PROBING M-THEORY DEGREES OF FREEDOM Bernard de Wit Second 2007 - PowerPoint PPT Presentation

PROBING M-THEORY DEGREES OF FREEDOM Bernard de Wit Second 2007 Workshop Galileo Galilei Institute, Firenze Utrecht University

Definition of M-Theory ? 11-dimensional supergravity ? toroidal compactifications thereof with E n ( n ) ( Z ) ? + Kaluza-Klein states (1/2-BPS) ? + branes + etcetera ? what about IIB theory ? Matrix theory ? Membrane theory We start from the (effective) field theory perspective with 32 supersymmetries (bottom up approach - unlike work by Englert, Nicolai, West, etc. ) work in progress with Hermann Nicolai and Henning Samtleben

｜ ｜ ｜ ｜ ｜ ｜ ⎨ ⎩ ⎧ ｜ ｜ ｜ ⎨ ⎩ ⎧ ｜ ↵ D=11 SUGRA 11 - ↵ IIB SUGRA 10 - 9 - ↵ central charges D=9 (2 , 1) ⊕ (1 , 1) massless 8 - states 7 - KK states 6 - 5 - extra KK states 4 - incomplete incomplete

⎨ ⎧ ⎩ ⎨ ⎧ ⎩ 11D - IIA - IIB PERSPECTIVE IIA momentum KK states + D0 branes KKA (2 , 1) IIB winding strings + D1 branes IIB momentum KK states KKB (1 , 1) IIA winding strings 9D SUGRA contains 2+1 gauge fields central charges Supermembrane ? Schwarz, 1996 Aspinwall, 1996 indication of higher-dimensional origin (without full decompactification) Abou-Zeid, dW, Lüst, Nicolai, 1999-2001

D=11 IIA D=9 IIB SO(1 , 1) ˆ G µ ν G µ ν g µ ν G µ ν 0 ˆ A µ 9 10 C µ 9 B µ G µ 9 − 4 G µ 9 , ˆ ˆ G µ 10 G µ 9 , C µ A α A α 3 µ µ 9 A µ ν 9 , ˆ ˆ A µ ν 10 C µ ν 9 , C µ ν A α A α − 1 µ ν µ ν ˆ A µ νρ C µ νρ A µ νρ A µ νρσ 2 � φ α φ α 0 G 9 10 , ˆ ˆ G 9 9 , ˆ φ , G 9 9 , C 9 G 10 10 exp( σ ) G 9 9 7 M BPS ( q 1 , q 2 , p ) = m KKA e 3 σ / 7 | q α φ α | + m KKB e − 4 σ / 7 | p | m 2 KKA m KKB ∝ T m

more generally: SUPERSYMMETRY ANTI-COMMUTATOR αβ P M + 1 αβ Z MN + 1 5! Γ MNP QR { Q α , ¯ Q β } = Γ M 2 Γ MN Z MNP QR αβ CENTRAL CHARGES (pointlike) 9 SL(2) × SO(1 , 1) SO(2) ( 2 , 1 ) ⊕ ( 1 , 1 ) 8 SL(3) × SL(2) U(2) ( 3 , 2 ) 7 E 4(4) ≡ SL(5) USp(4) 10 6 E 5(5) ≡ SO(5 , 5) USp(4) × USp(4) 16 → ( 4 , 4 ) 5 E 6(6) USp(8) 27 ⊕ 1 4 E 7(7) SU(8) 56 → 28 ⊕ 28 3 E 8(8) SO(16) 120 2 E 9(9) SO(16) 1 ⊕ 120 ⊕ 135 compare to vector fields!

CENTRAL CHARGES (stringlike) 9 SL(2) × SO(1 , 1) 2 8 SL(3) × SL(2) ( 3 , 1 ) 7 SL(5) 5 6 SO(5 , 5) 10 ⊕ 1 → ( 5 , 1 ) ⊕ ( 1 , 5 ) ⊕ ( 1 , 1 ) 5 E 6(6) 27 4 E 7(7) 63 3 E 8(8) 135 2 E 9(9) 135 compare to tensor fields! another perspective ........

GAUGINGS class of deformations of maximal supergravities gauging versus scalar-vector-tensor duality first: 3 space-time dimensions 128 scalars and 128 spinors, but no vectors ! E 8(8) ( R ) obtained by dualizing vectors in order to realize the symmetry solution: introduce 248 vector gauge fields with Chern-Simons terms N − 1 L CS ∝ g ε µ νρ A µ � P A ρ Q � M Θ MN N A ν ∂ ν A ρ 3 g f P Q Θ EMBEDDING TENSOR ‘invisible’ at the level of the toroidal truncation Nicolai, Samtleben, 2000

☞ ☞ another example: 5 space-time dimensions 42 scalars and 27 vectors, and no tensors ! to realize the symmetry E rigid 6(6) × USp(8) local introduce a local subgroup such as E 6(6) → SO(6) local × SL(2) inconsistent! Günaydin, Romans, Warner, 1986 vectors decompose according to: ( 15 , 1 ) + ( 6 , 2 ) 27 → ↵ charged vector fields must be (re)converted to tensor fields ! gauge group encoded into the α EMBEDDING TENSOR Θ M treated as spurionic order parameter ∈ E 6(6) probes new M-theory degrees of freedom α t α α X M = Θ M Θ M ↵ ↵ gauge group generators generators E 6(6)

The embedding tensor is subject to constraints ! P X P closure: [ X M , X N ] = f MN β Θ N γ f βγ P Θ P β t β N P Θ P α = f MN α = − Θ M α Θ M ↵ P ∈ E 6(6) X MN P X P [ X M , X N ] = − X MN contains the gauge group structure constants, but is P X MN not symmetric in lower indices, unless contracted with the embedding tensor !!!! supersymmetry: α ∈ 351 Θ M 27 × 78 = 27 + 351 + 1728 ( 351 × 351 ) s = 27 + 1728 + 351 ′ + 7722 + 17550 + 34398 (closure) dW, Samtleben, Trigiante, 2005

EMBEDDING TENSORS FOR D = 3,4,5,6,7 7 SL(5) 10 × 24 = 10 + 15 + 40 + 175 6 SO(5 , 5) 16 × 45 = 16 + 144 + 560 5 E 6(6) 27 × 78 = 27 + 351 + 1728 4 E 7(7) 56 × 133 = 56 + 912 + 6480 3 E 8(8) 248 × 248 = 1 + 248 + 3875 + 27000 + 30380 dW, Samtleben, Trigiante, 2002 characterize all possible gaugings group-theoretical classification universal Lagrangians applications in D = 3,4,5,7 space-time dimensions, in D=4, for N = 2,4,8 supergravities in D=3, for N = 1,...,6,8,9,10,12,16 supergravities de Vroome, dW, Herger, Nicolai, Samtleben, Schön, Trigiante, Weidner

digression: consider the representations appearing in ( 27 × 27 ) s = ( 27 + 351 ′ ) P = d I,MN Z P,I invariant tensor(s) d MNI : E 6(6) X ( MN ) � 27 two possible representations can be associated with the new index 351 ′ ′ + 1728 + 7722 27 × ( 27 × 27 ) s = 351 + 27 + 27 + 351 P = d MNQ Z P Q indeed: X ( MN ) ( 27 × 27 ) a = 351 from the closure constraint: Z MN Θ N α = 0 Z MN X N = 0 orthogonality → [ P Z Q ] N = 0 gauge invariant tensor X MN this structure is generic (at least, for the groups of interest) and we will exploit it later !

rather than converting and tensors into vectors and reconverting some of them them when a gauging is switched on, we introduce both vectors and tensors from the start, transforming into the representations and , respectively 27 27 extra gauge invariance ↵ M Λ P A Q µ − g Z MN Ξ µ N δ A M µ = ∂ µ Λ M − g X [ P Q ] Ξ M A µ M = ∂ µ A ν M + g X [ NP ] not fully covariant M − ∂ ν A µ N A ν P F µ ν M + g Z MN B µ ν N M = F µ ν introduce fully covariant field strength H µ ν to compensate for lack of closure: Q A [ µ P Ξ ν ] Q + g Z MN Λ P X P N Q B µ ν Q δ B µ ν M = 2 ∂ [ µ Ξ ν ] N − g X P N Ξ Ξ � S � P d P QS A [ µ R A ν ] P − g X RM Λ Q − g 2 d MP Q ∂ [ µ A ν ] because of the extra gauge invariance, the degrees of freedom remain unchanged upon switching on the gauging there will be a balanced decomposition of vector and tensor fields

Universal invariant Lagrangian containing kinetic terms for the tensor fields combined with a Chern-Simons term for the vector fields 1 Q + 1 2 i ε µ νρστ � � P � Q A σ R A τ S �� gZ MN B µ ν M Z MN L VT = D ρ B στ N + 4 d NP Q A ρ ∂ σ A τ 3 g X [ RS ] − 8 � M ∂ ν A ρ N ∂ σ A τ P 3 d MNP A µ + 3 P + 1 M A µ R � P A σ T ��� N A ν Q A ρ S A τ 4 g X [ QR ] ∂ σ A τ 5 g X [ ST ] this term is present for ALL gaugings there is no other restriction than the constraints on the embedding tensor dW, Samtleben, Trigiante, 2005 Can this be generalized?

⎭ ⎫ ⎬ ⎭ ⎫ ⎬ ｜ ｜ Non-abelian vector-tensor hierarchies Generalize the combined gauge algebra ☛ algebra closes on M ↵ non-closure α A µ Θ M M Λ P A µ Q − g Z M,I Ξ µ I M = ∂ µ Λ M − g X [ P Q ] δ A µ δ B µ ν I = 2 D [ µ Ξ ν ] I + · · · ☛ algebra closes on ↵ Z M,I B µ ν I non-closure J Φ µ ν J M δ B µ ν I = 2 D [ µ Ξ ν ] I + · · · − g Y IM Z M,I Y IN J = 0 J + 2 d I,MN Z N,J with J ≡ X MI Y IM M = 3 D [ µ Φ νρ ] I M + · · · δ S µ νρ I ☛ algebra closes on Y IM J S µ νρ J etcetera dW, Samtleben, 2005

explicit results are complicated: N + 1 � � P A ρ ] M ( ∂ ν A ρ ] N A ν Q ) H µ νρ I ≡ 3 D [ µ B νρ ] I + 2 d I,MN A [ µ 3 gX [ P Q ] J S µ νρ I J M + g Y IM Y IM M = g Λ N X NI J S µ νρ J M S µ νρ I M − g Λ N X NP P δ S µ νρ I M D ν Ξ ρ ] I + 3 ∂ [ µ A ν M Ξ ρ ] I M + 3 A [ µ + 3 D [ µ Φ νρ ] I − 2 g d I,NP Z P,J A [ µ M A ν N Ξ ρ ] J + 4 d I,NP Λ [ M A [ µ N ] ∂ ν A ρ ] J d J,P Q Λ Q A [ µ P + 2 g X NI M A ν N A ρ ] P Plumbing strategy: repair the lack of closure iteratively by introducing tensor gauge fields of increasing rank M − I − M − A µ → B µ ν → S µ νρ I → etc Λ M M Ξ µI Φ µ ν I encoded by the embedding tensor !

Leads to : rank ➯ 1 5 6 2 3 4 7 SL(5) 15 + 40 10 5 5 10 24 6 SO(5 , 5) 16 10 16 45 144 5 E 6(+6) 27 + 1728 27 27 78 351 4 E 7(+7) 133 + 8165 56 133 912 3 E 8(+8) 3875 + 147250 248 3875 Striking feature: rank D-2 : adjoint representation of the duality group dW, Samtleben, Nicolai, work in progress

rank ➯ 1 5 6 2 3 4 7 SL(5) 15 + 40 10 5 5 10 24 6 SO(5 , 5) 16 10 16 45 144 5 E 6(+6) 27 + 1728 27 27 78 351 4 E 7(+7) 133 + 8165 56 133 912 3 E 8(+8) 3875 + 147250 248 3875 Striking feature: rank D-1 : embedding tensor

rank ➯ 1 5 6 2 3 4 7 SL(5) 15 + 40 10 5 5 10 24 6 SO(5 , 5) 16 10 16 45 144 5 E 6(+6) 27 + 1728 27 27 78 351 4 E 7(+7) 133 + 8165 56 133 912 3 E 8(+8) 3875 + 147250 248 3875 Striking feature: rank D : closure constraint on the embedding tensor

rank ➯ 1 5 6 2 3 4 7 SL(5) 15 + 40 10 5 5 10 24 6 SO(5 , 5) 16 10 16 45 144 5 E 6(+6) 27 + 1728 27 27 78 351 4 E 7(+7) 133 + 8165 56 133 912 3 E 8(+8) 3875 + 147250 248 3875 Perhaps most striking: implicit connection between space-time Hodge duality and the U-duality group Probes new states in M-Theory!