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 on SU(3), Radicati, Cabibbo, Gell-Mann, Glashow, Phillips, Zumino(!), Gatto and others. The stamp of Coleman.
The KK reduction of d = 10 IIB sugra on S5 (KRN) (Kim, Romans and vN)
⊗
AdS5 S5 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 and Trieste 1984 (Coset Methods). 4. H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, Mass spectrum
of chiral ten-dimensional N = 2 supergravity on S5, Phys. Rev. D 32 (1985) 389. This is the subject of these lectures. Has become central for the AdS/CFT program. History (fun) History of dimensional reduction.
µ 1912: Gunnar Nordström proposes φ = 4πGT µ. 2 Natural extension of ∇ φ = 42πGρ after SR in 1905. 1913: He proposes uni cation of EM and gravity
Aµˆ(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 1 precession − 6 of Mercury.) 1921: Theodor Kaluza interchanges Maxwell and gravity: ! gµν(x) Aµ(x) gµˆνˆ(x, z) = Aν(x) φ(x) (d = 5 Einstein) = (d = 4 Einstein) + (d = 4 Maxwell) + ?
Also drops z. Puts det gµˆνˆ = −1. Note: In both cases one decomposes µˆ = (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 ∞ ιkz˙ MR P R . φ(x, z) = k=−∞ φk(x)e 4 2 2 X 2 ιkz˙ ∂ k R φ(x, z) = x + φ = x − 2 φ (x) e 1 ∂z R k 1 2 k 1 2 D 2 k 2 1938: Uni cation of gravity + strong + EM (+ weak) Klein proposes
ιez˙ Aµ(x) Bµ(x)e −−→ gµ5 = B∗(x)e−ιez˙ A (x) g = µ µ µˆνˆ 5×5 a matrix! −−→ g55 = 1
Bµ is Yukawa eld (strong ints) and Aµ is photon. Note: e = e(EM) = e(strong). From Einstein (d = 5) one gets (part of) nonabelian theory for SU(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: a (weak ints)! gµ5 = Bµσa + Aµ : SU(2) × U(1) ≥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 ( X α on Minternal g (x, z) = BI (x)K (z) µα µ Iα µ in Minkowski I The Killing vectors satisfy
α β L γ DαKβ + DβKα = 0 ; [KI ∂α,KJ ∂β] = fIJ KL∂γ Now equate
I β β β δgµα = δBµKIα = (∂µξ )gβα + (∂αξ )gµβ + ξ ∂βgµα .
Choose β P I β . Result (just substitute): ξ (x, z) = Λ (x)KI (z)
I I I J L I δBµ(x) = ∂µΛ (x) + fJLBµ (x)Λ (x) = (DµΛ) .
PvN (CNYITP) The KK program and Sugra Jun, 2016 4 / 29 History (fun) General expansions into (spherical) harmonics.
(( 1. I5 I5 I1 (((I1 . gµα(x, z) = hµα(x, z) = Bµ (x)Yα (z) + (Bµ(((x)DαY (z) For : Killing vectors. Degeneracy on is 1 . I5 = k = 1 Sn 2 n(n + 1) Gauge choice α , eliminates longitudinal harmonics. D(0)hµα = 0 2. (0) gαβ(x, z) = gαβ (z) + hαβ(x, z) 1 (0) γ hαβ(x, z) = h(αβ)(x, z) + n gαβ (z)h γ h (x, z) = ΦI14 (x)Y I14 (z) + ΦI5 (x)DY I5 (z) (αβ) αβ (α β) ((( I1 ((( I1 + (Φ(((x)(D(αDβ)Y (z) Gauge choice: α . I14 is transversal and traceless. D(0)h(αβ) = 0 Yαβ 3. For gravitinos: ψµˆ(x, z) = ψµ(x, z), ψα(x, z) . ± ± IL I I I+ ψµ(x, z) = ψµ (x)Ξ L (z); ψα(x, z) = ψ(α)(x, z) + χ (x)ταη (z). α The gauge Γ ψα = 0 leaves τα-term. ± ± ± ± I IL I I ψ (x, z) = ψ L (x)Ξ (z) + ψ L (x)D Ξ L (z). The lowest of (α) α √(α) I± ± Ξ 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 × The derivative lls the gap in dimensions fermion boson fermion boson 1 , 1 δ = ∂ × ([ ] − [√ ] = 2 [] = − 2 ). ↑ Dirac's leads to Susy! [δ(1), δ(2)]boson = 12∂ boson translation| {z } µ [1Q, 2Q] = (¯1γ 2) Pµ : “{Q, Q} = P ” superalgebras Local (x) → local translation → g.c.t. (GR) Local susy = Supergravity (Sugra).
m m n 2. Fields: gµν (or its square root eµ , eµ eν ηmn = gµν: Cartan). a gravitino(s): spin 1 spin 1 =spin 3 ψµ ⊗ 2 2 ⊕ · · · Other elds.
PvN (CNYITP) The KK program and Sugra Jun, 2016 6 / 29 The theoretical minimum of Sugra The theoretical minimum of Supergravity.
3.
a 1 a spin 3 + spin 2 δsusyψµ = ∂µ + more (gauge eld of susy) 2 κ not δ e m = κγ¯ mψ (δboson = fermion × ) susy µ µ spin 3 + spin 1 Newton constant: 2 16πG 2 κ = c4 . but ··· 4. Vielbeins (Cartan: repères mobiles) from Dirac equation (Wigner).
at space ¯ m µ with m n mn LDirac = −λγ δm∂µλ {γ , γ } = 2η . In curved space: {γµ(x), γν(x)} = 2gµν(x). µ m µ µ ν mn µν Ansatz: γ (x) = γ em (x). Then em en η = g (square root). m If eµ xed by gµν: teleparallelism (Einstein). µ Arbitrary em : local Lorentz rotation of frames (Cartan, Weyl).
µ n µ δlLem = lm (x)en .
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 D,E m µ L = −(det eµ )λγ¯ (x)∂µλ but also local Lorentz invariant if D,E,lL m µ L = −(det eµ )λγ¯ (x)Dµ(ω)λ 1 mn Dµ(ω)λ = ∂µλ + 4 ωµ γmγnλ δ λ = 1 λmnγ λ Whether or not lL 4 mn mn ωµ is an mn mn ← independent eld. δlLωµ = −Dµ(ω)λ ↓ mn mn m m0n n mn0 Dµλ = ∂µλ + ωµ m0 λ + ωµ n0 λ . 5. For free gravitinos One derivative (Dirac) Real (Majorana) RS 1 ¯ µνρ L = − 2 ψµγ ∂νψρ ¯ ¯ where ¯ † 0 and ¯ T . (ψµ)D = (ψµ)M (ψµ)D = ψµιγ˙ (ψµ)M = ψµ C PvN (CNYITP) The KK program and Sugra Jun, 2016 8 / 29 The theoretical minimum of Sugra The theoretical minimum of Supergravity.
Gauge invariance: δψµ = ∂µ. Also tree unitarity xes this. Die Eleganz, die überlasz ich den Schneidern. Action for d = 4 N = 1 Sugra:
LHE LRS z }| { z }| { 1 e ¯ µνρ L = − 2κ2 eR(e, ω) − 2 ψµγ Dν(ω)ψρ Minimal Einstein and local Lorentz covariantization.
√ m µνρ µ ν ρ [m n r] e = −g = det eµ ; γ (x) = em (x)en (x)er (x)γ γ γ mn ν µ 1 mn R(ω, e) = Rµν (ω)em en ; Dν(ω)ψρ = ∂νψρ + 4 ων γmγnψρ mn mn mn 1 Rµν = ∂µων − ∂νωµ ; δsusyψµ = κ Dµ(ω) m kn m kn 1 mn + ωµ kων − ων kωµ ; Dµ(ω) = ∂µ + 4 ωµ γmγn . = 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 that 1 in RS cancels m in HE xes δψµ = κ Dµ(ω) L δeµ L m m δeµ = κγ¯ ψµ! The remaining variations factorize: remaining 1 µνρσ δL( ) = − 2κ2 mnrs mn r κ mnr s κ2 ¯ s δωµ eν + 6 γ¯ Dµψν Dρ(ω)eσ − 4 ψργ ψσ | {z } Deser & Zumino: | FFN: {zmn } δω mn=··· ( rst order) ωµ (e,ψ) µ (second order) Superspace uses second order. µ α 7. Superspace: x (µ = 0, 1, 2, 3); θ (α = 1, 2); θ¯α˙ (α ˙ = 1, 2) Super elds: V∗(x, θ) (* Lorentz indices) µ µ µ µ Pµ: translations x → x + a . Local a (x) become g.c.t. (GR) α α α α Qα: translations θ → θ + . Local (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: 1 AdS5 ⊗ S5 (c = R ) AdS5 S5 10=5+5: (0) xµ zα (0) ⊗ gµν (x) gαβ (z) (0) 2 (0) (0) (0) (0) (0) 2 (0) (0) (0) (0) Rµνρσ = c gµρ gνσ − gµσ gνρ Rαβγδ = −c gαγ gβδ − gαδ gβγ (0) p (0) (0) p (0) Fµνρστ = c −g (x)µνρστ Fαβγδ = c g (z)αβγδ 2. Decompose all elds into background + quantum uctuations
(0) Φ∗? = Φ∗? (x or z) + ϕ∗?(x, z) 3. Expand P I I . The I are spherical ϕ∗?(x, z) = ϕ∗(x)Y? (z) Y? (z) harmonics (Legendre; Green in EM; for scalars on S2 in QM; for vectors on S2 in Jackson). 4. Substitute into linearized eld equations for ϕ. Collect all terms with the same Y?. Mixing of ϕ∗(x) occurs: NOT discussed below.
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 X B(x, z) = Bk(x)Y k(z) , k = 0, 1, 2, ···
k k Spectrum: − zY (z) = λs(n, k)Y (z) and degeneracy? (s for scalar) 0 1 2 2 2 2 Recall QM: Y 2= 1 ,Y = x +ιy, ˙ x − ιy,˙ z , Y = x + y − 2z , xy, ···. Choose monomial P¯k(¯x) in x¯µ (coords of Rn+1). Impose ¯P¯k(¯x) = 0 ⇒ µ µ α α α traceless. Go from x¯ to yˆ = (r, θ ). (Polar coords, θ =2z .) ds2 = dr2 + r2dΩ2 ! √ ˆ = √1 ∂ gˆgˆµν ∂ 1 0 gˆ ∂yˆµ ∂yˆν gˆµν(ˆy) = 2 k k 0 r gαβ(θ) P2¯ (¯x) = Pˆ (ˆy) chain rule& 0 = ¯P¯k(¯x) = ˆPˆk(ˆy). Use Pˆk(ˆy) = rkY k(θ). 2ˆ ˆk 2 1 n 1 k k 0 = P (ˆy) = rn ∂rr ∂r + r2 S(θ) r Y (θ) 2 k k 2 ⇒ − S(θ)Y (θ) = k(n + k − 1)Y (θ) .
PvN (CNYITP)2 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).
k k−2 ds(n, k) = #P¯ − #P¯ (for the trace) n + k n + k − 2 = − . k k − 2
k MR Scalars Y (z) on S5 k = 0 : a constant 12 20 k = 1 : x¯µ µ ν 1 µν 2 k = 2 : x¯ x¯ − n+1 δ x¯ . 5 6 .
1 k 1 2
PvN (CNYITP) The KK program and Sugra Jun, 2016 13 / 29 The nitty-gritty road: Explicit construction of spherical harmonics
Details: Vector harmonics on . (Jackson for ) Sn S2 Choose ¯k ¯k . Impose again Pµ (¯x) = 0, 0, P (¯x), 0, ··· , 0 ( ∂yˆν ˆPˆk(ˆy) = 0 0 = ¯P¯k = ˆPˆk(ˆy) ⇒ r µ µ ν ˆk ∂x¯ 2ˆPα (ˆy) = 0 2 % 2 just substitute 2 ˆk ∂x¯µ ¯k x¯µ ¯k k k Pr (ˆy) = ∂r Pµ = r Pµ = r ρ (θ) ˆk ∂x¯µ ¯k k+1 k Pα (ˆy) = ∂θα Pµ = r Vα (θ) .
ˆ ˆk 1 1 k+1 k k−1 k 0 = Pα = ∂r − r ∂r − r r Vα (θ) + r S(θ)Vα (θ) 2 k−1 k−1 k−12 + r ∂αρ(θ) + nlr Vα(θ) + r ∂αρ(θ) − Vα(θ) . BUT: k with α Vα(θ) = ∂ασ(θ) + Yα (θ) D Yα = 0. Then k k with . − SYα = λv(n, k)Yα λv(n, k) = k(n + k − 1) − 1, k = 1, 2, 3 2 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,··· ⊗ = ⊕ ⊕ ∂x¯µ ¯ 0 0 ∂yˆν Pµ(¯x) ρ Yα σ
dv(n, k) = (n + 1)ds(n, k) − ds(n, k + 1) − ds(n, k − 1) : Killing vectors: ν k=1 µ ν ν µ k = 1 µ = Yα =x ¯ ∂αx¯ − x¯ ∂αx¯ Degeneracy: 1 . 2 (n + 1)n : ν ρ k=2 1 µ ν ρ ν µ ρ k = 2 µ = Yα = 3 ∂αx¯ ((¯x x¯ )) − ∂αx¯ ((¯x x¯ )) MR 1 µ ρ ν ρ µ ν , Vectors on + 3 ∂αx¯ ((¯x x¯ )) − ∂αx¯ ((¯x x¯ )) Yα(z) S5 where µ ν µ ν 1 µν 2. ((¯x x¯ )) =x ¯ x¯ − n δ x¯
Note the Young symmetry 11 and the tracelessness % 64
4 15
k 1 2 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 1 : ±,k ( are Killing spinors ±). Later spin 3 . 2 Ξ (z) k = 0 η 2 Spin 1 : Begin again in n+1 with massless (nonchiral) Dirac 2 R equation in Cartesian coord: m µ ∂ ¯(k) . The γ δm ∂x¯µ ψ (¯x) = 0 ψ¯(¯x) have in one entry x¯µ1 ··· x¯µk and further traces. Examples: k = 0 : χ µ 1 µ k = 1 : x¯ − n+1 xγ/¯ χ µ ν 1 µ ν ν µ 2 µν µ ν k = 2 : x¯ x¯ − n+3 xγ/¯ x¯ + xγ/¯ x¯ +x ¯ δ χ ≡ x¯ x¯ cα;µν where χ is a constant spinor and c is gamma-traceless: µ β (γ )α cβ;µν = 0. Then go to polar (or other) coords yˆν = (r, θα); ψ¯(k)(¯x) = ψˆ(k)(ˆy) m µ ∂yˆν ∂ ˆ(k) 0 = γ δm ∂x¯µ ∂yˆν ψ (ˆy). 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: 1 2 3 . We need in general a suitable local Er = 1,Eθ = r, Eφ = r sin θ + ··· Lorentz rotation Λ(θ): Λ−1ψˆk(ˆy) ≡ ψk(ˆy). Insert unity ΛΛ−1 = I: −1 m µ ∂yˆν ∂ −1 ∂ k Λ Λ γ Λ δm ∂x¯µ ∂yˆν + Λ ∂yˆν Λ ψ (ˆy) = 0.
1 mn m λ m λ γmγn If L n = (e ) n, then Λ = e 4 , and −1 m m n −1 ∂ 1 mn Λ γ Λ = L nγ ;Λ ∂yˆν Λ = 4 ωµ γmγn. pure| {z gauge} Finally, n ∂yˆν µ m k γ ∂x¯µ δm L n Dνψ = 0.
| {zν } En ν m Given En nd L n (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(ˆy) = Λ−1(θ)ψˆk(ˆy) = Λ−1ψ¯k(¯x) = rkψk(θ) (because Λ is independent of r). Then " # 1 n 1 1 / k k γ ∂r + 2 r + r DS(θ) r ψ (θ) = 0
|{z} 1a from ωa
a For even : Use 1 0 I , a+1 0 −ισ˙ and with n + 1 γ = I 0 γ = ισ˙ a 0 ψk = Ξ+,k one gets Ξ−,k
/ ∓,k n ∓,k n ∓ι˙DSΞ = − k + 2 Ξ ⇒ λspinor(n, k) = ± k + 2 . For odd n + 1: Multiply by (1 ± ιγ˙ 1). Then (1 ∓ ιγ˙ 1)ψk = Ξ±,k and / ±,k n ±,k. Same result. DSΞ = −ι˙ k + 2 Ξ
PvN (CNYITP) The KK program and Sugra Jun, 2016 18 / 29 The nitty-gritty road: Explicit construction of spherical harmonics Spinor harmonics continued.
± MR Spinors Ξ (z) on S5
9 2 60 7 2 20 5 2 4
k 1 2
* 5 4 - 2 * 7 20 - 2 * 9 60 - 2
Young Tableau • • • • • : h n + k n + k − 1 i [ n ] dspinor(n, k) = − 2 2 k k − 1 |{z} factor 1/2 | {z } | {z } for #P¯(k) Dirac equation 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.
−1 1 θγ1γ2 1 φγ2γ3 1 χγ3γ4 1 ζγ4γ5 For polar coordinates: Λ = e 2 e 2 e 2 e 2 . (Euler angles: L = ··· e−φL23 e−θL12 ) : Killing spinors: ± ι˙ ±. On : k = 0 Dαη = ∓ 2 γαη S2
θ η± = eι˙φ/2 cos /2(1∓ι˙) 2 η+ I − sin θ/2(1±ι˙) Ξ±,k=0 = Λ−1(1∓ιγ˙ 1)χ = and θ η± = e−ι˙φ/2 sin /2(1∓ι˙) − II cos θ/2(1±ι˙) 2 η k = 1: Ξ±,k=1 =x ¯µ Λ−1χ − 1 x¯ν Λ−1γνΛ Λ−1γµΛ Λ−1χ |{z} n+1 Y k=1 | {z } | {z } | {z } η± rγ1 used −1 µ ∂x¯µ ν n Λ γ Λ= ∂yν E nγ x¯µ 1 1 D/ µ = r γ + r ( S x¯ ) 1 k=1 / 1 ± (KRN!) = n+1 nY ∓ DSY η .
PvN (CNYITP) The KK program and Sugra Jun, 2016 20 / 29 The nitty-gritty road: Explicit construction of spherical harmonics Spinor harmonics continued.
Spin 3 (gravitino on ): Combine vectors and spinors. 2 S5 ΞL,±,k = D Ξ±,k ;ΞT,±,k = (D x¯µ)¯xρ1 ··· x¯ρk (Σ±) β ψ± · a,α a α a,α a α ρ ρ | {z } β; 1 k Λ−1(θ) µ The constant spinors ψ are (anti) chiral if n + 1 = even (anti) symmetric in µ, ρ1, ··· , ρk gamma-traceless γµ ψ± = γρ1 ψ± = 0. ∓ µρ1···ρk ∓ µρ1···ρk Pauli matrices . Γ = 0 γa ;Γ = 0ιI ˙ ; γ± = (γ , ±ιI˙ ) . a γa 0 n+1 −ιI˙ 0 A a Identity (needed for spectrum): ± ∓ ± B ∓ Σ γA Σ = YA γB , A, B = 1, ··· , n + 1. In stereographic coordinates easy to check: b b 2zaz −2za ! B δa − 4R2+z2 4R2+z2 ± 2R ± ι˙z/ YA = b 2 2 ;Σ = √ · 2z 4R −z 2 2 4R2+z2 4R2+z2 4R + z
PvN (CNYITP) The KK program and Sugra Jun, 2016 21 / 29 The nitty-gritty road: Explicit construction of spherical harmonics Mass Spectrum on 1 . S5 e = 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 any −1 a 1 ab irrep. L dL = e Ka + ω Hab (Cartan-Maurer). De ne: 2 −1 −1 −1 a 1 ab . Y = L ⇒ dY = − L dL L = −e Ka − 2 ω Hab Y ←sph. harm. DαY = −KαY or Da Y = − Ka Y di.|{z} op. constant|{z} matrix
DbDaY = −[Db,Ka]Y − KaDbY ⇓ c c −ωb a KcY = −ωba DcY
DbDaY = KaKbY ab AB 1 ab SY = δ KaKb Y = δ TATB − 2 HabH 2 In any irrep R of G. − SY = C2(SO(6)) − C2(SO(5)) Y Decomposes into irreps Degeneracy2 of : dimension of Young tableau. Y of H, on which S acts. 2 PvN (CNYITP) The KK program and Sugra Jun, 2016 23 / 29 The Royal Road: Coset theory and Young tableaux Examples of Young tableau.
g1g2g3 f R : 1 # boxes = r Tensors f2 f3 of P 2 P 2 C2(R SO(n)) = −r(n − 1) − fi + gi . k z }| { Scalars: of SO(6); • of SO(5); k = 0, 1, 2, ···.
↑ 2 ) B B C2(SO(6)) = −k(5) − k + k YA = Yn+1 λs(n, k) = k(k − 4). C2(SO(5)) = 0 B + Ya ↓ k z }| { Vectors: of SO(6); of SO(5); k = 1, 2, 3, ···.
PvN (CNYITP) The KK program and Sugra Jun, 2016 24 / 29 The Royal Road: Coset theory and Young tableaux Examples continued.
k+1 z }| {
Antisym. tensors: of SO(6); of SO(5); k = 1, 2, 3.
k z }| { Symmetric tensors: 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.
• • • • R : • • Spinors • • of 1 P 2 P 2 C2(R SO(n)) = −rn − 8 n(n + 1) − fi + gi . Spin 1 : • • • • of ; of . 2 SO(6) • SO(5)
h 1 2 i h 1 i λspinor(n, k) = k × 6 + 8 × 6 × 5 + k − k − 8 × 5 × 4 2 10 = k + 5k + 8 k = 0, 1, 2. / / 1 / / 1 20 λ = − S = −DD + 4 R = −DD − 4 r2 / k,± 2 1 k,± ι˙DΞ = ±(k + 2 )Ξ .
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 Y M ⇒ extended sugra which contains both N = 1 d = 11 and IIB. M The Y are in 27 of E6 (recall: scalars from torus compacti cation form coset G/H = E6/USp(8)). R 5 27 µν MN Action: S = d xd Y e Rˆ + g (DµMMN ) Dν M 1 µν,M N more. − 4 MMN F Fµν + Symmetries: External gen. di. (EGD) and Internal gen. di. (IGD).
m ν m ν m δEGD(ξ)eµ (x, Y ) = ξ (x, Y )Dν eµ + (Dµξ )eν m m N m 1 M m Dµeν = ∂µeν − Aµ ∂N eν − 3 ∂M Aµ eν K δIGD(Λ)VM (x, Y ) = “LΛ”VM = Λ (x, Y )∂K VM M if VA ∈E6 then M N K L L also δ(Λ)VA ∈E6 + 6(P M ) L(∂K Λ )VN + λ(∂LΛ )VM N K α N K (P M ) L = (t ) M (tα) L (α = 1, ··· , 78) 1 N K 1 K N 5 NKP = 18 δM δL + 6 δM δL − 3 d dMLP | {z } | {z } | {z } needed for 2 usual term inv. tensors of E P =P 6
PvN (CNYITP) The KK program and Sugra Jun, 2016 27 / 29 New developments Exceptional Field Theory continued.
Closure: [LΛ1 , LΛ2 ] = L[Λ1,Λ2] De nes E bracket. Requires section constraint:
MNK d ∂N ∂K A = 0; MNK d ∂N A∂K B = 0. KK reduction: Make ansatz (sph. harm.)
−2 gµν(x, Y ) = ρ (Y )gµν(x) M −1 −1M N Aµ (x, Y ) = ρ (Y )UN (Y )Aµ (x) K L MMN (x, Y ) = UM (Y )UN (Y )MKL(x)
N −1 −1 N Consistency: EM (x, Y ) ≡ ρ (Y )U (Y )M must satisfy K N ∂ LEM EN = −XMN EK EM = EM ∂Y N .
PvN (CNYITP) The KK program and Sugra Jun, 2016 28 / 29 New developments Exceptional Field Theory continued.
M M N Consistency: for B (x, Y ) = EN (Y )B (x); M M N Λ (x, Y ) = EN (Y )Λ (x). M M N One gets δB = EN (Y )δB (x) (de nition) M M N K = LΛB = LEK (Y )Λ (x) EN (Y )B (x) K M N = Λ (x) LEK EN B (x)
| {z L }M require: −XKN EL then M K M N δB (x)=−Λ (x)XKN B (x). Two solutions of the section constraint yield
N = 1 d = 11 if E6(6) → Sl(6) × Gl(1); (Note: E6(6) is real.) m ¨¨ 27 → (y ,¨Ymn,y¯m) 6 + 15 + 6 IIB if E6(6) → Sl(5) × Sl(2). m mᨨ α 27 → (y ,ymα,¨y , y ) 5 + 10 + 10 + 2 Big Question: Is this Bookkeeping or Deep Physics?
PvN (CNYITP) The KK program and Sugra Jun, 2016 29 / 29