Null Reductions of M5-Branes

We perform a general reduction of an M5-brane on a spacetime that admits a null Killing vector, including couplings to background 4-form fluxes and possible twisting of the normal bundle. We give the non-abelian extension of this action and present its supersymmetry transformations. The result is a class of supersymmetric non-lorentzian gauge theories in 4+1 dimensions, which depend on the geometry of the six-dimensional spacetime. These can be used for DLCQ constructions of M5-branes reduced on various manifolds.


Introduction
The M5-brane is an interesting and important object in M-Theory for a variety of reasons. Its dynamics are described by a six-dimensional field theory with (2, 0) supersymmetry. For multiple M5-branes this is an interacting, strongly coupled superconformal field theory. However we currently lack a satisfactory understanding of this theory. Nevertheless a particularly fruitful application of M5-branes involves compactifying them on a manifold to obtain lower dimensional field theories. In this way many novel field theories have been identified as well as relations/dualities between them.
Recently we have studied null reductions of the M5-brane (a related abelian construction already appeared in [1] as well as in newer work [2]). In the simplest cases this leads to the construction of novel non-abelian field theories in (4+1)-dimensions with 24 (conformal) supersymmetries [3,4]. Due to the fact that one has fixed a particular null direction in the six-dimensional theory, the Lorentz group has been reduced from SO (1,5) to SO(4). However, they still admit a large bosonic spacetime symmetry, including a Lifshitz scaling, coming from the six-dimensional conformal group [5]. In this paper we extend this discussion to general null reductions of the M5-brane on a curved manifold.
Non-Lorentzian theories with Lifshitz scaling have received a great deal of attention, primarily from the perspective of their AdS dual geometry (for a review see [6]). While some supersymmetric Lifshitz theories have been explicitly constructed (for example see [7,8]) these often involve higher derivative terms, as is common in condensed matter systems. In contrast the field theories we obtain do not have higher derivatives but involve Lagrange multiplier constraints that reduce the dynamics to motion on a moduli space of anti-selfdual gauge fields [9,10], in line with the DLCQ description of the M5-brane [11,12]. Other classes of theories without Lorentz invariance but related to String/M-Theory have recently received attention in works such as [13][14][15][16][17][18].
Thus these more general null reductions should provide DLCQ-type descriptions of the field theories obtained by reducing M5-branes on other manifolds such as the Gaiotto theories [19]. Since there is no six-dimensional action based on non-abelian fields, the standard construction is to reduce the abelian theory and then find a suitable non-abelian extension that is compatible with supersymmetry. For example this was performed in [20] for the case of a general spacelike circle fibration. This was then followed by [21], who generalised this construction to include additional non-dynamical supergravity background fields. In this paper we will apply these constructions to the case of a null reduction. Although conceptually similar, reduction on a null direction is technically distinct and involves some interesting features. We will not consider the full background supergravity fields that were discussed in [21] however we will extend our results to backgrounds coming from fluxes in M-theory and a non-trivial connection on the normal bundle.
This paper is organised as follows. In section two we perform the general reduction of the abelian (2, 0) theory equations on a general spacetime with a null isometry. While the (2, 0) theory is based on a tensor multiplet, upon reduction we obtain vector fields. We then generalise the resulting action to a supersymmetric non-abelian gauge theory in section three. In section four we examine some special cases of the general reduction, and in section five we include couplings to background flux terms. Section six contains our conclusions and comments. Our conventions are summarised in the appendix, along with some formulae for the geometry.

The Abelian Dimensional Reduction
In this section we will reduce the equations of motion and supersymmetry variations of the abelian (2, 0) tensor multiplet on a six-dimensional manifold with metricĝ M N which admits a null Killing directionk M . We will use hats to denote six-dimensional geometrical objects throughout.

The Background
Consider a fixed curved background, i.e. there is no back-reaction on the metric from the matter fields. We will further only consider six-dimensional Lorentzian manifolds which admit a null killing vector fieldk In coordinates adapted to this isometry, (x + , x − , x i ) i ∈ {1, . . . , 4} it can be shown that the metric takes the general form (see also [22]) Here g ij is a Euclidean signature metric of a four-dimensional submanifold of the full sixdimensional spacetime. All components ofĝ M N are allowed to depend on x − and x i . The metric component g +− = −1 has been fixed using a suitable choice of the coordinate x − . This somewhat contrived choice of metric was chosen as it leads to the simpler inverse metricĝ It is important to note that this geometry is distinct from that invoked in [23], in which a spacelike circle is infinitely Lorentz boosted. Even if limits are examined carefully in that paper, one finds as the boost parameter goes to zero the length of the Killing vector is always positive. In contrast our Killing vector has length zero, as would be expected from a null reduction.
For the time being we do not consider any other background fields other than the metric, in section 5 off-brane fluxes are added.

Tensor Multiplet
The six-dimensional abelian N = (2, 0) tensor multiplet contains a self-dual 3-form, H =⋆H, (2.4) along with five scalar fields, X I , and a symplectic Majorana-Weyl spinor ψ. These fields transform in the trivial, fundamental, and spinor representations of the R-symmetry group SO(5) (or equivalently U Sp(4)) respectively. The supersymmetry transformations close up to the equations of motion: Here the supersymmetry parameter ǫ has opposite chirality under Γ 012345 to ψ, we make the choice Γ 012345 ǫ = ǫ and Γ 012345 ψ = −ψ.

Reducing H =⋆H
Let us first define the (4+1)-dimensional fields: In a trivial geometry these three fields are the independent components of the six-dimensional 3-form H, and F and G satisfy simple (anti-)self-duality constraints. Our task here is to see the implications of the six-dimensional self-duality condition for a general background.
In what follows we use the geometrical quantities associated to the four-dimensional manifold with metric g ij . In particular we define the fields F ij , G ij and F i − to have their indices raised by g ij . We also take with ε 1234 = 1. Along with the metric g ij , this allows us to define a four-dimensional Hodge star operator ⋆. To proceed it is convenient to work with forms, we define the one forms Written in forms the self-duality condition on F − is Applying ⋆ allows us to solve for H Eliminating H from the other relations we create two equations that depend only on F − , F, G along with the background fields σ, u, v and g. In particular we find these expressions simplify further to

DecomposingdH = 0
The exterior derivative is metric independent, so the results will hold for all backgrounds.
In components (2.14) Our construction has a x + isometry, so all fields are independent of x + . This gives an expression for each of the combinations of indices +−ij, +ijk, −ijk, ijkl Where we have written the 4 dimensional exterior derivative as d. The first and second expressions can be combined to give a simple five-dimensional Bianchi identity Implying that locally there exists (A − , A i ) such that The equations for G and F become Using the duality properties of F and G we can rewrite these equations in component form as respectively.

An Action
Lastly we wish to construct an action that reproduces these equations of motion, along with those of the scalars and fermions. In the latter cases a six-dimensional action already exists which can be trivially reduced to find an appropriate five-dimensional action. Somewhat remarkably the equations for F − , F and G can be derived from a Lagrangian density on a four-dimensional manifold with Euclidean signature, whose fields also depend on the 'time' coordinate x − . To this end we assume that F − and F arise from a potential (A − , A i ) as in (2.17). However we do not impose a potential for G but rather impose G = ⋆G 1 . Some trial an error shows that the equations motion (2.19) then arise from the lagrangian 21) and the k, l indices are raised with respect to g ij . Variation with respect to G immediately gives the anti-self-dual condition F = − ⋆ F. On the other hand varying A i and A − give (2.19) respectively. Inclusion of the scalars and fermions is easier, as there is a Lagrangian formulation for the free conformal case in any dimension; Performing the reduction by assuming ∂ + = 0, and inserting d = 6, we find Note that we have keep the fermionic terms andR in their six-dimensional form. In principle these can be computed from the expression (6.11), (6.13) and (6.6) found in the appendix. However expanding everything out in full detail for a general background leads to rather unwieldy expressions. Rather, we will provide more explicit expressions in various special cases below. It is helpful to introduce This derivative generally has torsion; One also finds that Putting all these together we can write the full abelian action as (2.28)

Supersymmetry and Non-Abelian Generalization
Next we want to show that the action (2.28) is supersymmetric. To this end we assume there there exists a solution to the conformal Killing spinor equation with ∂ + ǫ = 0. In particular this implieŝ which is a further condition that we must impose on the geometry. As it stands the action (2.28) is not invariant under the transformations that follow directly from (2.5). One problem is that the variation δG ij obtained from (2.5) is not self-dual off-shell. Thus we must adjust the algebra in a way that ensures δG ij is self-dual. A deeper issue is that although we impose the isometry ∂ + ψ = 0, this does not imply thatD + ψ = 0. For the bosonic fields this distinction does not cause a problem as both X I and H M N P do not couple to the spacetime connection (due to the fact that H M N P is anti-symmetric). But for ψ this leads to the Scherk-Schwarz-like mass term i 2ψM ψ in (2.28).
On-shell this is also not a problem as δH M N P in (2.5) contains terms involvingD + ψ which lead to the closure of the algebra and invariance of the equations of motion. However we find that theψM ψ term can only be made supersymmetric in general by modifying the variation of F −i and F ij in a way that means they are no longer closed. This in turn implies that a suitable expression for the supersymmetry variation of the gauge field cannot be defined. Since the existence of such a gauge field was crucial for the construction of the action, having no definable variation is not tenable.
Alternatively one might question why we start with the supersymmetry algebra (2.5) and not simply identify H = dB and impose H =⋆H as an equation of motion. However in this case one finds that G ij = 2∂ [i B j]− + ∂ − B ij and hence imposing an off-shell self-duality constraint on G ij and δG ij becomes non-trivial.
Thus to obtain a supersymmetric action after reduction on x + we find ourselves in a balancing act of finding off-shell expressions for δA − , δA i and δG ij = ⋆δG ij whenD + ψ = 0.

Correcting δG
The next problem is that δG is not self-dual off shell but to write the action we require that G is self-dual. A short calculation shows that where E(ψ) denotes the fermion equation of motion. Therefore we simply shift relabelling δ ′ G ij to δG ij gives us a self-dual δG.
Next the F ij F ij term, not present in the flat theory, must be accounted for in the supersymmetry transformations. We must use properties of F ij to shift δG ij in such a way to cancel the effects of this new term, whilst ensuring δG ij remains self-dual.
It is useful to note that a fermionic term of definite duality, e.g.ǭΓ + Γ ij ψ, can be used to build other terms of either the same or opposite duality (see Appendix A for for origin of these dualities). For instance inserting an additional Γ k will result in either a term of the same duality;ǭΓ + Γ k Γ ij ψ, or opposite duality;ǭΓ + Γ ij Γ k ψ. Furthermore we have the identity∇ [i F jk] = 0, allowing us to shift δG ij by any amount proportional to Γ ijk∇ k ψ so that δL shift by a total derivative. With this in mind we choose the shift which is self-dual by construction. A simple Gamma matrix manipulation shows the overall change to δL is By the modified Bianchi identity (2.27) for F under ∇, the first term is a total derivative so does not contribute. The shift (3.6) therefore cancels the term − 1 2 σδF ij F ij . Note that δG ij also has a term proportional to η, to account for terms arising from integration by parts.
Our corrected supersymmetry transformations read Again we have kept many of the fermionic terms in their six-dimensional form for notational simplicity. With these supersymmetry transformations we find that the action (2.28) is invariant up to terms arising from theψM ψ term. In other words we find δS = 0 if The implications of this constraint are explained in section 4.1.

Non-Abelian Theory
Our next task is to find a non-abelian extension of the abelian action found above which remains supersymmetric. After some trial and error we find that, assuming (3.9) holds, non-abelian extension is where all the fields now live in the adjoint of some gauge group. The supersymmetry transformations are where again we have leftR and the fermion derivatives in their six-dimensional form.

Twisting
We can also introduce an non-zero connection on the R-symmetry of the form and similarly forD M ǫ. This will allow us to introduce a twisting of the normal bundle. HereÂ M acts on X I in a representation of some subgroup Q of SO(5) andΩ IJ M provides a spinor embedding of Q into Spin (5).
Since this modification only affects the dynamics through derivatives of the scalars and fermions we can see its effect by modifying the matter part of the six-dimensional action to where T IJ is an invariant tensor of Q. This modification leads to 14) whereR M N IJ is the curvature ofΩ IJ M . Thus to obtain a supersymmetric reduction we must ensureD M ǫ =Γ M η, ∂ + ǫ = 0 and arrange for suitable choices of curvature and T IJ so that the terms in δS matter cancel. Indeed the usual role of twisting is to allow for solutions toD M ǫ = 0 on manifolds with non-vanishing curvature. For example in the case of a Riemann surface along x 3 , x 4 with normal directions X 6 , X 7 the first term vanishes and we can arrange to cancel the last two by taking and projecting on to spinors withΓ 34Γ 67 ǫ = ±ǫ, where the sign is chosen to correspond to solutions ofD M ǫ = 0.

Examples
In the previous section we constructed the non-abelian extension of the reduced M5-brane equations and their supersymmetry transformations. We left the fermion terms in a sixdimensional form as the complete expression in full generality is quite complicated and unenlightening. In this section we will evaluate some general classes of examples explicitly.

Obstruction fromM
In order to obtain a supersymmetric reduction we require in addition that (3.9), i.e. δψM ψ = 0, is satisfied. In addition the condition (3.2) ensures that We do not propose to give the general solutions to these conditions which place various restrictions on both ǫ and the background fields σ, u, v. For example if du is not anti-selfdual then the second equation implies that ǫ − = 0. Since there are no mass terms for the scalars (beyond the usual conformal coupling to the curvature) a physically well-motivated class of background that ensures (3.9) are those for which there is also no mass term for the fermions: This leads to the following conditions on the background fields With i v (·) denoting contraction with v. There are two natural solutions to these constraints: 2 Therefore from (4.1) we find In what follows we will only focus on these two cases so that we can be as explicit as possible. We emphasize that other solutions to the constraints (3.2) and (3.9) might also be possible.

Case 1:
Here the action is which is invariant under (4.7) For brevity we have left the six-dimensional Ricci scalar unexpanded, for completeness in terms of four-dimensional objects only this iŝ (4.8) with γ i jk the Christoffel symbols of the 4d metric. In the specific case of this metric being independent of x − this reduces toR = R . (4.9)

Case 2: u = 0
Here the action is since u is now zero F = F . Note also that since η = 0, we haveD M ǫ = 0 and henceR = 0. This action is invariant under the following transformations (4.11)

Flux Terms
In [21] the reduced M5-branes action is coupled to background supergravity fields such as a non-zero M-theory 4-formĜ µνρσ . 3 The presence of such a flux leads to Myers-like terms in the M5-brane effective action. In addition the fluxes modify the Killing spinor condition to: We need to find fluxes that are compatible with the condition ∂ + ǫ = 0. In particular applying the condition ∂ + ǫ = 0 to (5.1) for the choice µ = + leads to a purely algebraic constraint. For simplicity we will restrict our attention here to cases where this constraint is trivial: i.e.D + ǫ = 0 and there is no contribution in (5.1) from the fluxes for µ = +. Non-trivial cases arise in case 1 and require a cancellation betweenD + ǫ and the fluxes or twisting (and perhaps including additional restrictions on ǫ). These are better addressed on a case-by-case basis rather than our general discussion. Thus we restrict to case 2 (u = 0), whereD + ǫ = ∂ + ǫ = 0 and we only consider constant fluxes of the form with µ, ν, λ = +, −. In particular we find the possibilities C IJK , C IJk , C Ijk and C ijk . These are expected to lead to additional terms in the M5-brane effective action of the form: where m and m IJ are a mass terms which are linear in the fluxes. Starting with a general ansatz we find the only the following corrections to the action can be made supersymmetric: Along with this there additional terms in the supersymmetry transformations: δ → δ + δ ′ with and furthermore the Killing spinor equation is also modified tô which is in agreement with the eleven-dimensional supergravity Killing spinor equation (5.1). At first glance our result is somewhat surprising: we find no supersymmetric corrections possible for fluxes of the form C ijk or C IJk , no Myers-type flux term for C IJK and no bosonic mass terms at all. One way to see this strange behaviour is to note that the null theory can be obtained from a non-Lorentzian rescaling of familiar five-dimensional Yang-Mills theory [24]. Here one makes the rescaling of space and time according to and the matter fields by and then takes the limit ζ → 0, this is equivalent to [23]. One then makes the identification . The scaling of the supersymmetry parameter ǫ is fixed by requiring the fields scale the same way as their supersymmetry variations, this leads to [24] Let us now consider the form of S ′ that would arise from a spacelike reduction of the M5-brane in a non-vanishing supergravity flux (e.g. as in [21]): where again m IJ and m are linear in the fluxes. Examining the Killing spinor equation (5.1) one sees that we must scale the fluxes according to otherwise we encounter divergences or the fluxes are scaled away. As a result, the deformed action scales as, schematically, Thus in the limit ζ → 0 the only terms in S ′ that survive are precisely those in (5.4). The only exception is the Chern-Simons-like term which diverges, and therefore is not consistent with taking the limit.

Conclusions and Comments
In this paper we have performed a general reduction of the M5-brane along a null Killing direction. We then extended the result to a non-abelian theory. The result is a class of supersymmetric gauge theories in 4+1 dimensions but without Lorentz invariance. We also explored the effect of coupling of background supergravity fluxes to the M5-brane and twistings of the normal bundle. The results presented above include and generalise earlier results. In particular simply setting u = v = σ = 0 and g ij = δ ij recovers the flat space case [3], and setting u i = 1 2 Ω ij x j recovers the metric and action of of [4].
An interesting feature of this construction is how the information of H is encoded in a consistent way into the Lagrangian. Our isometry creates a natural split in the field; H is self-dual and closed, which is problematic for a Lagrangian. But here we find F is closed but with no self-duality constraint off-shell, whereas G satisfies a self-duality constraint but is not closed. On-shell the self-duality of G enforces anti-self-duality condition on F as it's equation of motion. In effect we have introduced a Lagrange multiplier, but without adding any new unphysical fields to our Lagrangian; H provides its own Lagrange multiplier. It would be interesting to explore how this construction ties in with the six-dimensional lagrangian approach of [25][26][27].
In case 2 G ij imposes the constraint F = − ⋆ F and therefore the dynamics is restricted to the space of anti-self-dual gauge fields on the four-dimensional submanifold. Such field configurations are then solved for by the ADHM construction in terms of moduli. The remaining part of the action leads to one-dimensional motion on the instanton moduli space [9,24]. This is in keeping with the various DLCQ proposals such as [11,12]. In case 1 G ij imposes the constraint F = − ⋆ F but here there are time-derivative terms and hence there is no simple reduction to motion on a moduli space.
The general form for the action includes an F ∧ F − ∧ v term which we can think of as a mixed Chern-Simons term between diffeomorphisms and gauge transformations. In particular for case 1 this term vanishes but in in case 2 we have u = 0 and so F ij = F ij . In this case if we let v (5) = v i dx i + σdx − then the metric admits a diffeomorphism v (5) → v (5) + d − (5)ω where ω depends on x i and x − . We can rewrite the terms involving F as where and (5) ω the Lagrangian shifts by a total derivative. Alternatively we can write in which case the gauge symmetry is only preserved up to a boundary term. We cannot write this term in a way which makes explicit both of these invariances simultaneously.
Thus we see that L cs mixes a five-dimensional diffeomorphism with the U (1) part of the gauge symmetry.
We hope that the results will be of use in studying the (2, 0) and related theories reduced on non-trivial manifolds through DLCQ-type constructions [11,12]. For example one could consider theories of class S [19] obtained by reduction of M5-branes on a Riemann surface Σ. Our results here should allow for a systematic construction in terms of motion on the moduli space of instantons on R 2 × Σ, i.e. Hitchin systems, coupled to scalars, fermions and possible additional data associated with singularities of Σ.
To avoid the confusion of whether or not Γ + means Γ plus or Γ plus minus , we will only use The relationŝ will be repeatedly used.

Appendix B: The Background
The vielbein (and inverse) for the metric are given byê with e i j being the vielbein for the four-dimensional metric g ij . Where u i and v i are defined to have their index raised by g ij , such that dot products are defined also with g ij . We also note thatĝ = det(ĝ M N ) = det(ê M N ) 2 det(η M N ) = − det(g ij ) . (6.12) Adding the fermions requires knowledge of the spin connection terms, the non zero terms of which arê where ω ijk is the four-dimensional spin connection for D i , the Levi-Civita connection for g ij on our euclidean submanifold.