, they are elements of u(2ℓ +
a a F × F L
1) u(2ℓ +1) u(4). Therefore, it is now clear that LL and LR in (2.17) are responsible
R a a
× ×
for generating the U(4) gauge symmetry in SO(4).
The SU(2) SU(2) = SO(4)-equivariance conditions stated at the beginning of this
× ∼
section can now be explicitly described as the fulfillment of the following conditions under
the adjoint actions of ωL and ωR.
[ωL,A ] = 0 = [ωR,A ],
a µ a µ
[ωL,AL]= ε AL,
a b abc c
[ωR,AR] = ε AR,
a b abc c
[ωL,AR] = 0 = [ωR,AL]. (2.24)
a b a b
It is necessary to find explicit parametrizations of A , AL and AR fulfilling these conditions.
µ a a
The adjoint actions of ωL and ωR expand in Clebsch-Gordan series as
1 1
2 (ℓ ,0) ( ,0) (ℓ ,0) ( ,0) = 4(0,0) 8(1,0) , (2.25)
L L
× ⊗ 2 ⊗ ⊗ 2 ⊕ ⊕···
(cid:20) (cid:21) (cid:20) (cid:21)
1 1
2 (0,ℓ ) (0, ) (0,ℓ ) (0, ) = 4(0,0) 8(0,1) , (2.26)
R R
× ⊗ 2 ⊗ ⊗ 2 ⊕ ⊕···
(cid:20) (cid:21) (cid:20) (cid:21)
6
wherethefactoroftwoineachlineaboveisduetothetwocopiesoftheIRRs(1 ,0)and(0, 1)
2 2
in LL and LR, respectively. Therefore the relavant part of the Clebsch-Gordan expansion
a a
takes the form
4(0,0) 8(1,0) 8(0,1). (2.27)
⊕ ⊕
The solution space for A is then 4-dimensional, whereas each of the solution spaces of AL
µ a
and AR are 8-dimensional.
a
It is not very hard to see that there are four invariants under the action of ωL and ωR.
a a
These are the three ‘idempotents”
XℓL 1 LL i1
QL = a ⊗ (ℓ2ℓR++11)/⊗2 a − 2 , Q†L = −QL, Q2L = −14(2ℓL+1)(2ℓR+1), (2.28)
L
1 XℓR LR i1
QR = (2ℓL+1)⊗ℓ +a1/⊗2 a − 2 , Q†R = −QR, Q2R = −14(2ℓL+1)(2ℓR+1),(2.29)
R
(XℓL 1 LL i1)(1 XℓR LR i1)
iQ Q = i a ⊗ (2ℓR+1)⊗ a − 2 (2ℓL+1)⊗ a ⊗ a − 2 ,
L R
(ℓ +1/2)(ℓ +1/2)
L R
(iQ Q ) = iQ Q , (iQ Q )2 = 1 , (2.30)
L R † − L R L R − 4(2ℓL+1)(2ℓR+1)
which are all [4(2ℓ +1)(2ℓ +1)]2 matrices and the identity matrix 1 .
L R − 4(2ℓL+1)(2ℓR+1)
These lead to the parametrization
1 1 i 1
A = aLQL+ aRQR+ b 1+ ic QLQR, (2.31)
µ 2 µ 2 µ 2 µ 2 µ
where a , b , c and d are all Hermitian U(1) gauge fields, and to the parametrizations
µ µ µ µ
1 1 1 1 1
AL = (χ +χ )[XL,QL]+ (χ +χ 1)QL[XL,QL]+i χ XL,QL + χ ωL
a 2 1 ′1 a 2 2 ′2− a 2 32{ a } 2 4 a
1 1 1 1 1
+2(χ1−χ′1)iQR[XaL,QL]+2(χ2−χ′2)iQRQL[XaL,QL]+i2χ′32iQR{XaLb,QL}+2χ′4iQRbωaL.
(2.32)
b b
1 1 1 1 1
AR = (λ +λ )[XR,QR]+ (λ +λ 1)QR[XR,QR]+i λ XR,QR + λ ωR
a 2 1 ′1 a 2 2 ′2− a 2 32{ a } 2 4 a
1 1 1 1 1
+2(λ1−λ′1)iQL[XaR,QR]+2(λ2−λ′2)iQLQR[XaR,QR]+i2λ′32iQL{XaRb,QR}+2λ′4iQLbωaR.
(2.33)
b b
Here χ , χ , λ and λ i = (1,2,3,4) are Hermitian scalar fields over , the curly brackets
i ′i i ′i M
denote anti-commutators throughout, and we have used
1 1
XL := XL, ωL := ωL,
a ℓ +1/2 a a ℓ +1/2 a
L L
1 1
XbR := XR, bωR := ωR. (2.34)
a ℓ +1/2 a a ℓ +1/2 a
R R
Let us also introduce the nbotation b
AL := A˜L +iQRA˜L
a a ′a
AR := A˜R+iQLA˜R (2.35)
a a ′a
for future convenience.
7
3 Reduction of the Yang-Mills Action over S2
F
Using the SU(2) SU(2)-equivariant gauge field in the action functional of the U(4) Yang-
×
Mills theory on S2 S2, we can explicitly trace it over the fuzzy spheres to reduce it
M⊗ F × F
to a theory on . It is quite useful to note the following identities
M
Q,[X ,Q] = 0, X ,[X ,Q] = 0, (sum over repeated a is implied), (3.1)
a a a
{ } { }
[Q, X ,Q ] = 0, [X , X ,Q ] = 0, (sum over repeated a is implied). (3.2)
a a a
{ } { }
which are valid for both the left and the right quantities and they significantly simplify the
calculations, since they greatly reduce the number of traces to be computed.
The reduced action has the form
1 1 1
S = + + VL + VR+ VL,R+a2VL+a2VR. (3.3)
LF LG g2 1 g2 1 g2 1 L 2 R 2
ZM L R LR
Each term in this expression is defined and evaluated below, while some details are relegated
to the appendix C.
3.1. The Field Strength Term
The field strength can be expressed as
1 1 i i
F = fL QL+ fRQR+ g 1 + h QLQR (3.4)
µν 2 µν 2 µν 2 µν 4 2 µν
where
fL = ∂ aL ∂ aL, fR = ∂ aR ∂ aR,
µν µ ν ν µ µν µ ν ν µ
− −
g = ∂ b ∂ b , h = ∂ c ∂ c (3.5)
µν µ ν ν µ µν µ ν ν µ
− −
The corresponding contribution to the Lagrangian is
1
:= Tr F F
LF 4g2 N µ†ν µν
(cid:16) (cid:17)
= 1 fL 2+ fR 2+ g 2+ h 2+ 2 fL fR
16g2 µν µν | µν| | µν| (2ℓ +1)(2ℓ +1) µν µν
L R
(cid:16)1(cid:12) (cid:12) (cid:12) (cid:12) 1
(cid:12) (cid:12) fR(cid:12)g (cid:12) fL h fL g fRh
−(2ℓ +1) µν µν − µν µν − (2ℓ +1) µν µν − µν µν
R L
2(cid:0) (cid:1) (cid:0) (cid:1)
g h . (3.6)
µν µν
−(2ℓ +1)(2ℓ +1)
L R
(cid:17)
3.2. The Gradient Term
The covariant derivatives are naturally expressed in two pieces
1 i 1
D ΦL = D (χ +χ )+QLD (χ +χ ) [XL,QL]+ ∂ χ XˆL,QL + ∂ χ ωˆL
µ a 2 µ 1 ′1 µ 2 ′2 a 4 µ 3{ a } 2 µ 4 a
1(cid:0) (cid:1) i 1
+iQR D (χ χ )+QLD (χ χ ) [XL,QL]+ ∂ χ XˆL,QL + ∂ χ ωˆL
2 µ 1− ′1 µ 2− ′2 a 4 µ ′3{ a } 2 µ ′4 a
(cid:18) (cid:19)
(cid:0) (cid:1) (3.7)
8
1 i 1
D ΦR = D (λ +λ )+QRD (λ +λ ) [XR,QR]+ ∂ λ XˆR,QR + ∂ λ ωˆR
µ a 2 µ 1 ′1 µ 2 ′2 a 4 µ 3{ a } 2 µ 4 a
1(cid:0) (cid:1) i 1
+iQL D (λ λ )+QRD (λ λ ) [XR,QR]+ ∂ λ XˆR,QR + ∂ λ ωˆR
2 µ 1− ′1 µ 2− ′2 a 4 µ ′3{ a } 2 µ ′4 a
(cid:18) (cid:19)
(cid:0) (cid:1) (3.8)
where we have (i = 1,2)
D χ = ∂ χ +ε a χ +ε c χ
µ i µ i ji µ j ji µ j
D χ = ∂ χ +ε a χ ε c χ . (3.9)
µ ′i µ ′i ji µ ′j ji µ ′j
−
with (i = 1,2).
The gradient term takes the form
:= L + R = Tr (D ΦL) (D ΦL)+(D ΦR) (D ΦR) , (3.10)
G G G µ a † µ a µ a † µ a
L L L N
(cid:16) (cid:17)
where
ℓ (ℓ +1) 1 1
L = L L 1+ (D χ )2+(D χ )2 + 1 (D χ )2
LG (ℓ +1/2)2 2(ℓ +1) µ 1 µ 2 − 2(ℓ +1) µ ′1
L " R R
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16)
(cid:0) (cid:1)
1ℓ (ℓ +1)(ℓ2 +ℓ 1/4) 1
+(D χ )2 + L L L L − (∂ χ )2+(∂ χ )2+ ∂ χ ∂ χ
µ ′2 # 4 (ℓL +1/2)4 " µ 3 µ ′3 (ℓR+ 21) µ 3 µ ′3#
(cid:17)
1 ℓ (ℓ +1) 1
L L
+ ∂ χ ∂ χ +∂ χ ∂ χ + ∂ χ ∂ χ +∂ χ ∂ χ
2(ℓ +1/2)3 µ 3 µ 4 µ ′3 µ ′4 2(ℓ +1) µ 3 µ ′4 µ ′3 µ 4
L R
(cid:20) (cid:21)
(cid:0) (cid:1)
1ℓ2 +ℓ +3/4 1
+ L L (∂ χ )2+(∂ χ )2+ ∂ χ ∂ χ . (3.11)
4 (ℓL +1/2)2 " µ 4 µ ′4 (ℓR + 21) µ 4 µ ′4#
ℓ (ℓ +1) 1 1
R = R R 1+ (D λ )2+(D λ )2 + 1 (D λ )2
LG (ℓ +1/2)2 2(ℓ +1) µ 1 µ 2 − 2(ℓ +1) µ ′1
R " L L
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16)
(cid:0) (cid:1)
1ℓ (ℓ +1)(ℓ2 +ℓ 1/4) 1
+(D λ )2 + R R R R− (∂ λ )2+(∂ λ )2+ ∂ λ ∂ λ
µ ′2 4 (ℓ +1/2)4 µ 3 µ ′3 (ℓ +1) µ 3 µ ′3
# R L
(cid:20) (cid:21)
(cid:17)
1 ℓ (ℓ +1) 1
R R
+ ∂ λ ∂ λ +∂ λ ∂ λ + ∂ λ ∂ λ +∂ λ ∂ λ
2(ℓ +1/2)3 µ 3 µ 4 µ ′3 µ ′4 2(ℓ +1) µ 3 µ ′4 µ ′3 µ 4
R L
(cid:20) (cid:21)
1ℓ2 +ℓ +3/4 1 (cid:0) (cid:1)
+ R R (∂ λ )2+(∂ λ )2+ ∂ λ ∂ λ . (3.12)
4 (ℓ +1/2)2 µ 4 µ ′4 2(ℓ +1) µ 4 µ ′4
R L
(cid:20) (cid:21)
It is useful to form the complex fields
χ = χ +iχ , χ¯ = χ iχ , λ = λ +iλ , λ¯ = λ iλ , (3.13)
1 2 1 2 1 2 1 2
− −
then the covariant derivatives are expressed as
D χ = ∂ χ+i(aL +c )χ, D χ = ∂ χ +i(aL c )χ ,
µ µ µ µ µ ′ µ ′ µ µ ′
−
D λ = ∂ λ+i(aR+c )λ, D λ = ∂ λ +i(aR c )λ . (3.14)
µ µ µ µ µ ′ µ ′ µ µ ′
−
We note that primed fields carry charge 1 under c .
µ
−
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