EXTENDED GAUGED SUPERGRAVITIES AND FLUXES

Transcription

1 EXTENDED GAUGED SUPERGRAVITIES AND FLUXES N s SO OS L U ST RAs U ST I s I T I Æs I L L Corfu, 7 September 2009 Bernard de Wit Utrecht University

2 Extended gauged supergravities and fluxes or Supersymmetric deformations of extended supergravities deformation parameters: charges ~ fluxes They can often be discussed in the context of M-Theory compactifications

3 11D IIA / IIB supergravity reduction on torus T ṇ reduction in presence of p-form fluxes Σ F (p) = C Σ torsion (geometric flux) de a = Tbc a e b e c nongeometric fluxes!!! 4D ungauged supergravity gauging 4D gauged supergravity Samtleben, Truncation of the infinite tower of KK states. The embedding of the gauged theory in the original theory differs from the embedding of the ungauged theory.

4 The possible gaugings may teach us something about BPS states of M-Theory that are not contained in the supergravity approximation TOPICS HIDDEN SYMMETRIES GAUGING AND GAUGE GROUP EMBEDDINGS HIERARCHY OF p-form FIELDS THE p-form HIERARCHY IN 4 SPACE-TIME DIMENSIONS MAXIMAL SUPERGRAVITIES LIFE AT THE END OF THE p-form HIERARCHY

5 HIDDEN SYMMETRIES The toroidal compactification of pure gravity (Kaluza-Klein) M D M d T n (D = d + n) g MN g µν + A µ n + g mn massless states: graviton, n gauge fields (KK photons), 1 scalar fields 2n(n + 1) infinite tower of massive graviton states resulting theory is invariant under the group GL(n) non-linearly realized on the scalars: the massive states carry KK photon charges GL(n) SO(n) charge lattice of KK tower: symmetry restricted to GL(n, Z)

6 Lower space-time dimensions do not follow the generic pattern: three space-time dimensions: the vector fields can be dualized to scalars (Hodge duality) massless: graviton (no states), 1 2n(n + 3) scalars symmetry non-linearly realized on the scalars SL(n + 1) SO(n + 1) Systematic features of toroidal compactifications: the rank of the invariance group increases with n when starting with scalars that parametrize a homogeneous target space, the target space remains homogeneous the presence of the massive states breaks the symmetry group to an arithmetic subgroup

7 Another example: graviton-tensor theory the symmetry of the resulting compactified theory depends sensitively on the original theory L D = 1 2 g R 3 ( ) 2 4 g [M B NP ] g MN g µν + A µ m + g mn B MN B µν + B mµ + B mn G SO(n, n; Z) massless states: graviton, tensor, 2n spin-1 states, and n 2 spinless states tower of massive graviton and tensor states

8 not the generic pattern in five, four and three space-time dimensions! e.g. upon including a dilaton in the original theory, one finds : d > 5 : G = R + SO(n, n; Z) (n, n) vectors d = 5 : G = R + SO(n, n; Z) (n, n) + 1 vectors d = 4 : G = SL(2; Z) SO(n, n; Z) (n, n) + 1 vectors d = 3 : G = SO(n + 1, n + 1; Z) 0 vectors GOAL: study all possible deformations induced by gauging subgroups of G The Hodge dilemma: to increase the symmetry dualize to lower-rank form fields the presence of certain form fields may be an obstacle to certain gauge groups what to do when the theory contains no (vector) gauge fields

9 Example: maximal supergravity in 3 space-time dimensions gauging versus scalar-vector-tensor duality Nicolai, Samtleben, scalars and 128 spinors, but no vectors! obtained by dualizing vectors in order to realize the symmetry E 8(8) (R) solution: introduce 248 vector gauge fields with Chern-Simons terms [ L CS g ε µνρ A M µ MN ν A N ρ 1 ] 3 g f P Q N A P Q Θ ν A ρ EMBEDDING TENSOR vectors invisible at the level of the toroidal truncation First: general analysis of gauge group embeddings.

10 GAUGING AND GAUGE GROUP EMBEDDINGS There are restrictions on the possible gaugings The gauge group must be a subgroup of the full rigid symmetry group of the Lagrangian and/or the equations of motion. Restrictions follow from the consistency of the combined p-form gauge transformations. They can also follow from supersymmetry. The restrictions are subtle! a gauge group may be a proper subgroup but can still not be realized for a certain ungauged Lagrangian.

11 Hence the field content is important But also the space-time dimension is relevant. In particular even and odd dimensions are different

12 Gauge group embeddings gauge a subgroup of with gauge fields A µ M G, the symmetry group of the ungauged theory transforming in some representation of G gauge group encoded into the EMBEDDING TENSOR Θ M α gauge group generators α X M = Θ M t α G generators Θ M α treated as a spurionic quantity, transforming under the action of G according to a product representation dw, Nicolai, Samtleben, Trigiante,

13 This representation branches into irreducible representations. Not all these representations are allowed!! (for instance, because of supersymmetry) Representation (linear) constraint EMBEDDING TENSORS FOR MAXIMAL SUPERGRAVITY IN D = 3,4,5,6,7 7 SL(5) = SO(5, 5) = E 6(6) = E 7(7) = E 8(8) = D G M α characterize all possible gaugings group-theoretical classification universal Lagrangians dw, Samtleben, Trigiante, 2002

14 Closure (quadratic) constraint closure: [X M, X N ] = f MN P X P Θ M β Θ N γ f βγ α = f MN P Θ P α = Θ M β t βn P Θ P α (X M ) γ α X MN P g Θ M α is invariant under the gauge group [X M, X N ] = X MN P X P X MN P contains the gauge group structure constants, but is in general not symmetric in lower indices, unless contracted with the embedding tensor!!!! Z M NP X (NP ) M Z M NP Θ M α = 0 Jacobi identity affected : X R [NP X M Q]R = 2 3 ZM R R[N X P Q]

15 in special basis: X P MN = f M problematic!! A µ M The gauge fields not involved in the gauging can still carry charges. This is known to be inconsistent! To see this: covariant derivative D µ = µ g A µ M X M Ricci identity [D µ, D ν ] = g F µν M X M field strength F M µν = µ A M ν ν A M µ + g X M NP A N P [µ A ν] anti-symmetric part

16 Palatini identity δf µν M = 2 D [µ δa ν] M 2 g Z M P Q δa [µ P A ν] Q NOT covariant indeed! options: try to enlarge/change the gauge group or... introduce an extra gauge transformation and MN introduce 2-form gauge fields B µν cancels the undesirable terms: δ Ξ A µ M = g Z M NP Ξ µ NP whose variation Z M NP F µν M H µν M = F µν M + g Z M NP B µν NP acts as an intertwining tensor between the gauge field representation and the 2-form field representation subtle: regard (NP ) as a single index, which does not map into the full symmetric tensor product!

17 This leads to, e.g. δb MN µν = 2 D [µ Ξ MN ν] 2 Λ M N H µν + 2 A M N [µ δa ν] g Y MN P RS P RS Φ µν H MN MN µνρ = 3 D [µ B νρ] M + 6 A [µ ν A N ρ] gx [P Q] N A P Q ν A ρ] etcetera where +g Y MN P RS C µνρ P RS P RS Φ µν P RS C µνρ new gauge parameter new tensor field Y MN P RS new covariant tensor proportional to the embedding tensor, orthogonal to Z M NP Potentially there are complete p-form representations

18 HIERARCHY OF p-form FIELDS this structure continues indefinitely A µ M B µν MN C µνρ MNP Λ M Ξ µ MN Φ µν MNP (p-form gauge fields) (transformation parameters) Z M NP Y MN P QR Y MNP QRST (intertwining tensors) The covariant intertwining tensors are all proportional to the embedding tensor and mutually orthogonal. The intertwining tensors have been determined by induction. dw, Samtleben, 2005 dw, Nicolai, Samtleben, 2008

19 Alternative deformations (digression) An obvious question is whether the gaugings discussed so far are the only viable deformations. While it is true that other deformations are known in supergravity, there are indications that these deformations are already incorporated in the present approach. H µν M = µ A ν M ν A µ M + g X NP M A [µ N A ν] P + g Z M NP B µν NP H µνρ MN = 3 D [µ B νρ] MN + 6 A [µ M ( ν A ρ] N gx [P Q] N A ν P A ρ] Q ) +g Y MN P RS C µνρ P RS O(g 0 ) : survives g = 0 limit (known from Einstein-Maxwell SG) Z M NP Θ M α = 0 = Θ = 0, Z 0 (Romans massive deformation)

20 At this point there is no Lagrangian yet. (There exist universal Lagrangians!) In the context of a Lagrangian the transformations of the gauge hierarchy are subject to change. Often the hierarchy breaks off at some point and higher rank forms do not appear in the Lagrangian (projection) The physical degrees of freedom are shared between the various tensor fields in a way which depends on the embedding tensor. studied/applied in D = 2,3,4,5,6,7 space-time dimensions in D=4, for N = 0,1,2,4,8 supergravities in D=3, for N = 1,...,6,8,9,10,12,16 supergravities by e.g.: Bergshoeff, Derendinger, de Vroome, dw, Herger, Hohm, Nicolai, Petropoulos, Ortin, Prezas, Riccione, Samtleben, Schön, Sezgin, Trigiante, Van Proeyen, van Zalk, Weidner, West, Zagermann, etc. Related work by, e.g.:d Auria, Ferrara, Hull, Louis, Micu, Reid-Edwards, Sommovigo, Vaula, etc.

21 Another example: 5 space-time dimensions 42 scalars and 27 vectors, and no tensors! in order 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! vectors decompose according to: 27 (15, 1) + (6, 2) linear constraint follows from supersymmetry: Günaydin, Romans, Warner, 1986 Θ M α = quadratic constraint follows from closure: charged vector fields must be (re)converted to tensor fields! ( ) s = dw, Samtleben, Trigiante, 2005

22 digression: consider the representations appearing in (27 27) s = ( ) X P (MN) = d I,MN Z P,I d MNI : E 6(6) invariant tensor(s) { 27 two possible representations can be associated with the new index (27 27) s = indeed: (27 27) a = 351 from the closure constraint: X (MN) P = d MNQ Z P Q anti-symmetric! Z MN Θ N α = 0 Z MN X N = 0 orthogonality X MN [P Z Q]N = 0 gauge invariant tensor this structure is generic!

23 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 27 and 27, respectively. δa M µ = µ Λ M g X M [P Q] Λ P A Q µ g Z MN Ξ µ N extra gauge invariance F µν M = µ A ν M ν A µ M + g X [NP ] M A µ N A ν P not fully covariant introduce fully covariant field strength H µν M = F µν M + g Z MN B µν N to compensate for lack of closure: δb µν M = 2 [µ Ξ ν]n g X Q P N A P [µ Ξ ν]q + g Z MN Λ P X Q P N B µν Q ( ) g 2 d MP Q [µ A P ν] g X P RM d P QS A R S [µ A ν] Λ Q because of the extra gauge invariance, the degrees of freedom remain unchanged (subtle) upon switching on the gauging there will be a balanced decomposition of vector and tensor fields

24 Universal invariant Lagrangian containing kinetic terms for the tensor fields combined with a Chern-Simons term for the vector fields L VT = 1 { [ ( 2 iεµνρστ gz MN P B µν M D ρ B στ N + 4 d NP Q A ρ σ A Q τ + 1 )] 3 g X [RS] Q A R S σ A τ Z MN 8 3 d MNP projects higher-p gauge transformations [ A µ M ν A ρ N σ A τ P g X [QR] M A µ N A ν Q A ρ R ( σ A τ P g X [ST ] P A σ S A τ T )]} zeroth order in the coupling constant! this term is present for ALL gaugings there is no other restriction than the constraints on the embedding tensor dw, Samtleben, Trigiante, 2005

25 The embedding tensor approach yields universal results for any theory of interest. Crucial: one works with complete duality representations of all the p-forms. Therefore there is a considerable redundancy of degrees of freedom which are controlled by the extra gauge invariances. There are also (unexpected) additional symmetries in the context of specific actions. The previous examples concerned odd space-time dimensions. Now we turn to even dimensions and consider D=4.

26 THE p-form HIERARCHY IN 4 SPACE-TIME DIMENSIONS Here the ungauged Lagrangian is not unique because of electric/magnetic duality Consider with n abelian gauge fields A µ Λ Field equations & Bianchi identities: [µ F νρ] Λ = 0 = [µ G νρ] Λ where G µν Λ = ε µνρσ L F ρσ Λ 2n-component vector of electric and magnetic fields and inductions: G µν M = ( Fµν Λ G µνλ ) Its rotations leave the field equations and Bianchi identities invariant!

27 ( ) F Λ ( ) F Λ ( ) ( ) U Λ Σ Z ΛΣ F Σ G Λ G Λ = W ΛΣ V Λ Σ G Σ The equations can be described on the basis of a new Lagrangian provided the rotation matrix is symplectic, ( ) 0 1 i.e. when it leaves the matrix Ω = invariant. 1 0 The new Lagrangian, which describes equivalent field equations and Bianchi identities, does not follow from straightforward substitution. Instead: L( F ) εµνρσ Fµν Λ GρσΛ = L(F ) εµνρσ F µν Λ G ρσλ Hamiltonian The Lagrangian does not transform as a function: L( F ) L(F ) but L(F ) εµνρσ F µν Λ G ρσλ does.

28 Invariance when L( F ) = L( F ) Electric groups ( Z = 0 ) : F Λ µν = U Λ Σ F µν Σ then L(U Λ Σ F Σ ) = L(F Λ ) 1 8 εµνρσ (U T W ) ΛΣ F µν Λ F ρσ Σ Electric gaugings δ local L = 1 8 εµνρσ Λ Λ X ΛΣΓ F µν Σ F ρσ Γ Peccei-Quinn function of coordinates non-abelian field strengths this requires an extra term L top = 1 3 g εµνρσ X ΛΣΓ A µ Λ A ν Σ ( ρ A σ Γ g X Ξ Γ A ρ Ξ A σ ) dw, Lauwers, Van Proeyen, 1985

29 The gauge generators should be consistent with the symplectic property of the electro/magnetic duality transformations: X M[N Q Ω P ]Q = 0 and are subject to a representation (linear) constraint: X (MN Q Ω P )Q = 0 = X (ΛΣΓ) = 0 2X (ΓΛ) Σ = X Σ ΛΓ X (ΛΣΓ) = 0 X (ΓΛ) Σ = X Σ ΛΓ hence, not in general anti-symmetric!

30 Consider also: X (MN) P = Z P MN = 1 2 ΩP R Θ R α t αm Q Ω NQ = Z P,α d αmn This leads to the definitions: d α MN (t α ) M P Ω NP Z M,α 1 2 ΩMN Θ N α = { Z Λα = 1 2 ΘΛα α Z Λ = 1 2 Θ Λ α magnetic electric 2-forms transform in adjoint representation Quadratic constraint: Z M α Θ M β d αp Q = 1 2 ΩMN Θ M β Θ N α d αp Q = 0 Possibly stronger version: Ω MN Θ M β Θ N α = 0 there exists a purely electric duality frame!

31 The Lagrangian: 1 - Define new electric and magnetic covariant field strengths: where H µν M = F µν M + gz M,α B µν α B µνα = d αmn B µν MN 2 - Include electric and magnetic gauge fields in the covariant derivatives and replace the (electric) field strengths by the modified ones given above. 3 - Add the following term to the Lagrangian: L top = 1 8 gεµνρσ Θ Λα ( B µνα 2 ρ A σλ + gx MNΛ A M ρ A N σ 1 4 gθ Λ β ) B ρσβ gεµνρσ X MNΛ A µ M A ν N ( ρ A σ Λ gx P Q Λ A ρ P A σ Q ) gεµνρσ X MN Λ A µ M A ν N ( ρ A σλ gx P QΛA ρ P A σ Q ) This represents the universal Lagrangian for any gauging. It depends on the embedding tensor whose constraints ensure its full gauge invariance!

32 4 - In principle the tensor fields can be integrated out. One then finds a conventional Lagrangian with electric gaugings written in an another electric/magnetic duality frame.

33 MAXIMAL SUPERGRAVITIES Apply the embedding tensor formalism to the maximal supergravities, with the duality group, the representations of the vector gauge fields and the embedding tensor as input. At this point, the number of space-time dimensions is not used! This purely group-theoretic analysis yields all the representations for the hierarchy of p-form fields.

34 Leads to : rank SL(5) SO(5, 5) 16 c s s s E 6(+6) E 7(+7) E 8(+8) Striking feature: rank D-2 : adjoint representation of the duality group dw, Samtleben, Nicolai, 2008 note: restricted representation, not the full symmetric tensor product

35 rank SL(5) SO(5, 5) 16 c s s s E 6(+6) E 7(+7) E 8(+8) Striking feature: rank D-1 : embedding tensor!

36 rank SL(5) SO(5, 5) 16 c s s s E 6(+6) E 7(+7) E 8(+8) Striking feature: rank D : closure constraint on the embedding tensor!

37 rank SL(5) SO(5, 5) 16 c s s s E 6(+6) E 7(+7) E 8(+8) Perhaps most striking: implicit connection between space-time electric/magnetic (Hodge) duality and the U-duality group Probes new states in M-Theory! Θ dial

38 M-theory implications: SL(5) SO(5, 5) 16 c s s s E 6(+6) E 7(+7) E 8(+8) The table coincides substantially with results based on several rather different conceptual starting points: M(atrix)-Theory compactified on a torus: duality representations of states Correspondence between toroidal compactifications of M-Theory and del Pezzo surfaces E11 decompositions

39 Algebraic Aspects of Matrix Theory on T n Elitzur, Giveon, Kutasov, Rabinovici, 1997 T n Based on the correspondence between super-yang-mills on and M-Theory on T n, a rectangular torus with radii R 1, R 2,... R n in the infinite-momentum frame. Invariance group consist of permutations of the R i combined with the T-duality relations ( i j k ) : R i l3 p R j R k R j l3 p R k R i R k l3 p R i R j l 3 p l6 p R i R j R k generate a group isomorphic with the Weyl group of E n(n) The explicit duality multiplets arise as representations of this group.

40 Example n=4 D=7 4 KK states on T n M 1 R i 6 2-brane states wrapped on T n M R jr k l 3 p j k 4 2-brane states wrapped on T n x 11 M R 11R i l 3 p 1 5-brane state wrapped on T n x 11 M R 11R 1 R 2 R 3 R 4 l 6 p the dimensions of these two multiplets coincide with those of the multiplets presented previously for vectors and tensors for higher n the multiplets are sometimes incomplete, because they are not generated as a single orbit by the Weyl group.

41 A Mysterious Duality Iqbal, Neitzke, Vafa, 2001 This cannot be a coincidence! It is important to uncover the physical interpretation of these duality relations. One possibility is that the del Pezzo surface is the moduli space of some probe in M-Theory. It must be a U-duality invariant probe... Such probe is the gauging encoded in the embedding tensor! E11 decomposition Based on the conjecture that E11 is the underlying symmetry of M-Theory. Decomposing the relevant E11 representation to dimensions D<11 yields representations that substantially overlap with those generated for the gaugings. West et. al., Bergshoeff et. al.,

42 LIFE AT THE END OF THE p-form HIERARCHY SL(5) SO(5, 5) 16 c s s s E 6(+6) E 7(+7) E 8(+8) It is possible to construct the hierarchy starting from the intermediate (D-3)-forms, assuming that they transform according to the conjugate of the representation associated with the vector fields. In this way one generates the (D-2)-, the (D-1)-, and the D-form fields, in accordance we the results found in the table. Note that the latter two forms are not related to any other forms by Hodge duality!

43 p-forms transforming in the conjugate of the representations of the 1-forms, the adjoint representation, the embedding tensor and the constraints: [D 3] C M = D [D 4] Φ [D 2] C α = D [D 3] Φ α M + Y [D 3] M Φ β α + Y [D 2] α,m Φ α M β [D 1] C M α = D [D 2] Φ M α + Y M α,p Q β [D 1] Φ P Q β [D] C MN α = D [D 1] Φ MN α + Y MN α,p QR β [D] Φ P QR β [D+1] C P QR α = D [D] Φ P QR α + intertwiners

44 closure constraint Q MN α δ M Θ N α = Θ M β δ β Θ N α intertwiners Y M α = Θ M α Y α,m β = δ α Θ M β Y M α,p Q β = δ δ Θ M α Q P Q β Y MN α,p QR β = δ M P Y N α,qr β + X P Q M δ N R δ β α + X P R N δ M Q δ β α X P α β δ N R δ M Q

45 Alternative form for the intertwiners (closer to the generic formulae that follow by induction) Y α,m β = t αm N Y N β X M β α, Y M α,p Q β = δ P M Y α,q β (X P ) Q β,m α, Y MN α,p QR β = δ P M Y N α,qr β (X P ) QR β,mn α orthogonality: Y Y Q MN α Y MN α,p QR β Q MN α = 0

46 What is the role of the higher form fields? This construction supports the following idea which has been worked out completely for three and four space-time dimensions: Regard the embedding tensor as a space-time field transforming in the appropriate representation, but not satisfying the quadratic closure constraint. Add the gauge invariant Lagrangian with (D-1)- and D-form fields: L = g ε µ 1µ 2 µ D C µ1 µ D 1 M α D µd Θ M α + g 2 ε µ 1µ 2 µ D C µ1 µ D MN α Q MN α dw, Samtleben, Nicolai, 2008 dw, van Zalk, 2009

47 Conclusions General gaugings of a large variety of theories can be constructed and studied in the framework of the embedding tensor technique, which, in principle, entails a hierarchy of p-forms. Maximal supergravity theories contain subtle information about M-Theory. This may be interpreted as an indication that supergravity needs to be extended towards string/mtheory.

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