the kaluza-klein program and supergravity and sugra talk van nieuwenhuizen...the kaluza-klein...
TRANSCRIPT
The Kaluza-Klein program
and Supergravity
Lectures at Erice, June 2016, at the occasion of the
40th Anniversary of Supergravity
Also: 50th Anniversary of my rst visit to Erice*
30th Anniversary of the paper I will discuss.
*1966 School on Strong and Weak Interactions, with lectures by Coleman onSU(3), Radicati, Cabibbo, Gell-Mann, Glashow, Phillips, Zumino(!), Gatto andothers. The stamp of Coleman.
The KK reduction of d = 10 IIB sugra on S5 (KRN)(Kim, Romans and vN)
S5AdS5
⊗
Contents:
1. History (fun)
2. The theoretical minimum of Sugra
3. General set-up of KK reductions
4. The nitty-gritty road: Explicit construction of spherical harmonics
5. The Royal Road: Coset theory and Young tableaux
6. New developments
References:
1. Du, Nilsson, Pope, Kaluza-Klein Supergravity, Phys. Rept. 130
(1986) 1. Focusses on S7 reduction of d = 11 sugra to d = 4.
2. Castellani, D'Auria, Fré, Supergravity and Superstrings:Textbook on geometric perspectives of sugra. Full of information(3 volumes).
3. Lectures by P. van Nieuwenhuizen at the Les Houches 1983 andTrieste 1984 (Coset Methods).
4. H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, Mass spectrumof chiral ten-dimensional N = 2 supergravity on S5, Phys. Rev. D32 (1985) 389. This is the subject of these lectures. Has becomecentral for the AdS/CFT program.
History (fun)
History of dimensional reduction.
1912: Gunnar Nordström proposes 2φ = 4πGTµµ.Natural extension of ∇2φ = 4πGρ after SR in 1905.
1913: He proposes unication of EM and gravity
Aµ(x, z) = Aµ(x), φ(x) . Drops z
(d = 5 Maxwell) = (d = 4 Maxwell) + (d = 4 gravity)
Ruled out in 1919: no light bending. (And −16 precession
of Mercury.)1921: Theodor Kaluza interchanges Maxwell and gravity:
gµν(x, z) =
(gµν(x) Aµ(x)Aν(x) φ(x)
)(d = 5 Einstein) = (d = 4 Einstein) + (d = 4 Maxwell) + ?
Also drops z. Puts det gµν = −1.
Note: In both cases one decomposes µ = (µ, 5).
PvN (CNYITP) The KK program and Sugra Jun, 2016 1 / 29
History (fun)
History of dimensional reduction (continued).
1926: O. Klein and V. Fock: Keep z in
φ(x, z) =∑∞
k=−∞ φk(x)eιkzR .
Dφ(x, z) =
[2x +
(∂∂z
)2]φ =
∑k
[(2x − k2
R2
)φk(x)
]eιkzR
1938: Unication of gravity + strong + EM (+ weak)Klein proposes
(gµν)
5×5=
gµ5 =
(Aµ(x) Bµ(x)eιez
B∗µ(x)e−ιez Aµ(x)
)a matrix!
g55 = 1
Bµ is Yukawa eld (strong ints) and Aµ is photon.Note: e = e(EM) = e(strong). From Einstein (d = 5) onegets (part of) nonabelian theory for SU(2)!
1 2k
1
4
MR
1 2
2
−−→
−−→
PvN (CNYITP) The KK program and Sugra Jun, 2016 2 / 29
History (fun)
History of dimensional reduction (continued).
1938: C. Möller: Charge independence of nuclear forces?Klein: gµ5 = Ba
µσa +Aµ : SU(2)× U(1) (weak ints)!
≥1940: P. Jordan, Y. Thiry, C. Brans, R. Dicke, · · · , strings, · · · :Keep the scalar(s) (dilaton, moduli).
1953: A. Pais and W. Pauli.
1954: Yang-Mills and Pauli.
PvN (CNYITP) The KK program and Sugra Jun, 2016 3 / 29
History (fun)
(Non)Abelian symmetries from Einstein transformations.
Set
gµα(x, z) =∑I
BIµ(x)KIα(z)
α onMinternal
µ in Minkowski
The Killing vectors satisfy
DαKβ +DβKα = 0 ; [KαI ∂α,K
βJ ∂β] = fIJ
LKγL∂γ
Now equate
δgµα = δBIµKIα = (∂µξ
β)gβα + (∂αξβ)gµβ + ξβ∂βgµα .
Choose ξβ(x, z) =∑
ΛI(x)KβI (z). Result (just substitute):
δBIµ(x) = ∂µΛI(x) + f IJLB
Jµ (x)ΛL(x) = (DµΛ)I .
PvN (CNYITP) The KK program and Sugra Jun, 2016 4 / 29
History (fun)
General expansions into (spherical) harmonics.
1. gµα(x, z) = hµα(x, z) = BI5µ (x)Y I5
α (z) +((((((((
(BI1µ (x)DαY
I1(z).
For I5 = k = 1: Killing vectors. Degeneracy on Sn is 12n(n+ 1).
Gauge choice Dα(0)hµα = 0, eliminates longitudinal harmonics.
2. gαβ(x, z) = g(0)αβ (z) + hαβ(x, z)
hαβ(x, z) = h(αβ)(x, z) + 1ng
(0)αβ (z)hγγ
h(αβ)(x, z) = ΦI14(x)Y I14αβ (z) +
ΦI5(x)D(αY
I5β) (z)
+((((
(((((((
ΦI1(x)D(αDβ)YI1(z)
Gauge choice: Dα(0)h(αβ) = 0. Y I14
αβ is transversal and traceless.
3. For gravitinos: ψµ(x, z) =(ψµ(x, z), ψα(x, z)
).
ψµ(x, z) = ψI±Lµ (x)ΞI
±L (z) ; ψα(x, z) = ψ(α)(x, z) + χI(x)ταη
I+(z).The gauge Γαψα = 0 leaves τα-term.
ψ(α)(x, z) = ψI±L (x)Ξ
I±Lα (z) + ψI
±L (x)D(α)Ξ
I±L (z). The lowest of
ΞI±L (z) are Killing spinors η±(z)
(a√
of a Killing vector).
PvN (CNYITP) The KK program and Sugra Jun, 2016 5 / 29
The theoretical minimum of Sugra
The theoretical minimum of Supergravity.
1. Rigid Susy:
δboson = fermion× εδfermion = ∂boson× ε
The derivative lls the gap in dimensions([fermion]− [boson] = 1
2 , [ε] = −12).
[δ(ε1), δ(ε2)]boson = ε1ε2∂︸ ︷︷ ︸translation
boson
[ε1Q, ε2Q] = (ε1γµε2)Pµ : “Q,Q = P” superalgebras
Local ε(x)→ local translation → g.c.t. (GR)
Local susy = Supergravity (Sugra).
2. Fields: gµν (or its square root eµm, eµ
meνnηmn = gµν : Cartan).
ψµa gravitino(s): spin 1 ⊗ spin 1
2=spin32 ⊕ · · ·
Other elds.
↑ Dirac's√
leads to Susy!
PvN (CNYITP) The KK program and Sugra Jun, 2016 6 / 29
The theoretical minimum of Sugra
The theoretical minimum of Supergravity.
3.
δsusyψµa = 1
κ∂µεa + more (gauge eld of susy)
δsusyeµm = κεγmψµ (δboson = fermion× ε)
Newton constant: κ2 = 16πGc4
.
spin 3
2 + spin 2not
spin 32 + spin 1but · · ·
4. Vielbeins (Cartan: repères mobiles) from Dirac equation (Wigner).
Lat spaceDirac = −λγmδµm∂µλ with γm, γn = 2ηmn .
In curved space: γµ(x), γν(x) = 2gµν(x).Ansatz: γµ(x) = γmem
µ(x). Then emµen
νηmn = gµν (square root).If eµ
m xed by gµν : teleparallelism (Einstein).Arbitrary em
µ: local Lorentz rotation of frames (Cartan, Weyl).
δlLemµ = lm
n(x)enµ .
PvN (CNYITP) The KK program and Sugra Jun, 2016 7 / 29
The theoretical minimum of Sugra
The theoretical minimum of Supergravity.
Dirac action is Einstein invariant if
LD,E = −(det eµm)λγµ(x)∂µλ
but also local Lorentz invariant if
LD,E,lL = −(det eµm)λγµ(x)Dµ(ω)λ
Dµ(ω)λ = ∂µλ+ 14ωµ
mnγmγnλ
δlLλ = 14λ
mnγmnλ
δlLωµmn = −Dµ(ω)λmn
Dµλmn = ∂µλ
mn + ωµmm′λ
m′n + ωµnn′λ
mn′ .
5. For free gravitinosOne derivative (Dirac)Real (Majorana)
LRS = −12 ψµγ
µνρ∂νψρ
(ψµ)D = (ψµ)M where (ψµ)D = ψ†µιγ0 and (ψµ)M = ψTµC.
Whether or notωµmn is anindependent eld.←↓
PvN (CNYITP) The KK program and Sugra Jun, 2016 8 / 29
The theoretical minimum of Sugra
The theoretical minimum of Supergravity.
Gauge invariance: δψµ = ∂µε. Also tree unitarity xes this.Die Eleganz, die überlasz ich den Schneidern.Action for d = 4 N = 1 Sugra:
L =
LHE︷ ︸︸ ︷− 1
2κ2 eR(e, ω)
LRS︷ ︸︸ ︷− e
2 ψµγµνρDν(ω)ψρ
Minimal Einstein and local Lorentz covariantization.
e =√−g = det eµ
m ; γµνρ(x) = emµ(x)en
ν(x)erρ(x)γ[mγnγr]
R(ω, e) = Rµνmn(ω)em
νenµ ; Dν(ω)ψρ = ∂νψρ + 1
4ωνmnγmγnψρ
Rµνmn = ∂µων
mn − ∂νωµmn ; δsusyψµ = 1κDµ(ω)ε
+ ωµmkων
kn − ωνmkωµkn ; Dµ(ω)ε =(∂µ + 1
4ωµmnγmγn
)ε.
= gauge curvature for SO(3, 1).
PvN (CNYITP) The KK program and Sugra Jun, 2016 9 / 29
The theoretical minimum of Sugra
The theoretical minimum of Supergravity.
6. Second- versus rst-order formalism. Requiring thatδψµ = 1
κDµ(ω)ε in LRS cancels δeµm in LHE xes
δeµm = κεγmψµ! The remaining variations factorize:
δL(remaining) = − 12κ2 ε
µνρσεmnrs(δωµ
mneνr + κ
6 εγmnrDµψν
)︸ ︷︷ ︸Deser & Zumino:
δωµmn=··· (rst order)
(Dρ(ω)eσ
s − κ2
4 ψργsψσ
)︸ ︷︷ ︸
FFN: ωµmn(e,ψ)(second order)
Superspace uses second order.
7. Superspace: xµ(µ = 0, 1, 2, 3); θα(α = 1, 2); θα(α = 1, 2)Superelds: V∗(x, θ) (* Lorentz indices)Pµ: translations x
µ → xµ + aµ. Local aµ(x) become g.c.t. (GR)Qα: translations θ
α → θα + εα. Local εα(x) become local susy!Mmn: local Lorentz rotations (frames, falling lifts)Superspace Supergravity is Supersymmetric General Relativity.
PvN (CNYITP) The KK program and Sugra Jun, 2016 10 / 29
General set-up of KK reductions
General set-up of KK.
1. Find a solution of classical eld eqs. Our case: AdS5 ⊗ S5 (c = 1R)
AdS5 S5
g(0)µν (x) g
(0)αβ (z)
R(0)µνρσ = c2
(g
(0)µρ g
(0)νσ − g(0)
µσ g(0)νρ
)R
(0)αβγδ = −c2
(g
(0)αγ g
(0)βδ − g
(0)αδ g
(0)βγ
)F
(0)µνρστ = c
√−g(0)(x)εµνρστ F
(0)αβγδε = c
√g(0)(z)εαβγδε
2. Decompose all elds into background + quantum uctuations
Φ∗? = Φ(0)∗? (x or z) + ϕ∗?(x, z)
3. Expand ϕ∗?(x, z) =∑ϕI∗(x)Y I
? (z). The Y I? (z) are spherical
harmonics (Legendre; Green in EM; for scalars on S2 in QM; forvectors on S2 in Jackson).
4. Substitute into linearized eld equations for ϕ. Collect all termswith the same Y?. Mixing of ϕ∗(x) occurs: NOT discussed below.
10=5+5:
xµ zα⊗
PvN (CNYITP) The KK program and Sugra Jun, 2016 11 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Details of KK: Scalar harmonics on Sn (n = 5).
One complex scalar eld B(x, z) in d = 9 + 1 IIB sugra, only quantum
B(x, z) =∑
Bk(x)Y k(z) , k = 0, 1, 2, · · ·
Spectrum: −2zYk(z) = λs(n, k)Y k(z) and degeneracy? (s for scalar)
Recall QM: Y 0 = 1 , Y 1 = x+ ιy, x− ιy, z , Y 2 = x2 + y2 − 2z2, xy, · · ·.Choose monomial P k(x) in xµ (coords of Rn+1). Impose 2P k(x) = 0⇒traceless. Go from xµ to yµ = (r, θα). (Polar coords, θα = zα.)
ds2 = dr2 + r2dΩ2
gµν(y) =
(1 0
0 r2gαβ(θ)
)2 = 1√
g∂∂yµ√ggµν ∂
∂yν
P k(x) = P k(y)
0 = 2P k(x) = 2P k(y). Use P k(y) = rkY k(θ).
0 = 2P k(y) =(
1rn∂rr
n∂r + 1r2 2S(θ)
)rkY k(θ)
⇒ −2S(θ)Y k(θ) = k(n+ k − 1)Y k(θ) .
chain rule
PvN (CNYITP) The KK program and Sugra Jun, 2016 12 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Young tableaux for scalars.
: symmetric and traceless irrep of SO(n+ 1).
ds(n, k) = #P k −#P k−2 (for the trace)
=
(n+ k
k
)−(n+ k − 2
k − 2
).
k = 0 : a constant
k = 1 : xµ
k = 2 : xµxν − 1n+1δ
µν x2
...
1 2k
5
12
MR
1
6
20
Scalars Y k(z) on S5
PvN (CNYITP) The KK program and Sugra Jun, 2016 13 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Details: Vector harmonics on Sn. (Jackson for S2)
Choose P kµ (x) =(
0, 0, P k(x), 0, · · · , 0). Impose again
0 = 2P kµ =∂yν
∂xµ2P kν (y)⇒
2P kr (y) = 0
2P kα(y) = 0
P kr (y) = ∂xµ
∂r Pkµ = xµ
r Pkµ = rkρk(θ)
P kα(y) = ∂xµ
∂θα Pkµ = rk+1V k
α (θ) .
0 = 2P kα =(∂r − 1
r
)(∂r − 1
r
)rk+1V k
α (θ) + rk−12S(θ)V kα (θ)
+ rk−1∂αρ(θ) + nlrk−1Vα(θ) + rk−1(∂αρ(θ)− Vα(θ)
).
BUT: Vα(θ) = ∂ασ(θ) + Y kα (θ) with DαYα = 0.
Then −2SYkα = λv(n, k)Y k
α with λv(n, k) = k(n+ k− 1)− 1, k = 1, 2, 3.
just substitute
PvN (CNYITP) The KK program and Sugra Jun, 2016 14 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Vector harmonics continued.
Young tableau:
⊗ = ⊕↓ k=1,2,3,···
⊕∂xµ
∂yν Pµ(x) ρ′ Yα σ′
dv(n, k) = (n+ 1)ds(n, k)− ds(n, k + 1)− ds(n, k − 1)
k = 1: Killing vectors:νµ
= Y k=1α = xµ∂αx
ν − xν∂αxµ
Degeneracy: 12(n+ 1)n.
k = 2:ν ρµ
= Y k=2α = 1
3
[∂αx
µ((xν xρ))− ∂αxν((xµxρ))]
+13
[∂αx
µ((xρxν))− ∂αxρ((xµxν))],
where ((xµxν)) = xµxν − 1nδµν x2.
1 2k
4
11
MR
15
64
Vectors Yα(z) on S5
Note the Young symmetryand the tracelessness
PvN (CNYITP) The KK program and Sugra Jun, 2016 15 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Details: Spinor harmonics.
First spin 12 : Ξ±,k(z) (k = 0 are Killing spinors η±). Later spin 3
2 .
Spin 12: Begin again in Rn+1 with massless (nonchiral) Dirac
equation in Cartesian coord: γmδmµ ∂∂xµ ψ
(k)(x) = 0. Theψ(x) have in one entry xµ1 · · · xµk and further traces.Examples:
k = 0 : χ
k = 1 :(xµ − 1
n+1/xγµ
)χ
k = 2 :[xµxν − 1
n+3
(/xγµxν + /xγν xµ + x2δµν
)]χ ≡ xµxνcα;µν
where χ is a constant spinor and c is gamma-traceless:
(γµ)αβcβ;µν = 0.
Then go to polar (or other) coords yν = (r, θα);ψ(k)(x) = ψ(k)(y)
0 = γm(δm
µ ∂yν
∂xµ
)∂∂yν ψ
(k)(y).
PvN (CNYITP) The KK program and Sugra Jun, 2016 16 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Spinor harmonics continued.
We need more! In Rn+1 with polar coordinates:(ds)2 = dr2 + r2dθ2 + r2 sin2 θdφ2 + · · ·. Then a diagonal vielbein:E1r = 1, E2
θ = r, E3φ = r sin θ + · · ·. We need in general a suitable local
Lorentz rotation Λ(θ): Λ−1ψk(y) ≡ ψk(y). Insert unity ΛΛ−1 = I:
Λ(
Λ−1γmΛ)(
δmµ ∂yν
∂xµ
)(∂∂yν +
(Λ−1 ∂
∂yν Λ))
ψk(y) = 0.
If Lmn = (eλ)mn, then Λ = e14λmnγmγn , and
Λ−1γmΛ = Lmnγn ; Λ−1 ∂
∂yν Λ = 14 ωµ
mn︸ ︷︷ ︸pure gauge
γmγn.
Finally,
γn(∂yν
∂xµ δmµLmn
)︸ ︷︷ ︸
Enν
Dνψk = 0.
Given Enν nd Lmn (or vice-versa for stereographic coords).
PvN (CNYITP) The KK program and Sugra Jun, 2016 17 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Spinor harmonics continued.
Use ψk(y) = Λ−1(θ)ψk(y) = Λ−1ψk(x) = rkψk(θ) (because Λ isindependent of r). Then[
γ1(∂r + n
21r︸︷︷︸
from ωa1a
)+ 1
r/DS(θ)
]rkψk(θ) = 0
For even n+ 1: Use γ1 =(
0 II 0
), γa+1 =
(0 −ισaισa 0
)and with
ψk =(
Ξ+,k
Ξ−,k
)one gets
∓ι /DSΞ∓,k = −(k + n
2
)Ξ∓,k ⇒ λspinor(n, k) = ±
(k + n
2
).
For odd n+ 1: Multiply by (1± ιγ1). Then (1∓ ιγ1)ψk = Ξ±,k and/DSΞ±,k = −ι
(k + n
2
)Ξ±,k. Same result.
PvN (CNYITP) The KK program and Sugra Jun, 2016 18 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Spinor harmonics continued.
1 2k
- 92
- 72
- 52
52
72
92
MR
4
20
60
4*
20*
60*
Spinors Ξ±(z) on S5
Young Tableau • • • • • :
dspinor(n, k) =[(n+ k
k
)︸ ︷︷ ︸
#P (k)
−(n+ k − 1
k − 1
)︸ ︷︷ ︸Dirac equation
]2[n
2]︸︷︷︸
factor 1/2for ψ+,ψ−
PvN (CNYITP) The KK program and Sugra Jun, 2016 19 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Spinor harmonics continued.
For polar coordinates: Λ−1 = e12θγ1γ2
e12φγ2γ3
e12χγ3γ4
e12ζγ4γ5
. (Eulerangles: L = · · · e−φL23e−θL12)
k = 0: Killing spinors: Dαη± = ∓ ι
2γαη±. On S2:
Ξ±,k=0 = Λ−1(1∓ιγ1)χ =
η±I = eι
φ/2(
cos θ/2(1∓ι)− sin θ/2(1±ι)
)η±II = e−ι
φ/2(
sin θ/2(1∓ι)cos θ/2(1±ι)
)
2 η+
and2 η−
k = 1:
Ξ±,k=1 = xµ︸︷︷︸Y k=1
(Λ−1χ
)︸ ︷︷ ︸
η±
− 1n+1 x
ν(
Λ−1γνΛ)
︸ ︷︷ ︸rγ1
(Λ−1γµΛ
)︸ ︷︷ ︸
usedΛ−1γµΛ= ∂xµ
∂yνEνnγn
= xµ
rγ1+ 1
r( /DS x
µ)
Λ−1χ
= 1n+1
[nY k=1 ∓
(/DSY
1)]η±. (KRN!)
PvN (CNYITP) The KK program and Sugra Jun, 2016 20 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Spinor harmonics continued.
Spin 32(gravitino on S5): Combine vectors and spinors.
ΞL,±,ka,α = DaΞ±,kα ; ΞT,±,ka,α = (Dax
µ)xρ1 · · · xρk (Σ±)αβ︸ ︷︷ ︸
Λ−1(θ)
ψ±
β;ρ1 ρkµ
·
The constant spinors ψ are(anti) chiral if n+ 1 = even(anti) symmetric in µ, ρ1, · · · , ρkgamma-traceless γµ∓ψ
±µρ1···ρk = γρ1
∓ ψ±µρ1···ρk = 0.
Γa =(
0 γaγa 0
); Γn+1 =
(0 ιI−ιI 0
); γ±A = (γa,±ιI) .
Identity (needed for spectrum):
Σ±γ∓AΣ± = YABγ∓B , A,B = 1, · · · , n+ 1.
In stereographic coordinates easy to check:
YAB =
(δab − 2zazb
4R2+z2−2za
4R2+z2
2zb
4R2+z24R2−z2
4R2+z2
); Σ± =
2R± ι/z√4R2 + z2
·
Pauli matrices
PvN (CNYITP) The KK program and Sugra Jun, 2016 21 / 29
The nitty-gritty road: Explicit construction of spherical harmonics
Mass Spectrum on S5
(e = 1
R
).
PvN (CNYITP) The KK program and Sugra Jun, 2016 22 / 29
The Royal Road: Coset theory and Young tableaux
The Royal Road: Cosets and Young tableaux.
S5 : G/H = SO(6)/SO(5). Coset representatives: L(z) ∈ SO(6) in anyirrep. L−1dL = eaKa + 1
2ωabHab (Cartan-Maurer). Dene:
Y = L−1 ⇒ dY = −(L−1dL
)L−1 =
(−eaKa − 1
2ωabHab
)Y .
DαY = −KαY or Da︸︷︷︸di. op.
Y = − Ka︸︷︷︸constant matrix
Y←sph. harm.
DbDaY = −[Db,Ka]Y −KaDbY
⇓−ωbca KcY = −ωbacDcY
DbDaY = KaKbY
2SY =(δabKaKb
)Y = δABTATB − 1
2HabHab
−2SY =[C2(SO(6))− C2(SO(5))
]Y
Degeneracy of Y : dimension of Young tableau.
∣∣∣∣∣ In any irrep R of G.Decomposes into irrepsof H, on which 2S acts.
PvN (CNYITP) The KK program and Sugra Jun, 2016 23 / 29
The Royal Road: Coset theory and Young tableaux
Examples of Young tableau.
TensorsR :
g1g2g3f1f2f3
# boxes = r
C2(R of SO(n)) = −r(n− 1)−∑f2i +
∑g2i .
Scalars:
k︷ ︸︸ ︷of SO(6); • of SO(5); k = 0, 1, 2, · · ·.
C2(SO(6)) = −k(5)− k2 + k
C2(SO(5)) = 0
λs(n, k) = k(k − 4).
Vectors:
k︷ ︸︸ ︷of SO(6); of SO(5); k = 1, 2, 3, · · ·.
YAB =
↑Yn+1
B
+ YaB
↓
PvN (CNYITP) The KK program and Sugra Jun, 2016 24 / 29
The Royal Road: Coset theory and Young tableaux
Examples continued.
Antisym. tensors:
k+1︷ ︸︸ ︷of SO(6); of SO(5); k = 1, 2, 3.
Symmetric tensors:
k︷ ︸︸ ︷of SO(6); of SO(5);
k = 2, 3, · · ·.
PvN (CNYITP) The KK program and Sugra Jun, 2016 25 / 29
The Royal Road: Coset theory and Young tableaux
Examples continued.
SpinorsR :
• • • •• •• •
C2(R of SO(n)) = −rn− 18n(n+ 1)−
∑f2i +
∑g2i .
Spin 12: • • • • of SO(6); • of SO(5).
λspinor(n, k) =[k × 6 + 1
8 × 6× 5 + k2 − k]−[
18 × 5× 4
]= k2 + 5k + 10
8 k = 0, 1, 2.
λ = −2S = − /D /D + 14R = − /D /D − 1
420r2
ι /DΞk,± = ±(k + 12)Ξk,± .
PvN (CNYITP) The KK program and Sugra Jun, 2016 26 / 29
New developments
Exceptional Field Theory. (O. Hohm, H. Samtleben, arXiv:1312.0614)
Idea: Introduce new coordinates YM ⇒ extended sugra which containsboth N = 1 d = 11 and IIB.
The YM are in 27 of E6 (recall: scalars from torus compactication formcoset G/H = E6/USp(8)).
Action: S =∫d5xd27Y e
[R+ gµν(DµMMN )
(DνMMN
)− 1
4MMNFµν,MFµνN + more].
Symmetries: External gen. di. (EGD) and Internal gen. di. (IGD).
δEGD(ξ)eµm(x, Y ) = ξν(x, Y )Dνeµ
m + (Dµξν)eν
m
Dµeνm = ∂µeν
m −ANµ ∂Neνm − 13∂MA
Mµ eν
m
δIGD(Λ)VM (x, Y ) = “LΛ”VM = ΛK(x, Y )∂KVM
+ 6(PNM )KL(∂KΛL)VN + λ(∂LΛL)VM
(if VA
M∈E6 then
also δ(Λ)VAM∈E6
)(PNM )KL = (tα)NM (tα)KL (α = 1, · · · , 78)
= 118δM
NδLK︸ ︷︷ ︸
needed for P2=P
+ 16δM
KδLN︸ ︷︷ ︸
usual term
− 53dNKP dMLP︸ ︷︷ ︸
inv. tensors of E6
PvN (CNYITP) The KK program and Sugra Jun, 2016 27 / 29
New developments
Exceptional Field Theory continued.
Closure: [LΛ1 ,LΛ2 ] = L[Λ1,Λ2]
Denes E bracket.
Requires section constraint:
dMNK∂N∂KA = 0;
dMNK∂NA∂KB = 0.
KK reduction: Make ansatz (sph. harm.)
gµν(x, Y ) = ρ−2(Y )gµν(x)
AµM (x, Y ) = ρ−1(Y )U−1N
M (Y )AµN (x)
MMN (x, Y ) = UMK(Y )UNL(Y )MKL(x)
Consistency: EMN (x, Y ) ≡ ρ−1(Y )U−1(Y )MN must satisfy
LEMEN = −XMNKEK
(EM = EMN ∂
∂Y N
).
PvN (CNYITP) The KK program and Sugra Jun, 2016 28 / 29
New developments
Exceptional Field Theory continued.
Consistency: for BM (x, Y ) = ENM (Y )BN (x);ΛM (x, Y ) = ENM (Y )ΛN (x).
One gets δBM = ENM (Y )δBN (x) (denition)
= LΛBM = LEK(Y )ΛK(x)
(ENM (Y )BN (x)
)= ΛK(x)
(LEKEN
M)︸ ︷︷ ︸
require: −XKNLELM then
δBM (x)=−ΛK(x)XKNMBN (x).
BN (x)
Two solutions of the section constraint yield
N = 1 d = 11 if E6(6) → Sl(6)×Gl(1); (Note: E6(6) is real.)27→ (ym,
Ymn,ym6 + 15 + 6
)
IIB if E6(6) → Sl(5)× Sl(2).
27→ (ym,ymα,ymα,y
α
5 + 10 + 10 + 2)
Big Question: Is this Bookkeeping or Deep Physics?PvN (CNYITP) The KK program and Sugra Jun, 2016 29 / 29