On the Scalar Manifold of Exceptional Supergravity

0.12 MB·English

by  Sergio L. Cacciatori

CERN-PH-TH/2012-023
On the Scalar Manifold of Exceptional
Supergravity
Sergio L. Cacciatori1,4, Bianca L. Cerchiai2,4, and AlessioMarrani3
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n 1DipartimentodiScienzeedAltaTecnologia,
a
Universita` degliStudidell’Insubria,ViaValleggio11,22100Como,Italy
J
[email protected]
1
3
2DipartimentodiMatematica,

] Universita` degliStudidiMilano,ViaSaldini50,20133Milano,Italy
h [email protected]
t
-
p 3PhysicsDepartment,TheoryUnit,CERN,
e
CH1211,Geneva23,Switzerland
h
[ [email protected]

1 4INFN,SezionediMilano
v
ViaCeloria,16,20133Milano,Italy
7
6
6
6
.
1
0
2
Abstract
1
v: WeconstructtwoparametrizationsofthenoncompactexceptionalLiegroupG=E7(−25),basedona
i fibrationwhichhasthemaximalcompactsubgroupK = E6×U(1) asafiber.ItiswellknownthatGplays
X animportantroleintheN =2d=4magicexceptionalsupZe3rgravity,whereitdescribestheU-dualityof
r thetheoryandwherethesymmetricspaceM= G givesthevectormultiplets’scalarmanifold.
a K
First,bymakinguseoftheexponentialmap,wecomputearealizationof G,thatisbasedontheE
K 6
invariantd-tensor,andhenceexhibitsthemaximalpossiblemanifest[(E ×U(1))/Z ]-covariance.This
6 3
providesabasisforthecorrespondingsupergravitytheory,whichistheanalogueoftheCalabi-Vesentini
coordinates.
ThenwestudytheIwasawadecomposition.ItsmainfeatureisthatitisSO(8)-covariantandtherefore
it highlights the role of triality. Along the way we analyze the relevant chain of maximal embeddings
whichleadstoSO(8).
Itisworthnoticingthatbeingbasedonthepropertiesofa“mixed”Freudenthal-Titsmagicsquare,the
wholeprocedurecanbegeneralizedtoabroaderclassofgroupsoftypeE .
7
TalkgivenattheXVIIEuropeanWorkshoponStringTheory,heldattheUniversityofPadua,September5-9,2011
1 The “mixed” magic square and the 56 of the Lie algebra e
7( 25)
−
ExceptionalLiegroupsactassymmetriesinmanyphysicalsystems. Inparticular,noncompactformsofthe
groupE enterasU-dualityofd= 3andd = 4supergravitytheories. Herewefocusonthe = 2d = 4
7
N
magicexceptionalsupergravity,wheretherelevantrealformisG=E .
7( 25)
As the first step we needto constructthe Lie algebrae . To th−is aim, we are goingto followthe
7( 25)
techniqueoutlinedinSec. 7of[1],whichisbasedonthenon−-symmetric“mixed”magicsquare[2,3,4]:
Table1:The“mixed”magicsquare
R C H O
R SO(3) SU(3) USp(6) F
4( 52)
C SU(3) SU(3) SU(3) SU(6) E −
6( 78)
H Sp(6,R) SU⊕(3,3) SO (12) E −
S ∗ 7( 25)
O F E E E −
S 4(4) 6(2) 7( 5) 8( 24)
− −
TherowsandthecolumnscontainthedivisionalgebrasoftherealnumbersR,thecomplexnumbersC,
thequaternionsH,theoctonionsOandtheirsplitformsH andO .
S S
ThentheTitsformulagivestheLiealgebra correspondingtorowAandcolumnBas[4]:
L
(A,B)=Der(A) Der(J (B))∔(A J (B)). (1)
L ⊕ 3 ′⊗ ′3
Here, the symbol denotes direct sum of algebras, whereas ∔ stands for direct sum of vector spaces.
Furthermore, Der m⊕eans the linear derivations, J (B) denotes the rank-3 Jordan algebra on B, and the
3
priming amounts to considering only traceless elements. One of the main ingredients entering in the last
termistheLieproduct,whichextendsthemultiplicationtoA J (B). ItsexplicitexpressionforA=H
andB=Ocanbefounde.g.in[5]. ′⊗ ′3 S
FortheLiealgebraofE theTitsformula(1)yields:
7( 25)
−
e = (H ,O)=Der(H ) Der(J (O))∔(H J (O))=sl(2,R) f ∔(H J (O)).(2)
7(−25) L S S ⊕ 3 ′S⊗ ′3 ⊕ 4 ′S ⊗ ′3
The second step is to identify the subalgebra K generating the maximal compact subgroup
K :=(E U(1))/Z of E . This can be achieved by using the Tits formula (1) once more
6( 78) 3 7( 25)
tocompute−them×anifestlyf -covaria−ntexpressionfore :
4 6( 78)
−
e = (C,O)= (R,O)∔(i J (O))=Der(J (O))∔(i J (O))=f ∔(i J (O)),(3)
6(−78) L L ⊗ ′3 3 ⊗ ′3 4 ⊗ ′3
wherewearepickingtheonlyimaginaryuniti H whichsatisfiesi2 = 1. Thus,weobtain:
S
∈ −
K=ad Der(J (O))∔(i J (O)), (4)
i⊕ 3 ⊗ ′3
with ad H the adjoint action of i, generating the maximal compact subgroup U(1) of the group
i S
SL(2,R)a∈ppearingin(2).
An explicitconstructionofthematricesφ , I = 1,...,78, realizingthe e subalgebrainits irre-
I 6( 78)
duciblerepresentationFund = 27hasbeenperformede.g. inSec. 2.1of[6]b−ymakinguseof(3)andof
theexplictexpressionoff initsirrep. Fund=26previouslycomputedin[7].
4( 52)
Finally,byputtingtogeth−eralltheseingredients,wefindthatanexplicitsymplecticrealizationoftheLie
algebrae initsirreduciblerepresentationFund=56isasfollows[8].
7( 25)
Thegenera−torsofthemaximalcompactsubgroupK (antihermitianmatrices):
φI −→027 027 −→027
T T
 −→0 0 −→0 0 
27 27
e : Y = , I =1,...,78; (5)
6( 78) I
−  027 −→027 −φTI −→027 
 T T 
 −→0 0 −→0 0 
 27 27 
 
1
√i6I27 −→027 027 −→027
 −→0T i 3 −→0T 0 
27 − 2 27
u(1): Y = ; (6)
79  027 −→0q27 −√i6I27 −→027 
 
 −→0T27 0 −→0T27 i 32 
 q 
Thegeneratorsofthecoset =G/K (hermitianmatrices):
M
027 −→027 2iAα i√2−→eα
1 −→0T27 0 i√2−→eTα 0 
Y = , α=1,...,27; (7)
α+79 2 −2iAα −i√2−→eα 027 −→027 
 
 −i√2−→eTα 0 −→0T27 0 
 
027 −→027 2Aα √2−→eα
−
1 −→0T27 0 √2−→eTα 0 
Y = , α=1,...,27. (8)
α+106
2 −2Aα √2−→eα 027 −→027 
 √2−→eTα 0 −→0T27 0 
 
HereIn isthen n identitymatrix,027 isthe27 27nullmatrix,−→0n isthezerovectorinRn, and
e ,α=1,...,27,is×thecanonicalbasisofR27. ×
−→α
ThematricesA aredefinedintermsofthed-tensorofthe27ofE . Thereisacubicform,which
α 6( 78)
isdefinedforanyj ,j ,j J (O)as[9,10,11]: −
1 2 3 3
∈
1 1 1
Det(j ,j ,j ):= Tr(j j j )+ Tr(j )Tr(j )Tr(j ) Tr(j )Tr(j j )+cyclicperm. ,(9)
1 2 3 1 2 3 1 2 3 1 2 3
3 ◦ ◦ 6 −6 ◦
(cid:16) (cid:17)
where is the product in J (O). By choosing a basis j of J (O) normalized as j ,j :=
◦ 3 { a}a=1,...,26 ′3 h a bi
Tr(j j ) = 2δ , a completionto a basis forJ (O) can beobtainedbyaddingj = 2I . Thenthe
a ◦ b ab 3 27 3 3
matricesAα’sare 27 27symmetricmatrices, whosecomponents,explicitlycomputediqn[5], satisfythe
×
followingrelation[9]:
3 1
(A )β = Det(j ,j ,j )=: d , (10)
α γ 2 α γ β √2 αγβ
where d = d is the totally symmetric rank-3 invariant d-tensor of the 27 of of E , with a
αγβ (αγβ) 6( 78)
normalization suitable to match Det(j ,j ,j ) given by (9). Whenever the choice of the b−asis j is
α γ β α
{ }
exploitedinordertodistinguishtheidentitymatrixfromthetracelessones,thed ofE hasamaximal
αβγ 6
manifestlyF -invarianceonly. However,itiscrucialto pointoutthat, beingexpressedonlyin terms
4( 52)
ofthe invarian−td-tensor, theresult(10) doesnotdependonthe particularchoiceof the basis j . Thus,
α
theexpressionsofY (7)andofY (8) exhibitthemaximalmanifestcompact[(E {U(1}))/Z ]-
α+79 α+106 6 3
×
covariance.
AcoupleofremarksonthepropertiesofthematricesY ’sareinorder.Thefirstisthattheysatisify:
A
Y usp(28,28), A=1,...,133. (11)
A
∈
Moreover, in order to guarantee that the period of the maximal torus in the E subgroup equals 4π, the
6
standardchoice for the periodof the spin representationsof the orthogonalsubgroups[7, 6], the matrices
2
1
YA’s are orthonormalizedas Y,Y′ 56 := Tr(YY′) with signature( 79,+54). As a consequence,the
h i 12 −
components(A )β :=A arenormalizedasA Aηβγ =5δη.
α γ αβγ αβγ α
This is consistent with the normalization of the d-tensor (of E ) adopted e.g. in [12], which is
6( 26)
dictatedbytheexpressionf(z) := 1d zαzβzγ fortheKa¨hler-inv−ariant( X0 2-rescaled)holomorphic
3! αβγ
prepotentialfunctioncharacterizingspecialKa¨hlergeometry(seee.g.[13,14,15],andRefs. therein).
(cid:0) (cid:1)
2 Manifestly [(E U(1))/Z ]-covariant Construction of the Coset
6 3
×
M
In this Section we construct a manifestly [(E U(1))/Z ]-covariant parametrization of the symmetric
6 3
×
space = E7(−25) . As we have seen in the previous Sec. 1, it is generated by the matrices
Y ,M(7)and(E(86()−w78i)t×hUI(1=))/1Z,3...,54.Throughtheexponentialmapping,itcanbedefinedasfollows:
79+I
27
:=exp x Y +y Y , withx R,y R, forα=1,...,27. (12)
α 106+α α 79+α α α
M ! ∈ ∈
α=1
X
Inordertomakethecomplexstructureof manifest,itisconvenienttointroducethefollowingcom-
M
plexlinearcombinationsofthematrices:
1 1
ζ := (Y +iY ), ζ¯ := (Y iY ) (13)
α 79+α 106+α α 79+α 106+α
√2 √2 −
togetherwiththecorrespondingcomplexlinearcombinationsoftheparameters:
1 1
z := (y +ix ), z¯ := (y ix ), (14)
α α α α α α
√2 √2 −
whichallowstorewrite(12)as
27
:=exp z¯ ζ +z ζ¯ . (15)
α α α α
M !
α=1
X
27
Byintroducingthe27dimensionalcomplexvectorz := z ~e ,describingthescalarfields,andthe28
α α
×
α=1
X
27
√2 z¯ A z
α α
−
28matrix := α=1 ,theexpressionfor (15)enjoysthesimpleform:
A X M
 zT 0 
 
 
Sh(√ )
Ch( †) A†A
AA A √
:=exp 0 A = p A†A . (16)
M A† 0 !  Sh(pAA†)A† Ch(√A†A) 
 † 
 AA 
ThisisaHermitianmatrix,ofthesamefpormasthefinitecosetrepresentativeworkedout[16]forthesplit
(i.e. maximally non-compact)counterpart = E7(7) , which is the scalar manifold of maximal
= 8,D = 4supergravity,associatedtoJM(NO=8). HoSwUe(8v)e/rZ,2while isreal,becauseof(11) is
3 S =8
N MN M
anelementofUSp(28,28).
3
By using the machinery of special Ka¨hler geometry (see e.g. [13, 14, 15], and Refs. therein), the
symplecticsectionsdefiningthesymplecticframeassociatedtothecosetparametrizationintroducedabove
canbedirectlyreadfrom(16):
uΛ(z,z) v (z,z)
i iΛ
=: , (17)
M  
viΛ(z,z) ui (z,z)
Λ
 
where the symplectic index Λ = 0,1,...27 (with 0 pertaining to the = 2, D = 4 graviphoton), and
N
i=α,28. Thus,thesymplecticsectionsread(seee.g.[17,15]andRefs. therein;subscript“28”omitted):
fΛ : = 1 (u+v)Λ = fΛ,fΛ := LΛ,LΛ =exp 1K XΛ,XΛ ; (18)
i √2 i α Dα 2 Dα
(cid:16) (cid:17) (cid:16) (cid:17) (cid:18) (cid:19)(cid:16) (cid:17)
i 1
h : = (u v) = h ,h := M ,M =exp K F ,F , (19)
iΛ −√2 − iΛ α|Λ Λ Dα Λ Λ 2 Dα Λ Λ
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where istheKa¨hler-covariantdifferentialoperator,
D
:= LΛ,M T =exp 1K XΛ,F T (20)
Λ Λ
V 2
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
isthesymplecticvectorofKa¨hler-covariantlyholomorphicsections,and
K := ln i XΛF XΛF (21)
Λ Λ
− −
h (cid:16) (cid:17)i
istheKa¨hlerpotentialdeterminingthecorrespondinggeometry. Amoreexplicitexpressionfor(16)would
be needed in order to check that the prepotential F does not exist (i.e., 2F = XΛF = 0 [18]) in the
Λ
symplectic frame we have just introduced, which can be considered the analogue of the Calabi-Vesentini
basis[19,18],whosemanifestcovarianceisthemaximalone.
3 The Iwasawa Decomposition and the role of triality
Now we are goingto find anotherparametrizationfor the coset , providedby the Iwasawa decomposi-
M
tion. Inthiscasethemaximalmanifestcovarianceisbrokendowntoa subgroupSO(8), thusprovidinga
manifestlytriality-symmetricdescription.
The manifold has rank 3, which means that the maximal dimension of the intersection between a
M
CartansubalgebraofE andthegeneratorsof is3. Inparticular,wecanpick3suchgeneratorsto
7( 25)
bethediagonalgenerators−oftheJordanalgebraJ (OM)itself,namelyh =Y ,h =Y andh =Y .
3 1 123 2 132 3 133
The followingstep is to determinea basis of 54 3 = 51 positive rootsλ+, i = 1,...,51 with
W+ − i
respecttoH . ThentheIwasawadecompositionofthecoset isdefinedas:
3
M
51
:=exp(x h +x h +x h )exp( y λ+). (22)
M 1 1 2 2 3 3 i i
i=1
X
Asanticipated,oneofitsmainfeaturesisthatsincetheelementsh ,h , h h commutewitha28-
1 2 3 3
∈
dimensionalsubalgebraso(8), theIwasawaparametrizationof exhibitsamaximalmanifestcovariance
M
given by SO(8). Therefore, the 51-dimensional linear space Λ generated by the positive roots is
+ +
W
invariantunderthe(adjoint)actionofSO(8),anditdecomposesintoirreps. ofSO(8)as:
Λ =13+82+82+82, (23)
+ v c s
whichisamanifestlytriality-symmetricdecomposition.Inparticular,atthelevelofalgebrasso(8)=tri(O)
withtheautomorphismgroupAut(t(O))=Spin(8)ofthenormedtrialityovertheoctonionsO[20].
4
Itisworthremarkingthattheappearanceofthesquareforthethree8irreps. in(23)isaconsequenceof
thecomplex(specialKa¨hler)structureofthecoset .
M
Moreover,itshouldbeobservedthattheSO(8)enteringin(23)canbeidentifiedas:
SO(8) (SO(10) U(1)) F . (24)
4( 52)
⊂ × ∩ −
This can be und(cid:2)erstood by noticing that it ca(cid:3)n be obtained from both the following chains of maximal
symmetricembeddings[21]:
E7( 25) E6( 78) U(1)′ SO(10) U(1)′ U(1)′′ SO(8) U(1)′ U(1)′′ U(1)′′′(25)
− ⊃ − × ⊃ × × ⊃ × × ×
and
E7( 25) E6( 78) U(1)′ F4( 52) U(1)′ SO(9) U(1)′ SO(8) U(1)′. (26)
− ⊃ − × ⊃ − × ⊃ × ⊃ ×
Inthelastlineof(25)thefirsttwoU(1)factorshavethephysicalmeaningof“extra”T-dualitiesgenerated
bytheKaluza-Kleinreductions,respectivelyD =5 D =4,andD =6 D =5.
DenotingwithsubscriptsU(1)-charges,theadjoi→ntirrep.133ofE →branchesaccordingto(25)as
7( 25)
(seee.g.[22]): −
133 = 78 +1 +27 +27
0 0 2 ′+2
−
= 1 +16 +16 +45 +1
0,0 0, 3 ′0,+3 0,0 0,0
−
+1 +10 +16 (27)
2,+4 2, 2 2,+1
− − − −
+1+2, 4+10+2,+2+16′+2, 1
− −
= 1 +8 +8 +8 +8
0,0,0 c,0, 3,1 s,0, 3, 1 c,0,+3, 1 s,0,+3,+1
− − − −
+1 +8 +8 +28 +1
0,0,0 v,0,0,+2 v,0,0, 2 0,0,0 0,0,0
−
+1 +1 +1 +8 +8 +8
2,+4,0 2, 2,+2 2, 2, 2 v, 2, 2,0 c, 2,+1,+1 s, 2,+1, 1
− − − − − − − − − − −
+1 +1 +1 +8 +8 +8 .
+2, 4,0 +2,+2, 2 +2,+2,+2 v,+2,+2,0 c,+2, 1, 1 s,+2, 1,+1
− − − − −
4 Final Remarks
Itisveryinterestingtoremarkthatbeingbasedonlyonthealgebraicpropertiesofthe“mixed”Freudenthal-
Tits magic squarein Table 1, the constructionof the basis with the maximalpossible covariance(16) and
the computationof the Iwasawa decomposition(22) describedhere can be bothgeneralized[8] at least to
abroaderclassofminimallynon-compact,simplegroupsoftypeE [23]. Moreover,italsoturnsoutthat,
7
like for , in all these cases the maximal covariance (at least at the Lie algebra level) of the Iwasawa
M
decompositionisgivenbytheautomorphismalgebraofthecorrespondingnormedtriality[20].
Acknowledgements
TheworkofB.L.C.hasbeensupportedinpartbytheEuropeanCommissionundertheFP7-PEOPLE-IRG-
2008GrantNo. PIRG04-GA-2008-239412StringTheoryandNoncommutativeGeometrySTRING.
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