Five-dimensional Yang-Mills-Einstein supergravity on orbifolds: Parity assignments

0.29 MB·English

by  Sean McReynolds

hep-ph/0506057
Five-dimensional Yang-Mills-Einstein supergravity on orbifolds:
Parity assignments
Sean McReynolds∗
Physics Department
Pennsylvania State University
University Park, PA. 16802, USA
We discuss the options for parity assignments in (on-shell) N = 2 five-dimensional Yang-Mills-
Einstein supergravity theories (YMESGTs) coupled to tensor and hypermultiplets on the orbifold
spacetime M ×S1/Z . Along the lines of orbifold-GUTs, we allow for general breaking of the
4 2
five-dimensionalgaugegroupattheorbifoldfixedpoints. Wethenextendthediscussiontothecase
wheretheorbifold is S1/(Z ×Z ). Wedonotpresumetheexistenceoffieldswithsupportonlyat
2 2
fixedpoints. Asinthefamiliarcaseof(rigid)super-Yang-Millstheoriesonsuchorbifolds,onlybulk
hypermultipletscan lead to chiral multiplets in complex representations of the gauge group on the
6
boundaries. Massless chiralmultipletscomingfrombulkvectorortensormultipletscanpotentially
0
be used as Higgs supermultiplets, though a “doublet-triplet” splitting via parity assignments is
0
not available for the tensor sector. We also find parity assignments for objects other than fields
2
that appear in the Lagrangian, which will partially determine the structure of interactions of the

n boundary theories. Assigning odd parities to the scalar sector of vector/tensor multiplets requires
a the four-dimensional boundary moduli spaces to lie on the boundary of the classical K¨ahler cone,
J whichcorrespondstocollapsedCalabi-Yau2-cyclesattheorbifoldfixedpointsinacompactification

6 of eleven-dimensional supergravity. There is an ambiguity in how to effect odd parity for the field-
independentCIJK tensorofthe5Dtheory,whichmayadmitaclassicalinterpretationasCalabi-Yau
6 4-cycles collapsing to either 2- or 0-cycles.
v
7 PACSnumbers:
5
0
6 I. INTRODUCTION (GUTs).
0
5 Since an orbifold spacetime is singular, field theories
0 are not well-defined on it, which requires some further
Phenomenological field-theoretic model building has
/ interpretation (in the downstairs picture, the boundary
h recently refocused on scenarios in which the universe
is sharp). Supergravity admits solitonic solutions that
p appears higher-dimensional above some energy scale to
- couldultimatelybeinterpretedastheboundariesofthese
p examine new electroweak and supersymmetry breaking theories; these solutions are domain walls with some
e schemes [1] and new strong-electroweak unification sce-
thickness, smoothing out the singular nature of sharp
h narios [2]. In [3], it was pointed out that the estimated
boundaries. Supergravity in orbifold scenarios is inter-
:
v scaleofstrong-electroweakunificationwasaroundtheen- esting for other reasons: as the familiar argument goes,
i ergy scale where a Kaluza-Klein type universe may not
X the gauge couplings in supersymmetric versions of the
be able to be approximated by a 4D theory, in which
Standard Model unify [7] at a scale close to the scale at
r casegrandunificationwouldoccur inhigher dimensions.
a which quantum gravitational effects are expected to be
In [4], it was suggested that the size of an extra dimen-
non-negligible. Since there is a significant interpolation
sion could be much larger (TeV scale) within the frame-
involved in these suggestive results, one might look for
work of perturbative string theory (one of the motiva-
unification of gauge and gravitational couplings. There-
tions was to tie this scale to the = 1 supersymme-
N fore, the next step in bottom-up model building is to
try breaking scale). Subsequently, the Horava-Witten
consider supergravity versions of the interesting scenar-
(HW) scenario [5] and Randall-Sundrum (RS) scenar-
ios.
ios [6] served as the most recent revival of interest in
intermediate scale five-dimensional spacetimes in which Orbifold constructions have been performed many
thegroundstatespacetimeisasingularspacethatisiso- times in the literature for both rigidly and locally su-
morphic to a spacetime with boundaries. In particular, persymmetric field theories; for examples of the former,
the “orbifold” spacetime S1/Γ (upstairs picture) see[1,2];forthelatter,see[8–12]. Thegenericresultisa
4
M ×
correspondstoamanifoldwithtwoboundariesseparated theorywith4Dboundarieswith1/2supersymmetryand
by a 5D bulk (downstairs picture). This spacetime can brokengaugegroup;thelowenergydescriptionisa4Def-
resolve a number of issues in the supersymmetric Stan- fective theory for each boundary. However, a systematic
dardModelandsupersymmetricGrandUnifiedTheories classification of the types of boundary conditions avail-
able via parity assignments has not been performed for
Yang-Mills-Einstein supergravity theories (YMESGTs)
coupled to vector, hyper-, and tensor multiplets (usu-
∗Electronicaddress: [email protected] ally, orbifold supergravity theories are considered in the
2
context of HW or (original) RS type scenarios in which sistency of local coordinate transformations, supersym-
the StandardModelgauge,andsometimes matter,fields metry transformations and gauge transformations. The
aresupportedonlyonthe boundaries). Inthispaper,we choice of Γ reflects different classes of boundary condi-
aim to provide a more complete list of options, includ- tions from the downstairs point of view. We will first
ing acarefullookatthe tensorsector,forthe lowenergy considerthesimplestcaseΓ=Z ,whichresultsinathe-
2
spectrumviaparityassignmentsinthesimplecaseofthe ory with equivalent spectra and interactions at the two
S1/Z orbifold. We do not presume boundary-localized Γ fixed points.
2
field content, which would follow from the appropriate ThechoiceofthewayZ actsonquantitiesinthethe-
2
couplingof3-branesontheorbifoldfixed-planes(orother oryreflectsaparticularsetofconsistentboundarycondi-
masslessfields arisinginanM-theoreticframework). We tions. TheZ actsasreflectionsontheS1 coveringspace
2
will then extend our results to the case S1/(Z Z ). [ πR, πR] (where πR πR ),with fixed points at
2 2
Some of the results are generic to theories with×super- −0 , πR . To leave{−the s}pa≡ce{ } S1/Z invariantun-
4 2
{ } { } M ×
Yang-Mills coupled to hypermultiplets, while others are der this action, the coordinate functions, basis vectors,
unique to supergravity. While orbifold-GUTsareamain basis 1-forms, and metric components have
motivation,the resultsarenotrestrictedtothesescenar-
P(xµ; ∂ ; dxµ)=+1 P(x5; ∂ ; dx5)= 1
ios. As a novel example of 5D GUT in the framework of µ 5 −
supergravity,weillustratesomeoftheparityassignments P(gˆµν; gˆ55)=+1 P(gˆµ5)= 1,
−
using an SU(5,1) gauging.
where P(Φ) denotes the Z parity of the object Φ. The
In the next section, we will make generic remarks 2
fixed planes are non-oriented.
about field theory on orbifolds. In section III, we list
Fieldscarryinginternalquantumnumbersaresections
the =1 supermultiplets that have propagatingmodes
oans tdNhiectoartbeidfolbdyficxoendspisltaennetsoZft-hpearsiptyaceatsismigenMme4n×tsS.1/ZW2e, osufcahfiabesirtbuuatnidolne,,iwtimthakspesacseetnismeefobreitnhgetahcetiboansoefsZpa2cteo.bIne
2 lifted from the base space to the total space [13]. There
first discuss the pure Yang-Mills-Einstein and hyper-
are a number of ways to perform this lift, corresponding
multiplet sectors in turn. The tensor sector, discussed tovariousclassesofboundaryconditions. Justasthe Z
in section IIIC, is subtler and has not been discussed 2
actiononthecoveringspaceS1resultsinasingularspace
in the literature much. In section IV, we list the Z2- S1/Z , the Z action on the total space will, in general,
parityassignmentsforobjectsotherthanfieldsappearing 2 2
change the structure of the fibers over the base space.
in the Lagrangian; in particular, spacetime- and field-
Objects other than fields appearing in the Lagrangian
independent quantities may be assigned odd parities,
generallycarryrepresentationindicesofthegaugegroup.
without the need to assume that there is a formulation
Such quantities are structures appearing in the gauge
of the theory in which an odd-parity field is responsible
bundle, andare therefore generallyaffected by modifica-
for this. In section VI, we extend the results to the case tions of the gauge bundle resulting after Z action (even
in which the fifth dimension is S1/(Z Z ), which can 2
2× 2 if they are field independent). This gives meaning to
be more phenomenologically interesting. We end with assigning these objects Z parities.
conclusions(sectionVII)andsomefuturedirections(sec- Althoughphysicalstate2son 4 S1/Z mustbeeven
tion VIII). Appendix A contains some conventions used under Z -action, the field opeMrato×rs can2carry even or
in this paper, while appendix B contains a discussion of 2
odd parity. We will take the expansion of an odd parity
the fermionic parity assignments.
field on the orbifold to have an nth term of the form
Φ(n)(xµ,x5)=A Φ(n)(xµ)sin(nx5/R)
− n −
(1)
II. SUPERGRAVITY ON S1/Γ +B Φ(n)(xµ)ǫ(x5)cos(nx5/R),
n −
In modeling five-dimensional spacetimes with four- where ǫ(x5) is +1 for ( πR,0) and 1 for (0,πR); the
− −
dimensional boundaries, we can choose a particular con- Φ(n)(xµ)areeven;andA , B arenormalizationfactors.
− n n
structionusingaspacetimeoftheformM S1/Γ,where To avoid δ2 factors in the Lagrangian (δ being the
4
×
Γisadiscretegroupthatactsnon-freelyonthecircle[32]. Dirac distribution), bosonic fields cannot haveǫ(x5) fac-
The orbifold S1/Γ,whichhastwo4Dfixedplanes, tors in the above expansion. Therefore, we impose the
4
M ×
is isomorphicto a groundstatespacetime ,where condition B = 0 for odd bosonic fields, which therefore
4 n
M ×I
is a closed interval so that this is a spacetime with vanish on the orbifold fixed planes.
I
two disjoint 4D boundaries. Instead of considering the On the other hand, the equations of motion and ki-
5D theories with these spacetime boundaries (the down- netictermsforfermionicfieldsinvolveasinglederivative,
stairs picture), it is often convenient to compactify the so ǫ(x ) factors are allowed (they will give rise to δ(x )
5 5
5D theoryon S1, followedby assignmentof Γ-parities to factors in the equations of motion and upstairs picture
quantities in the theory (the upstairs picture). Since the Lagrangian). Therefore, fermionic fields on S1/Z are
2
Lagrangian (density) must be even under Z , there are not well-defined on the upstairs picture fixed planes.
2
constraintsontherelativeparitiesofthe fieldsandother Odd-parity field-independent objects CI1...In carry-
J1...Jn
objects. Further constraints are imposed by the con- ing gauge indices can generally be redefined as either
3
ǫ(x5)CI1...In or κ(x5)CI1...In, where CI1...In now has definite, which means that the manifold is Riemannian.
even paJr1i..t.yJ,nand κ(x5) isJ1...Jn J1...Jn The equation (ξ)=k (k R+) defines a family of real
V ∈
hypersurfaces, and in particular
0 for x5 = πR
1 for πR−<x5 <0 (ξ)=1 (4)
κ(x5)=−0 for x−5 =0 (2) V
+1 for 0<x5 <πR defines a real nV-manifold corresponding to the scalar
manifold . The functions hI and hI that appear in
However, consistency may require one or the other. fermionicMterRms of the Lagrangian and sx˜usy transforma-
tions are directly proportionalto ξI and ξI , re-
|V=1 ,x˜|V=1
spectively; the hI are essentially embedding coordinates
III. SUPERMULTIPLETS APPEARING IN of in the ambient space. In [14], it was shown that,
R
Γ=Z2 CASE whMen (4) holds, the CIJK may be put in the “canonical
form”
A. Pure Yang-Mills-Einstein supergravity
1
C =1, C = δ ,
000 0ij ij
−2 (5)
An = 2 5D Maxwell-Einstein supergravity theory
N C =0, C =arbitrary,
(MESGT) [14] consists of a minimal supergravity multi- 00i ijk
plet and n vector supermultiplets. The total field con-
V whereI =(0,i),i=1,...,n . Wedenotetherestriction
tent is V
of the ambient space metric to as
R
M
emˆ,Ψi,AI,λip˜,φx˜ ,
{ µˆ µˆ µˆ } a◦ =a .
IJ IJ V=1
|
where I = (0,1,...,n ) labels the graviphoton and vec-
V
tor fields from the n vector multiplets; i = (1,2) is an The metric of the scalarmanifold is then the pullback of
V
SU(2)R index; and p˜ = (1,...,nV) and x˜ = (1,...,nV) the restricted ambient space metric to MR:
label the fermions and scalars from the n vector mul-
tiplets. The scalar fields parametrize an nV-dimensional g = 3 a◦ hI hJ.
V x˜y˜ 2 IJ ,x˜ ,y˜
real Riemannian manifold , so the indices p˜,q˜,...
R
M
and x˜,y˜,... may also be viewed as flat and curved in- Both of these metrics are positive definite due to the
dices of R, respectively. The supersymmetry parame- constraint >0.
ters ǫi, tMhe gravitini Ψi, and the spin-1/2 fields λp˜i are The C V tensor completely determines the MESGT
µ IJK
5Dsymplectic-Majoranaspinors(seeappendixA),which Lagrangian [14]. Therefore, the global symmetry group
can be written in 2-component spinor notation as of the Lagrangian is given by the symmetry group, G,
of this tensor, along with automorphisms of the = 2
η ζ N
ǫ1 = ǫ2 = superalgebra; that is, the symmetry group of the La-
eζ∗ eη∗
grangianis G SU(2) . Since G consists of symmetries
(cid:18) (cid:19) (cid:18)− (cid:19) R
×
of the full Lagrangian,they are symmetries of the scalar
sectorinparticular,andthereforeisometriesofthescalar
α β
Ψ1 = µ Ψ2 = µ (3) manifold : G Iso( )(theSU(2) actionistriv-
µ eβ∗ µ eα∗ MR ⊂ MR R
(cid:18) µ (cid:19) (cid:18)− µ (cid:19) ial on the scalars). The full Lagrangian, however, is not
necessarily invariant under the full group Iso( ).
R
M
δp˜ γp˜ A subgroup K G may then be gauged if nV +1
λp˜1 = eγp˜∗ λp˜2 = eδp˜∗ . dim[K]. The vecto⊂r fields decompose into ≥
(cid:18) (cid:19) (cid:18)− (cid:19)
Introducing (nV + 1) parameters ξI(φ) depending on nV+1=adj(K)⊕non-singlets(K)⊕singlets(K).
the scalar fields, we define a cubic polynomial (ξ) =
C ξIξJξK, where C is a constant rank-V3 sym- Under infinitesimal K-transformations parametrized by
IJK IJK αI, the bosonic fields transform as:
metric tensor. This polynomial determines a symmetric
rank-2 tensor 1
δ AI = ∂ αI +αJfI AK
1 ∂ ∂ α µˆ −g µˆ JK µˆ (6)
a (ξ)= ln (ξ).
IJ −3∂ξI ∂ξJ V δαφx˜ =αIKIx˜,
TheparametersξI canbeinterpretedascoordinatefunc- where Kx˜ are a set of n Killing vectors on the scalar
I V
tions for an(n +1)-manifold,calledthe ambientspace. manifoldparametrizedbytheφx˜;andfI andαI vanish
V JK
The tensora , whichmayhaveindefinite signature,de- if any index corresponds to a spectator vector field [33].
IJ
fines a metric on this space. However, the coordinates Now the C must be a rank-3 symmetric invariant
IJK
are restricted via (ξ)>0 so that the metric is positive ofK. IfK iscompact,thenonlyC (see(5))mustbea
ijk
V
4
rank-3 symmetric invariant. The 5D bosonic YMESGT to a 4D local Lorentz invariance. The four local symme-
Lagrangianis [14] tries that are broken can be used to fix four degrees of
freedom in the fu¨nfbein. Splitting µˆ = (µ,5), we choose
eˆ−1 bos = the parametrizationfor the fu¨nfbein to be [14]
L
1Rˆ 1 a◦ I Jµˆνˆ 1g D φx˜Dµˆφy˜
− 2 − 4 IJ FµˆνˆF − 2 x˜y˜ µˆ eˆmˆ = e−σ2emµ 2eσCµ .
+ 6eˆ√−16CIJKǫµˆνˆρˆσˆλˆ{FµˆIνˆFρˆJσˆAKλˆ (7) µˆ (cid:18) 0 eσ (cid:19)
+ 3gFI AJ(fK ALAM) Since gˆµˆνˆ =eˆmµˆˆ eˆnνˆˆηmˆnˆ,
2 µˆνˆ ρˆ LM σˆ λˆ
3
+ g2(fJ AGAH)(fK ALAF)AI ,
5 GH νˆ ρˆ LF σˆ λˆ µˆ}
gˆ =e−σg +4e2σC C
µν µν µ ν
where hats indicate five-dimensional quantities and eˆ is gˆ =e2σ (9)
the determinant of the fu¨nfbein. The AI are abelian 55
vector fields and [34] µ gˆµ5 =2e2σCµ.
FµˆIνˆ ≡2∂[µˆAIνˆ] Furthermore, let AIµˆ = (AIµ, AI). Under infinitesimal lo-
I FI +2gAI AJ calcoordinatetransformationsofthecompactcoordinate
Fµˆνˆ ≡ µˆνˆ [µˆ νˆ] parameterizedbyξ5(xµ), the 4Dfields AI andC trans-
Dµˆφx˜ ≡∂µˆφx˜+gKIx˜AIµˆ. form as µ µ
Thesupersymmetrytransformationsofthebosonicfields
δ AI = ∂ ξ5AI
are ξ5 µ − µ (10)
δ C = 2∂ ξ5,
1 ξ5 µ − µ
δeˆmˆ = ǫ¯iΓmΨ
µˆ 2 µi
with the remaining 4D bosonic fields being invariant.
δAI = 1hIǫ¯iΓ λp˜+ i√6hIΨˆ¯iǫ (8) One can interpret ξ5(xµ) as a parameter for local U(1)
µˆ − 2 p˜ µ i 4 µˆ i
transformations,forwhichC isagaugefield. Notethat
µ
i
δφx˜ =2fp˜x˜ǫ¯iλpi˜, Uth(e1)vetcrtaonrsffioerlmdsatAioIµnst.raTnosfporrompenrolyn-dtirmiveianlsliyonuanldlyerretdhuecsee
the theory, one must make field redefinitions in order to
wherethefx˜aren -beinoftherealscalarmanifold .
p˜ V MV obtain U(1) (KK)-invariantfields
Dimensional reduction of 5D =2 YMESGT
N
Inthe“upstairs”orbifoldconstruction,onestartswith
a 5D theory, and compactifies on S1. It is sufficient for AIµ →AIµ+2CµAI.
our purposes to use the dimensionally reduced theory,
consisting of those fields satisfying ∂ Φ=0. The dimen- The dimensionally reduced bosonic Lagrangian be-
5
sional reduction breaks the 5D local Lorentz invariance comes [15]
e−1 = 1R 3 a◦ D h˜IDµh˜J 1e−2σ a◦ D AIDµAJ
DR IJ µ IJ µ
L − 2 − 4 − 2
(1e3σ+eσ a◦ AIAJ)C Cµν 1eσ a◦ I µνJ eσ a◦ AI J Cµν
− 2 IJ µν − 4 IJ FµνF − IJ Fµν (11)
e−1 4
+ C ǫµνρσ(AI J K +2AIAJ KC + AIAJAKC C )
2√6 IJK FµνFρσ Fµν ρσ 3 µν ρσ
g2P,
−
where h˜I eσhI, and and
≡
D AI ∂ AI +gAJfI AK (13)
µ ≡ µ µ JK
P ≡ 43e−3σ a◦IJ (AKfKILhL)(AMfMJNhN), (12) DCµh˜I ≡ 2∂∂µh˜IC+.gAJµfJIK˜hK ((1145))
µν [µ ν]
≡
5
Just as AI was redefined to be KK-invariant, we make e.g.); the fields in (3) can be written at the boundaries
µ
the further redefinitions either as left-chiral fermions with their right-chiral con-
jugates:
Ψi Ψi +ΨiC (16)
µ → µ 5˙ µ δp˜ 0
Γµ → Γµ+Γ5˙Cµ, (17) λp˜1 = 0 , λp˜2 = eδp˜∗ ,
(cid:18) (cid:19) (cid:18)− (cid:19)
sothatΨi andΓ arenowKK-invariant. Thedimension-
µ µ or right-chiral fermions with their left-chiral conjugates,
allyreducedsusytransformationsofthebosonicfieldsare
which we denote with a bar:
1
δ′emµ = 2¯ǫiΓmΨ(µ4i) λ¯p˜1 = eγ0p˜∗ , λ¯p˜2 = γ0p˜ .
1 1 √6 (cid:18) (cid:19) (cid:18) (cid:19)
δAI = hIǫ¯iΓ λp˜+ ie−σΨ¯iǫ ( h˜I +AI)
µ − 2 p˜ µ i 2 µ i 2 Assuming for simplicity that the only boundary-
1 √6 propagating vector fields are gauge fields for the bulk
δAI = − 2hIp˜ǫ¯iΓ5˙λpi˜+ 4 iψ¯iǫihI (18) gauge group K, consistent parity assignments in the up-
stairs picture allow the following boundary multiplets in
1
δeσ = ¯ǫiΓ5ψ the downstairs picture:
i
2
δC = 1e−σǫ¯iΓ5Ψ Multiplet Representation Type
µ µi
2 g , Ψ K singlet
µν µ α
i { }
δφx = 2faxǫ¯iλai, {Aαµ, λρi} adj[Kα] R
λ¯pi, za R [K/K ] R
V α
{ }
where Ψi = (Ψi,ψi); δ′ denotes the “bare” susy trans- Ψ , z0 K singlet
µˆ µ { 5˙ } α
formationfromfivedimensionsplusalocalLorentztrans-
formationtomaintaintheconditioneˆm =0;andwehave where we’ve split the index I =(0,α,a); x˜=(x,χ); and
identified the four-dimensional gravit5i˙ni to be p˜= (p,ρ). We have denoted the surviving gauge group
on the boundaries as K , and R [K/K ] is the repre-
α V α
Ψ(4) en Ψ + 1(Γn)−1Γ5Ψ . (19) sentation of the coset space elements. The value of n′ in
µi ≡ µ{ ni 2 5i} α =1,...,n′ and a = (n′+1),...,(nV +1) is arbitrary
fornow,sothereiscompletefreedominchoosingparities
The dimensionallyreducedLagrangiancanbewrittenin
forvectormultiplets;wewillconsiderthismorecarefully
terms of a K¨ahler scalar manifold with complex scalars
in section V. The second to last multiplet consists of
1 i a chiral multiplet in a real representation and its CPT
zI AI + h˜I. (20) conjugate. The case in which there are K-singlet vector
≡ 3 √2
fields propagating on the boundaries is straightforward.
More details of the dimensional reduction to 4D = 2 What happens when a non-compact group is gauged
N
supergravity can be found in [15]. in five dimensions? If the non-compactgaugefields were
assigned even parity, then a non-compact gauge group
Boundary propagating multiplets
It’s clear from appendix B that the action of Z on would appear in the 4D theory. However, there would
2
the supersymmetry spinors ǫi necessarily requires half not be the proper degrees of freedom to give a ground
state with compact gauge symmetry since the scalar
of the components to be odd, so that the original eight
degrees of freedom AI needed to form massive = 1
supersymmetry currents will be broken to four on the
N
vector multiplets must have odd parity. Therefore, the
boundaries, so that there is at most = 1 susy there.
Intermsofsymplectic-MajoranaspinoNrsǫi,theZ action non-compact gauge fields must be assigned odd parity.
2
We will then get = 1 chiral multiplets in the coset
is represented as
N
K/H, with H the maximal compact subgroup of K.
iΓ5ǫ1 and iΓ5ǫ2. Thisis anovelwayofobtaininga4DHiggssector,along
−
the lines of previous Higgs-gauge unifications in higher
(The4-componenteigenspinorsoftheZ2actionarelinear dimensions [16], since there will be a distinct scalar
combinations of the two symplectic-Majorana spinors.) potential for these scalars.
The zero mode susy parameters can be written in the
upstairs picture as (see table ?? in appendix B)
η ǫ(x5)ζ
ǫ1 = ǫ(x5)eζ∗ , ǫ2 = eη∗ , B. Hypermultiplet couplings [17]
(cid:18) (cid:19) (cid:18) − (cid:19)
andsodon’thaveawell-definedlimitonthefixed-planes. A colection of n hypermultiplets in five dimensions
H
In the downstairs picture, on the other hand, fermions consist of 2n fermions and 4n real scalars, the lat-
H H
will have a well-defined limit at the boundaries (see [8] ter parametrizing a quaternionic n -manifold with
H Q
M
6
tangent space groupUSp(2n ) SU(2) . We write the Multiplet Representation Type
H R
×
hypermultiplets as {ζ1n, qX1} RH[Kα] R or C
ζA,qX˜ , {ζ2n˜, qX2} RH[Kα] R or C
{ }
where X˜ = 1,...,4nH are the curved indices of MQ; where we have further split X = (X1,X2) with X1 =
and A = 1,...,2nH are flat, USp(2nH) indices. The 1,...,nH and X2 = nH + 1,...,2nH. We get a left-
4n -beinfX˜ relatescalarmanifoldcurvedandflatspace chiral multiplet and its CPT conjugate. Here R [K ] is
H iA H α
metrics the decomposition of R [K] under the group K K,
H α
⊂
and is the real-dimensionalrepresentation(see appendix
g fX˜fY˜ =ǫ C ,
X˜Y˜ iA jB ij AB A for conventions).
where i,j = 1,2 are SU(2) indices. Note that, in Example
R
contrast to the case of vector multiplets, the scalars Consider the “unified” MESGT with SU(5,1) global
form 2nH SU(2)R-doublets, while the 2nH fermions are symmetry group [18, 19], whose vector fields are in
SU(2)R-singlets.[35]In 2-componentspinornotation,we 1-1 correspondence with the traceless elements of the
write the fermions as LorentzianJordanalgebraJC [18]. Thetheoryis“uni-
(1,5)
fied” in the sense that all of the vector fields of the
ζA
ζA = 1 . (21) 5D theory (including the bare graviphoton) furnish the
ζ2A ! adj[SU(5,1)]. TheCIJK tensorisarank-3symmetricin-
variantof the globalsymmetry group,so its components
If non-trivial isometries of are gauged, they act on
Q are proportional to the d-symbols of SU(5,1).
M
the scalars as
As in [19], we can now couple hypermultiplets whose
δ qX˜ =αIKX˜(q), (22) scalars parametrize the quaternionic manifold
α I
where the KIX˜ are the Killing fields on the quaternionic E7
scalarmanifold;andαI arethesamelocaltransformation Q = (25)
M SO(12) SU(2)
parameters as in the pure YMESGT sector. The susy ×
transformations for the scalars are
to the MESGT based on JC , gauging the common
δqX˜ =−iǫ¯iζAfiXA˜. (23) SU(5,1) subgroup. Then th(1e,5)five-dimensional ground
state would have at most an SU(5) U(1) gauge group
The 5D bosonic hypermultiplet Lagrangian (coupled to ×
coupled to hypermultiplets in the 1 5 10. We may
a YMESGT) is ⊕ ⊕
then make parity assignments by associating each type
1 of index with an SU(5) representation:
eˆ−1 = g qX˜ µqY˜ 2g2V ViA, (24)
Lhyper −2 X˜Y˜Dµˆ D − iA
where α a 0 n n˜
√6 adj[SM] SU(5)/SM 5 ¯5 1 1 ¯5 10 c.c.
ViA hIKX˜fAi ⊕ ⊕ ⊕ ⊕
≡ 4 I X˜
The4Dlowenergyeffectivetheoryofeachboundarywill
and the K-covariantderivative is
have an = 1 supergravity multiplet; SM gauge mul-
D qX˜ qX˜ +gAIKX˜(q), tiplets; wNeak doublet and color triplet chiral multiplets
µ ≡Dµ µˆ I
both with a scalar potential; and left-chiral matter mul-
where µ is the covariant derivative associated with the tiplets (including a sterile fermion multiplet) along with
D
tangent space Lorentz and USp(2nH) SU(2) connec- their right-chiral conjugates. There are also the generic
×
tions. The dimensional reduction of the bosonic La- singlet left- and right-chiral multiplets coming from the
grangian is straightforwardand will not be quoted. 5D supergravitymultiplet, as well as chiralmultiplets in
Boundary propagating multiplets the symmetric space SU(5)/SM.
Half of the hypermultiplet field content is required to Remark:
have odd parity; therefore, let’s split the index X˜ = Atleastwithinthecategoryofhomogeneousquaternionic
(X,χ), with X =1,...,2n and χ=2n +1,...,4n . manifolds,alloccurencesofhypermultiplets inthe5and
H H H
We let qX be the even parity fields, and qχ the odd 10 of SU(5) come from spaces admitting an SO(10)
fields. Similarly, we spit the index A = (n,n˜) with isotropysubgroup under which these =2 hypermulti-
N
n = 1,...,n and n˜ = n + 1,...,2n . If we cou- plets join a singlet to form the 16 (see [19]). In the pre-
H H H
ple hypermultiplets in the quaternionic-dimensionalrep- vious example, we made a choice to truncate the =1
N
resentationR [K]ofthegaugegrouptoa5DYMESGT, left-chiralmultipletsinthe16ofSO(10)(andtheirright-
H
the multiplets with boundary propagatingmodes will be chiral conjugates in the 16).
7
C. Tensor multiplet couplings The new or modified terms in the bosonic 5D La-
grangianinvolving are [20]
When a MESGT with n abelian vector multiplets is
V
gauged,the symmetrygroupoftheLagrangianisbroken = eˆ a◦ BM BNµˆνˆ eˆ a◦ I BMµˆνˆ
to the gauge group K G. The n +1 vector fields LT −4 MN µˆνˆ − 2 IM Fµˆνˆ
V
decompose into K-reps ⊂ + 1 ǫµˆνˆρˆσˆλˆΩ BM BN
4g MN µˆνˆ∇ρˆ σˆλˆ
nV+1=adj(K)⊕non-singlets(K)⊕singlets(K). eg φx˜ µˆφy˜+ 1 C ǫµˆνˆρˆσˆλˆBM BN AI,
Such a gauging requires the non-singlet vector fields to − 2 x˜y˜Dµˆ D 2√6 MNI µˆνˆ ρˆσˆ λˆ
be dualized to anti-symmetric tensor fields [20] satisfy-
whereφx˜ arescalarsinbothvectorandtensormultiplets,
ing a field equation that serves as a “self-duality” con-
and
straint [21] (thus keeping the number of degrees of free-
dom the same):
φx˜ =∂ φx˜+Kx˜AI,
Dµˆ µˆ I µˆ
BM =cMǫ ρσλ∂ BN + , (26)
µν N µν [ρ σλ] ··· with Kx˜ the Killing vectors on the vector/tensor scalar
I
wherecM hasdimensionsofinversemass;squarebrackets manifold. The 5D field equations for the BM are (in
N µˆνˆ
denote anti-symmetricpermutations;andellipsesdenote terms of forms)
terms involving other fields. A tensor field does not re-
quire an abelian invariance to remain massless. ⋆DBM =gΩMN a◦ I˜,
We have already discussed the scalar sector of a pure MI˜H
5D YMESGT. When tensor multiplets are coupled the where I˜= FI . (28)
scalar manifold is again a real Riemannian space, but H BM !
which cannot be decomposed globally as a product of
“vector” and “tensor” parts. We can, of course, iden- The presence of non-trivially charged tensors also intro-
tify an orthogonal frame of scalars at each point of the duces a scalarpotentialP(T) that wasnotpresentin the
manifold: the vector multiplets are associated with the case of pure YMESGTs. The term in the Lagrangian
combinationhIφx˜ atagivenpoint,whilethetensormul- is [20]
x˜
tiplets are associated with the independent combination
hxM˜ φx˜. Similarly, the combination of fermions hIp˜λp˜i are LP(T) =−2g2eˆWp˜Wp˜,
associatedwithvectormultiplets,while hMp˜ λp˜i withten- √6 (29)
sor multiplets. We will write φx˜ and φm˜ to denote the where Wp˜=− 8 hpM˜ ΩMNhN.
scalar partners of the vector and tensors, respectively,
at any given point of the scalar manifold. Similarly, we
write λp˜i and λℓ˜i as the fermionic partners of the vec- Dimensional reduction
In the dimensional reduction, we parametrize the ten-
tor and tensor fields, respectively. It is then implicitly
sor field as
understoodthatthe meaning ofthis notationis givenby
the above discussion.
0 A˜M
Whentensorsarepresent,wewilluseindicesI,J,Kfor BM = − ν ,
5D vector fields and M,N,P for 5D tensor fields. Then µˆνˆ A˜Mµ BµMν !
n tensor multiplets are
T
where tildes have been used to help distinguish from 4D
BM,λℓ˜i,φm˜ . vectorsarising as components of 5D vectors. The result-
{ µˆνˆ } ingLagrangiancontainingbothA˜M andBM isanalogous
µ µν
To be consistent with the gauge symmetry, the compo- to the 1st order formulation of the Freedman-Townsend
nents of the C-tensor are constrained to be [20]: model [23]. One can obtain a 2nd order formulation in
terms ofthe A˜M by using the Euler-Lagrangeequations,
√6 µ
C = Ω ΛP which appear as constraints relating these fields in the
IMN 2 NP IM (27) dimensionally reduced theory. However, in the case of
CMNP =0, CMIJ =0, dimensionalreduction,thereisanobstruction[24]toob-
tainingalocal 2ndorderLagrangianintermsoftheBM.
where Ω is antisymmetric and ΛP are symplec- µν
NP IM This will not be present in the case of the orbifold.
tic K-representation matrices appearing in the K- The ξ5 transformations of the dimensionally reduced
transformationofthe tensor fields: δαBµMν =αIΛMINBµNν. fields BM and A˜M are
Furthermore, C must be a rank-three symmetric K- µν µ
IJK
invariant tensor. Note: We are assuming a compact or
δ BM =∂ ξ5A˜M ∂ ξ5A˜M
non-semisimplegaugegroupK;see [22]formoregeneral ξ5 µν µ ν − ν µ (30)
couplings where C =0. δ A˜M =0.
MIJ 6 ξ5 µ
8
We must therefore make a field redefinition 1. Odd Parity ΩMN
BM BM 4C A˜M (31)
µν → µν − [µ ν] If the self-duality relation is used to express all tensor
fields in terms of the vectors A˜M, the mass of the A˜M
so that the BM are now KK-invariant. The full di- µ µ
µν is non-vanishing at the orbifold fixed points. However,
mensional reduction of tensor-coupled YMESTs is given
thereareinsufficientfermionicdegreesoffreedomtoform
in [24]. The new or modified terms in the 5D bosonic
massive = 1 vector multiplets. Therefore, we must
Lagrangianare use the ENuler-Lagrange equations to write A˜M BM.
µ → µν
e−1 (4) = e−2σa◦ (D AI)A˜µM 1e−2σa◦ A˜MA˜µN There is an obstruction to doing this with the dimen-
L − IM µ − 2 MN µ sionally reduced 1st order Lagrangianobtained by using
− 34a◦I˜J˜(Dµh˜I˜)(Dµh˜J˜)+ 2e√−16CMNIǫµνρσBµMνBρNσAI KthKe-tinovpaorlioagnictafiletlderrmedoeffintihteionfosrm[24Ω],MinNpCaµrνtiAc˜uMµlaA˜rNνd,uebutot
the problem is avoided in the orbifold theory since the
e−1 e−1
+ ǫµνρσΩ C A˜MA˜N + ǫµνρσΩ BMD A˜N obstruction vanishes on the 4D fixed planes.
g MN µν ρ σ g MN µν ρ σ Once the 2nd order Lagrangian with tensors is ob-
1eσa◦ BMBNµν 1eσa◦ ( I +2C AI)BMµν tained, the BµMν can then be Hodge dualized to scalars
− 4 MN µν − 2 IM Fµν µν BM by adding a term of the form
g2P,
− e−1∆ =ǫµνρσΩ M D BN (38)
(32) L MNBµνρ σ
to the Lagrangian (if this step is performed before inte-
where
grating over the fifth dimension, one needs a δ(x5) fac-
DµA˜Mν ≡ ∂µA˜Nν +gAIµΛNIPA˜Pν (33) toonr)t,hewshcearlearDs,ρainsdtheMga=ug3e!c∂ovBarMian.t derivative acting
h˜I˜ ∂ h˜I˜+gAIMI˜ h˜K˜. (34) Bµνρ [µ νρ]
Dµ ≡ µ µ IK˜ The multiplets that will propagateon the fixed planes
are
The total scalar potential, P, is now
Multiplet Representation Type
P =2e−σWp˜Wp˜
λ¯ℓi, zM N N R
+ 3e−3σa◦ (AIMI˜ hK˜)(AJMJ˜ hL˜), (35) { } ⊕
4 I˜J˜ IK˜ JL˜ Thatis,thereareleft-chiralmultipletsinarealrepresen-
tation along with the CPT conjugate right-chiral multi-
where Wp˜ is defined in eqn (29), and
plets.
Example
MJ˜ = fIJK 0 , (36) Consider the “unified” 5D MESGT based on the
(I)K˜ 0 ΛNIM ! Lorentzian Jordan algebra JC , whose global symme-
(1,5)
try group is SU(5,1) [18]. We can couple this the-
and I˜=(I,M).
ory to hypermultiplets whose scalars parametrize the
Parity assignments of tensor sector particular scalar manifold (25). If we gauge the com-
Since (30) is only true for KK-transformations con- mon SU(5) U(1) SU(5,1) subgroup, we will get
nected to the identity, the A˜M are not necessarily even SU(5) U(1×) gauge m⊂ultiplets, along with tensor multi-
µ
under Z2 action. However,the abovedo lead to the con- plets in×the 5 ¯5 and hypermultiplets in the 1 5 10.
⊕ ⊕ ⊕
straint This is then similar to the ground state theory in the
SU(5,1) gauging example before, but with some impor-
P(A˜M)= P(BM),
µ − µν tant differences; one of which being that the 5D bare
graviphotonA0 does nottakepartingaugingthe isome-
componentwise. These two fields do not describe inde- µˆ
tries of the scalar manifold. We can make the assign-
pendent propagating degrees of freedom since they are
ments
related by a constraint equation (coming from the fact
that the 5D tensors satisfied a “self-duality” field equa- α a 0 M n n˜
tion (26) reducing the number of propagating modes): adj[SM U(1)] SU(5)/SM 1 5 ¯5 1 ¯5 10 c.c.
× ⊕ ⊕ ⊕
BµMν =cMN(⋆DA˜N)µν +··· , (37) The propagating modes along the fixed planes will be
SU(3) SU(2) U(1)2gaugefields;weakdoublet(Higgs)
wherecM isproportionaltoΩMP a◦ ;⋆istheHodgeop- chiral m×ultiplet×s; color triplet chiralmultiplets; and left-
N PN
erator;andthedotsindicatetermsinvolvingotherfields. chiralmatter multiplets (including asterilefermionmul-
There are two classes of assignments we can make, char- tiplet) with their CPT conjugates. Again, there is also
acterized by the parity of the symplectic form Ω on the generic singlet spin-1/2 multiplet coming from the
MN
the vector space spanned by the 5D tensors. 5D supergravity multiplet, and chiral multiplets in the
9
symmetric space SU(5)/SM. All of these multiplets are assigned odd parity. The field dependent objects are the
tree-level massless, while the scalars in the 5 ¯5 have a restrictedambientspacemetric a◦ (φ) andscalarmani-
⊕ IJ
potential term. foldmetricsg (φ)andg (q);thehI(φ);thescalarviel-
xy XY p
bein fp(φ) and fX(q); the Killing vectors on the scalar
x iA
manifoldKI(φ)andKI (q). These vanish when assigned
2. Even Parity for ΩMN odd parity dxue to the foXrmofthe bosonic fieldexpansion
in eqn (1).
If Ω has even parity, then the Z action acts re-
MN 2 Pure YMESGT
ducibly on the symplectic vector space and projects out
In[14],itwasshownthattheC definingaMESGT
IJK
halfofthebosonicfields. Consistencywithsupersymme-
may be put in a “canonical” basis satisfying the positiv-
try requires that we use the field equation (37) to elimi-
ity of = C ζIζJζK (see (5)). The parity assign-
nate BM from the Lagrangian. Splitting M = ( , ¯) V IJK
µν M M mentsofthecomponentsaredeterminedbyrequiringthe
twhietnh hMav=e a1,m.a.s.s,nmTa/t2riaxnd M¯ = nT/2,...,nT, the A˜Mµ pio=ly(nαo,ma)i,alwVe htoavbee invariant under Z2 action. Splitting
M e−2σ a◦ . Even Odd
MN MN
≡
C C C C C C C C
000 0ab 0αβ abc aαβ 0aα αβγ αab
Therefore,then /2vectors,n spin-1/2fields,andn /2
T T T
scalars form n /2 massive =1 vector multiplets. There is freedom in choosing ǫ(x5) or κ(x5) as the jump-
T
N
Themultiplets withpropagatingmodes onthebound- ing function for odd components.
aries are Consistency of the infinitesimal gauge transforma-
tions (6) require parity assignments for the fI and
JK
Multiplet Representation Type αI(x) to be
AM, λℓ¯i, hM NR R
{ } Even Odd
Thenotationfortherepresentationmeansthatthegauge fα fα fa fa
βγ ab bc αβ
groupatthefixedpointsmustsupportareal N(whereas
f0 f0 f0
the 5D gauge grouphad to support a complex N). Let’s aβ α0 αβ
αβ α0 αb
illustrate this with an example.
Example wherefI vanishesifanyoftheindicescorrespondto5D
JK
Theminimalexampleinwhichoneisleftwithagroup spectator vector fields [36]; and permutations of the in-
containing SM is where the 5D gaugegroupis SU(10) dices have the same parity. The (upstairs picture) gauge
×
U(1). Starting with the “unified” MESGT defined by transformation parameters are subject to an expansion
the LorentzianJordanalgebraJC andwith SU(N,1) on S1/Z just as in (1). Consistency of the 5D gauge
(1,10) 2
global symmetry of the Lagrangian, we can gauge the algebra requires that odd fI be redefined by ǫ(x5)fI ,
JK JK
SU(10) U(1) subgroup, yielding tensors in the 10 where the fI are now even [37].
JK
× ⊕
10. If the symplectic form has even parity, then the Thecomponentsoftherestrictedambientspacemetric
orbifold conditions require the group to be broken to at and scalar manifold metric have parities determined by
leastSO(10) U(1),underwhichwehavemassivevector therequirementthatthe lineelementsofthosespacesbe
×
multiplets in the (real) 10. This theory will appear as a preserved:
ground state of the 4D = 1 theory with gauge group
(SO(10) U(1))⋉ T10,Nanalogous to the = 2 case Even Odd
discussed×in [24]. There are also chiral muNltiplets from a◦αβ a◦ab a◦aβ a◦0α
the broken gauge multiplets forming the 54, along with a◦ a◦
00 0a
their CPT conjugates.
g g g
xy χψ χy
Consistency of the gauge transformations (6) determine
IV. OBJECTS OTHER THAN FIELDS the parities of the scalar manifold Killing fields:
Even Odd
There are field-dependent and independent objects
Kx Kx Kx
thatappearintheLagrangianandsupersymmetrytrans- α 0 a
formations that carry Z2 parity. In particular, the field K0χ Kaχ Kαχ
independent objects are the CIJK tensor defining the Thenon-zerocomponentsKχ,whichareasetofsections
a
MESGT that exists prior to gauging; the structure con-
ofthe normalbundle overthe 4D scalarmanifold, deter-
stants fJIK and transformation parameters αI(x) of the mine the form of the scalar potential involving the φx
5D gauge group; and the symplectic tensor ΩMN and and Aa at the fixed points:
transformation matrices ΛM in the tensor coupled the-
IN
ory. Thesecontainanimplicit ǫ(x5)orκ(x5)factorwhen g KχKψAaAb, (39)
χψ a b
10
whereg isthemetricdeterminedbythenormalbundle where we have split ℓ˜=(ℓ,ℓ¯) and m˜ =(m,m¯).
χψ
connection. Hypermultiplet couplings
Finally,thefunctionshI andhIp;thescalarvielbeinfxp; The parity assigmentsfor the Killing vectorsand viel-
and the functions hI =hIfp are required to satisfy bein of the quaternionic scalar manifold are required to
x p x
be
Even Odd
Even Odd
h0 ha hα
KX1 KX2 KX1 KX2
h0 ha hα ha hα α a a α
p p ρ ρ p fX1 fX1 fX1 fX1
fp fρ fp fρ 1A1 2A2 2A1 1A2
x χ χ x fX2 fX2 fX2 fX2
h0 ha hα ha hα 2A1 1A2 1A1 2A2
x x χ χ x
Tensor couplings
V. PARITY ASSIGNMENTS OF THE SCALAR
In the tensor-coupled theory, parity assignments de-
SECTOR
pend on the parity of the symplectic form Ω . The
MN
parities of the additional C-tensor components must be
In the previous sections, we listed the boundary mul-
tiplets by “orbifolding” 5D YMESGTs coupled to ten-
Even Odd sor multiplets. It’s clear, for example, that 5D vector
Even Odd multiplets should yield 4D boundary vector multiplets
CMN¯α CMNα CM¯N¯α C C whentheassociatedscalarmanifoldembeddingfunctions
MNa MNα
CMNa CM¯N¯a CMN¯a P(Ω )= Ω hI(φ) are assignedodd parity, so that the parities of the
MN MN
P(ΩMN)=+ΩMN − hI induce parity assignments for the physicalscalars φx˜.
However,itisnotimmediatelyclearhow theassignments
Consistencyofthegaugetransformationsrequiretherep-
are transmitted from one to the other. That is, how are
resentation matrices to satisfy
thephysicalscalarstruncatedinlightofthefactthatthe
‘odd’scalarpartnersofthe‘even’vectorsAαarethecom-
Even Odd µˆ
ΛM ΛM¯ ΛM¯ ΛM Even Odd binationshhαx˜iφx˜,where hxα˜ ∝hα,x˜. We willconsiderthis
αN αN¯ αN αN¯ using two examples based on symmetric “very special”
ΛM0¯N ΛM0N¯ ΛM0¯N¯ ΛM0N ΛMαN ΛM0N ΛMaN real scalar manifolds.
ΛM¯ ΛM ΛM¯ ΛM P(Ω )= Ω The first example is of the “generic Jordan” family of
aN aN¯ aN¯ aN MN − MN
P(Ω )=+Ω Maxwell-Einstein supergravity theories (MESGTs) with
MN MN
symmetric scalar manifolds of the form [14]
Ifoddgaugetransformationparametersαa areexpanded
as in eqn (1), odd ΛM ǫ(x5)ΛM for consistency of SO(nV 1,1)
the gauge algebra; thIeNre→lation C IN Ω ΛP from MR =SO(1,1)× SO(n − 1) ,
eqn (27) then requires odd C MNIǫ∝(x5)MCP IN. Note V −
IMN IMN
→ where n is the number of vector multiplets. The cubic
that when Ω has odd parity, the set of coefficients V
MN
◦ polynomialforthetheoryinthe absenceofanorbifoldis
cM ΩMP a areodd; a choicethatwasmade in[10].
NA∼s in the pPuNre YMESGT case, the ambient space and V =CIJKξIξJξK, where
scalar manifold line elements should be preserved under 1
the Z2 action so that C000 =1, C00i =0, C0ij =−2δij,
1 1
Even Odd C111 = , C1ab =+ δab,
◦ ◦ ◦ Even Odd √2 √2
a◦MN a◦M¯N¯ ◦ aMN◦¯ a◦aM a◦αM with a,b = 2,...,n . On S1/Z , the C
a a a a V 4 2 IJK
M¯α Ma Mα M¯a gx˜y˜ gmn gx˜m canhave odd components satisMfying×jumping conditions,
g g g
x˜y˜ mn x˜m P(Ω )= Ω though we take h0 to always have even parity.
MN MN
P(ΩMN)=+ΩMN − Forexample,assigningoddparityto h1 andC (and
1ij
redefining C ǫ(x5)C ), the polynomial is
Finally, the functions hM(φ) and hM; the vielbein fℓ ; 1ij → 1ij
ℓ m
and the functions hIx =fxℓhIℓ satisfy 2 32 3 ǫ(x5)
= (ξ0)3 ξ0δ ξiξj (ξ1)3
ij
V 3 − 2 − √2
Even Odd Even Odd (cid:18) (cid:19) (cid:20)
hM(φ) hM¯(φ) hM(φ) +3ǫ(x5)ξ1[(ξ2)2+ +(ξnV)2] .
hMm hMm¯¯ hMm¯ hMm¯ hMm √2 ··· (cid:21)
hMℓ hMℓ¯˜ hMℓ˜ hMℓ¯ hMℓ hMℓ¯ Restricting to the scalar hypersurface = 1, the terms
P(ΩMN)=+ΩMN P(ΩMN)=−ΩMN withh1 ≡ξ1|V=1vanish(sinceh1isapaVrity-oddfunction