(1,0) Superconformal Models in 6D and Non-abelian Tensor Multiplets

Transcription

1 (1,0) Superconformal Models in 6D and Non-abelian Tensor Multiplets Robert Wimmer, YITP H. Samtleben, E. Sezgin and R.W. arxiv: H. Samtleben, E. Sezgin, L. Wulff and R.W. to appear

2 Introduction & Motivation

3 M-theory DOF & AdS7/CFT6 Correspondence M-theory (11D SUGRA) provides extended objects: M2- and M5-branes. Recently, quite some progress in the understanding of the effective dynamics of multiple M2-branes (BLG & ABJM 08). Finding a field theory for multiple M5-branes seems to be much harder: 1.) Non-abelian Gauge Couplings for Tensor Multiplets? (2-form Gauge Potentials ) B µν 2.) (2,0) Superconformal Symmetry? (no free parameter/weak coupling parametrization) Ad 1.): Circumvent no-go theorems (Bekaert, Hennaux, Sevrin) by introducing also gauge fields and 3-form potential Tensor Hierarchy C µνρ Ad 2.): Situation similar to M2-branes ( N = 8 SO(4) BLG accident ). ABJM: N M2-branes on R 8 /Zk N = 6 but k =1,2,...weak coupling!!! 6D: Don t hope for N = (2,0) accident Study N = (1,0) models

4 Tensor Hierarchy

5 Field Content M5-brane zero mode fluctuation (broken susy) Low Energy Theory: Bµν sd (2,0) Tensor: + fermions Φ A=1,...5 (1,0) Tensor: B sd φ χ i=1,2 µν Vector: A µ Y ij λ i gauge 3-Form: C µνρ (1,0) Hyper: q I, q I=1,2 + ferm. Abelian δb = dλ (1), δa = dλ (0) + Λ (1) F = F B Covariant (abelian invariant) Hodge Duality: B (p) B (4 p) A r, B I, C r

6 Field Content M5-brane zero mode fluctuation (broken susy) Low Energy Theory: Bµν sd (2,0) Tensor: + fermions Φ A=1,...5 (1,0) Tensor: B sd φ χ i=1,2 µν Vector: A µ Y ij λ i gauge 3-Form: C µνρ (1,0) Hyper: q I, q I=1,2 + ferm. Abelian δb = dλ (1), δa = dλ (0) + Λ (1) F = F B Covariant (abelian invariant) Hodge Duality: B (p) B (4 p) A r, B I, C r

7 Non-abelian: (de Wit, Nicolai, Samtleben 05/08 Bergshoeff, Gunaydin, Sezgin 07-10) Input 1: Input 2: Overcome no-go theorem extend configuration space A r B I C r...1-form... 2-form... 3-form Tensor Hierarchy Each form in some representation of the gauge group Define Generalized Field Strengths : F r := F r + h r IB I YM H I := DB I + d I rs K rs + g Ir C r all 2-forms made of A, B all 3-forms made of A, B, C with h r I, g rs, d r pq being invariant tensors of the gauge group Need to find δa r, δb I, δc r such that...

8 Non-abelian: (de Wit, Nicolai, Samtleben 05/08) Input 1: Input 2: Overcome no-go theorem extend configuration space A r B I C r...1-form... 2-form... 3-form Tensor Hierarchy Each form in some representation of the gauge group Define Generalized Field strengths : F r := F r + h r IB I YM H I := DB I + d I rs K rs + g Ir C r {}}{ d + A {}}{ da A, A 3 all 2-forms made of A, B all 3-forms made of A, B, C with h r I, g rs, d r pq being invariant tensors of the gauge group Need to find δa r, δb I, δc r such that...

9 Non-abelian: (de Wit, Nicolai, Samtleben 05/08) Input 1: Input 2: Overcome no-go theorem extend configuration space A r B I C r...1-form... 2-form... 3-form Tensor Hierarchy Each form in some representation of the gauge group Define Generalized Field strengths : F r := F r + h r IB I YM H I := DB I + d I rs K rs + g Ir C r all 2-forms made of A, B all 3-forms made of A, B, C with h r I, g rs, d r pq being invariant tensors of the gauge group Need to find δa r, δb I, δc r such that...

10 Condition: The generalized field strengths must transform covariantly: δf r =Λ r sf s,δh I =Λ I JH J There is a (unique) solution in terms of n = 0, 1, 2 - form gauge parameters : Λ r, Λ I (1), Λ (2)r δf r : δh I : δa r = D Λ r h r I Λ I (1) δb I = D Λ I (1) ( di pq 2Λ p F q A p δa q ) g Ir Λ (2) r δc r = D Λ (2) r + b I rs { F s Λ I (1) +Λs H I + B I δa s di pq A s A p δa q} +... This defines a consistent tensor gauge symmetry, the algebra closes. The invariance of the group tensors ( h r I,...) gives non-linear algebraic conditions, for which we have solutions.

11 The ellipsis denote terms with g Ir (...) =0. The generators X r in the two representations have to be of the form: (X r ) t s = f t rs + d I rs h t I, (X r ) J I =2h s Id J rs g Js b Isr 2-forms are charged only if h s Id J rs 0 or g Js b Isr 0 The various group tensors ( g Ir,...) have to satisfy certain conditions, in particular h r Ig Is =0, h r IX. The invariance conditions ( δg Ir r =0 =0,...) give multi-linear and quadratic conditions. Complex system: δg Ir =0: g (f + hd+ gb) =0! δd I rs =0: h (dd) fd d g b! =0 g Iu b Ju(r d J st) =0 magic identity δb Irs =0: h (db)+fb+ gb! =0 δf rs t =0: f [rs u f t]u v! = 1 3 f [rs u d I r]u hv I Curvatures transform covariantly & symmetry algebra closes. Note: so far no dynamics.

12 Supersymmetry & Superconformal Field Equations

13 Closure of SUSY Coupling of a single uncharged (1,0) tensor multiplet to YM was described in (Bergshoeff, Sezgin, Sokatchev 96). Construct susy transformation such that they close into translations and gaug-tensor gauge transformations: [δ ɛ1,δ ɛ2 ] = ξ µ µ + δ Λ + δ Λ(1) + δ Λ(2) Note: depend on the fields Λ, Λ (1), Λ (2) A r,b I, φ, C r YM Multiplet: δa r µ = ɛγ µ λ r, δλ ir = 1 8 γµν Fµνɛ r i 1 2 Y ij r ɛ j, δy ij r = ɛ (i γ µ D µ λ j)r pure YM closes off-shell into transl. + ordinary gauge transf.

14 Closure of SUSY Coupling of a single uncharged (1,0) tensor multiplet to YM was described in (Bergshoeff, Sezgin, Sokatchev 96). Construct susy transformation such that they close into translations and gaug-tensor gauge transformations: [δ ɛ1,δ ɛ2 ] = ξ µ µ + δ Λ + δ Λ(1) + δ Λ(2) Note: depend on the fields Λ, Λ (1), Λ (2) A r,b I, φ, C r YM Multiplet: δa r µ = ɛγ µ λ r, δλ ir = 1 8 γµν Fµνɛ r i 1 2 Y ij r ɛ j hr Iφ I ɛ i, δy ij r = ɛ (i γ µ D µ λ j)r +2h r I ɛ (i χ j)i still closes off-shell but into gauge-tensor gauge transf.

15 Closure of SUSY Coupling of a single uncharged (1,0) tensor multiplet to YM was described in (Bergshoeff, Sezgin, Sokatchev 96). Construct susy transformation such that they close into translations and gaug-tensor gauge transformations: [δ ɛ1,δ ɛ2 ] = ξ µ µ + δ Λ + δ Λ(1) + δ Λ(2) Note: depend on the fields Λ, Λ (1), Λ (2) A r,b I, φ, C r YM Multiplet: δa r µ = ɛγ µ λ r, δλ ir = 1 8 γµν Fµνɛ r i 1 2 Y ij r ɛ j hr Iφ I ɛ i, δy ij r = ɛ (i γ µ D µ λ j)r +2h r I ɛ (i χ j)i still closes off-shell but into gauge-tensor gauge transf.

16 Closure of SUSY Coupling of a single uncharged (1,0) tensor multiplet to YM was described in (Bergshoeff, Sezgin, Sokatchev 96). Construct susy transformation such that they close into translations and gaug-tensor gauge transformations: [δ ɛ1,δ ɛ2 ] = ξ µ µ + δ Λ + δ Λ(1) + δ Λ(2) Note: depend on the fields Λ, Λ (1), Λ (2) A r,b I, φ, C r YM Multiplet: δa r µ = ɛγ µ λ r, δλ ir = 1 8 γµν F r µνɛ i 1 2 Y ij r ɛ j hr Iφ I ɛ i, δy ij r = ɛ (i γ µ D µ λ j)r +2h r I ɛ (i χ j)i Tensor Multiplet: still closes off-shell but into gauge-tensor gauge transf. δφ = ɛχ, δχ i = 1 48 γµνρ H µνρ ɛ i γµ µ φɛ i, δb µν = ɛγ µν χ closes on-shell with Hµνρ =0.

17 Closure of SUSY Coupling of a single uncharged (1,0) tensor multiplet to YM was described in (Bergshoeff, Sezgin, Sokatchev 96). Construct susy transformation such that they close into translations and gaug-tensor gauge transformations, on-shell: [δ ɛ1,δ ɛ2 ] = ξ µ µ + δ Λ + δ Λ(1) + δ Λ(2) Note: depend on the fields Λ, Λ (1), Λ (2) A r,b I, φ, C r YM Multiplet: δa r µ = ɛγ µ λ r, δλ ir = 1 8 γµν F r µνɛ i 1 2 Y ij r ɛ j hr Iφ I ɛ i, δy ij r = ɛ (i γ µ D µ λ j)r +2h r I ɛ (i χ j)i Tensor Multiplet: still closes off-shell but into gauge-tensor gauge transf. δφ I = ɛχ I, δχ ii = 1 48 γµνρ H I µνρɛ i γµ D µ φ I ɛ i 1 2 di rsγ µ λ ir ɛγ µ λ s, B I µν = ɛγ µν χ I, closes on-shell + gauge-tensor gauge transf.

18 Closure of SUSY Coupling of a single uncharged (1,0) tensor multiplet to YM was described in (Bergshoeff, Sezgin, Sokatchev 96). Construct susy transformation such that they close into translations and gaug-tensor gauge transformations, on-shell: [δ ɛ1,δ ɛ2 ] = ξ µ µ + δ Λ + δ Λ(1) + δ Λ(2) Note: depend on the fields Λ, Λ (1), Λ (2) A r,b I, φ, C r YM Multiplet: δa r µ = ɛγ µ λ r, δλ ir = 1 8 γµν F r µνɛ i 1 2 Y ij r ɛ j hr Iφ I ɛ i, δy ij r = ɛ (i γ µ D µ λ j)r +2h r I ɛ (i χ j)i Tensor Multiplet: still closes off-shell but into gauge-tensor gauge transf. δφ I = ɛχ I, δχ ii = 1 48 γµνρ H I µνρɛ i γµ D µ φ I ɛ i 1 2 di rsγ µ λ ir ɛγ µ λ s, B I µν = ɛγ µν χ I, closes on-shell + gauge-tensor gauge transf.

19 3-Form: C µνρ r = b Irs ɛγ µνρ λ s φ I Field Equations The closure of the susy algebra puts the system on-shell (it s not a bug, its a feature!), but does not introduce new conditions on the various constant group tensors. However, susy needs those conditions. special Fierz identities! 1.) [ δ 1, δ 2 ] on A r, λ ir, Y ij and φ I Note: only A r sees δ Λ(1),δ Λ(2) 2.) [ δ 1, δ 2 ] on χ ii χ ii - e.o.m.: /Dχ ii =... 3.) [ δ 1, δ 2 ] on B I H I - e.o.m.: H I =... H I µνρ = d I rs λ r γ µνρ λ s, /Dχ ii = 1 2 di rsf r στ γ στ λ is +2d I rsy ij r λ s j + ( d I rsh s J 2b Jsr g Is) φ J λ ir, D µ D µ φ I ( = 1 2 di rs F r µν F µν s 4 YijY r ij s +8 λ r γ µ D µ λ s) 2 ( b Jsr g Is 8d I rsh s ) λr J χ J 3 d I rsh r Jh s K φ J φ K

20 4.) [ δ 1, δ 2 ] on C r YM - Multiplet e.o.m.: ( ) g Kr b Irs Yij s φ I 2 λ s (i χi j) =0 g Kr ( b Irs F s µν φ I 2 λ s γ µν χ I) = 1 4! ε µνλρστ g Kr (4) λρστ H r g Kr b Irs ( φ I /Dλ s i / Dφ I λ s i ) = g Kr b Irs ( 1 4 F s χ I i HI λ s i Y s ij χ ji hs Jφ I χ J i di uv γ µ λ u i λ s γ µ λ v) This system of e.o.m. transforms consistently into itself under susy (isolated cubic fermion terms cancel due to special Fierz + magic identities!). Novel system of susy non-abelian couplings for multiple (1,0) tensor multiplets in 6D No dimensionful parameter (classical superconformal symmetry). 3-form puts YM on-shell and is dual to vector field 1 st order e.o.m. The actual model depends on the choice of the gauge group/representation and assoc. invariant tensors solutions for the various conditions!

21 A first model Choosing a semisimple compact gauge group with Lie-algebra the various conditions are satisfied by: I, r... adjoint, g ri δ rs... Cartan-Killing G h s r 0, d p rs d rst g pt, b p rs f prs g SUSY vacua: h r I φ I 0 =0 g Kr b Irs φ I 0 φ ad 0 diag( 0 δ r s,k r δ r s ) With φ I := φ I 0 + ϕ I one obtains for the linearized field equations: / χ ir =0, dc r =0, ( db r + C r ) =0, ϕ r =0, / χ i r +2k r λ i r =0, da r 1 k r dc r =0, / λ i r =0, Y r ij =0 Can gauge B=0, SD closed 3-form only, equivalent descript. of tensor multiplet Not direct sum of tensor and gauge multiplet non-decomposable 3-form communicates DOF between tensor and YM multiplet.

22 Models with an Action

23 Action Existence of an action requires a metric for kin. terms. In addition, the various group tensors have to satisfy additional conditions: η IJ h r I = η IJ g Jr, d I rs = 1 2 ηij b Jrs, η IJ d I p(q dj rs) =0 This simplifies the conditions but also implies: indefinite metric!!! η IJ g Ir g Js =0 L = 1 8 (D µφ I ) χ I /Dχ I b Irsφ I ( F r µνf µν s 4Y r ijy ij s +8 λ r /Dλ s) 1 96 (HI µνρ) b Irs λ r H I λ s 1 4 b Irs λ s F r χ I + b Irs Y r ij λ is χ ji (b Jsrg s I 4b Isr g s J) φ I λr χ J b Irsg r Jg s K φ I φ J φ K 1 48 L top 1 24 b Irsb I uv λ r γ µ λ u λs γ µ λ v Self duality equ. imposed consistently after deriving 2 nd order equ. (like IIB SUGRA). Problem of a different category. 1 st order YM equ. are not contracted with g Ir. VEV of scalar field acts as inverse Yang- Mills coupling, no dimensionful coupling.

24 topological term: M 7 L top M 7 ( birs F r F s H I H I DH I ) Nonlinear conditions for the various group tensors f t rs, g Ir, h I r,... imply that there is only a single possible (overall) coupling constant. Requirement of gauge invariance might give quantization conditions for the dimensionless coupling constant (see Chern-Simons). Conformal Symmetry The action is invariant under the following conformal transformations: δ conf Φ= L ξ Φ+ w Φ ΩΦ (µ ξ ν) =Ω η µν Φ=( φ I,B I,χ I,A r,y ij,λ r,c r ) w Φ = (2, 0, 5/2, 0, 2, 3/2, 0) Note: [ Conf, SUSY ] = special SUSY Superconformal Symmetry w/

25 dv L top = 6 gi r C r H I + b Irs B I F r F s b Irs h r Jh s K B I B J B K + B I [ h s I b J sub Jvr A u A v da r (b Irsf r pq +4b Jqs X J pi ) f s uv A p A q A u A v] 1 10 f up s b J qsb Jvr A p A q A u A v da r

26 topological term: M 7 L top M 7 ( birs F r F s H I H I DH I ) Nonlinear conditions for the various group tensors f t rs, g Ir, h I r,... imply that there is only a single possible (overall) coupling constant. Requirement of gauge invariance might give quantization conditions for the dimensionless coupling constant (see Chern-Simons). Conformal Symmetry The action is invariant under the following conformal transformations: δ conf Φ= L ξ Φ+ w Φ ΩΦ (µ ξ ν) =Ω η µν Φ=( φ I,B I,χ I,A r,y ij,λ r,c r ) w Φ = (2, 0, 5/2, 0, 2, 3/2, 0) Note: [ Conf, SUSY ] = special SUSY Superconformal Symmetry w/

27 Gauge Groups & Representations

28 Semisimple Groups The A r B I Stückelberg coupling is determined by rk(h r I) n V, n T. Choose basis/decomposition A r =(A α,a a ),B I =(B b,b a ) such that: [h] r I = [ δ a b ] r I = h r I g Is =0 [g] Ir = [ ga β g a b 0 0 ] Ir For semisimple Lie groups the remaining conditions reduce to: f rs γ = f rs c = 1 2 [ ] γ fαβ [ ] T α T, β Lie algebra d c rs = 1 2 g [ ] T α T β & representations: adj... (T ad α ) β γ = f αβ γ R, R... T α,t α δ(b a rs, b a rs, d c rs) =0 [X α ] s t = [ T ad α T α ] [X α ] I J = [ t T α, g a r b b rs =2d a sb T α ] T αc b = g r c b b rα, g r a b b ra =0

29 Models For a given nontrivial representation R, the different models are classified by the choice of the representation R, under which vector and tensor multiplets V =(A, λ i,y ij ), T =(B, χ i,φ) transform. R is trivial Rep.: d c αβ = d c δ αβ, g ri =0=b Irs C decouples YM off-shell! Fields: F r = (V α ) ad, (V a, T a ) R, (T a ) sing { F α DA a 1 2 T βb a A β A b + B a =: B a, H I = { Ha = db a + d a δ αβ K αβ DB a Coupling of uncharged (Bergshoeff, Sezgin, Sokatchev 96) and charged (1,0) tensor multiplet to off-shell YM multiplet. Need susy dynamics for YM.

30 R is the adjoint: [g] Ir = [ ] β δα Ca decouples Fields: (V α, T α,c α ) ad, (V a, T a ) R, Some of the invariant tensors d γ rs,b a bc,b α bγ, b a bγ, which are otherwise unconstrained may exist for special groups and representations R. The so far considered explicit models do not provide an action.

31 R is Contragredient rep.: T α = Tα t 1 g Ir = [ ] 0 b δa 0 0, η IJ = [ ] 0 b δa δ a b 0, [b] = 2 [d] d abc = d (abc), d aβγ = d a(βγ), d abγ = d (ab)γ Fields: (V α ) ad, (V a, T a ) R, (T a,c a ) R F r = { F α B a, H I = { Ha = DB a + d ars K rs + C a := C a DB a E.o.m.: (DB a ) ( ) = T αb a λα γ µνρ λ b e.t.c.

32 Conclusions and Outlook We have constructed a general class of 6D (1,0) superconformal models with non-abelian vector and tensor multiplets: generically needs non-dynamical 3-form dual of gauge field models parametrized by dimensionless constant group tensors Action only for special cases, i.g. only e.o.m. Perturbatively defined only in the conformally broken phase, scalar VEV acts as inverse YM coupling Questions? Solution w/o 3-form YM multiplet off-shell. Introduce 1 st order equations in supersymmetric way? Models with action: does large gauge symmetry help to remove ghosts? Anomaly cancellation? Including hypermultiplets relation to M5 branes? Background with (1,0) symmetry & charged tensors? Reduction to 4/5D SYM, SD string solutions... e.c.t.

33 Conclusions and Outlook We have constructed a general class of 6D (1,0) superconformal models with non-abelian vector and tensor multiplets: generically needs non-dynamical 3-form dual of gauge field models parametrized by dimensionless constant group tensors Action only for special cases, i.g. only e.o.m. Perturbatively defined only in the conformally broken phase, scalar VEV acts as inverse YM coupling Questions? Solution w/o 3-form YM multiplet off-shell. Introduce 1 st order equations in supersymmetric way? Models with action: does large gauge symmetry help to remove ghosts? Anomaly cancellation? Including hypermultiplets relation to M5 branes? Background with (1,0) symmetry & charged tensors? Reduction to 4/5D SYM, SD string solutions... e.c.t. Thank You!

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