A nonabelian M5 brane Lagrangian in a supergravity background
AA nonabelian M5 brane Lagrangianin a supergravity background
Andreas Gustavsson
Physics Department, University of Seoul, 13 Siripdae, Seoul 130-743 Korea( [email protected])
Abstract
We present a nonabelian Lagrangian that appears to have (2 ,
0) superconformal symmetryand that can be coupled to a supergravity background. But for our construction to work, we haveto break this superconformal symmetry by imposing as a constraint on top of the Lagrangianthat the fields have vanishing Lie derivatives along a Killing direction. a r X i v : . [ h e p - t h ] J un Introduction
Finding the nonabelian M5 brane Lagrangian is a long-standing problem, but at the sametime it has also been clear for a long time that a unique classial nonabelian Lagrangianfor a selfdual tensor field with manifest (2 ,
0) superconformal symmetry can not exist[11], [7] and we will review the argument below. With the discovery of the M2 braneLagrangians [1], [2], [3] a new hope was that also the M5 brane Lagrangian may be foundif one relaxes some of the symmetries that should be present in the classical Lagrangianin the same spirit as one did for the ABJM Lagrangian [3] of multiple M2 branes thatpreserves only a subgroup of the SO (8) R-symmetry group. The worldvolume theory offlat M2’s has the bosonic symmetry group of AdS × S . Since S is a Hopf fiber bundleover CP there is a way of breaking its isometry group SO (8) down to SU (4) × U (1)corresponding to this Hopf fibration and it is only this latter R-symmetry that is manifestin the ABJM Lagrangian. For the M5’s on the other hand, we have the bosonic symmetrygroup of AdS × S but here S is not a Hopf circle-bundle so for the M5’s it may bebetter to consider an orbifolding of the AdS space which reduces the Lorentz symmetryrather than the R-symmetry. We will not attempt to orbifold AdS in this paper, butwe will consider a nonabelian theory that breaks the Lorentz symmetry at the classicallevel of the Lagrangian. More generally we will present a candidate Lagrangian for M5’son Lorentzian six-manifolds that has at least one Killing vector field that corresponds toan isometry direction. We break translational symmetry along this isometry direction bykeeping only the zero modes of the fields in this direction. This isometry direction can bequite general. It can be fibered over a five-manifold. It can be a compact circle directionor it can be a noncompact direction. It can be of any signature, timelike, spacelike or null.The only thing that we demand is that all fields have vanishing Lie derivates along thisisometry direction. This approach to the M5’s has been studied previously [8], [7], [10] butin this paper we generalize these results and obtain a nonabelian Lagrangian coupled tosupergravity background fields. This is a generalization of the abelian M5 brane coupledto a supergravity background fields that was studied in [4].If we put Lie derivatives to zero in one spatial direction, then the theory will becomefive-dimensional and this should be nothing but 5d SYM coupled to supergravity fields[5], although expressed in a six-dimensional language. However, the Killing vector fieldcan be of any signature and in particular it can be light-like [8], [9]. So our Lagrangian ismore general than the Lagrangian of 5d SYM. But at the quantum level the distinctionbetween 5d SYM and the M5 brane is blurred since we do not understand whether thesecould in fact be just different faces of one and the same theory [18], [19]. Our Lagrangianappears to have 6d (2 ,
0) superconformal symmetry. But this symmetry is broken at the2lassical level as we shall constrain the fields to have vanishing Lie derivatives in onedirection. The hope is that this broken symmetry will be restored in the quantum theoryand that instanton particles of 5d SYM will give us those missing momentum modes alongthe isometry direction. Supersymmetry variations for such a theory have been foundpreviosly for special cases. First for flat R , in [8] and then for (1 ,
0) superconformalsymmetry on generic circle-bundle manifolds in [10]. The corresponding Lagrangian forthis supersymmetric system has been unknown for some time. Recently there was aninteresting suggestion for such a Lagrangian in flat R , in [7] based on a construction ofa Lagrangian for a selfdual tensor field that had appeared in [13], [14].In [11] it was argued that we shall not attempt to write down a Lagrangian for aselfdual tensor field since the partition function for a selfdual tensor field when put on aneuclidean six-manifold that has three-cycles is not unique. If the partition function is notunique, then a path integral argument suggests that also the Lagrangian can also not beunique. So we shall not look for a unique Lagrangian for the selfdual tensor field. Insteadwe may start with quantizing a nonchiral theory and at the end perform a holomorphicfactorization to select a partition function for the M5 brane theory.In this paper we will go against this philosophy, at least naively, and instead we willconsider the Lagrangian for a selfdual tensor field that was found in [13], [14]. It appearsthat this Lagrangian can be supersymmetrized and then it might also have applicationsto the M5 brane system [7], [6].The objection rised by the paper [11] to the study of Lagrangians for selfdual tensorfields, can be avoided when there is a Killing direction in the six-manifold that mightselect one partition function as special compared to the many other partition functionsthat may also appear. For instance, if this Killing vector is timelike, then we may puttime along this Killing vector and use Hamiltonian quantization that will give us a uniquepartition function. The canonical example for this approach is the M5 brane on a flatsix-torus where Hamiltonian quantization selects for us a unique the partition function,among several candidate partition functions, that is the one that happens to also bemodular invariant [12]. In fact our Lagrangian, that depends on a choice of Killing vectorfield, may also fit well with the idea of [11] after all, because from this work one isjust discouraged to go looking for a unique Lagrangian for the selfdual tensor field. OurLagrangian is not necessarily unique. If there are several Killing vector fields then there isone Lagrangian for each choice of ‘preferred’ Killing vector field that is used to constructour Lagrangian. This is in the same spirit as that of Hamiltonian quantization, but heregeneralized to Killing vectors that can be either timelike, spacelike or null leading to moregeneral quantizations than the usual Hamiltonian quantization that applies only for the3ase of a timelike Killing vector.Any proposed Lagrangian for the M5’s can be put to the following tests. The first andsimplest test of any candidate (1 ,
0) supersymmetric Lagrangian is whether this can beenhanced to (2 , ,
0) at the quantum level. Another test is whether any attempted (2 , ,
0) supersymmetrycan be enhanced to (2 ,
0) superconformal symmetry. Finally one may test whether a givencandidate Lagrangian can be consistently coupled to the eleven-dimensional supergravitybackground fields while preserving superconformal symmetry.In this paper we will present a Lagrangian that appears to pass all of these tests, butthis is not entirely correct because for this construction to work we need to impose as aconstraint on top of the Lagrangian that the Lie derivatives of all the fields vanish along aKilling direction and thus we need to break some of the superconformal symmetry at theclassical level. But a breaking of some of the spacetime symmetries at the classical levelof a Lagrangian is precisely what we should expect as that enables us to have a classicalLagrangian description of the M5’s that is not unique, but depends on a choice of Killingvector.
Following [7], [13], [14] we introduce a selfdual tensor field H + MNP . This is an auxiliarytensor three-form field whose role in the Lagrangian is as a Lagrange multiplier fieldthat implements the selfduality condition on another three-form field that we will denoteas g MNP . Part of g MNP is a three-form h MNP with the wrong sign kinetic term in theLagrangian. For abelian gauge group this three-form is a field strength of a two-formgauge potential b MN , so that h MNP = 3 ∇ [ M b NP ] . For the nonabelian generalization wewill not present an explicit realization of h MNP in terms of some nonabelian two-formgauge potential. This is one of the longstanding mysterious aspects of the theory ofmultiple M5 branes, the mystery of what exactly would be the nonabelian two-form. Wewill not try to answer this question here. But we will postulate the the infinitesimalvariation can be presented as δh MNP = 3 D [ M δb NP ] (2.1)4or some infinitesimal nonabelian two-form variation δb MN . Let us assume that h MNP v P = − F MN . Let us define the gauge algebra valued one-form Y T := ε MNP RST [ h MNP , F RS ] (2.2)Dualizing (2.2) we get [ h [ MNP , F RS ] ] = − ε MNP RST Y T Moreover v M [ h [ MNP , F RS ] ] = 0 since v M F MN = L v A N = 0 and we assume that h MNP v P = − F MN . This is realized by taking Y T ∼ v T . Conversely, if Y T has another componentnot parallel to v T then we get ε MNP RST Y T v M (cid:54) = 0. So we have Y M = v M Y for some gauge algebra valued zero-form Y . We now get D M ( v M Y ) = v M D M Y = L v Y = 0once we impose the gauge fixing condition v M A M = 0. We get zero because we constrainthe Lie derivative along v M of all fields to vanish, so in particular L v Y = 0. Using theBianchi identity D [ T F RS ] = 0 we now get ε MNP RST [ F RS , D T h MNP ] = 0 (2.3)This does does not necessarily imply that D [ T h MNP ] = 0 since to derive (2.3) we haveassumed that F RS = − h RST v T , so F RS can not be varied independently from h MNP . If v (cid:54) = 0 then we have the projection operators P NM = δ NM − v v M v N Q NM = 1 v v M v N that enable us to decompose h MNP = h (cid:48) MNP − F [ MN v P ] where h (cid:48) MNP = P QP h MNQ
Now h (cid:48) MNP and F MN can be varied independently from each other. Also, we notice that ε MNP RST [ F RS , D T ( F MN v P )] = ε MNP RST [ F RS , F MN ] ∇ T v P = 05imply because ε MNP RST = ε P RMNST . Then we have left ε MNP RST [ F RS , D T h (cid:48) MNP ] = 0and since F RS is independent from h (cid:48) MNP , we conclude that D [ T h (cid:48) MNP ] = 0Hence D [ T h MNP ] = 12 F MN w P T (2.4)where we define w MN = ∇ M v N − ∇ N v M To formulate the supersymmetry variations, we find it convenient to introduce an in-finitesimal variation δB MN := − δb MN . When this variation is a supersymmetry variation,then this is given by δB MN = i ¯ ε Γ MN ψ . But this is just the infinitesimal variation, andwe do not introduce nonabelian gauge potentials b MN nor B MN in this paper, only theirinfinitesimal variations.The Lagrangian is a sum of two terms, L = L b + L m where the gauge field part is L b = 124 h MNP + 16 H + MNP g − MNP + 16 h − MNP w MNP + 124 w MNP + λ − MNP G + MNP + 148 ε MNP QRS F MN W P QRS − v ε MNP RST (cid:18) A M ∇ N A P − ie A M A N A P (cid:19) w RS v T and the matter field part is L m = −
12 ( D M φ A ) + i ψ Γ M D M ψ − µ AB φ A φ B + e ψ Γ M Γ A [ ψ, φ A ] v M + e v φ A , φ B ] + i ψ Γ MNP Γ A ψT AMNP where we have defined g − MNP = h − MNP + w − MNP + 6 T AMNP φ A G + MNP = H + MNP + h + MNP + w + MNP G + MNP is a supersymmetry singlet and g − MNP = 0 is the selfduality equation ofmotion we get by varying the selfdual field H + MNP in the Lagrangian. We present theexplicit form of the mass matrix µ AB in equation (3.1).The supersymmetry variations are δφ A = i ¯ ε Γ A ψδA M = δB MN v N δH + MNP = − δh + MNP − δw + MNP δh MNP = − D [ M δB NP ] δB MN = i ¯ ε Γ MN ψδW MNP Q = − e ¯ ε Γ MNP Q Γ A [ ψ, φ A ]We define w MNP = W MNP Q v Q that is a three-form with selfdual and antiselfdual components whose supersymmetryvariations are δw + MNP = e ε Γ Q Γ MNP [ ψ, φ A ] v Q δw − MNP = − e ε Γ MNP Γ Q [ ψ, φ A ] v Q The supersymmetry variation of the fermions is δψ = 112 Γ MNP εH + MNP + Γ M Γ A εD M φ A − A ηφ A − ie M Γ AB ε [ φ A , φ B ] v M Neither L b nor L m is supersymmetric by themselves, and only the sum is supersymmetric.The supersymmetry parameter satisfies the conformal Killing spinor equation D M ε = Γ M η −
18 Γ A Γ RST Γ M εT AMNP (2.5)It should be noted that this equation implies that Γ M D M ε = η since Γ M Γ RST Γ M = 0.Here T AMNP is a supergravity background tensor field, carrying in addition an R-symmetryvector index A = 1 , ...,
5. This tensor field is antiselfdual, since the spinors are chiral,Γ ε = − ε Γ ψ = ψ where Γ = Γ is the 6d chirality gamma matrix. All our gamma matrices areeleven-dimensional, so in particular the gamma matrices for the Lorentz group and theR-symmetry group anticommute, { Γ M , Γ A } = 0.7he theory also couples to the supergravity background R-gauge field V ABM throughthe covariant derivatives that acts on the matter fields as D M ψ = ∇ M ψ − ie [ A M , ψ ] + 14 V ABM Γ AB ψD M φ A = ∇ M φ A − ie [ A M , φ A ] + V ABM φ A where ∇ M is the geometric covariant derivative that only involves the Christoffel symbol,and e is an electric charge, which eventually will be fixed to some value of order one dueto selfduality. But to determine the exact value of e will require considerations that gobeyond just classical supersymmetry so we will keep this as a free parameter here. Allour fields transform in the adjoint representation of the gauge group. But this maybe canbe made more general if one can find a nonabelian gerbe structure for our theory. L m We make the ansatz L m = −
12 ( D M φ A ) + i ψ Γ M D M ψ − µ AB φ A φ B + a ψ Γ M Γ A [ ψ, φ A ] v M + b φ A , φ B ] + ic ψ Γ MNP Γ A ψT AMNP and for the supersymmetry variation of ψ we make the ansatz δψ = 112 Γ MNP εH + MNP + Γ M Γ A εD M φ A − A ηφ A − id M Γ AB ε [ φ A , φ B ] v M while for the other fields we let those vary according to the what we stated before. Thenwe compute the supersymmetry by adopting the convention that we make integrations byparts in such a way that δψ does not appear in anyone of the terms and discard boundaryterms. This will uniquely determine the variation as δ L m = D M φ A δφ A + i ¯ ψ Γ M D M δψ − µ AB φ A δφ B − ie [ δA M , φ A ] D M φ A + e ψ Γ M [ δA M , ψ ]+ a ¯ ψ Γ M Γ A [ δψ, φ A ] v M + a ψ Γ M Γ A [ ψ, δφ A ] v M + b [ φ A , φ B ][ φ A , δφ B ]+ ic ψ Γ MNP Γ A δψT AMNP
We now pick the commutator terms from this variation and postpone the study of all therest to later. Let us also study the cubic term in fermi-fields later. Then we will for now8ocus on the following terms in the variation of the matter fields Lagrangian( δ L m ) comm = 12 ¯ ψ Γ A Γ MN ε [ eF MN − aH + MNP v P , φ A ]+(4 a − d ) ¯ ψ Γ AB Γ M η [ φ A , φ B ] v M − d ψ Γ AB Γ C Γ MNP Γ Q ε [ φ A , φ B ] T CMNP v Q − cd ψ Γ C Γ AB Γ MNP Γ Q ε [ φ A , φ B ] T CMNP v Q + i (cid:0) adv − b (cid:1) ¯ ε Γ B ψ [ φ A , [ φ A , φ B ]]+ d ψ Γ MN Γ AB ε [ φ A , φ B ] ∇ M v N In addition to these terms, we also get the terms d ¯ ψ Γ M Γ N Γ AB ε [ D M φ A , φ B ] v N + e ¯ ψ Γ M Γ N Γ A Γ B ε [ D N φ B , φ A ] v M and another such commutator term comes from − ( δD M ) φ A D M φ A = − e ¯ ψ Γ MN ε [ D M φ A , φ A ] v N so the sum of all these terms for a = d = e just becomes a couple of Lie derivatives,2 e ¯ ψ Γ AB ε [ L v φ A , φ B ] − e ¯ ψε [ L v φ A , φ A ]So we can now conclude that we shall pick a = ed = eb = e v To determine the value of c requires some more work. This comes about by putting4 a − d = 2 e and then by looking at the term2 e ¯ ψ Γ AB Γ M η [ φ A , φ B ] v M and by using the Killing spinor equation to extract from this term the following term2 e ¯ ψ Γ AB (cid:18) v M D M ε + d e Γ MN ∇ M v N (cid:19) [ φ A , φ B ]+ e ψ Γ AB Γ C Γ MNP Γ Q ε [ φ A , φ B ] T CMNP v Q Let us make the following ansatz for the Lie derivative of a spinor field, L v χ = v P D P χ + α P Q χ ∇ M v N v ¯ ψ = v P D P ¯ ψ − α ψ Γ P Q ∇ P v Q Then for the vector field T M = ¯ ψ Γ M χ we get L v T M = v P D P T M + α T Q ( ∇ M v Q − ∇ Q v M )For this to agree with the Lie derivative of a vector field we shall have α = 1 ∇ M v N + ∇ N v M = 0so we must now require that v M is a Killing vector field. Since the Lie derivative that wewant here is L v ε = v M D M ε + 14 Γ MN ε ∇ M v N we clearly see that we shall choose d = e . This second term that got generated throughthe usage of the Killing spinor equation now combines with the two other terms to giveus − e ψ Γ AB Γ C Γ MNP Γ Q ε [ φ A , φ B ] T CMNP v Q − ce ψ Γ C Γ AB Γ MNP Γ Q ε [ φ A , φ B ] T CMNP v Q + e ψ Γ AB Γ C Γ MNP Γ Q ε [ φ A , φ B ] T CMNP v Q Thus for c = 1, we get the commutator [Γ AB , Γ C ] = − [ A δ B ] C and the three termscollapse to e ψ Γ A Γ MNP Γ Q ε [ φ A , φ B ] T BMNP v Q We now use the symplectic Majorana properties to write¯ ψ Γ B Γ MNP Γ Q ε = ¯ ε Γ Q Γ MNP Γ B ψ and then we recall that δw + MNP = e ε Γ Q Γ MNP Γ A [ ψ, φ A ] v Q to write this result as e ε Γ Q Γ MNP Γ A ψ [ φ A , φ B ] T BMNP v Q = e ε Γ Q Γ MNP Γ A [ ψ, φ A ] φ B T BMNP v Q = δw + MNP T BMNP φ B
10e now recall that δH + MNP = − δh + MNP − δw + MNP
By considering the abelian type of terms below, we will discover that the above variationcombines with those abelian terms into δ L m = − δ (cid:0) H + MNP T BMNP φ B (cid:1) + ... Let us now write down the cubic terms in fermi-fields,( δ L m ) cubic = ie ψ Γ M [¯ ε Γ MN ψv N , ψ ] + ia ψ Γ M Γ A [ ψ, ¯ ε Γ A ψ ] v M This is identically zero for a = e by a Fierz identity that we derive in the appendix.We now turn to the abelian terms, by which we refer to as those terms that will survivealso when we put all the commutators to zero. Abelian terms arise from the followingterms in δ L m , ( A
1) = D φ A δφ A ( A
2) = i ¯ ψ Γ M D M δψ ( A
3) = − µ AB φ A δφ B ( A
7) = i ψ Γ MNP Γ A δψT AMNP
We now extract all the abelian terms that will appear in each of these terms,( A
1) = D φ A i ¯ ε Γ A ψ ( A a ) = − i ·
12 ¯ ψ Γ M Γ RST Γ UV W Γ M Γ A εT AUV W H + RST ( A b ) = − i ε Γ NP ψD M H + MNP ( A c ) = − i ¯ ψ Γ A Γ M ηD M φ A ( A d ) = − i ψ Γ RST Γ M Γ A Γ B εT BRST D M φ A ( A e ) = − i ¯ ε Γ A ψD φ A ( A f ) = e ψ Γ A Γ MN ε [ F MN , φ A ]( A g ) = i ψ Γ A Γ MN εW ABMN φ B ( A h ) = 4 i ¯ ψ Γ A Γ M ηD M φ A ( A i ) = 4 i ¯ ψ Γ A (Γ M D M η ) φ A ( A a ) = − i ·
12 ¯ ψ Γ UV W Γ RST Γ A εT AUV W H + RST A b ) = − i ψ Γ MNP Γ Q Γ A Γ B εT AMNP D Q φ B ( A c ) = − i ¯ ψ Γ MNP Γ A Γ B ηT AMNP φ B We now find that the following terms cancel,0 = ( A
1) + ( A e )0 = ( A c ) + ( A h )Now we will expand out ( A i ) by using5Γ M D M η = − R ε + 18 Γ MN Γ AB εW ABMN −
34 Γ A Γ MNP ηT AMNP −
18 Γ A Γ UV W Γ M εD M T AUV W that is a direct consequence of (2.5) as we show in the appendix. Here W ABMN is a fieldstrength of the R-gauge field as defined in (B.1). Then we get( A ia ) = − R i ¯ ε Γ A ψ ( A ib ) = i
10 ¯ ψ Γ A Γ BC Γ MN εW BCMN φ A ( A ic ) = − i ψ Γ A Γ B Γ RST ηT BRST φ A ( A id ) = − i
10 ¯ ψ Γ A Γ B Γ RST Γ M ε ( D M T BRST ) φ A Now we collect terms as follows,( A g ) + ( A ib ) = i ψ Γ MN (cid:18) Γ E δ F G + 15 Γ G Γ EF (cid:19) εW EFMN φ G = − i ψ Γ MN (cid:18) δ GE −
15 Γ G Γ E (cid:19) Γ F εW EFMN φ G ( A c ) + ( A ic ) = − i ¯ ψ Γ MNP (cid:18) Γ A Γ B + 35 Γ B Γ A (cid:19) ηT AMNP φ B = − i ¯ ψ Γ MNP (cid:18) δ AB −
15 Γ A Γ B (cid:19) ηT BMNP φ A We also get ( A d ) + ( A b ) = − i ψ Γ RST Γ M Γ A Γ B εT BRST D M φ A − i ψ Γ MNP Γ Q Γ A Γ B εT AMNP D Q φ B = − i ψ Γ RST Γ M εT ARST D M φ A We conclude that the contribution to the supersymmetry variation of the matter fieldsLagrangian that comes from the abelian terms is given by δ L m = ¯ ψ (cid:18) δ AB −
15 Γ A Γ B (cid:19) χ B φ A − iR ψ Γ A εφ A + µ AB i ¯ ψ Γ A εφ B I thank Dongsu Bak for that he carried out a similar computation to this one for the abelian M5brane in an unfinished separate project several years ago. χ A = − i MN Γ B εW ABMN − i Γ MNP ηT AMNP + i UV W Γ M εD M T AUV W
For this variation to vanish we shall take µ AB = R δ AB − D AB (3.1)where D AB is a symmetric tensor that satisfies χ A −
15 Γ A Γ B χ B = Γ B εD AB We can also see that D AB shall be traceless by contracting both sides with Γ A from theleft.To better understand the variation of the matter field Lagrangian, we will now studythe following term in δ L b Lagrangian L T = H + MNP T AMNP φ A Its has the following supersymmetry variation δ L T = − i ¯ ε Γ NP ψD M (cid:0) T AMNP φ A (cid:1) + i ¯ ε Γ A ψH + MNP T AMNP
We are now interested in the first term that we expand out in two terms − i ¯ ε Γ NP ψ (cid:0) D M T AMNP (cid:1) φ A − i ¯ ε Γ NP ψT AMNP D M φ A The second term cancels ( A d ) + ( A b ) by using the fact that T AMNP is antiselfdual andthe first term combines with ( A id ) to give the last term in χ A as χ Alast = 3 i Γ MN εD P T AMNP
Then finally the term ( A
8) = i ¯ ε Γ A ψH + MNP T AMNP combines with other terms into a cancelation,( A a ) + ( A a ) + ( A
8) = 0which uses the gamma matrix identity − ·
12 Γ M Γ RST Γ UV W Γ M − ·
12 Γ
UV W Γ RST = 12 (cid:18) g RST,UV W + 16 Γ
RST UV W (cid:19) We discovered this gamma matrix identity by using GAMMA [21].
RST UV W = ε RST UV W
Γ and that Γ ε = − ε so that when this acts on ε it will generate a projection onto the selfdual part of T AUV W which is zero.We are left with ( A b ) and we have added one term that we need to subtract again.Combining this with the commutator term obtained previosuly, we are now ready to writedown our final result for the variation of L m . It is given by δ L m = − δB NP D M H + MNP − δ (cid:0) H + MNP T AMNP φ A (cid:1) L b Let us begin by making a supersymmetry variation of (cid:101) L b given by (cid:101) L b = 124 h MNP + 16 H + MNP (cid:0) h − MNP + w − MNP + 6 T AMNP φ A (cid:1) + 16 h − MNP w + MNP + 124 w MNP + λ − MNP (cid:0) H + MNP + h + MNP + w + MNP (cid:1) + 148 ε MNP QRS F MN W P QRS where we omit the Chern-Simons term. Here δH + MNP = − δh + MNP − δw + MNP
We get δ (cid:101) L b = 112 h + MNP δh MNP + 112 h − MNP δh MNP − (cid:0) h − MNP + w − MNP (cid:1) ( δh MNP + δw MNP )+ 16 H + MNP ( δh MNP + δw MNP )+ 112 w + MNP δw MNP + 112 w − MNP δw MNP + δ (cid:0) H + MNP T AMNP φ A (cid:1) + 16 h − MNP δw MNP + 16 w + MNP δh MNP + 148 ε MNP QRS δ ( F MN W P QRS )The coefficients of selfdual and antiselfdual components now conspire so that we obtainseveral terms that are wedge products between three-forms, δ (cid:101) L b = δ (cid:0) H + MNP T AMNP φ A (cid:1) + 112 · ε MNP RST h RST δh MNP
14 112 · ε MNP RST w RST δw MNP + 16 H + MNP δ ( h MNP + w MNP )+ 112 · ε MNP RST w RST δh MNP + 148 ε MNP QRS δ ( F MN W P QRS ) (4.1)We now expand out the term in the second line δ L = 112 · ε MNP RST h RST δh MNP = 124 ε MNP RST D M h RST δB NP Now we use (2.4) and we get δ L = − ε MNP RST F MR w ST δB NP To proceed we want neither F MN nor w MN to have any component in the v M direction.This is solved for F MN by imposing the gauge fixing condition v M A M = 0 and by de-manding L v A M = 0 since this implies that F MN v N . For w MN we need to assume that v M v M is constant, which implies that our six-manifold shall be a K-contact manifold,since only then do we also get w MN v N = 0. This is easy to see. First we note that L v v N = v M ∇ M v N + ( ∇ N v M ) v M and then we use the Killing equation ∇ M v N + ∇ N v M = 0on the second term, and we see that it cancels the first term so L v v N = 0. Next we notethat v M w MN = v M ∇ M v N − v M ∇ N v M = L v v N − ∇ N ( v M v M ) and this vanishes only if v M v M is constant. As now no component in the direction of v M comes from neither F MR nor from w ST it must come from δB NP . So we can replace δB NP → Q SP δB NS = δA N v P /v , δ L = − v ε MNP RST F MR w ST v P δA N This variation is now precisely canceled by the variation of the Chern-Simons term L CS = − v ε MNP RST tr (cid:18) A M ∇ N A P − ie A M A N A P (cid:19) w RS v T so that we have δ L CS + δ L = 0The term in the third line in (4.1) is worrisome as it can not be canceled by any otherterm. Fortunately it is identically zero as the following detailed computation shows, ε MNP RST w RST δw MNP = ε MNP RST W RST U δW MNP V v U v V − eε MNP RST W RST U ¯ ε Γ MNP V [ ψ, φ A ] v U v V = − eW RST U ¯ ε Γ RS [ ψ, φ A ] v U v T = 0where we have used the gamma matrix identityΓ MNP V Γ MNP RST = 18Γ [ RS δ T ] V After all these considerations, our result collapses to δ L b = δ (cid:0) H + MNP T AMNP φ A (cid:1) + 16 H + MNP δ ( h MNP + w MNP )+ 112 · ε MNP RST w RST δh MNP + 124 ε MNP RST D M (cid:0) δB NQ v Q (cid:1) W P RST + 148 ε MNP QRS F MN δW P QRS
The two terms on the third line cancel up to a Lie derivative, ε MNP RST δh RST w MNP + 124 ε MNP RST D M (cid:0) δB NU v U (cid:1) W P RST = i
24 ¯ ε Γ MNP Q ψ (cid:0) v S D S W MNP Q + 4 D Q (cid:0) W MNP S v S (cid:1) − v S D Q W MNP S (cid:1) = i
24 ¯ ε Γ MNP Q ψ L v W MNP Q
Putting this Lie derivative to zero as a constraint that we impose on top of the Lagrangian,we can now write the variation of the Lagrangian as δ L b = δ (cid:0) H + MNP T AMNP φ A (cid:1) + 16 H + MNP δh MNP − ε MNP RST H + MNP δW RST Q v Q + 148 ε MNP RST F MN δW P RST
We will now argue that the two last terms cancel upon using the constraint F MN = (cid:0) H + MNP + 6 T AMNP φ A (cid:1) v P (4.2) The gamma matrix relations that are used here are ε MNP QRS Γ RS = 2Γ MNP Q
Γ5Γ [ P QRS v M ] = Γ P QRS v M − [ P QR | M | v S ]
16o this end we start by making the following observation that if we define selfdual partsof W MNP Q as W ± MNP Q = 12 (cid:18) W MNP Q ± ε [ MNP RST W | RST | Q ] (cid:19) then we can write the last term in the Lagrangian in the following form148 ε MNP QRS F MN W P QRS = 124 ε MNP QRS F MN W − P QRS
This is a consequence of W MNP P = 0 that follows if one assumes that W MNP Q is totallyantisymmetric in all four indices. Now we use the constraint (4.2) and then this termbecomes proportional to (cid:16) H + P QR − T − P QRA φ A (cid:17) v S δW − P QRS = H + P QR v S δW − P QRS so the upshot is that by using (4.2) we have148 ε MNP QRS F MN δW P QRS = 148 ε MNP QRS H + MNU v U δW P QRS and now it is easy to see that this cancels against the term − ε MNP RST H + MNP δW RST Q v Q by noting the following identity3!4! H +[ MNQ δW RST P ] = 2!4! H +[ MN | Q | δW RST P ] − H +[ MNP δW RST ] Q and the fact that the left-hand side is identically zero because we antisymmetrize in sevenindices, each of which takes six different values. So we are left with the variation δ L b = δ (cid:0) H + MNP T AMNP φ A (cid:1) + 12 δB NP D M H + MNP so that this cancels the variation of L m , δ L b + δ L m = 0 We will derive the on-shell Bianchi identity for H + MNP that is required for on-shell closureof the supersymmetry variations when we act twice with supersymmetry variations on H + MNP . We will show that it arises as an equation of motion that we derive from theLagrangian L = L b + L m . This is thus a consistency check.17 he equation of motion for A M Varying A M we get0 = 124 ε MNP QRS (cid:18) D N W P QRS − v F [ NQ w RS v P ] (cid:19) − ie (cid:2) D M φ A , φ A (cid:3) − e { ¯ ψ, Γ M ψ } Let us dualize the equation of motion,5 (cid:18) D [ M W NP QR ] − v F [ MP w QR v N ] (cid:19) = ε MNP QRS (cid:16) ie [ D S φ A , φ A ] + e { ¯ ψ, Γ S ψ } (cid:17) (5.1)Contracting with v R we get4 (cid:18) D [ M w NP Q ] − F [ MN w P Q ] (cid:19) − L v W MNP Q = ieε MNP QRS [ D S φ A , φ A ] v R + e ε MNP QRS { ¯ ψ, Γ S ψ } v R (5.2) The equation of motion for b M N
Varying h MNP according to our postulated rule, δh MNP = 3 D [ M δb NP ] , we get D M (cid:0) h MNP + 2 H + MNP + 2 w + MNP + 12 λ − MNP (cid:1) = 12 ε MNP RST F RS w T M (5.3)
The equation of motion for H + M N P
Varying H + MNP we get the selfduality equation of motion h − MNP + w − MNP + 6 T AMNP φ A + 6 λ − MNP = 0 (5.4)
The equation of motion for λ − M N P
Varying λ − MNP we get a constraint that relates H + to h + + w + , H + MNP = − h + MNP − w + MNP
This constraint is supersymmetry invariant by itself.
The equation of motion for W M N P Q
Varying W MNP Q we get H + UV T v T − F UV − h − UV T v T + 112 ε UV T MNP w MNP v T − λ − UV T v T = 018hen if we use (5.4), then this equation reduces to F UV = (cid:0) H + UV T + 6 T AUV T φ A (cid:1) v T (5.5)To see this, we need to establish that the remaining terms cancel. Namely we need toestablish that w − UV T v T + 112 ε UV T MNP w MNP v T = 0but this is an identity that collapses to w UV T v T = W UV T R v T v R = 0 by using the defintion w − UV T = 12 (cid:18) w UV T − ε UV T MNP w MNP (cid:19) of the antiselfdual part.We now notice that h + w − T A φ A is selfdual, which means that h + w − T A φ A = h + + w + because T A is antiselfdual. Then we can use this in the constraint H + = − h + − w + toget H + + 6 T A φ A = − h − w which means that we can express (5.5) as F MN = − h MNP v P (5.6)where we have used that w MNP v P = W MNP Q v P v Q = 0 for the nonchiral w MNP . Theequation (5.6) is invariant under supersymmetry variations up to a Lie derivative that weconstrain to be zero. Namely the variation of the right-hand side is − δh MNP v P = 3( D [ M δB NP ] ) v P = 2 D [ M (cid:0) δB N ] P v P (cid:1) + v P D P δB MN − D [ M v P δB N ] P = δF MN + L v δB MN If we eliminate λ − MNP from (5.4) and insert that into (5.3) then we get D M (cid:0) h + MNP − h − MNP + 2 H + MNP + 2( w + MNP − w − MNP ) − T − MNPA φ A (cid:1) = 14 ε MNP RST F RS w T M D [ M (cid:16) h RST ] + 2 H + RST ] + 2 w RST ] + 12 T ARST ] φ A (cid:17) = 14 F [ RS w T M ] Now we use the Bianchi identity D [ M h RST ] = 0 and we get D [ M (cid:16) H + RST ] + 6 T AMNP ] φ A (cid:17) = 14 F [ RS w T M ] − D [ M w RST ] and finally we use the equation of motion for A M obtained in (5.2) and we arrive at theon-shell Bianchi identity D [ M (cid:16) H + NP Q ] + 6 T ANP Q ] φ A (cid:17) = − ie ε MNP QRS [ D S φ A , φ A ] v R − e ε MNP QRS { ¯ ψ, Γ S ψ } v R that is the equation of motion that is required in order to close the supersymmetryvariations on H + MNP as was originally shown in [8], but here this equation of motion wasderived from the Lagrangian.Our computation is the same in spirit as that in [7], but it differs in the details. In [7] inplace of our h MNP there appears instead expressions directly in terms of a nonabelian two-form b MN (using our notation). This is of course more attractive than our approach since itmakes the equations explicit. However, their nonabelian two-form gauge potential appearsin places where we would not expect that a gauge potential would appear explicitly, inthe Lagrangian and in the supersymmetry variation of W MNP Q . Those quantities shalltransform gauge covariantly, which is why we have chosen to set up the things in a differentway from [7].Since unlike [7] we have allowed v M to have a nonvanishing derivative, reflected inhaving a nonvanishing two-form w MN , this has led us to discover a new Chern-Simonsterm A ∧ F ∧ v ∧ w . In [20] it was shown that if one puts M5 brane on S × M for someeuclidean three-manifold M and if one performs dimensional reduction on S (possibly asquashed S , which would be reflected in having a nontrivial rescaling between our w MN and v M ), one gets a complex Chern-Simons theory on M . It seems plausible that our realChern-Simons term could be somehow related to this complex Chern-Simons theory oneuclidean M . In our computation we have assumed Lorentzian signature, so it seems likewe would not get a complex Chern-Simons in our Lorentzian computation. This needs tobe studied further. 20 The relation with the nonchiral Lagrangian
If we integrate out λ − MNP , then that will amount to replacing H + MNP with − h + MNP − w + MNP in the Lagrangian. If we do that, then we can recast the Lagrangian in the form L b = − g MNP − · ε MNP RST g MNP C RST + 12 w MNP T AMNP φ A + 148 ε MNP QRS F MN W P QRS where C MNP = w MNP + 6 T AMNP φ A .If we truncate to the sector W MNP Q = 0 by hand in this Lagrangian, then we recoverthe traditional nonchiral Lagrangian [11] (cid:101) L b = − F MNP + 112 · ε MNP RST F MNP C RST where F MNP = − h MNP + C MNP and where only the selfdual part of h MNP is coupled tothe three-form field C MNP = − T AMNP φ A Putting W MNP Q = 0 is not a consistent truncation in the nonabelian case because thesupersymmetry variation of W MNP Q is nonzero. But it is a consistent truncation in theabelian case and there this nonchiral action (cid:82) (cid:16) (cid:101) L b + L m (cid:17) is fully supersymmetric oncewe replace δψ = Γ MNP εH + MNP + ... with δψ = Γ MNP εH MNP + ... where H MNP := − h MNP .The role of W MNP Q is to promote the constraint (5.6) to an equation of motion, whichhas the advantage that we can derive the equations of motion by varying the fields A M and h MNP as independent fields in the Lagrangian.
Here we assume that the supersymmetry parameter is commuting and compute δ on eachfield. Since we have introduced many auxiliary fields with no accompanying fermionicauxiliary fields, we do not necessarily expect closure on all these auxiliary fields. Closure on φ A δ φ A = − iS M D M φ A − i ¯ εηφ A − i ¯ ε Γ AB ηφ B − ie [ φ A , Λ]21here the gauge parameter is Λ = − i ¯ ε Γ M Γ A εφ A v M Closure on A M δ A M = − iS T (cid:0) H + MNT + 6 T AMNT φ A (cid:1) v N + D M Λ − i ¯ ε Γ A Γ M ( L v ε ) φ A We have closure up to a gauge transformation if we impose the constraint F MN = (cid:0) H + MNP + 6 T AMNP φ A (cid:1) v P (7.1)This constraint is consistent with what we found in (5.5) and in (4.2), so now we havefound this constraint by three different computations, thereby making it rather convincingthat it must be correct. Closure on W M N P Q ? Using the equation of motion (5.1) we get δ W MNP Q = 5 iS R D [ M W NP QR ] = − iS R D R W NP QM + 4 iS R D [ M W NP Q ] R = − iS R D R W NP QM − i ( D [ M S R ) W NP Q ] R +4 iD [ M (cid:0) W NP Q ] R S R (cid:1) which we can write as δ W MNP Q = − i L AS W MNP Q + D [ M λ NP Q ] where λ NP Q = 4 iW NP QR S R and L AS is a Lie derivative where gauge covariant derivatives are used. We were unableto show that the second term is a gauge symmetry of the Lagrangian. However, we mayeliminate this problem by simply integrating out W MNP Q that will impose the constraint(7.1). 22
Deriving the fermionic equation of motion fromselfduality
We would like to show that we get the fermionic equation of motion by making a super-symmetry variation of the selfduality equation of motion (cid:0) h MNP + w MNP + 6 T AMNP φ A (cid:1) − = 0 (8.1)if we vary h MNP according to the rule δh MNP = 3 D [ M δb NP ] . We use δh − MNP = − i D Q (cid:0) ¯ ε Γ MNP Γ Q ψ (cid:1) = − i ε Γ MNP Γ Q D Q ψ + i
16 ¯ ε Γ Q Γ RST Γ A ψT ARST δw − MNP = − e ε Γ MNP Γ Q [ ψ, φ A ] v Q Making a supersymmetry variation of (8.1), we then get − i Γ MNP Γ Q D Q ψ + i Q Γ RST Γ MNP Γ Q Γ A ψT ARST − e Γ MNP Γ Q [ ψ, φ A ] v Q + 12 iT AMNP Γ A ψ = 0Now contracting from the left with Γ MNP and usingΓ
MNP Γ MNP = − MNP Γ Q Γ RST Γ MNP Γ Q = 144Γ RST we get i Q D Q ψ + 18 i Γ RST Γ A ψT ARST + 120 e Γ Q [ ψ, φ A ] v Q + 12 i Γ MNP Γ A ψT AMNP = 0Then i Γ Q D Q ψ + i MNP Γ A ψT AMNP + e Γ Q [ ψ, φ A ] v Q = 0which agrees with the fermionic equation of motion that we obtain from the matter fieldsLagrangian.We have assumed that the infinitesimal variation of h MNP is on the form δh MNP =3 D [ M δb NP ] and we have seen that this assumption takes the selfduality equation of motionto the expected fermionic equation of motion. This thus seems like the correct assumptionfor the infinitesimal variation of h MNP . Acknowledgements
I would like to thank Neil Lambert for explaining his ideas in his recent work to me,and Ulf Gran for assistance on how to use his computer program GAMMA [21]. Thiswork was supported in part by NRF Grant 2020R1A2B5B01001473 and NRF Grant2020R1I1A1A01052462. 23
Derivation of the Fierz identity
For the M5 brane we have the following Fierz identity for two anticommuting fermions ψ a and ψ b where a, b, ... are adjoint gauge group indices, ψ a ¯ ψ b = (cid:18) −
116 ¯ ψ b Γ M ψ a Γ M + 116 ¯ ψ b Γ M Γ A ψ a Γ M Γ A + 1192 ¯ ψ b Γ MNP Γ AB ψ a Γ MNP Γ AB (cid:19) P − where P − = (1 − Γ). Then we getΓ
P Q ψ a ¯ ψ b Γ Q ψ c = 316 ( ¯ ψ b Γ M ψ a )Γ P M ψ c + 516 ( ¯ ψ b Γ P ψ a ) ψ c + 316 ( ¯ ψ b Γ M Γ A ψ a )Γ P M Γ A ψ c + 516 ( ¯ ψ b Γ P Γ A ψ a )Γ A ψ c + 1192 ( ¯ ψ b Γ RST Γ AB ψ a )Γ RST Γ P Γ AB ψ c Γ A ψ a ¯ ψ b Γ P Γ A ψ c = 516 ( ¯ ψ b Γ M ψ a )Γ P M ψ c −
516 ( ¯ ψ b Γ P ψ a ) ψ c −
316 ( ¯ ψ b Γ M Γ A ψ a )Γ P M Γ A ψ c + 316 ( ¯ ψ b Γ P Γ A ψ a )Γ A ψ c − ψ b Γ RST Γ AB ψ a )Γ RST Γ P Γ AB ψ c Adding these, we get the following identityΓ
P Q ψ a ( ¯ ψ b Γ Q ψ c ) + Γ A ψ a ( ¯ ψ b Γ P Γ A ψ c ) = 12 (cid:0) Γ P Q ψ c ( ¯ ψ b Γ Q ψ a ) + Γ A ψ c ( ¯ ψ b Γ P Γ A ψ a ) (cid:1) So when we contract the gauge indices a, b, c with totally antisymmetric structure con-stants f abc of the gauge group, we get (cid:0) Γ P Q ψ a ( ¯ ψ b Γ Q ψ c ) + Γ A ψ a ( ¯ ψ b Γ P Γ A ψ c ) (cid:1) f abc = 0and this is precisely the identity we need for the cubic terms in the supersymmetryvariation of the Lagrangian to vanish. B A consequence of the Killing spinor equation
From (2.5) we have6Γ M D M η = Γ M Γ N D M D N ε = D M ε + R ε − W ABMN Γ MN Γ AB ε and D M ε = Γ M D M η −
34 Γ A Γ RST ηT ARST −
18 Γ A Γ P QR Γ M εD M T AP QR W ABMN = ∂ M V ABN − ∂ N V ABM (B.1)is the field strength of the R-gauge field background potential V ABM . By taking these tworesults together, we get5Γ M D M η = − R ε + 18 Γ MN Γ AB εW ABMN −
34 Γ A Γ MNP ηT AMNP −
18 Γ A Γ UV W Γ M εD M T AUV W
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