As we have seen, the metric d s Ω 2 in (2.6) is conformal to 2n-dimensional Minkowski space, and hence can be realised as the conformal boundary of Lorentzian AdS_2n+1. Indeed, a particular slicing of AdS_2n+1 that makes this form for the conformal boundary manifest has long been known in the literature [12]. Let us review this construction now, with a focus using this holographic perspective to probe the form of the conformal algebra on the boundary.

Let Z ^a , a = 0, 1, … , n be a set of (n + 1) complex coordinates, and η _ab = diag (−1, 1, … , 1). Then, when constrained to η a b Z a Z ̄ b = − 1 , (2.10) the Z ^a provide coordinates on Lorentzian AdS_2n+1, with metric given by d s 2 = η a b d Z a ⁡ d Z ̄ b , (2.11) suitably pulled back to solutions of (2.10).

Next, we can parameterise solutions to the constraint (2.10) with 2n + 1 real coordinates (y, x ⁺, x ⁻, x ⁱ ). We have ² Z 0 = e i x + / 2 R cosh y 2 + 1 2 e y / 2 i R x − + 1 4 | x ⃗ | 2 Z n = e i x + / 2 R sinh y 2 − 1 2 e y / 2 i R x − + 1 4 | x ⃗ | 2 Z A = 1 2 e i x + / 2 R e y / 2 M x ⃗ + i R Ω x ⃗ A , A = 1 , … , n − 1 , (2.12) where here M is the orthogonal matrix appearing in (2.4).

These coordinates provide a description for AdS_2n+1 as a one-dimensional fibration over a non-compact form of n-dimensional complex projective space, sometimes denoted CP ̃ n . In this construction, x ⁺ is the coordinate along the fibre. The metric (2.11) now takes the form d s 2 = − 1 4 R 2 d x + + R 2 e y d x − + 1 2 Ω i j x j d x i 2 + d s CP ̃ n , (2.13) where d s CP ̃ n = 1 4 d y 2 + 1 4 e y d x i d x i + R 2 4 e 2 y d x − + 1 2 Ω i j x j d x i 2 , (2.14) can be identified as the metric on CP ̃ n , with isometry group SU(1, n). Then, projecting orthogonally to the orbits of ∂/∂x ⁺, we land precisely on CP ̃ n with metric d s CP ̃ n , as claimed.

To go to the conformal boundary, we now restrict to a surface of constant y, and take y large. It is then clear that as we do so, the metric approaches the form d s 2 → 1 4 e y − 2 d x + d x − + 1 2 Ω i j x j d x i + d x i d x i , (2.15) thus recovering the form of the metric d s Ω 2 .

Finally, let us discuss symmetries. Each isometry in the bulk, described by some Killing vector field, corresponds to a conformal symmetry on the boundary. The full set of such symmetries form the algebra s o ( 2,2 n ) . It will be useful for what follows, however, to identify the subalgebra of boundary conformal symmetries that commute with translations along the x ⁺ direction. We see that this subalgebra can be identified with the subalgebra of bulk isometries that commute with translations along the fibre. It is hence given by u ( 1 ) ⊕ s u ( 1 , n ) , where u ( 1 ) describes translation along the fibre, while s u ( 1 , n ) forms the algebra of isometries of the CP ̃ n transverse to the fibres.

Each continuous spacetime symmetry of a conformal field theory on Minkowski space is generated by an operator G, with the set of all such operators forming the algebra s o ( 2,2 n ) under commutation. We take the conventional basis, made up of translations P ̂ μ , Lorentz transformations M ̂ μ ν , dilatation D ̂ and special conformal transformations K ̂ μ .

Each operator G in turn correspond to a conformal Killing vector G _∂ of the metric d s M 2 . Explicitly, these are i P ̂ μ ∂ = ∂ ̂ μ ω ̂ = 0 i M ̂ μ ν ∂ = x ̂ μ ∂ ̂ ν − x ̂ ν ∂ ̂ μ ω ̂ = 0 i D ̂ ∂ = x ̂ μ ∂ ̂ μ ω ̂ = 1 i K ̂ μ ∂ = x ̂ ν x ̂ ν ∂ ̂ μ − 2 x ̂ μ x ̂ ν ∂ ̂ ν ω ̂ = − 2 x ̂ μ , (2.16) with indices raised and lowered with the Minkowski metric η _μν . Each of these vector fields G _∂ then satisfies L i G ∂ η = 2 ω ̂ η , where L i G ∂ is the Lie derivative along iG _∂ , and the Weyl factors ω ̂ are as given.

Their non-vanishing commutators are i M ̂ μ ν , P ̂ ρ = η ν ρ P ̂ μ − η μ ρ P ̂ ν i P ̂ μ , D ̂ = P ̂ μ i M ̂ μ ν , K ̂ ρ = η ν ρ K ̂ μ − η μ ρ K ̂ ν i K ̂ μ , D ̂ = − K ̂ μ i M ̂ μ ν , M ̂ ρ σ = η ν ρ M ̂ μ σ + η μ σ M ̂ ν ρ − η μ ρ M ̂ ν σ − η ν σ M ̂ μ ρ i P ̂ μ , K ̂ ν = 2 M ̂ μ ν − η μ ν D ̂ . (2.17) We can then perform the coordinate transformation (2.2) followed by the Weyl rescaling to arrive at the metric g as in (2.6). The operators { P ̂ μ , M ̂ μ ν , D ̂ , K ̂ μ } still generate the theory’s spacetime symmetries, and the corresponding vector fields are also conformal Killing vectors of the metric g , albeit with shifted Weyl factors given by ω = ω ̂ − 1 2 R tan x + 2 R i G ∂ + . Then, for each vector field, L i G ∂ g = 2 ω g .

It is now straightforward to see that translations along x ⁺ are an isometry ³ of the metric g. In terms of the original Minkowski symmetry generators, this is realised by the combination P + ≔ P ̂ + + 1 4 Ω i j M ̂ i j + 1 8 R 2 K ̂ − → i P + ∂ = ∂ + . (2.18) Then, given some conformal field theory on 2n-dimensional Minkowski space, we can perform a Kaluza-Klein on the x ⁺ interval. At the level of the symmetry algebra, this amounts to choosing a basis for the space of local operators which diagonalises P ₊. The resulting operators are Fourier modes on the x ⁺ interval. They fall into representations of the centraliser of P ₊ within s o ( 2,2 n ) , which we call h .

A basis for the subalgebra h in terms of the generators (2.16) is given by P + = P ̂ + + 1 4 Ω i j M ̂ i j + 1 8 R 2 K ̂ − → i P + ∂ = ∂ + H = P ̂ − → i H ∂ = ∂ − P i = P ̂ i + 1 2 Ω i j M ̂ j − → i P i ∂ = 1 2 Ω i j x j ∂ − + ∂ i B = 1 2 R Ω i j M ̂ i j → i B ∂ = R Ω i j x i ∂ j J α = 1 2 L i j α M ̂ i j → i J α ∂ = L i j α x i ∂ j T = D ̂ − M ̂ + − → i T ∂ = 2 x − ∂ − + x i ∂ i G i = M ̂ i + − 1 4 Ω i j K ̂ j → i G i ∂ = x i ∂ + + 1 2 Ω i j x − x j − 1 8 R − 2 x j x j x i ∂ − + x − ∂ i + 1 4 2 Ω i k x k x j + 2 Ω j k x k x i − Ω i j x k x k ∂ j K = 1 2 K ̂ + → i K ∂ = x i x i ∂ + + 2 x − 2 − 1 8 R − 2 x i x i 2 ∂ − + 1 2 Ω i j x j x k x k + 2 x − x i ∂ i , (2.19)

where the J ^α are absent for n = 1, 2, and otherwise α = 1, … , n ²−2n. Here, the L i j α are constant matrices that are explained below.

Then, these vector fields are indeed conformal Killing vector fields of g , as L i G ∂ g = 2 ω g with Weyl factor given by T : ω = 1 G i : ω = 1 2 Ω i j x j K : ω = x − , (2.20) and vanishing for the other generators.

Let us identify the subalgebra of pure rotations within h , and in particular identify the matrices L ^α . First, the case of n = 1 is somewhat trivial, as we have no spatial directions, and so no rotations to start with. Similarly straightforward is n = 2, whereby we can simply take B = 1 2 R Ω i j M ̂ i j as the single generator of the rotation subalgebra s o ( 2 ) , which is easily seen the commute with P ₊ and hence survive the reduction.

So let us take n ≥ 3. We may, a priori, consider a general spatial rotation of the form A i j M ̂ i j for any (2n − 2) × (2n − 2) matrix A with A _ij = −A _ji , forming s o ( 2 n − 2 ) ⊂ s o ( 2,2 n ) . Then, it is clear from the form of P ₊ that this rotation commutes with P ₊ and thus lies within h precisely if [A,Ω] _ij = A _ik Ω _kj −Ω _ik A _kj = 0. It then follows from the relation (2.4) that A must be similar to an element of s o ( 2 n − 2 ) ∩ s p ( 2 n − 2 ) ≅ u ( 1 ) ⊕ s u ( n − 1 ) . Hence, the set of matrices A form a (2n − 2)-dimensional representation of u ( 1 ) ⊕ s u ( n − 1 ) . In particular, we can take Ω itself to span the u ( 1 ) factor, while we write L ^α , α = 1, … , n ²−2n form the generators of s u ( n − 1 ) . We have [ L α , L β ] i j = f γ α β L i j γ for f γ α β the structure constants of s u ( n − 1 ) . Note, by construction, the matrices ( Ω L α ) i j = Ω i k L k j α are symmetric and traceless for each α.

Thus, for n ≥ 3 the total rotation subalgebra is u ( 1 ) ⊕ s u ( n − 1 ) ⊂ h , spanned by {B, J ^α }. For example, for the first non-trivial case n = 3, one can show from the relation (2.4) that all choices of the 4 × 4 matrices Ω _ij fall into two classes: those that are anti-self-dual, and those that are self-dual. These two cases correspond to det(M) = +1 and det(M) = −1, respectively. Then, for anti-self-dual (self-dual) Ω _ij , one can choose for their L i j α the self-dual (anti-self-dual) ’t Hooft matrices.

Finally, let us state the commutation relations for the algebra h . The commutators of the rotation subalgebra span{B, J ^α } both amongst themselves and with the other generators can be summarised as follows. As we have seen, {B, J ^α } form a basis for u ( 1 ) ⊕ s u ( n − 1 ) . The commutation relations descend from those of the L ^α , so that i B , J α = 0 i J α , J β = f γ α β J γ . (2.21) The remaining generators are sorted into “scalar” generators S = { P + , H , T , K } which are inert under rotations, i [ B , S ] = 0 , i [ J α , S ] = 0 , and otherwise “vector” generators V i = { P i , G i } which transform as i B , V i = − R Ω i j V j , i J α , V i = − L i j α V j , (2.22) All remaining commutators are found to be i G i , P j = − δ i j P + − 1 2 Ω i j T + 1 2 R n + 1 n − 1 δ i j B + β i j α J α , i T , H = − 2 H , i T , K = 2 K , i H , P i = 0 , i K , H = − T , i H , G i = P i , i G i , G j = − Ω i j K , i K , P i = − G i , i T , P i = − P i , i K , G i = 0 , i T , G i = G i , i P i , P j = − Ω i j H . (2.23) where the coefficient in front of B in the commutator i [G _i , P _j ] holds down to n = 2. Further, we denote by β i j α the constants such that 1 2 δ j k Ω i l + δ i k Ω j l − δ j l Ω i k − δ i l Ω j k − 2 n − 1 δ i j Ω k l = β i j α L k l α . (2.24) One can show that this equation can always be uniquely solved for the β i j α , for any choice of the basis L i j α for s u ( n − 1 ) , and further that β i j α = β j i α and β i i α = 0 .

Following the discussion in Section 2.1, we identify h = u ( 1 ) ⊕ s u ( 1 , n ) . This splitting is made explicit by adjusting the rotation B to twist along the x ⁺ interval. In detail, we define B ̃ ≔ B − 2 R n − 1 n + 1 P + . (2.25) Then, { H , P i , B ̃ , J α , T , G i , K } form a basis for s u ( 1 , n ) , while P ₊ generates the u ( 1 ) factor.

At the level of symmetries, the presence of conformal symmetry in the relativistic theory manifests itself as an enhancement of the Poincaré algebra to the conformal algebra. The analogous statement in non-relativistic theories is an enhancement of the inhomogeneous Galilean algebra—or rather, its central extension, the Bargmann algebra—to the Schrödinger algebra. Let us denote by Schr(d) the Schrödinger algebra governing the non-relativsitic conformal dynamics of a particle in d spatial dimensions.

Then, Schr(d) is realised precisely as the centraliser of a null translation within the conformal algebra s o ( 2 , d + 2 ) of R 1 , d + 1 . The single central element of Schr(d), often interpreted as particle number, is realised by this null translation.

Recall, we defined the subalgebra h = u ( 1 ) ⊕ s u ( 1 , n ) ⊂ s o ( 2,2 n ) as the centraliser of the generator P ₊. It is clear that in the limit that R → ∞, the coordinate transformation (2.2) and subsequent Weyl rescaling become trivial, and as such P ₊ degenerates to become simply a null translation. Indeed, this is also evident from the form of P ₊ in terms of the conventional conformal generators, as in (2.19), where we see that as R → ∞, we have P + → P ̂ + .

Hence, in the limit R → ∞, the subalgebra h must become some subalgebra of Schr ( 2 n − 2 ) ⊂ s o ( 2,2 n ) . Note that h needn’t give us the whole Schrödinger algebra, since there may be elements within s o ( 2,2 n ) that only commute with P ₊ strictly in the R → ∞ limit. Indeed, this is precisely what happens. It is evident that strictly in the R → ∞ limit, the spatial 2-form Ω _ij drops out entirely, and thus the breaking of the rotation subalgebra s o ( 2 n − 2 ) → u ( 1 ) ⊕ s u ( n − 1 ) does not occur. One can indeed show that in taking the limit R → ∞ and adding back in by hand the rotations broken by Ω _ij at finite R, we do indeed recover the Schrödinger algebra Schr (2n − 2).

Things therefore work smoothly at the level of the algebra. However, given a theory admitting the Ω-deformed non-relativistic conformal symmetry u ( 1 ) ⊕ s u ( 1 , n ) , there is an additional step we should take in order to recover the correct global Schrödinger group. In particular, a particle interpretation requires that the particle number N has discrete eigenvalues, corresponding in turn to a compactification of the null direction as x ⁺ ∼ x ⁺ + 2πR ₊ for some R ₊, which by a Lorentz boost is seen to be unphysical.

A convenient way to arrive at this setup—which from the 2n-dimensional perspective coincides with that of DLCQ—is to first introduce an orbifold. In particular at finite R the orbifold restricts to operators that are periodic but with period 2πR/K along the x ⁺ direction for some K ∈ N . Equivalently, we project onto the Hilbert subspace of states with P ₊ eigenvalue in K R Z . Now, taking K, R → ∞ while holding their ratio R ₊≔R/K fixed, we do indeed arrive at null-compactified Minkowski space but in such a way as to keep the Kaluza-Klein tower fixed. As required, we arrive at Schr (2n − 2), with particle number N identified by ⁴ P + = − k R N → − R + N , which does indeed have integer eigenvalues.

Indeed, this precise DLCQ limit of a u ( 1 ) ⊕ s u ( 1 , n ) theory has been performed explicitly in the case n = 3, both at the level of actions [13] as well as correlators [9].

A deep and powerful result tool in the study of relativistic conformal field theory is the operator-state map, relating on one hand conformal primary operators, and on the other, eigenstates of the Hamiltonian of the theory on a sphere. An analogous map exists in conventional non-relativistic conformal field theories [1], which relates primary operators—defined in a way entirely analogous to the above—to eigenstates of the Hamiltonian augmented by a harmonic potential.

We will now show that construction applies in an almost identical way to the u ( 1 ) ⊕ s u ( 1 , n ) theories of this work. Said another way, we verify that this operator-state map is not spoilt by the Ω-deformation that parameterises our departure form the Schrödinger algebra of conventional non-relativistic CFT. Indeed, one may recover from our construction the familiar map of Nishida-Son in the Schrödinger limit as outlined in Section 3.1.

We approach the construction of our operator-state map from the perspective of automorphisms of the symmetry algebra, a well-established point of view in relativistic CFTs which has also recently been formulated for non-relativistic CFTs governed by the Schrödinger group [4].

Given some operator Φ(0) at the origin, we may define a state | Φ 〉 = Φ 0 | 0 〉 . (3.3) Next, let us perform a Wick rotation in the symmetry algebra, defining D = −iT. Then, if Φ(0) has scaling dimension Δ under T, then D | Φ 〉 = D Φ 0 | 0 〉 = D , Φ 0 | 0 〉 = − i T , Φ 0 | 0 〉 = Δ Φ 0 | 0 〉 = Δ | Φ 〉 , (3.4) and thus |Φ⟩ has eigenvalue Δ under D. Then, just as with operators, we can use the ladder operators {H, K} and {P _i , G _i } to raise and lower the D eigenvalue of |Φ⟩. For instance, DH |Φ⟩ = (Δ + 2)H |Φ⟩, while DG _i |Φ⟩ = (Δ−1)G _i |Φ⟩.

We can consider | O 〉 specifically for a primary operator O . This state then sits at the bottom of a semi-infinite tower of operators, since K | O 〉 = 0 and G i | O 〉 = 0 .

Thus, we have on one hand primary operators and their descendants, all with definite scaling dimension, and on the other hand, eigenstates of the operator D = −iT. Let us now however explore an alternative frame, related by a similarity transform on the Hilbert space and space of operators. As we shall see, this transformation, which can be seen as a non-relativstic analogue of the operator-state map of relativistic CFT, relates the spectra of D with that of a combination of the form ∼ H + K . In many physical examples, one can thus study the spectrum of D by instead studying the dynamics of particles trapped in a confining potential provided by K [1, 2]. We now show that this operator-state map present in Schrödinger invariant theories generalises to the theories studied here.

So let us consider transformed states and operators given by | Φ ̄ 〉 = e − μ H e 1 2 μ K | Φ 〉 , Φ ̄ = e − μ H e 1 2 μ K Φ e − 1 2 μ K e μ H , (3.5) for some constant μ. Note, this transformation is clearly consistent with the identification (3.3). In particular, for a primary operator O we have | O ̄ 〉 = e − μ H e 1 2 μ K O 0 | 0 〉 = e − μ H O 0 | 0 〉 , (3.6) as is familiar from the usual non-relativistic operator-state map [1]. Then, this defines an alternative map between on one hand the primary operators O and their descendants, and on the other, towers of eigenstates of D ̄ generated by acting with the ladder pairs { H ̄ , K ̄ } and { P ̄ i , G ̄ i } .

Explicitly, the transformed operators under (3.5) are D ̄ = μ H + 1 μ K H ̄ = 1 4 μ μ H − 1 μ K + i T K ̄ = − μ μ H − 1 μ K − i T P ̄ i = 1 2 P i + 1 μ i G i G ̄ i = i μ P i − 1 μ i G i , (3.7) while the remaining generators, the rotations and central charge, transform trivially as B ̄ = B , J ̄ α = J α and P ̄ + = P + . A primary state | O ̄ 〉 then satisfies D ̄ | O ̄ 〉 = Δ | O ̄ 〉 P ̄ + | O ̄ 〉 = p + | O ̄ 〉 B ̄ | O ̄ 〉 = − r O B | O ̄ 〉 J ̄ α | O ̄ 〉 = − r O J α | O ̄ 〉 G ̄ i | O ̄ 〉 = 0 K ̄ | O ̄ 〉 = 0 , (3.8) while acting with H ̄ and P ̄ i generates towers of descendents.

Up to normalisation these operators (3.7) take the same form as in conventional non-relativistic CFT [1, 5], and thus automatically satisfy the same algebra in the R → ∞ limit.

If we assume unitarity in the original Minkowskian theory, then all states will have non-negative norm. Just as is the case of Lorentzian CFTs, we can use this assumption to place constraints on the eigenvalues of certain operators. The original Minkowskian symmetry generators were all Hermitian operators, but since we are interested in states quantised in the analogue of radial quantisation, we should instead consider the barred generators of (3.7). Simple Hermiticity of the original generators implies for the barred generators the following reality conditions D ̄ † = D ̄ H ̄ † = − 1 4 μ 2 K ̄ K ̄ † = − 4 μ 2 H ̄ P ̄ i † = 1 2 μ i G ̄ i G ̄ i † = 2 μ i P ̄ i . (3.9) So let us now consider the primary state | O ̄ 〉 with data { Δ , p + , r O [ B ] , r O [ J α ] } . Then, we have | H ̄ | O ̄ 〉 | 2 = 〈 O ̄ | H ̄ † H ̄ | O ̄ 〉 = − 1 4 μ 2 〈 O ̄ | K ̄ , H ̄ | O ̄ 〉 = 1 4 μ 2 〈 O ̄ | D ̄ | O ̄ 〉 = Δ 4 μ 2 ⟨ O ̄ | O ̄ ⟩ . (3.10) If the theory is unitary, all states have non-negative norm, implying that for primary states Δ ≥ 0 . (3.11) Since H ̄ and P ̄ i raise Δ, this bound clearly holds for all descendants as well. Let us similarly consider | P ̄ i | O ̄ 〉 | 2 = i 2 μ 〈 O ̄ | G ̄ i , P ̄ i | O ̄ 〉 = 1 2 μ 〈 O ̄ | − P ̄ + + 1 2 R n + 1 n − 1 B ̄ + β i i α J ̄ α | O ̄ 〉 , (3.12) where i is not summed over. But let us now sum over i, so as to exploit the tracelessness of β i j α . Then, we arrive at ∑ i | P ̄ i | O ̄ 〉 | 2 = n − 1 μ − p + − 1 2 R n + 1 n − 1 r O B ⟨ O ̄ | O ̄ ⟩ , (3.13) where recall that r O [ B ] ∈ 1 2 Z is the charge of O under a particular U (1) rotation subgroup, while p ₊ is its central charge, taking values in 1 R Z in theories descended from 2n-dimensional CFTs. Again imposing non-negative norms, we require M R ≔ − p + − 1 2 R n + 1 n − 1 r O B ≥ 0 . (3.14) In particular, if p + ∈ 1 R Z then M is rational. We can think of M as playing a role analogous to particle number in conventional NRCFTs.

Note, we see that a scalar primary must have p ₊ ≤ 0. It is interesting to note that this condition appears to be manifestly realised in known supersymmetric interacting gauge theory examples of s u ( 1,3 ) theories [9], in a rather novel way. In particular, in such theories, P ₊ is identified as instanton charge, and the dynamics are constrained such that only anti-instantons, corresponding to p ₊ ≤ 0, are allowed to propagate [10].

We can use the positivity of Δ and M to improve our bound for Δ. In particular, for any primary with M > 0, consider the norm [2]. H ̄ − R 2 M ∑ i P ̄ i P ̄ i | O ̄ 〉 2 ≥ 0 . (3.15) This leads to the inequality n − 1 Δ Δ − n − 1 ≥ 4 n − 1 M 2 + ∑ α r O J α 2 , (3.16) the right hand side is manifestly semi-positive, and we have already shown Δ is too, so one arrives at Δ ≥ n − 1 . (3.17) for any primary with M > 0. Since we have 2n − 2 spacial dimensions we see that, despite the Ω deformation, this bound agrees with the usual bound for theories with a Schrödinger symmetry algebra [2].

In six-dimensions the only choices for relativistic superconformal algebras are D (4, 1) and D (4, 2) corresponding to o s p ( 8 * | 2 ) and o s p ( 8 * | 4 ) respectively. In the following we cover the latter, the Bosonic part of which is s o ( 6,2 ) ⊕ s o ( 5 ) R , which we demonstrate is broken to u ( 1 ) ⊕ o s p ( 6 | 4 ) by the conformal compactification. It is straightforward to extend our discussion to o s p ( 8 * | 2 ) → u ( 1 ) ⊕ o s p ( 6 | 2 ) .

In Minkowski signature we choose conventions where all Bosonic generators are Hermitian, as before. Their commutation relations are the same as in Section 2.2. The R-symmetry generators have the standard form i R ̂ I J , R ̂ K L = δ J K R ̂ I L + δ I L R ̂ J K − δ I K R ̂ J L − δ J L R ̂ I K . (4.1) with I ∈ {1, … , 5}. The Fermionic generators are six-dimensional symplectic-Majorana-Weyl Fermions. The reality condition as applied above is Q ̂ α A = i Ω A B Γ 0 α β C β γ Q ̂ γ B † , (4.2) and similar for S ̂ , again this is in Minkowski signature, and the dagger here is not transposing spin indices (spinors see it as just complex conjugation). Ω ^AB and C ^αβ are the five and six-dimensional charge conjugation matrices, with A ∈ {1, … , 4} and α ∈ {1, … , 8}. ⁵ The Q ̂ and S ̂ have opposite chirality under Γ_* = Γ₀₁₂₃₄₅. Γ * Q ̂ = − Q ̂ , Γ * S ̂ = S ̂ . (4.3) Again we wish to find the maximal subalgebra of all elements that commute with the element P ₊, defined in terms of the six-dimensional (hatted) operators as P + = P ̂ + + 1 4 Ω i j M ̂ i j + 1 8 R 2 K ̂ − . (4.4) We find that 3/4 of the supercharges commute with P ₊. Precisely which set of supercharges this is depends on whether Ω _ij is self-dual or anti-self-dual; without loss of generality, let us choose the latter case. Then, letting a ± subscript denote chirality under Γ₀₅, the commuting supercharges are Q − A ≔ Γ − Q ̂ − A S + A ≔ Γ + S ̂ + A Θ − A ≔ 1 4 R Ω i j Γ i j Q ̂ A + 1 R Γ − S ̂ A . (4.5) The alternative case, where Ω _ij is self-dual, is found simply by swapping all Γ₀₅ chiralities. Then, their commutation relations with the bosonic generators are i Q − , H = 0 i S + , H = − Γ + Q − i Θ − , H = 0 i Q − , P i = 0 i S + , P i = R Ω i j Γ + j Θ − i Θ − , P i = 1 2 R Γ i Q − i Q − , B = 0 i S + , B = 0 i Θ − , B = 1 4 R Ω i j Γ i j Θ − i Q − , J α = 1 4 L i j α Γ i j Q − i S + , J α = 1 4 L i j α Γ i j S + i Θ − , L α = 0 i Q − , T = Q − i S + , T = − S + i Θ − , T = 0 i Q − , G i = − R Ω i j Γ j Θ − i S + , G i = 0 i Θ − , G i = 1 4 R Γ − i S + i Q − , K = − 1 2 Γ − S + i S + , K = 0 i Θ − , K = 0 i Q − A , R I J = 1 2 Γ ̃ I J A B Q − B i S + A , R I J = 1 2 Γ ̃ I J A B S + B i Θ − A , R I J = 1 2 Γ ̃ I J A B Θ − B , (4.6) while we have anti-commutators i Q − α A , Q − β B = − 4 Γ − Π − C − 1 α β Ω A B H i S + α A , S + β B = − 8 Γ + Π + C − 1 α β Ω A B K i Θ − α A , Θ − β B = − 2 Γ − Π + C − 1 α β Ω A B P + − B / R − 1 / 4 Ω i j Γ − Γ i j Π + C − 1 α β Γ ̃ I J Ω − 1 A B R I J i Q − α A , S + β B = − 2 Γ − Γ + Π + C − 1 α β Ω A B T + Γ ̃ I J Ω − 1 A B R I J − 1 / 2 Γ i j Γ − Γ + Π + C − 1 α β Ω A B L i j α J α i Q − α A , Θ − β B = − 2 R Ω i j Γ − Γ j Π + C − 1 α β Ω A B P i i S + α A , Θ − β B = − 2 R Ω i j Γ + Γ − Γ j Π + C − 1 α β Ω A B G i , (4.7) where we have defined the projectors Π_± = 1/2 (1 ±Γ_*).