E6(6)–covariance in type IIB
Oscar Varela
Harvard
Franz Ciceri, Bernard de Wit, O.V., to appear
Miami 2014, Lago Mar Resort, Fort Lauderdale 22 Dec 2014 The problem
Maximal (ungauged) supergravities in lower dimensions enjoy invariance under exceptional symmetry groups: the so-called U-dualities.
d-dimensional (ungauged) supergravities arise upon toroidal reduction of D-dimensional supergravities.
But D-dimensional supergravity doesn’t exhibit any exceptional symmetry in any obvious way.
A long-standing question has therefore been: where do these symmetries come from?
Oscar Varela (Harvard) E6(6) in IIB 2 / 19 The solution
In other words, do the U-dualities hide somewhere in D-dimensions, or are they just an artifact of the dimensional reduction?
The solution is: yes, they hide in D-dimensions! I’ll show that the D-dimensional theories admit a reformulation where the exceptional symmetry is manifest.
This reformulation can be achieved at the expense of waiving local D-dimensional Lorentz covariance: only d-dimensional local Lorentz covariance is manifest.
The fields retain dependence on all D coordinates: the theory is thus D-dimensional, albeit with manifest d-dimensional local Lorentz covariance only.
Oscar Varela (Harvard) E6(6) in IIB 3 / 19 Outline
1 Higher-dimensional realisation of local vs global symmetries
2 The E6(6)–covariant formulation of IIB supergravity
3 Final comments
Oscar Varela (Harvard) E6(6) in IIB 4 / 19 Local vs global in high D Local vs global in high D
1 Higher-dimensional realisation of local vs global symmetries
2 The E6(6)–covariant formulation of IIB supergravity
3 Final comments
Oscar Varela (Harvard) E6(6) in IIB 5 / 19 Local vs global in high D Symmetries in supergravity
Maximally supersymmetric supergravity in dimension d < 11 is invariant under a global symmetry group G (the U-duality group), and a local symmetry group H (the R-symmetry group), with H ⊂ G. The group G ”is in the E series”.
The fermions lie in linear representations of H.
The vectors and tensors lie in linear representations of G.
The scalars feel both G and H: they lie in the coset G/H, so that G acts non-linearly on them.
Oscar Varela (Harvard) E6(6) in IIB 6 / 19 Local vs global in high D D-dim supergravity with local H invariance
The question of whether D = 11 supergravity admitted a reformulation with manifest local R-symmetry H was answered positively long ago.
de Wit and Nicolai showed in the 80’s that D = 11 supergravity admits a reformulation with manifest local SU(8) invariance (the R-symmetry of d = 4), at the expense of only allowing the D = 11 theory to exhibit a local SO(1, 3) invariance, rather than SO(1,10).
They found hints of E7(7) (the G of d = 4) in their formulation, but not a conclusive
proof that D = 11 supergravity coud be reformulated with E7(7) symmetry.
Oscar Varela (Harvard) E6(6) in IIB 7 / 19 Local vs global in high D D-dim supergravity with global G invariance?
The challenge since has been to reformulate the D-dimensional supergravities with manifest global G invariance. Many subsequent developments in string/M-theory can be understood in this context:
Generalised geometry (see talk by Erik Plauschinn)
Double field theory
Exceptional field theory
In this talk: I’ll follow an approach that stays as close as possible to the original formulation of type IIB.
Oscar Varela (Harvard) E6(6) in IIB 8 / 19 E6(6) in IIB
The E6(6)–covariant formulation of IIB supergravity
1 Higher-dimensional realisation of local vs global symmetries
2 The E6(6)–covariant formulation of IIB supergravity
3 Final comments
Oscar Varela (Harvard) E6(6) in IIB 9 / 19 E6(6) in IIB IIB and D = 5 N = 8 supergravities
IIB SU(1, 1) U(1) D = 5 N = 8 E6(6) USp(8)
A α EM 1 1 eµ 1 1 α M AMN 2 1 Cµ 27 1
AMNP QR 1 1 Cµν M 27 1 scalars 3 − 1 1 scalars 78 − 36 = 42
1 ψM 1 2 ψM 1 8 3 λ 1 2 λ 1 48
For the scalars, vielbeine can be defined in IIB, φα, in the 2 of SU(1,1), and in D = 5 AB N = 8, VM , in the 27 of E6(6) and 27 of USp(8).
In D = 5 N = 8: E6(6)–covariant vector-tensor hierarchy manifest in the embedding tensor formalism.
Oscar Varela (Harvard) E6(6) in IIB 10 / 19 E6(6) in IIB The strategy
Split the D = 10 spacetime into 5 + 5–dimensions. Recall SU(1, 1) =SL(2, R). Then:
For the bosons:
SO(1, 9) × SL(2, R) −→ SO(1, 4) × SO(5) × SL(2, R)
−→ SO(1, 4) × SL(6, R) × SL(2, R)
−→ SO(1, 4) × E6(6)
For the fermions:
Spin(1, 9) × U(1) −→ Spin(1, 4) × USp(4) × U(1)
−→ Spin(1, 4) × SU(4) × U(1)
−→ Spin(1, 4) × USp(8) ,
But very non-obvious how to implement these!
Oscar Varela (Harvard) E6(6) in IIB 11 / 19 E6(6) in IIB Kaluza-Klein-like decompositions
Split the bosonic fields according to SO(1, 9) × SL(2, R) −→ SO(1, 4) × SO(5) × SL(2, R), e.g. −1/3 α m a A ∆ eµ Bµ em EM = , a 0 em
α and similarly for AMN and AMNPQ. Focusing here only on vectors and tensors:
vectors SO(5) × SL(2, R) tensors SO(5) × SL(2, R)
A m EM −→ Bµ (5, 1) − − α α α AMN −→ Aµm (5, 2) Aµν (1, 2)
AMNPQ −→ Aµmnp (10, 1) Aµνmn (10, 1) TOTAL: 25 12
Oscar Varela (Harvard) E6(6) in IIB 12 / 19 E6(6) in IIB Field counting
But we expect 27 vectors and 27 tensors, in the 27 and 27 of E6(6). Bring in the dual fields: Aα , the dual of Aα , M1...M6 MN the dual graviton,
AMNPQ is self-dual, so it shouldn’t give any further contribution.
vectors SO(5) × SL(2, R) tensors SO(5) × SL(2, R)
A m EM −→ Bµ (5, 1) − − α α α AMN −→ Aµm (5, 2) Aµν (1, 2)
AMNPQ −→ Aµmnp (10, 1) Aµνmn (10, 1)
Aα −→ Aα (1, 2) Aα (5, 2) M1...M6 µmnpqr µνmnpq dual graviton −→ − − Aµνm;mnpqr (5, 1) TOTAL: 27 27
Oscar Varela (Harvard) E6(6) in IIB 13 / 19 E6(6) in IIB Further redefinitions
Extremely non-trivial, non-linear redefinitions are needed so that the supersymmetry variations of these comply with the E6(6) vector-tensor hierarchy. For example:
20 β 5 β γ δ Cµ αmnpqr = Aα µmnpqr + 3 i εαβ Aµ [m Anpqr] − 6 εαβ εγδ A [mn Aµ p A qr]
M So we end up with vectors Cµ and tensors Cµν M in the 27 and 27 of E6(6). Further dualising the SO(5) indices:
m mn I M Cµ ,Cµ ,Cµ αm ,Cµ α = Cµ ,Cµ αI = Cµ
αm α αI Cµν m ,Cµν mn ,Cµν ,Cµν = Cµν I ,Cµν = Cµν M in agreement with
SL(2)×SO(5) SL(2)×SL(6) (1, 5) + (1, 10) + (2, 5) + (2, 1) ←− (1, 15) + (2, 6) ←− 27
Oscar Varela (Harvard) E6(6) in IIB 14 / 19 E6(6) in IIB 27-bein and fermions
We have also managed to rearrange
the scalars into a vielbein in the 27 of E6(6)
A the external components of the gravitino into ψµ , in the 8 of USp(8)
ABC the internal components of the gravitino ψa and the dilatino λ into χ , in the 48 of USp(8):
ABC I hAB Ci I7 hAB Ci IJ7 hAB Ci χ = (Γ ) ϕI − (Γ ) (Γ7ϕI ) + (Γ ) φIJ ,
where ΓI , I = 1,..., 6, are SU(4) = SO(6) gamma matrices and
A A 3 A A A 3 A ϕa = 0 , ϕ6 = 8 i λ , φab = 0 , φa6 = 8 i ψa .
Oscar Varela (Harvard) E6(6) in IIB 15 / 19 E6(6) in IIB Supersymmetry variations
All of these redefined fields transform under supersymmetry into one another, following supersymmetry rules analog to those of D = 5 N = 8.
For example:
M AC B ABC M δCµ = 2 iΩ¯ ¯C ψµ + ¯C γµχ VAB ,
√ 4 AB ¯ C N P C δCµν M = 5 5 VM 2 ψ[µA γν] ΩBC − i¯χABC γµν + 2 dMNP C[µ δCν]
Here, dMNP is the E6(6) cubic invariant, which we can read off!
Oscar Varela (Harvard) E6(6) in IIB 16 / 19 Final comments Final comments
1 Higher-dimensional realisation of local vs global symmetries
2 The E6(6)–covariant formulation of IIB supergravity
3 Final comments
Oscar Varela (Harvard) E6(6) in IIB 17 / 19 Final comments Consequence 1: uniqueness
Our results imply that here exists a unique maximally supersymmetric theory with local SO(1, d − 1) covariance, regardless of whether it is formulated in d-dimensions (the natural choice) or in D-dimensions.
This is, after all, not so surprising: the maximal supersymmetry algebra containing so(1, d − 1) is unique (up to central extension).
Oscar Varela (Harvard) E6(6) in IIB 18 / 19 Final comments Consequence 2: consistent truncations
While the low-d ungauged supergravities arise from standard dimensional reduction, the story is not so obvious for gauged supergravities.
Our formalism clears the way to the investigation of which gauged low-d supergravities arise from high D upon consistent truncation in general.
In particular, we find explicit consistent truncation formulae for type IIB on S5 down to D = 5 N = 8 SO(6)-gauged supergravity. This is the relevant for application in the best-known example of (large N) AdS/CFT.
Oscar Varela (Harvard) E6(6) in IIB 19 / 19