Lightlike conformal reduction of 6d (1 , 0) theories

We study 6d (1 , 0) superconformal theories. These have a natural lightlike conformal Killing vector, the Dirac current. We perform a conformal dimensional reduction along the Dirac current down to five-dimensions in such a way that we always preserve at least two real supercharges.


Introduction
For the superconformal M5 brane in absence of supergravity background fields we have the 6d conformal Killing spinor equation which restricts the six-dimensional Lorentzian worldvolume geometry of the M5 brane.
Here Γ M are 6d gamma matrices, ε and η are four-component spacetime spinors of opposite Weyl chiralities, and each transform as a four-component spinor under the Sp(2) = SO(5) R-symmetry rotations.There is a conjectured duality between the M5 brane theory and a theory we get by dimensional reduction along a circle [1], [2].This suggests that we may study the M5 brane theory using a 5d theory, which has an ordinary gauge field and hence a nonabelian generalization.If v M is a spatial Killing vector that generates the circle, then supersymmetries that remain in 5d satisfy in addition to (1.1).This additional equation (1.2) may not be satisfied by any of the solutions to (1.1) in which case all supersymmetries are broken by the dimensional reduction.If all supersymmetries are broken, it becomes difficult to check if the duality between the 6d and 5d theories still holds.Without supersymmetry it is hard to do quantum computations in the 5d theory, which is nonrenormalizable.On the other hand, the full 6d superconformal symmetry should be present in the 5d quantum theory if the duality between the 6d and 5d theories is correct.The full superconformal symmetry may not be a symmetry of the classical 5d Lagrangian.But we expect that it is a hidden symmetry in the quantum theory that emerges when we sum over all instanton sectors.For an M5 brane on R 1,5 there is an OSp(8|2) superconformal symmetry whose bosonic subalgebra is SO(2, 6) × Sp (2).Here SO (2,6) is the conformal symmetry and Sp(2) = SO (5) is the R-symmetry.These bosonic symmetries correspond to isometries of AdS 7 ×S 4 , which is the dual geometry in the AdS-CFT correspondence.By compactifying one spatial direction we get R 1,4 ×S 1 .Upon dimensional reduction to 5d super-Yang-Mills on R 1,4 the conformal symmetry SO (2,6) is broken to the Poincare symmetry ISO (1,4).The M5 brane on R 1,5 has 32 real supercharges, parametrized by 16 real Poincare supersymmetry parameters ε P and 16 special conformal supersymmetry parameters η.The solutions to (1.1) are given by ε = ε P + Γ M ηx M .Upon circle compactification, only the 16 Poincare supercharges corresponding to ε P survive.
In the above two examples the geometry of the M5 brane was conformally flat.We may also consider other kinds of geometries, for example R 1,1 ×TaubNUT, that admit conformal Killing spinors.If for some geometry there are n solutions to the 6d conformal Killing spinor equation (1.1), then because of the hidden four-component R-symmetry spinor index, there are in total 4n real supercharges.In flat space we have n = 4 + 4 = 8.But we will now continue to refer to the case of 4n real supercharges on a curved spacetime as (2,0) supersymmetry.The smallest amount of supersymmetry we can have is (1,0) supersymmetry corresponding to Sp(1) = SU (2) R-symmetry and 2n real supercharges.
A complete classification of Lorentzian six-manifolds that admit conformal Killing spinors with Sp(2) R-symmetry is still lacking.But for Sp(1) R-symmetry, a complete classification of Lorentzian six-manifolds that have conformal Killing spinors is found in [10].The difficulty with extending this classification to Sp(2) R-symmetry of the (2,0) theory, is related to the fact that the Dirac current εΓ M ε in (2,0) theory is not necessarily lightlike.It is lightlike only if the matrix εΓ M γ A ε has rank one.Here γ A denote the gamma matrices of the Sp(2) = SO(5) R-symmetry where the index ranges over A = 1, 2, 3, 4, 5.
In this paper we will take the R-symmetry to be Sp(1) = SU (2) where we have the 6d conformal Killing spinor equation Again Γ M are 6d gamma matrices, and we display the SU (2) R-symmetry indices I, J, ... = 1, 2 explicitly.We have an invariant antisymmetric tensor ε IJ and its inverse ε IJ that we, by a convention, use to rise and lower indices by acting from the right, ψ I = ψ J ε JI and ψ I = ψ J ε JI .Here ψ I represents any field or parameter that carries the index I.
We will show that we may reduce a Lorentzian 6d (1, 0) superconformal theory with 2n real supercharges down to a 5d supersymmetric Yang-Mills theory while preserving at least two real supercharges, without imposing the extra condition (1.2) on the supersymmetry parameter.We pick one pair of supersymmetry parameters ε I for I = 1, 2, and construct the Dirac current V M = εI Γ M ε I which will always be a lightlike conformal Killing vector [10], [11].We then perform a conformal reduction along V M .
If we pick another solution to (1.3) then we get another Dirac current and by conformal reduction we get another supersymmetric 5d theory.Therefore, in the most generic case, we may only preserve two real supercharges, namely those which correspond to our choice of two supersymmetry parameters ε I for I = 1, 2 that we use to define Our conformal reduction is a generalization of an ordinary dimensional reduction to the case when the vector field along which we dimensionally reduce is a conformal Killing vector field.For an ordinary Killing vector field v M we have the isometry condition of the metric tensor g M N , Under dimensionsional reduction along this isometry, we put for all the fields, here collectively denoted as Φ.For our conformal Killing vector field V M = εI Γ M ε I , we instead have where Ω = ∇ P V P = 12ε I η I .What stands on the right-hand side may be thought of as an infinitesimal Weyl transformation of the metric tensor if we make Ω infinitesimal.A general infinitesimal Weyl transformation of the metric may be written as where σ is an arbitrary infinitesimal scalar field parametrizing the Weyl transformation.The Weyl weight of the metric is ∆ g = 2 so we may write To perform dimensional reduction along a conformal Killing vector, a natural guess seems to be that we should constrain the fields, collectively denoted as Φ, as where ∆ Φ is the Weyl weight of the field under consideration.That is δΦ = ∆ Φ σΦ under an infinitesimal Weyl transformation.In order to preserve supersymmetry under the dimensional reduction, it turns out that we need to also add an R-rotation term.
To motivate this R-rotation, we note that by a straightforward computation that can be found in an appendix in [11], we have where R J I is a generator of SU (2) R-symmetry that we define as the following traceless part As we will show in this paper, a supersymmetry preserving dimensional reduction is achieved by modifying (1.4) by adding an R-rotation term as for all fields that rotate under R-symmetry.Here we display this for a field Φ I in the fundamental representation of SU (2) with one index I downstairs, but this generalizes straightforwardly to any field, or composite field, in any R-symmetry representation by taking tensor products of the fundamental representation.We notice that Ω changes under a Weyl transformation whereas R I J is invariant.This may be seen by noting the Weyl transformation rules, We may use these to show that These transformations show that we may make a Weyl transformation with transformation parameter σ that satisfies such that a conformal Killing vector V M becomes an ordinary Killing vector with respect to the Weyl transformed metric.
To make this discussion more explicit, we may choose a coordinate system where the metric takes the form ds 2 = e −2σ(x + ) dx + dx − + ... We may then perform a Weyl transformation that removes the conformal factor, Then we have a Killing vector V = ∂ ∂x + and hence we can make a periodic identification x + ∼ x + + 2πR and that is true even if σ(x + ) is not periodic.After the Weyl transformation, the supersymmetric dimensional reduction amounts to putting for all fields collectively denoted Φ I (and as before, for brevity we just put one index I downstairs, but this can be generalized).If we have the Weyl transformation rules of the metric and the field, then for the original metric, the vector field V = V = ∂ + becomes a conformal Killing vector field, and (1.5) becomes which is easily shown to be equivalent with (1.6) using the above Weyl transformation.By noting that R I J is covariantly constant, and hence also constant as it carries no spacetime indices (for a derivation of this fact, we refer to an appendix in [11]), we can integrate (1.7) around the circle to get the global identification had not been constant along the circle, then we would have had a path-ordered exponent, but since R J I indeed is constant that simplifies to an ordinary exponent.We thus find a periodic boundary condition that is twisted by both the Weyl symmetry and the R-symmetry.The R-symmetry is a global symmetry, but we can gauge this R-symmetry by coupling the theory to a background R-symmetry gauge field.Then for an R-gauge field that is locally pure gauge, we can gauge it to zero.But this may involve a gauge parameter that is not necessarily periodic and this gauge parameter gives rise to the R-symmetry twisted boundary condition [12], [13], [14].The Weyl symmetry is already a local symmetry, and hence it is a gauge symmetry, and therefore we may also allow for a Weyl symmetry twist in our boundary condition.
Ordinary (or untwisted) dimensional reduction along a lightlike direction amounts to a discrete light cone quantization, or DLCQ for short.If we apply DLCQ to the M5 brane theory, it is conjectured to reduce to a supersymmetric quantum mechanics on the moduli space of instantons [15], [16], [17].This has also been extended to 6d (1,0) superconformal theories [18].It would be interesting to see how our conformal dimensional reduction with an R-symmetry twist would modify this DLCQ proposal.
In section 2 we present the result, the Lagrangian and the supersymmetry variations.In section 3 we show that the conformal reduction commutes with the supersymmetry variations and hence it is a consistent truncation.In section 4 we show that the Lagrangian is invariant under the proposed supersymmetries.In the appendices we collect derivations of various geometric formulas.

Lightlike conformal reduction
Lightlike dimensionsal reduction of the abelian selfdual tensor field H M N P has been studied in [19], [21].We assume that we have two lightlike conformal Killing vectors V M and U M whose inner product is denoted V M U M = N and will be assumed to be everywhere nonzero (and hence we may choose N > 0 by convention).We define We invert these relations as where tildes indicate quantities whose contractions with U M and V M are zero.We might put a tilde on K M for free, since K M = K M .The antisymmetric tensor induces an antisymmetric tensor in the four transverse directions, The selfduality H M N P = 1 6 ε M N P QRS H QRS leads to and that we use to eliminate H M N P .Following [19], [21] we take antiselfduality of G M N offshell and then the selfduality of F M N follows from a Lagrangian where G M N acts as a Lagrange multiplier, For a nonabelian gauge group we do not have this starting point at a selfdual H M N P but we can keep F M N , G M N and K M that we promote to nonabelian fields and search for a Lagrangian that has (1, 0) superconformal symmetry.There is a vector multiplet with the bosonic fields the gauge potential A M and a real scalar ϕ and the fermionic fields are two fermions λ I subject to a symplectic Majorana condition.The gauge field and the scalar carry 3 + 1 degrees of freedom, and the fermions carry 8 degrees of freedom.There may depending on the formulation also be extra auxiliary fields, such as here for example the G M N that carry no physical degrees of freedom.To this vector multiplet we may couple an arbitrary number r of hypermultiplets whose bosonic fields are 2r scalar fields q AI subject to a reality condition and r fermions ψ A , also subject to a reality condition.
Here A = 1, ..., r labels the different hypermultiplets.There are 2r bosonic degrees of freedom and 4r fermionic degrees of freedom.For convenience we will assume that the hypermultiplet fields are in the adjoint represenation of the gauge group.The Lagrangian is given by the sum Here we define which satisfies We show these properties Appendix B. The normalization of the Chern-Simons term L CS is such that its infinitesimal variation is given by δω(A) We will show that this Lagrangian is invariant under where all fields are restricted by the conformal reduction (2.3) and (2.4).There is a coupling constant e that appears in various commutator terms and it appears in the gauge covariant derivatives of matter fields (collectively denoted by Φ) as For the purpose of showing that the classical Lagrangian is supersymmetric, the value of e is not essential.Its value becomes important only in the quantum theory.The supersymmetry parameters ε I satisfy the conformal Killing spinor equation The Dirac current is a lightlike conformal Killing vector.For V M to be nonzero we shall take ε I to be commuting spinors.We have the following result, where we put The derivation of (2.2) can be found in the appendix in [11].
The Dirac conjugate is defined as where In flat Minkowski spacetime the chirality matrix is defined as Γ = Γ 012345 , and it is numerically the same in curved spacetime.The symplectic Majorana condition reads The 6d charge conjugation matrix C has the properties These relations can be used to show that (ε I η J ) * = −ε I η J .Since R IJ = R JI (as a direct consequence of R I I = 0) we can see that (R J I ) * = −R I J and hence this is an anti-hermitian generator of SU (2).
If we assume that Ω = 0 then we must have η I = 0 and then V M becomes a Killing vector and (2.2) becomes L V ε I = 0 in which case this simply says that those supersymmetries survive under an ordinary lightlike dimensional reduction along V M .
We now ask ourselves if this familiar situation can be generalized to the case when Ω is not zero.As we discussed in the Introduction, we shall impose (1.5) on all the fields, properly interpreted for fields in any representation of the R-symmetry.For the fields in the (1, 0) theory, we have Φ = (A M , ϕ, λ I , G M N , K M , q AI , ψ A ). Then for the tensor multiplet fields, we shall impose and for the hypermultiplet fields, we shall impose As we showed in [11] once we have V M and (1, 0) supersymmetry, we necessarily will also have another lightlike conformal Killing vector U M .In order to show that the Lagrangian is superconformal, it is necessary to assume that their Lie derivatives commute, 3 Commuting Lie derivatives with supersymmetries We begin by showing that the supersymmetry variations commute with the conformal reduction by showing that on all fields Φ in the tensor and hypermultiplets.This shows that the conformal reduction is consistent with the supersymmetry variations.
We begin by checking if the conformal reduction is consistent with the variation We act on both sides with L V , which yields Ωδq AI − 8R I J δq AJ and we see that indeed 3.2 Checking δψ A = ...

We now check the variation
First we compute Next we shall compute We begin by computing the Lie derivative acting the first term, which is a spinorial quantity (since the vector and R indices are fully contracted), so we use the usual formula for a Lie derivative acting on a spinorial quantity.Moreover, by V M A M = 0, we can use a gauge covariant derivative in the Lie derivative for free.Thus we compute Now we look at the term The second term combines with the last term above, switching We get Now we apply the conformal reduction, and then we find that the Weyl weights add up to − 1 12 + 1 3 + 1 6 = 5 12 and we get In the process we have noted that R I J is covariantly constant [11].Now we shall add the Lie derivative acting on the second term, In the Appendix we show that By applying conformal reduction on L V q AI = − Ω 3 q AI − 8R I J q AJ and by noting that L V F M N = 0, we get So by summing the two terms, we get We shall next compute Let us first study the supersymmetry variation itself, where Since the gauge potential is a conformal weight zero, we will replace gauge covariant derivatives with geometric covariant derivatives, or alternatively consider abelian gauge group.The generalization to nonabelian case is then obtained by simply replacing back the gauge covariant derivatives, but there would be no changes in the computation that follows other than notational, since we have L V A M = 0 and V M F M N = 0 under the conformal reduction.We now get We may discard the last line from the variation because selfduality will be preserved when acting on both sides with L V trivially.Basically T M N = δB M N and the last line is nothing but δH − M N P = 0 which shall be respected by acting on it with L V .So we descend to That is, Now we act on both sides with L V .Then The first line is zero by the fact that L V T M N = 0 and [L V , L U ] = 0. Let us put After some cancelations, we get Now we use that L V T M = 0 and get which we can write as We conclude that 3.4 Checking δϕ = ...

First we compute
We now apply conformal reduction, and find that the Weyl weights add up as 1 2 − 5 2 = −2 and we get and hence 3.5 Checking δA M = ...

First we compute
Next we compute for the vielbein and L V V M = 0 together with conformal reduction of λ I and εI .This leads to the following numerology 1  12 + 2 6 − 5 12 = 0 and we get The computation of the second line proceeds in an analogous manner as we did the computation for L V δψ A .The first line is computed using The Weyl weights add up to 1 2 − 3 = − 5 2 In summary, we get It is also interesting to note in this context that U M /N and V M /N are conformal singlets, as we show in Appendix A.

The superconformal Lagrangian
When showing that the Lagrangian is superconformal we may follow the computation in the Appendix of [21].Below we will highlight what changes in that computation as we change from ordinary lightlike to conformal lightlike Killing vectors.We get and collecting all terms that involve G M N coming from the variation of the matter part we find The last lines vanishes when imposing the conformal reduction condition Next collecting terms involving F M N from the matter part, we find Adding the corresponding term in δL A they combine into That we cancel by the Chern-Simons term.At last we collect terms involving K M from the matter part, Adding the corresponding contribution from δL A we get by summing them up which is zero by the conformal reduction condition After cancelling most remaining terms in a standard manner, what remains then are two contributions that we will write as δL = δL tensor m + δL hyper m .We will now look at each of these contributions in turn.First where the second line comes from varying A M in the scalar field kinetic term.Simplifying we get We note V M A M = 0, so what we find inside the commutator is L V ϕ = − Ω 3 ϕ and hence the commutator is vanishing, so we get The first term is rewritten using L The second term is rewritten as using (that we derive in Appendix C) as Adding up the contributions, we find In summary then, we have shown that the Lagrangian is superconformal δL = 0 up to boundary terms.Let us now discuss these boundary terms.When we check the supersymmetry of the Lagrangian we performed integrations by parts to bring derivatives acting on fermions to instead act on bosonic fields, as our convention.In the process there appears boundary terms.These are on the form where Then we notice that Since K has Weyl scaling dimension −6 we find that The upshot is that D M V M K = 0 and so reduces to boundary term of a five-dimensional theory as the component of the original K M along V M does not give any contribution.

Singularities and boundaries
Our discussion so far has been concerned with the local aspects of the dimensional reduction.In the end we need to integrate the Lagrangian density over the six-manifold, where we constrain the Lie derivative of the fields along the conformal vector field V as We may encounter a singularity if the vector field V vanishes on a submanifold.To see how such a singularity may arise explicitly, we will study a conformally flat spacetime with the metric for i = 1, 2, 3, 4 and where σ can be an arbitrary function of all six coordinates.Then the solutions to the conformal Killing spinor equation are given by where ε I0 and η I are constant bosonic spinors that correspond to the Poincare and the special conformal supercharges respectively.The Dirac current is given by Here we define Γ M g = e −σ Γ M where Γ M are the Minkowski space gamma matrices.If we expand the Dirac current in components, then we get Here we define |x| 2 = 2x + x − + x i x i .From this expansion we can see that the terms that appear in V = V M ∂ M correspond to a translation, a dilatation, a Lorentz transformation and a special conformal transformation, respectively.We may also expand One may now explicitly see that V M does not depend on σ whereas Ω depends on σ, which reflects the general fact that V M is Weyl invariant while Ω is not.If for a given Ω as computed with the flat Minkowski metric we can solve the equation with respect to σ then V becomes a Killing vector in the metric (5.2) with this particular solution σ.First let us select two Poincare supercharges corresponding to ε I = ε I0 that we want to preserve under the dimensional reduction.Then Ω = 0 and the Dirac current V M is a Killing vector.If we choose the two supercharges such that V M = δ M + then we can perform the dimensional reduction by imposing the constraints that L V = ∂ + vanishes on all fields.For the 6d tensor multiplet scalar field we start with the action By periodically identifying x + ∼ x + + 1 and imposing the constraint ∂ + ϕ = 0 we obtain the dimensionally reduced action If we assume that the metric of the 5d theory is ds 2 = −c 2 (dx − ) 2 + dx i dx i , then the corresponding action would have been We recover our action (5.4) in the nonrelativistic limit c → ∞.The (2, 0) supersymmetric extension of (5.4) was obtained in [20], [21].
Let us next select two special conformal supercharges ε I = Γ M η I x M that we want to preserve under the dimensional reduction.If we assume that ηI Γ M η I = δ M + , then the conformal Killing vector has the components where r 2 = x i x i .This vector field vanishes along the lightlike line parametrized by x + where x − = 0 and r = 0 and our aim is to explore what implications that will have for the dimensionally reduced theory.Let us examine the scalar field ϕ in the (1, 0) tensor multiplet whose scaling dimension is ∆ ϕ = −2.The constraint equation for ϕ is given by Explicitly it becomes or, if we make the field redefinition φ = (x − ) 2 ϕ, then The action in flat Minkowski space is given by After the field redefinition, the action becomes after we drop one boundary term by assuming that the field drops off sufficiently fast at the lightlike infinities x + → ±∞.We may use the constraint to substitute into this action, where we now put t = x − .Then the action becomes where G αβ is the metric on a unit three-sphere.Let us define a five-dimensional field as whose action is Here g 2 Y M is the Yang-Mills coupling constant, which is proportional to the compactification radius of the orbit that is generated by the vector field V M .For this step we thus assume that we have performed a compactification of the six-manifold by making a periodic identification as we travel a certain distance (2π times the compactification radius) along the Killing vector field V M .We may for example start out from the hypersurface x + = 0 at a point (t, x i ) on the hypersurface, and then we travel a certain distance along the integral line of the vector field to a new point (x ′+ , t ′ , x ′i ).At this new point, the metric is exactly the same as it was where we started at (0, t, x i ) because we were traveing along a Killing direction.We can therefore identify the two points (0, t, x i ) ∼ (x ′+ , t ′ , x ′i ) and that is one way to generate our compactified spacetime manifold.We could pick another hypersurface, but the end result, the compactified manifold, will become the same.
The justification for our five-dimensional action is as follows.We can recover the field φ(x + , t, x i ) by solving the equation of motion that we derive from S 5d for the field u(x − , x i ) if we also use the constraint equation that determines how the field depends on x + .All information about the dynamics of the zero mode φ is thus captured by the field u(t, x i ) and its action S 5d .
By solving the equation ( 5.3) we find the Weyl rescaled metric with respect to which V M is a Killing vector.One may also check explicitly that with respect to this metric, V M as given above, indeed satisfies the Killing vector equation We perform dimensional reduction along V M by imposing the constraint Here we notice that ϕ acquires a corresponding Weyl factor (x − ) 2 under the Weyl transformation of the metric.We may now also understand the appearance of the factor of 1 that appears in the action (5.6) as a measure factor that comes from the Weyl rescaled metric g Let us now look at a different five-dimensional action, where we have added a kinetic energy term for the scalar field.Such an action can be understood as an action of a theory that lives on a Lorentzian five-manifold with the metric Our action in (5.7) is recovered from this action by taking the nonrelativistic limit c → ∞.
The metric in both 6d and in 5d have a singularity at x − = 0.This might be interpreted as a lightlike boundary in the 6d theory, or as an initial time slice in the reduced 5d theory.It would be interesting to work out the supersymmetric extension of this action and check that the action is supersymmtric and preserves the two supercharges ε I = Γ M η I x M .We could also consider the case where we keep one Poincare supersymmetry ε I0 for say I = 1, and one special conformal supersymmetry η I for say I = 2.In that case there will be a nonvanishing R-rotation R I J that is Weyl invariant.Then we need to in addition twist all fields that are charged under the R-symmetry by a corresponding term in the constraint equations that we impose under the dimensional reduction.The form of the vector field V M changes depending on what supersymmetries we want to preserve and that will result in another conformally flat metric where V M becomes a Killing vector field.
More generally, by following our procedure, we may obtain a supersymmetric Lagrangian in five dimensions that should be closely related to the six-dimensional theory.The five-dimensional theory could provide us with a window into this six-dimensional theory.The ability to preserve some supersymmetry under the dimensional reduction might be useful in order to probe various aspects of the six-dimensional theory for various six-dimensional geometries.For example one could try to use such Lagrangians in five dimensions for supersymmetric localization.
The original (2, 0) R-symmetry is SO(5) but that is now broken down to SU (2) that acts on the Weyl spinor index I and the other SU (2) acts on the Weyl spinor index A.But now we may allow for an arbitrary number k of hypers, by letting A = 1, ..., 2k and then we have an Sp(k) flavor symmetry rotating those hypers.Since the original SO(5) R-symmetry is now gone, we hope that recycling of index A will not cause any confusion.
We will now choose Then it can be shown that [11] Γ The supersymmetries become M ε I V M = 0 This results in a new Lagrangian that is given by L = L A + L CS + L tensor m + L hyper m where L A and L CS are as before and where D M λ I + e 2 λI Γ M [λ I , ϕ]V M