A nonabelian M5 brane Lagrangian in a supergravity background

We present a nonabelian Lagrangian that appears to have $(2,0)$ superconformal symmetry and that can be coupled to a supergravity background. But for our construction to work, we have to break this superconformal symmetry by imposing as a constraint on top of the Lagrangian that the fields have vanishing Lie derivatives along a Killing direction.


Introduction
Finding the nonabelian M5 brane Lagrangian is a long-standing problem, but at the same time it has also been clear for a long time that a unique classial nonabelian Lagrangian for 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 brane Lagrangians [1], [2], [3] a new hope was that also the M5 brane Lagrangian may be found if one relaxes some of the symmetries that should be present in the classical Lagrangian in the same spirit as one did for the ABJM Lagrangian [3] of multiple M2 branes that preserves only a subgroup of the SO(8) R-symmetry group. The worldvolume theory of flat M2's has the bosonic symmetry group of AdS 4 × S 7 . Since S 7 is a Hopf fiber bundle over CP 2 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 manifest in the ABJM Lagrangian. For the M5's on the other hand, we have the bosonic symmetry group of AdS 7 × S 4 but here S 4 is not a Hopf circle-bundle so for the M5's it may be better to consider an orbifolding of the AdS 7 space which reduces the Lorentz symmetry rather than the R-symmetry. We will not attempt to orbifold AdS 7 in this paper, but we will consider a nonabelian theory that breaks the Lorentz symmetry at the classical level of the Lagrangian. More generally we will present a candidate Lagrangian for M5's on Lorentzian six-manifolds that has at least one Killing vector field that corresponds to an isometry direction. We break translational symmetry along this isometry direction by keeping only the zero modes of the fields in this direction. This isometry direction can be quite general. It can be fibered over a five-manifold. It can be a compact circle direction or 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 this isometry direction. This approach to the M5's has been studied previously [8], [7], [10] but in this paper we generalize these results and obtain a nonabelian Lagrangian coupled to supergravity background fields. This is a generalization of the abelian M5 brane coupled to a supergravity background fields that was studied in [4].
If we put Lie derivatives to zero in one spatial direction, then the theory will become five-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 field can be of any signature and in particular it can be light-like [8], [9]. So our Lagrangian is more general than the Lagrangian of 5d SYM. But at the quantum level the distinction between 5d SYM and the M5 brane is blurred since we do not understand whether these could in fact be just different faces of one and the same theory [18], [19]. Our Lagrangian appears to have 6d (2, 0) superconformal symmetry. But this symmetry is broken at the classical level as we shall constrain the fields to have vanishing Lie derivatives in one direction. The hope is that this broken symmetry will be restored in the quantum theory and that instanton particles of 5d SYM will give us those missing momentum modes along the isometry direction. Supersymmetry variations for such a theory have been found previosly for special cases. First for flat R 1,5 in [8] and then for (1, 0) superconformal symmetry on generic circle-bundle manifolds in [10]. The corresponding Lagrangian for this supersymmetric system has been unknown for some time. Recently there was an interesting suggestion for such a Lagrangian in flat R 1,5 in [7] based on a construction of a 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 a selfdual tensor field since the partition function for a selfdual tensor field when put on an euclidean six-manifold that has three-cycles is not unique. If the partition function is not unique, then a path integral argument suggests that also the Lagrangian can also not be unique. So we shall not look for a unique Lagrangian for the selfdual tensor field. Instead we may start with quantizing a nonchiral theory and at the end perform a holomorphic factorization 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 will consider the Lagrangian for a selfdual tensor field that was found in [13], [14]. It appears that this Lagrangian can be supersymmetrized and then it might also have applications to the M5 brane system [7], [6].
The objection rised by the paper [11] to the study of Lagrangians for selfdual tensor fields, can be avoided when there is a Killing direction in the six-manifold that might select one partition function as special compared to the many other partition functions that may also appear. For instance, if this Killing vector is timelike, then we may put time along this Killing vector and use Hamiltonian quantization that will give us a unique partition function. The canonical example for this approach is the M5 brane on a flat six-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 be modular invariant [12]. In fact our Lagrangian, that depends on a choice of Killing vector field, may also fit well with the idea of [11] after all, because from this work one is just discouraged to go looking for a unique Lagrangian for the selfdual tensor field. Our Lagrangian is not necessarily unique. If there are several Killing vector fields then there is one Lagrangian for each choice of 'preferred' Killing vector field that is used to construct our Lagrangian. This is in the same spirit as that of Hamiltonian quantization, but here generalized to Killing vectors that can be either timelike, spacelike or null leading to more general quantizations than the usual Hamiltonian quantization that applies only for the case of a timelike Killing vector.
Any proposed Lagrangian for the M5's can be put to the following tests. The first and simplest test of any candidate (1, 0) supersymmetric Lagrangian is whether this can be enhanced to (2,0). There are several attempted nonabelian M5 brane Lagrangians in the literature that do not appear to pass this test [16], [17], [15] although that does not rule out the more exotic possibilty (actually realized by ABJM theory) that supersymmetry could get enhanced to (2, 0) at the quantum level. Another test is whether any attempted (2, 0) Lagrangian in flat space can be put on curved space and whether (2, 0) supersymmetry can be enhanced to (2, 0) superconformal symmetry. Finally one may test whether a given candidate Lagrangian can be consistently coupled to the eleven-dimensional supergravity background fields while preserving superconformal symmetry.
In this paper we will present a Lagrangian that appears to pass all of these tests, but this is not entirely correct because for this construction to work we need to impose as a constraint on top of the Lagrangian that the Lie derivatives of all the fields vanish along a Killing direction and thus we need to break some of the superconformal symmetry at the classical level. But a breaking of some of the spacetime symmetries at the classical level of a Lagrangian is precisely what we should expect as that enables us to have a classical Lagrangian description of the M5's that is not unique, but depends on a choice of Killing vector.

The supersymmetric Lagrangian
Following [7], [13], [14] we introduce a selfdual tensor field H + M N P . This is an auxiliary tensor three-form field whose role in the Lagrangian is as a Lagrange multiplier field that implements the selfduality condition on another three-form field that we will denote as g M N P . Part of g M N P is a three-form h M N P with the wrong sign kinetic term in the Lagrangian. For abelian gauge group this three-form is a field strength of a two-form gauge potential b M N , so that h M N P = 3∇ [M b N P ] . For the nonabelian generalization we will not present an explicit realization of h M N P in terms of some nonabelian two-form gauge potential. This is one of the longstanding mysterious aspects of the theory of multiple M5 branes, the mystery of what exactly would be the nonabelian two-form. We will not try to answer this question here. But we will postulate the the infinitesimal variation can be presented as for some infinitesimal nonabelian two-form variation δb M N . Let us assume that h M N P v P = −F M N . Let us define the gauge algebra valued one-form for some gauge algebra valued zero-form Y . We now get Now h M N P and F M N can be varied independently from each other. Also, we notice that and since F RS is independent from h M N P , we conclude that where we define To formulate the supersymmetry variations, we find it convenient to introduce an infinitesimal variation δB M N := −δb M N . When this variation is a supersymmetry variation, then this is given by δB M N = iεΓ M N ψ. But this is just the infinitesimal variation, and we do not introduce nonabelian gauge potentials b M N nor B M N in this paper, only their infinitesimal variations.
The Lagrangian is a sum of two terms, L = L b + L m where the gauge field part is and the matter field part is Here G + M N P is a supersymmetry singlet and g − M N P = 0 is the selfduality equation of motion we get by varying the selfdual field H +M N P in the Lagrangian. We present the explicit form of the mass matrix µ AB in equation (3.1).
The supersymmetry variations are that is a three-form with selfdual and antiselfdual components whose supersymmetry variations are The supersymmetry variation of the fermions is 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 Here T A M N P is a supergravity background tensor field, carrying in addition an R-symmetry vector index A = 1, ..., 5. This tensor field is antiselfdual, since the spinors are chiral, where Γ = Γ 012345 is the 6d chirality gamma matrix. All our gamma matrices are eleven-dimensional, so in particular the gamma matrices for the Lorentz group and the R-symmetry group anticommute, The theory also couples to the supergravity background R-gauge field V AB M through the covariant derivatives that acts on the matter fields as 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 due to selfduality. But to determine the exact value of e will require considerations that go beyond just classical supersymmetry so we will keep this as a free parameter here. All our fields transform in the adjoint representation of the gauge group. But this maybe can be made more general if one can find a nonabelian gerbe structure for our theory.

The supersymmetry variation of L m
We make the ansatz and for the supersymmetry variation of ψ we make the ansatz while for the other fields we let those vary according to the what we stated before. Then we compute the supersymmetry by adopting the convention that we make integrations by parts in such a way that δψ does not appear in anyone of the terms and discard boundary terms. This will uniquely determine the variation as We now pick the commutator terms from this variation and postpone the study of all the rest to later. Let us also study the cubic term in fermi-fields later. Then we will for now focus on the following terms in the variation of the matter fields Lagrangian In addition to these terms, we also get the terms and another such commutator term comes from so the sum of all these terms for a = d = e just becomes a couple of Lie derivatives, So we can now conclude that we shall pick To determine the value of c requires some more work. This comes about by putting 4a − 2d = 2e and then by looking at the term and by using the Killing spinor equation to extract from this term the following term Let us make the following ansatz for the Lie derivative of a spinor field, Then for the vector field T M =ψΓ M χ we get For this to agree with the Lie derivative of a vector field we shall have so we must now require that v M is a Killing vector field. Since the Lie derivative that we want here is we clearly see that we shall choose d = e. This second term that got generated through the usage of the Killing spinor equation now combines with the two other terms to give us Thus for c = 1, we get the commutator [Γ AB , Γ C ] = −4Γ [A δ B]C and the three terms collapse to We now use the symplectic Majorana properties to writē and then we recall that We now recall that δH + M N P = −δh + M N P − δw +

M N P
By considering the abelian type of terms below, we will discover that the above variation combines with those abelian terms into Let us now write down the cubic terms in fermi-fields, 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 survive also when we put all the commutators to zero. Abelian terms arise from the following terms in δL m , We now extract all the abelian terms that will appear in each of these terms, We now find that the following terms cancel, Now we will expand out (A2i) by using that is a direct consequence of (2.5) as we show in the appendix. Here W AB M N is a field strength of the R-gauge field as defined in (B.1). Then we get Now we collect terms as follows, We conclude that the contribution to the supersymmetry variation of the matter fields Lagrangian that comes from the abelian terms is given by 1 where For this variation to vanish we shall take where D AB is a symmetric tensor that satisfies We can also see that D AB shall be traceless by contracting both sides with Γ A from the left.
To better understand the variation of the matter field Lagrangian, we will now study the following term in δL b Lagrangian Its has the following supersymmetry variation We are now interested in the first term that we expand out in two terms The second term cancels (A2d) + (A7b) by using the fact that T A M N P is antiselfdual and the first term combines with (A2id) to give the last term in χ A as We discovered this gamma matrix identity by using GAMMA [21].
Then we notice that Γ RST U V W = ε RST U V W Γ and that Γε = −ε so that when this acts on ε it will generate a projection onto the selfdual part of T A U V W which is zero. We are left with (A2b) 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 write down our final result for the variation of L m . It is given by The supersymmetry variation of L b Let us begin by making a supersymmetry variation of L b given by where we omit the Chern-Simons term. Here The coefficients of selfdual and antiselfdual components now conspire so that we obtain several terms that are wedge products between three-forms, We now expand out the term in the second line Now we use (2.4) and we get To proceed we want neither F M N nor w M N to have any component in the v M direction. This is solved for F M N by imposing the gauge fixing condition v M A M = 0 and by de- For w M N 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 M N v N = 0. This is easy to see. First we note that 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 M R nor from w ST it must come from δB N P . So we can replace δB N P → Q S P δB N S = δA N v P /v 2 ,

This variation is now precisely canceled by the variation of the Chern-Simons term
The term in the third line in (4.1) is worrisome as it can not be canceled by any other term. Fortunately it is identically zero as the following detailed computation shows, where we have used the gamma matrix identity After all these considerations, our result collapses to The two terms on the third line cancel up to a Lie derivative, 3 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 We will now argue that the two last terms cancel upon using the constraint The gamma matrix relations that are used here are To this end we start by making the following observation that if we define selfdual parts of W M N P Q as then we can write the last term in the Lagrangian in the following form This is a consequence of W M N P P = 0 that follows if one assumes that W M N P Q is totally antisymmetric in all four indices. Now we use the constraint (4.2) and then this term becomes proportional to so the upshot is that by using (4.2) we have and now it is easy to see that this cancels against the term

Equations of motion
We will derive the on-shell Bianchi identity for H + M N P that is required for on-shell closure of the supersymmetry variations when we act twice with supersymmetry variations on H + M N P . We will show that it arises as an equation of motion that we derive from the Lagrangian L = L b + L m . This is thus a consistency check.

The equation of motion for A M
Varying A M we get Let us dualize the equation of motion,

The equation of motion for b M N
Varying h M N P according to our postulated rule, δh M N P = 3D [M δb N P ] , we get 3)

The equation of motion for H +M N P
Varying H +M N P we get the selfduality equation of motion The equation of motion for λ −M N P Varying λ −M N P we get a constraint that relates H + to h + + w + , This constraint is supersymmetry invariant by itself.

The equation of motion for W M N P Q
Varying W M N P Q we get Then if we use (5.4), then this equation reduces to To see this, we need to establish that the remaining terms cancel. Namely we need to establish that Then we can use this in the constraint H + = −h + − w + to get which means that we can express (5.5) as where we have used that w M N P v P = W M N P Q v P v Q = 0 for the nonchiral w M N P . The equation (5.6) is invariant under supersymmetry variations up to a Lie derivative that we constrain to be zero. Namely the variation of the right-hand side is

The on-shell Bianchi identity
If we eliminate λ − M N P from (5.4) and insert that into (5.3) then we get where we may now notice how the coefficients in the Lagrangian conspire so that this becomes nonchiral three-forms once we dualize the three-form expression in the parentesis. We then get Now we use the Bianchi identity D [M h RST ] = 0 and we get and finally we use the equation of motion for A M obtained in (5.2) and we arrive at the on-shell Bianchi identity that is the equation of motion that is required in order to close the supersymmetry variations on H + M N P as was originally shown in [8], but here this equation of motion was derived from the Lagrangian.
Our computation is the same in spirit as that in [7], but it differs in the details. In [7] in place of our h M N P there appears instead expressions directly in terms of a nonabelian twoform b M N (using our notation). This is of course more attractive than our approach since it makes the equations explicit. However, their nonabelian two-form gauge potential appears in places where we would not expect that a gauge potential would appear explicitly, in the Lagrangian and in the supersymmetry variation of W M N P Q . Those quantities shall transform gauge covariantly, which is why we have chosen to set up the things in a different way from [7].
Since unlike [7] we have allowed v M to have a nonvanishing derivative, reflected in having a nonvanishing two-form w M N , this has led us to discover a new Chern-Simons term A ∧ F ∧ v ∧ w. In [20] it was shown that if one puts M5 brane on S 3 × M 3 for some euclidean three-manifold M 3 and if one performs dimensional reduction on S 3 (possibly a squashed S 3 , which would be reflected in having a nontrivial rescaling between our w M N and v M ), one gets a complex Chern-Simons theory on M 3 . It seems plausible that our real Chern-Simons term could be somehow related to this complex Chern-Simons theory on euclidean M 3 . In our computation we have assumed Lorentzian signature, so it seems like we would not get a complex Chern-Simons in our Lorentzian computation. This needs to be studied further.

The relation with the nonchiral Lagrangian
If we integrate out λ − M N P , then that will amount to replacing H + M N P with −h + M N P −w + M N P in the Lagrangian. If we do that, then we can recast the Lagrangian in the form where C M N P = w M N P + 6T A M N P φ A . If we truncate to the sector W M N P Q = 0 by hand in this Lagrangian, then we recover the traditional nonchiral Lagrangian [11]  The role of W M N P Q is to promote the constraint (5.6) to an equation of motion, which has the advantage that we can derive the equations of motion by varying the fields A M and h M N P as independent fields in the Lagrangian.

Closure of supersymmetry variations
Here we assume that the supersymmetry parameter is commuting and compute δ 2 on each field. Since we have introduced many auxiliary fields with no accompanying fermionic auxiliary fields, we do not necessarily expect closure on all these auxiliary fields.

Closure on φ
where the gauge parameter is We have closure up to a gauge transformation if we impose the constraint This constraint is consistent with what we found in (5.5) and in (4.2), so now we have found this constraint by three different computations, thereby making it rather convincing that it must be correct.

Closure on W M N P Q ?
Using the equation of motion (5.1) we get which we can write as where λ N P Q = 4iW N P QR S R and L A S is a Lie derivative where gauge covariant derivatives are used. We were unable to show that the second term is a gauge symmetry of the Lagrangian. However, we may eliminate this problem by simply integrating out W M N P Q that will impose the constraint (7.1).
Making a supersymmetry variation of (8.1), we then get Now contracting from the left with Γ M N P and using Γ M N P Γ M N P = −120 Γ M N P Γ Q Γ RST Γ M N P Γ Q = 144Γ RST we get i120Γ Q D Q ψ + 18iΓ RST Γ A ψT A RST + 120eΓ Q [ψ, φ A ]v Q + 12iΓ M N P Γ A ψT A M N P = 0 Then iΓ Q D Q ψ + i 4 Γ M N P Γ A ψT A M N P + eΓ Q [ψ, φ A ]v Q = 0 which agrees with the fermionic equation of motion that we obtain from the matter fields Lagrangian.
We have assumed that the infinitesimal variation of h M N P is on the form δh M N P = 3D [M δb N P ] and we have seen that this assumption takes the selfduality equation of motion to the expected fermionic equation of motion. This thus seems like the correct assumption for the infinitesimal variation of h M N P .

A 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, where P − = 1 2 (1 − Γ). Then we get Adding these, we get the following identity So when we contract the gauge indices a, b, c with totally antisymmetric structure constants f abc of the gauge group, we get Γ P Q ψ a (ψ b Γ Q ψ c ) + Γ A ψ a (ψ b Γ P Γ A ψ c ) f abc = 0 and this is precisely the identity we need for the cubic terms in the supersymmetry variation of the Lagrangian to vanish.
B A consequence of the Killing spinor equation is the field strength of the R-gauge field background potential V AB M . By taking these two results together, we get