HALF-BPS QUOTIENTS IN M-THEORY: ADE WITH A TWIST

arXiv:0909.0163v1 [hep-th] 1 Sep 2009

PAUL DE MEDEIROS, JOSÉ FIGUEROA-O’FARRILL, SUNIL GADHIA,
AND ELENA MÉNDEZ-ESCOBAR
Abstract. We classify Freund–Rubin backgrounds of eleven-dimensional supergravity of
the form AdS4 ×X 7 which are at least half BPS — equivalently, smooth quotients of theround 7-sphere by finite subgroups of SO(8) which admit an (N > 3)-dimensional subspaceof Killing spinors. The classification is given in terms of pairs consisting of an ADE subgroupof SU(2) and an automorphism defining its embedding in SO(8). In particular we find novel
half-BPS quotients associated with the subgroups of type Dn≥6 , E7 and E8 and their outerautomorphisms.

1. Introduction and contextualisation
Recent progress in the AdS/CFT correspondence for M2-branes [1] makes desirable a precise dictionary between the near-horizon geometries of supersymmetric M2-branes and theirdual three-dimensional superconformal field theories. The emerging picture deriving from thepioneering work of Bagger and Lambert [2, 3], Gustavsson [4] and Aharony, Bergman, Jafferis and Maldacena [5] is that the dual theories are superconformal Chern–Simons theories
coupled to matter. Superconformal field theories in three-dimensions are invariant under theorthosymplectic Lie superalgebra osp(N |4) [6], where 1 ≤ N ≤ 8. As reviewed for example in[7], which contains a more comprehensive bibliography, superconformal Chern–Simons theories
can be labelled by a pair consisting of a metric Lie algebra and a unitary representation, witha precise condition on the type of representations allowed/required for each value of N . For
N > 3 the representation theoretic data determining the superconformal Chern–Simons theory is very constrained and a classification is possible. This classification is very concrete for
N > 4 and not yet completely explicit in the case of N = 4; although there is a clear algorithm
how to construct all such theories. It is therefore for the N > 3 theories that we expect thedictionary to be easiest to establish. As a preliminary step in this programme, one needs aclassification of the possible near-horizon geometries at least for N > 3 and the purpose of thisnote is precisely to complete this classification.
As explained, for example, in [8, 9], the near-horizon geometry of a supersymmetric M2brane configuration admitting a superconformal field theory is of the form AdS4 ×X 7 , where(X, g) is a riemannian 7-dimensional manifold admitting real Killing spinors. This is the nearhorizon geometry of an M2-brane background where the transverse space to the membraneis the metric cone C(X) of X. As a riemannian manifold, C(X) = R+ × X with metricgC = dr2 + r2 g, where r > 0 is the coordinate on R+ . Bär’s cone construction [10] saysthat (X, g) admits real Killing spinors if and only if (C(X), gC ) admits parallel spinors. If
X is complete, then a theorem of Gallot’s [11] says that C(X) is either flat or irreducible.
The irreducible holonomy representations in eight dimensions follow from Berger’s list and
M. Wang [12] determined which of them admit parallel spinors. The well-known answer is thatan irreducible, simply-connected, riemannian eight-dimensional manifold admitting parallelspinors must have precisely one of the following holonomy groups: Spin(7), SU(4) or Sp(2),the last two corresponding to Calabi–Yau 4-folds and hyperkähler 8-manifolds, respectively.
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DE MEDEIROS, FIGUEROA-O’FARRILL, GADHIA, AND MÉNDEZ-ESCOBAR

Letting N denote the dimension of the space of parallel spinors (of, say, positive chirality): wefind that Spin(7)-holonomy manifolds have N = 1, whereas Calabi–Yau 4-folds have N = 2
and hyperkähler manifolds have N = 3. If the manifolds are not simply connected, then ofcourse N may be lower. It follows that for N > 3 the holonomy has to be reducible, but thenGallot’s theorem says that the cone must be flat, hence a quotient of R8 . As the notationsuggest, the integer N in this paragraph is the same as in the previous paragraph. It canalso be interpreted in terms of the fraction of supersymmetry preserved by the near-horizon
geometry of the M2-brane solution, which is N8 . The geometric construction of the osp(N |4)
algebra starting from the supergravity background was given in [13].
In terms of the 7-dimensional manifold X, the reduced holonomy of the cone implies the∂,existence of parallel differential forms which, when contracted with the Euler vector r ∂rgive rise to geometric objects on X satisfying a number of identities defining certain wellknown classes of geometries: manifolds of weak G2 holonomy, Sasaki-Einstein and 3-Sasaki 7manifolds, respectively, for the irreducible cones. If the cone is flat, then X is locally isometricto the round 7-sphere S 7 . We will assume that X is smooth. Therefore the manifolds X for
which N > 3 are quotients S 7 /Γ where Γ < SO(8) is a finite group acting freely on S 7 in sucha way that the quotient is spin and admits and (N > 3)-dimensional space of Killing spinors.
There is some previous literature on this problem, which is reviewed in [9, §3.1] (and references therein). The case of S 7 /Γ a lens space has also been studied in [14] and [15] and more
recently also in [16] emphasising the rôle of the spin structure. This corresponds to Γ a finite
cyclic group and all examples of N > 5 are of this form. In [9, §3.1] there is a brief discussionof N = 5 examples and also some N = 4 lens spaces. In this note we complete the case of
N = 4 and hence the N > 3 classification.
The organisation of this note is very simple. After a brief introduction to the concrete
mathematical problem in Section 2, we discuss the quotients S 7 /Γ with N > 3 in turn, startingwith N = 8 and ending with N = 4, this latter case taking up the bulk of the note. We end bysummarising the results in Table 6. In particular, there are examples consisting of twisted (ina sense to be made precise below) embeddings of the binary dihedral, binary octahedral andbinary icosahedral groups which appear to be new.
2. Sphere quotients with Killing spinors
We now describe the classification of smooth quotients S 7 /Γ of the unit sphere in R8 by
finite subgroups Γ < SO(8), for which the space of real Killing spinors has dimension N > 3.
Bär’s cone construction reduces the classification of these quotients to the classification offinite subgroups Γ < Spin(8) acting freely on the unit sphere in the vector representation
Λ1 and leaving invariant (pointwise) an N -dimensional subspace in either of the half-spinor
representations ∆± . The subgroups of SO(8) which act freely on the unit sphere have beenclassified by Wolf [17] and their lifts to Spin(8) have been classified in [18]. Every lift is a choiceof spin structure on the quotient and this choice will manifest itself very concretely as signs we
can attach to the generators of Γ consistently with the relations. This is to be expected, since
spin structures on S 7 /Γ are classified by
H 1 (S 7 /Γ, Z2 ) ∼= Hom(Γab , Z2 ) ,= Hom(Γ, Z2 ) ∼= Hom(π1 (S 7 /Γ), Z2 ) ∼ab(1)where the abelianisation Γ of Γ is the quotient of Γ by its commutator subgroup [Γ, Γ]consisting of elements of the form aba−1 b−1 for a, b ∈ Γ.
Let Γ < Spin(8) be a subgroup not containing −1. This condition guarantees that it maps
isomorphically under Spin(8) → SO(8) to some subgroup (also denoted) Γ < SO(8). Assume
that it acts freely on S 7 ⊂ Λ1 . The quotient S 7 /Γ, which is thus smooth and spin, is said tobe of type (N+ , N− ), where N± = dim ∆± Γ .

HALF-BPS QUOTIENTS IN M-THEORY

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There are two approaches one can take to attack this problem. As in [18], one can startfrom Wolf’s list of smooth S 7 quotients, analyse the possible lifts to Spin(8) and compute thedimension of the invariant subspace in the spinor representation. Alternatively, one can startfrom finite subgroups of Spin(8), not containing −1 and which leave invariant a subspace ofspinors of the requisite dimension and then check whether the action on S 7 is free. This latterapproach works well for N > 3, but perhaps not so well for smaller values of N .
Indeed, suppose that Γ < Spin(8), not containing −1, leaves invariant an N -dimensional
subspace V ⊂ ∆+ and let ∆+ = V ⊕ V ⊥ . Let SO(V ⊥ ) denote the subgroup of SO(∆+ ) whichacts trivially on V . Then Γ belongs to the preimage of SO(V ⊥ ) in Spin(8), which is isomorphic
to Spin(8 − N ). The group Spin(8) acts transitively on the grassmannian of N -planes in ∆+ ,whence any two SO(V ⊥ ) subgroups will lift to Spin(8 − N ) subgroups of Spin(8) which areconjugate, and quotients by conjugate subgroups are isometric. It is therefore enough to fix
one Spin(8 − N ) subgroup of Spin(8) once and for all. In summary, the problem of classifying
quotients of type (N, 0) is tantamount to classifying finite subgroups of a fixed Spin(8 − N )subgroup of Spin(8) whose action on the unit sphere of the vector representation Λ1 is free.
The reason this method works well for N > 3 is that we have good control over the finite
subgroups of Spin(2) ∼= U(1), Spin(3) ∼= Sp(1) and hence also of Spin(4) ∼= Sp(1) × Sp(1).
We now proceed to list the smooth quotients with N > 3. One first fixes an N -dimensional
subspace V ⊂ ∆+ and identifies the Lie subalgebra of so(8) which leaves V pointwise invariant.
This calculation is easily performed in the Clifford algebra Cℓ(8) relative to an explicit matrix
realisation, for which it is useful to use Mathematica. Exponentiating in Cℓ(8) determines aSpin(8−N ) subgroup whose action on the unit sphere S 7 ⊂ Λ1 in the vector representation canthen be easily investigated. Our Clifford algebra conventions are as follows: the generators ofCℓ(8) are γi , for i = 1, 2, . . . , 8 and satisfy γi2 = −1 in our conventions. We use the shorthandγij···k = γi γj · · · γk for i, j, . . . , k distinct. We will employ an explicit realisation based on theoctonions in which γi are real, skewsymmetric 16 × 16 matrices whose entries take only thevalues 0, ±1.
It will be convenient to introduce the following elements in Cℓ(8):i = 21 (γ12 + γ34 )i′ = 21 (γ56 + γ78 )j = 21 (γ13 − γ24 )j′ = 21 (γ57 − γ68 )′k = 21 (γ14 + γ23 )k(2)= 21 (γ58 + γ67 )Each of i, j, k squares to −P− and similarly with primes, where P− = 12 (1 − γ1234 ) andP−′ = 12 (1 − γ5678 ). These elements span an sp(1) ⊕ sp(1) Lie subalgebra of the so(8) Lie
algebra spanned by the γij , which the exponential maps surjectively onto an Sp(1) × Sp(1) Lie
subgroup of SO(8) where all the finite groups we shall consider lie.

3. N = 8
The (unquotiented) 7-sphere possesses a unique spin structure admitting the maximal number of Killing spinors of either sign of Killing’s constant, so it has type (8, 8), since dim ∆± = 8.
The real projective space RP7 is the quotient of the sphere by a Z2 subgroup of SO(8) which
sends x ∈ R8 to −x. This subgroup admits two inequivalent lifts to Spin(8): given by ±γ9 ,where γ9 is the chirality operator in Cℓ(8). This means that RP7 has two inequivalent spin
structures: one of type (8, 0) and the other of type (0, 8). Any other quotient will have type
(N, 0) or (0, N ) for some N < 8. Up to change of orientation, we may consider only thosequotients of type (N, 0).

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4. N = 7
There is no quotient with N = 7. This is proved by an argument employed in [19] in a
related context. Suppose Γ < Spin(8) leaves invariant (pointwise) a 7-dimensional subspace of∆+ , then so will some element 1 6= g ∈ Γ. Now g generates a cyclic subgroup of order > 1,
lying inside some circle group inside a maximal torus. The only such freely acting circle groups
have as orbits the fibres of the canonical fibration S 7 → CP3 . That group preserves a complex
structure on ∆+ , whence the invariant subspace is a complex subspace and hence always ofeven dimension.
5. N = 6Groups Γ leaving invariant a six-dimensional subspace of ∆+ are contained in a circle subgroup of Spin(8) whose Lie algebra is conjugate to the u(1) generated by γ12 + γ34 + γ56 + γ78 ,where we view the spin group and its Lie algebra as sitting inside the Clifford algebra Cℓ(8).
The commutant of u(1) is u(1) ⊕ su(4), where su(4) ∼= so(6) is the Lie algebra of the Spin(6)subgroup of Spin(8) associated to the six-dimensional vector space of invariant spinors. The
finite subgroups of a circle group are cyclic and it is easy to check that they act freely on S 7 :they rotate all elementary 2-planes by the same angle. Alternatively, thinking of Λ1 as C4 ,they act by multiplying each component by the same phase, whence only the origin is fixed,
but the origin does not lie on the unit sphere.
6. N = 5Groups Γ leaving invariant (pointwise) a five-dimensional subspace of ∆+ are contained ina Spin(3) subgroup of Spin(8) whose action on the vector representation is obtained as follows.
It will prove convenient to employ the isomorphism Spin(3) ∼= Sp(1) with the unit quaternions.
Consider the action of Sp(1) on H by left quaternion multiplication: if u ∈ Sp(1) and q ∈ H,
then u · q = uq. Thinking of H as a four-dimensional real vector space with basis 1, i, j, k, we
get an embedding Sp(1) ֒→ SO(4) given explicitly by
ζ1 −ζ2 −ζ3 −ζ4 ζ2
ζ1 −ζ4ζ3  ,ζ1 + ζ2 i + ζ3 j + ζ4 k 7→ 
(3)
 ζ3
ζ4
ζ1 −ζ2 ζ4 −ζ3
ζ2
ζ1and we can them embed SO(4) diagonally into SO(4) × SO(4) ⊂ SO(8). In other words,
thinking of R8 as H2 , if u ∈ Sp(1) is a unit quaternion, then its action on (q1 , q2 ) ∈ H2 issimply via left multiplication: u · (q1 , q2 ) = (uq1 , uq2 ). It is clear that this action is free on thesphere, as the only point (q1 , q2 ) which has a nontrivial stabilizer is the origin, which does notlie on the unit sphere.
The finite subgroups of Sp(1) are classified by the ADE Dynkin diagrams and tabulated inTable 1. The cyclic groups corresponding to An≥2 actually leave invariant a six-dimensional
subspace and are precisely the groups considered above in the section on N = 6. The cyclic
group of order 2 corresponding to A1 actually acts trivially on ∆+ and is the group whose
quotient is the real projective space discussed in the section on N = 8.
7. N = 4Groups Γ leaving invariant (pointwise) a four-dimensional subspace of ∆+ are containedin a Spin(4) ∼= Sp(1) × Sp(1) subgroup of Spin(8) whose action on the vector representation
is defined in a very similar way to the case of N = 5, except that we have two copies ofSp(1), each embedding in SO(4) as in (3) and then embedding SO(4) × SO(4) into SO(8) inHALF-BPS QUOTIENTS IN M-THEORY

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Table 1. Finite subgroups of Sp(1)Dynkin
An
Dn≥4
E6
E7
E8

Name
Zn+1
2D2(n−2)
2T
2O
2I

Order
n+1
4(n − 2)24
48
120

Presentationt tn+1 = 1
s, t s2 = tn−2 = (st)2s, t s3 = t3 = (st)2s, t s3 = t4 = (st)2s, t s3 = t5 = (st)2

the obvious way. In terms of quaternions, (u1 , u2 ) ∈ Sp(1) × Sp(1) acts on (q1 , q2 ) ∈ H2 byleft multiplication: namely, (u1 , u2 ) · (q1 , q2 ) = (u1 q1 , u2 q2 ). Now consider a finite subgroupΓ < Sp(1) × Sp(1). It will consist of pairs (u1 , u2 ) of unit quaternions acting via simultaneousleft multiplication on pairs of quaternions. It is clear that for this action to be free on thesphere, it is necessary and sufficient that there be no elements in Γ of the form (1, u) or (u, 1),
other than the identity (1, 1). To see what this means, let’s digress to review Goursat’s theoryof subgroups of the direct product of two groups [20, §4.3].
Let A, B be groups and let C < A× B be a subgroup. Let L = π1 (C) < A and R = π2 (C) <B be the images of C under the cartesian projections π1 : A × B → A and π2 : A × B → B,
respectively. Alternatively, L < A consists of those a ∈ A such that there exists b ∈ B with(a, b) ∈ C, and similarly R < B consists of those b ∈ B such that there exists a ∈ A with(a, b) ∈ C. Let L0 < L consist of those a ∈ L such that (a, 1) ∈ C, and let R0 < R consist of
those b ∈ R such that (1, b) ∈ C. It is easy to see that L0 ⊳ L and R0 ⊳ R are normal subgroups
and moreover that the embedding of C in L × R is precisely the graph of an isomorphism
L/L0 ∼= R/R0 . Indeed, given a ∈ L one sends it to some b ∈ R such that (a, b) ∈ C. Thereis no unique such b, but any two are in the same R0 coset, whence this process defines a
map L → R/R0 , which is surjective by the definition of R and can be shown to be a grouphomomorphism. The kernel of this map is clearly L0 , which proves the desired isomorphism.
This result is known as Goursat’s lemma.
Now back to our geometric problem, we have a finite subgroup Γ of Sp(1) × Sp(1) withprojections L, R which are finite subgroups of Sp(1). We argued that for the action on thesphere to be free the subgroups L0 and R0 have to be trivial, whence Goursat’s lemma saysthat L ∼= R and hence that Γ embeds in L × L as the graph of an automorphism τ of L. Of
course, taking τ to be the identity, we recover precisely the groups discussed in the section
on N = 5. Similarly, if τ is an inner automorphism, hence induced by conjugation, then thisgives conjugate subgroups in SO(8) and hence the quotient is isometric to that when τ is theidentity, leading us back to the case of N = 5. Hence in order to obtain a true N = 4 quotient
we need only consider outer automorphisms1 of the finite subgroups of Sp(1). It is however
not guaranteed that outer automorphisms will give rise to inequivalent quotients. The relevant
condition is whether or not the effect of the automorphism can be undone via conjugation
by SO(8) or by Spin(8) in the spinor representation. This seems to boil down to whether the
automorphism in question can be realised by conjugation in SO(4). In the quaternionic picture,
an element u ∈ Sp(1) is embedded into SO(H) ∼= SO(4) as the linear transformation q 7→ uq,
for q ∈ H. The element [(uL , uR )] ∈ SO(H) acts via q 7→ uL qūR , for (uL , uR ) ∈ Sp(1) × Sp(1),so it acts on linear transformations by conjugation: the endomorphism (q 7→ uq) in SO(H)transforms into q 7→ uL (u(ūL quR ))ūR = uL uūL q, whence u 7→ uL uūL gets conjugated inSp(1). The question is thus whether the normaliser N (Γ) of Γ in Sp(1) acts on Γ in such a
1We must own up to a slight abuse of notation, by which automorphisms which are not inner will be referred

to as outer, even though strictly speaking outer automorphisms are equivalence classes of automorphisms modulo
inner automorphisms.

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DE MEDEIROS, FIGUEROA-O’FARRILL, GADHIA, AND MÉNDEZ-ESCOBAR

way that it surjects on the full automorphism group. If it does then all automorphisms are
realised as conjugation in SO(8) and hence the quotients are isometric. For the case of 2T theouter automorphisms are in fact induced by conjugating with the normaliser, but this is not
the case for 2O nor 2I.
In summary, the possible N = 4 quotients are of the following form. Let Γ < Sp(1) bea finite group and let ι : Γ → SO(4) denote the restriction to Γ of the embedding definedby equation (3). If α ∈ Aut(Γ) is an automorphism, we will let ια : Γ → SO(4) denote theembedding defined by ια (u) = ι(α(u)). Putting the two together, we have an embeddingΓ ֒→ SO(8) defined by the sequence of embeddings
ι×ιαΓ −−−−→ SO(4) × SO(4) −−−−→ SO(8),(4)

the second one being the obvious one. In other words, if u ∈ Γ, then its action on (q1 , q2 ) ∈ H2 ∼=R8 is u · (q1 , q2 ) = (uq1 , α(u)q2 ). If α is the identity we will call this the diagonal embeddingof Γ in SO(8), otherwise we will say the embedding is twisted (by α). As mentioned above if
α is an inner automorphism, then the twisted embedding is conjugate in SO(8) to the diagonalembedding and hence give rise to isometric quotients. Hence to classify the possible N = 4
quotients we will first determine the possible outer automorphisms of Γ, then we will determine
the possible lifts to Spin(8) of the subgroup of SO(8) obtained by embedding Γ as in equation(4) and for each such outer automorphism and each such lift we will calculate the dimensionof the space of invariant spinors.
This latter task will be accomplished using character formulae. To compute N± = dim ∆± Γ ,the dimensions of the Γ-invariant subspaces of chiral spinors, we use the projection formulaX
1(5)χ∆± (u) ,N± = |Γ|u∈Γ

where χ∆± (u) is the trace of the group element u in the representation ∆± , respectively. We
can work directly in the Clifford algebra Cℓ(8) ∼= R(16) and by going to an explicit matrix
realisation for Cℓ(8) where the γi are real 16 × 16 matrices, then the group elements u will also
be 16 × 16 matrices. Then their characters are given simply byχ∆± (u) = Tr uΠ± ,(6)where Π± = 12 (1 ± γ9 ) are the projectors corresponding to the chiral spinor representations.
Since characters are class functions, the calculation is further simplified by summing notover the group but over conjugacy classes:
X1(7)|[u]| × χ∆± (u) ,N± = |Γ|conjugacy classes [u]where |[u]| is the size of the conjugacy class of u.
We will perform this analysis type by type, starting with the cyclic subgroups.
7.1. An−1 . For the group of type An−1 , with n ≥ 2, which is cyclic of order n, all automorphisms are outer, since the group is abelian. An automorphism is determined uniquely bywhat it does to a generator, and being an automorphism it has to send it to another generator.
Hence the group of automorphisms is the group Z×n of multiplicative units in Zn ; that is, those
integers in the range {1, . . . , n − 1} which are coprime to n. The order of Z×n is the value φ(n)of Euler’s totient function.
r
i2π/n
Let r ∈ Z×of Zn
n act on Zn by sending a generator t to t . Then the generator t = e
lifts to Spin(8) ⊂ Cℓ(8) as
2πt̂ = σ exp(i + ri′ ) ,(8)n

HALF-BPS QUOTIENTS IN M-THEORY

7

in terms of the elements of Cℓ(8) introduced in equation (2), and where σ is a sign satisfying
σ n = 1, whence if n is odd then σ = 1. This is precisely what one expects from the isomorphismsin (1). Here Zn is abelian, whence the spin structures in any (smooth, at least) quotient are
in one-to-one correspondence with(∼ Z2 , if n is even;
Hom(Zn , Z2 ) =(9){1}, if n is odd.
Hence there are two spin structures if n is even (labelled by σ above) and only one if n is odd.
Using an explicit matrix realisation for Cℓ(8) we can compute the characters of the spinor
representations:
2p(r + 1)π2p(r − 1)π
p
p
p
(10)
+ cos
χ∆+ (t ) = Tr∆+ (t̂ ) = 2σ 2 + cos
n
n
and


2pπ
2prπ
.
+ cos
χ∆− (tp ) = Tr∆− (t̂p ) = 4σ p cos
n
n

(11)

Let us first compute N− . The projection formula (5) says that
N− =



n−1
1X p
2pπ
2prπ
+ cos
4σ cos
n p=0
n
n
n−1

=


−i2π
−i2rπ
−i2rπ
i2π
2 X
(σe n )p + (σe n )p + (σe n )p + (σe n )p .
n p=0Now, each of the sums is geometric:
n−1
X
p=0

zp =

( n

z −1
z−1

n

z 6= 1z=1,(12)

and moreover the expression for z is such that z n = 1. Hence the sum gives either 0 if z 6= 1i2πxor else n. We need to identify the cases with z = 1. If σ = 1, this happens when e n = 1, forx = ±1, ±r, which requires x = 0 (mod n). Clearly this can never happen, whence if σ = 1,
we find that N− = 0. If σ = −1, which requires n even and hence r odd, z = 1 happens wheni2πxe n = −1, which requires 2x/n to be an odd integer. Clearly if n > 2 this cannot happen,
since r is coprime to n, but if n = 2, then this happens in all four cases x = ±1, ±r. In thiscase we have N− = 8, which corresponds to the other spin structure on RP7 .
To compute N+ we again use the projection formula (5) to obtain
n−1
2p(r − 1)π2p(r + 1)π
1X p
+ cos
2σ 2 + cos
N+ =
n p=0
n
n
n−1

=


−i2(r−1)πi2(r+1)π−i2(r+1)πi2(r−1)π
1 X pnn)p + (σe n )p + (σe)p
4σ + (σe n )p + (σen p=0The sums are again geometric, whence the need to identify the cases when the summand is 1.
Looking first at the first summand, we see that if σ = 1 this contributes 4 to N+ , whereas if
σ = −1, which implies that n is even, the first sum cancels. We may summarise this by saying

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DE MEDEIROS, FIGUEROA-O’FARRILL, GADHIA, AND MÉNDEZ-ESCOBAR

the first sum gives a contribution of 2(1 + σ). The rest of the sums depend on what r is. Forr generic, the exponential sums vanish, e.g.,
n−1
X

(σe

p=0

i2(r±1)πn)p =σ n ei2(r±1)π − 1ei2(r±1)π
n

−1

=0,

(13)

since σ n = 1. Hence the generic such quotient has N+ = 4 for the positive spin structure and
N+ = 0 for the negative spin structure. For some values of r, however, the exponentials sum
might contribute.
Let us first of all consider the case of σ = 1. We have to distinguish the cases n > 2 andn = 2. If n > 2 then if r = ±1 (mod n), we have that precisely two of the exponential sums
contribute an extra 2 to N+ , bringing it up to N+ = 6. Notice that for n = 3, 4, 6 there areno other possible values of r than these. If n = 2, then r is odd, since it is coprime to n, andhence r ± 1 is even, in which case all four exponential sums contribute equally yielding N+ = 8.
Of course, r = 1 is the diagonal embedding and r = −1 corresponds to a twisted embedding
which is conjugate (by j, say) in Sp(1) to the diagonal one, whence the quotients are isometric.
Finally let us consider the case of σ = −1, which requires n to be even. Then the exponential
sums will contribute provided that r ± 1 = n2 (mod n). Since r has to be coprime to n, it has
to be odd, which then forces n to be a multiple of 4. In this case precisely two of the sums
contribute and we find N+ = 2.
In summary, to obtain an N+ = 4 quotient we need to take σ = 1 and r 6= ±1 (mod n).
7.2. Dn+2 . This is the binary dihedral group 2D2n , for n ≥ 2, defined abstractly in terms ofgenerators and relations as
2D2n = s, t s2 = tn = (st)2

,

(14)

or explicitly in terms of quaternions by s = j and t = eiπ/n , with central element s2 = tn =(st)2 = −1. The group has order 4n. There are n + 3 conjugacy classes which are listed alongwith their sizes and the orders of their elements in Table 2, which also displays the charactersof the spinorial representations. The notation ℓ(p, 2n) in that table means the least common
multiple of p and 2n.
Automorphisms are uniquely determined by their action on generators, provided that theirimages still satisfy the relations. In addition, automorphisms must preserve the order of theelements and must also map conjugacy classes to conjugacy classes. By definition, inner automorphisms preserve the conjugacy classes, so if an automorphism does not then it must beouter. These considerations nail down the outer automorphisms once the conjugacy classes
are enumerated, as we have done. First we can consider the automorphisms which fix s andtransform t 7→ tr where (r, 2n) = 1. Notice that tr still obeys (tr )n = −1, since r is odd, andthat (str )2 = −1, whence s and str are again generators. Since the conjugacy class {t, t−1 }gets mapped to {tr , t−r }, this automorphism is outer for r 6= 1, 2n−1. Also we can consider the
automorphism which fixes t and sends s 7→ st. We could consider a more general automorphism
sending s 7→ stq but up to inner automorphisms (which fix t) only q = 1 need be considered.
This is an automorphism because (st)2 = (st2 )2 = −1, so that the relations are satisfied. It isclearly outer, since s and st belong to different conjugacy classes. These two automorphisms
commute and clearly they give rise to a group of outer automorphisms isomorphic to Z×
2n × Z2 .
We observe that the element jeiπ/2n ∈ Sp(1) conjugates st back to s at the price of sending t
to t−1 , which is an equivalent generator. Hence we can always ignore the Z2 factor from thepoint of view of obtaining quotients with N = 4. In other words, from now on we will consider
the outer automorphisms which leave s fixed and send t 7→ tr for r ∈ Z×2n \ {±1}.

HALF-BPS QUOTIENTS IN M-THEORY

9

Depending on the parity of n, we have either 2 or 4 possible spinorial lifts of 2D2n . This
follows from the isomorphisms in (1) once we understand the abelianisation of 2D2n . It is
not hard to see that the commutator subgroup is the cyclic subgroup generated by t2 , which
has order n. This means that the abelianisation has order 4 and hence can either be cyclic or
isomorphic to Klein’s Viergruppe, a.k.a. Z2 × Z2 . To determine which, we argue as follows. If n
is even, then −1 is in the commutator subgroup, so modulo the commutator subgroup the four
elements 1, s, t and st surject onto the abelianisation: since both s and t have order 2 modulo
the commutator subgroup, this four-element group is isomorphic to Z2 × Z2 . On the other
hand, if n is odd, then −1 is not in the commutator subgroup, but modulo the commutator
subgroup t = −1. Hence the four elements which surject to the abelianisation are ±1 and ±s,
which is a cyclic group of order 4 with generator s. Finally, using (1) and the isomorphisms
Hom(Z4 , Z2 ) ∼
= Z2

and

Hom(Z2 × Z2 ) ∼= Z2 × Z2 ,(15)

we expect to see two signs labelling the lifts of 2D2n to Spin(8) when n is even and only one
sign when n is odd.
Indeed, let us first consider the lift to Cℓ(4). In the notation of equation (2), the following
generate a group of Cℓ(4) isomorphic to 2D2n :ππandt̂ = τ exp i ,(16)ŝ = σ exp j2nwhere σ and τ are signs, but with τ n = 1. The lifts to Spin(8) corresponding to the embedding
twisted by r ∈ Z×
2n is therefore given byππŝ = σ exp (j + j′ )andt̂ = τ exp (i + ri′ ) ,(17)2nwith σ, τ signs again with τ n = 1. It is now straightforward to compute the spinorial charactersby using an explicit realisation of Cℓ(8). The results are given in Table 2, in which p =1, ..., n − 1.
Table 2. Conjugacy classes and spinorial characters of 2D2n
Class
1
−1
s
st
tp

Size
1
1
n
n
2

Order
1
2
4
4
ℓ(2n, p)/p

χ∆+
8
8
4σ
4στ
+ cos p(r+1)π)2τ p (2 + cos p(r−1)π
n
n

χ∆−
8
−8
0
0
prπ
4τ p (cos pπ
n + cos n )Using the projection formula (7), we determine N± . First of all, as a check of our calculations, we should obtain that N− = 0. This is very similar in spirit to the computation of N−
in the cyclic case. Indeed, one has
!
n−1

X
pπ
prπ 
1
p
4τ cos8−8+0+0+2
+ cos
N− =
4n
n
n
p=1
n−1


1 X  iπ/n p(τ e) + (τ e−iπ/n )p + (τ eirπ/n )p + (τ e−irπ/n )p
n p=1


1 τ n + eiπ/n ττ n + e−iπ/n τ−eirπ τ n + eirπ/n τ−e−irπ τ n + e−irπ/n τ
=
+
+
+
.
n 1 − τ eiπ/n
1 − τ e−iπ/n1 − τ eirπ/n
1 − τ e−irπ/n=10DE MEDEIROS, FIGUEROA-O’FARRILL, GADHIA, AND MÉNDEZ-ESCOBAR

Using that r is odd and that τ n = 1, we can rewrite this expression as
1 + e−iπ/n τ
1 + eirπ/n τ
1 + e−irπ/n τ
1 1 + eiπ/n τ
+
+
+
N− =
n 1 − τ eiπ/n
1 − τ e−iπ/n1 − τ eirπ/n
1 − τ e−irπ/n
1 1 + eiπ/n τ
eiπ/n τ + 1 1 + eirπ/n τ
eirπ/n τ + 1
=
= 0,
+
+
+
n 1 − τ eiπ/nτ eiπ/n − 1 1 − τ eirπ/nτ eirπ/n − 1

where in the last line we multiplied the second term top and bottom by τ eiπ/n and the fourthterm by τ eirπ/n and used that τ 2 = 1.
Moving on to the computation of N+ , the projection formula says that

!
n−1
X
p(r − 1)πp(r + 1)π1p4τ 2 + cos
8 + 8 + 4nσ + 4nστ +
+ cos
N+ =
4n
n
n
p=1


n−1
1X p
p(r − 1)πp(r + 1)π= σ(1 + τ ) ++ cos
τ 2 + cos
n p=0
n
n
n−1

= (1 + σ)(1 + τ ) +i(r−1)πi(r+1)πi(r+1)πi(r−1)π1 X
(τ e n )p + (τ e− n )p + (τ e n )p + (τ e− n )p .
2n p=0For generic values of r, the exponential sums vanish:
n−1
X

(τ e±

p=0

i(r±1)πn)p =τ n e±i(r±1)π − 1
τ e±i(r±1)π
n

−1

=0,

(18)

since τ n = 1 and r is odd since it is coprime to 2n. In these cases we have N+ = (1 + σ)(1 + τ ),whence N+ = 4 if σ = τ = 1 and N+ = 0 otherwise. For some values of r, for whichi(r±1)π= 1, then two of the corresponding exponential sums will contribute a total of 1 to
τe n
N+ .
Let us first consider τ = 1. Then we have extra contributions wheneveri(r ± 1)π
= 1,
(19)
exp
n

whence if r = ±1 (mod 2n), then we have N+ = 5 for σ = 1 and N+ = 1 for σ = −1. It should
be pointed out that for n = 2 and n = 3 there are no r ∈ Z×
2n which do not obey r = ±1
(mod 2n), whence there are no N+ = 4 quotients for these values of n. Finally let us consider
the case of τ = −1, which forces n to be even. Then we get extra contributions wheneveri(r ± 1)πexp
= −1 ,(20)n

which implies that r ± 1 is an odd multiple of n, hence r = n ± 1. In this case we have that
N+ = 1 regardless the value of σ.
In summary, we get N+ = 4 for σ = τ = 1 and r 6= ±1 (mod 2n), which requires n ≥ 4. We
get N+ = 5 for σ = τ = 1 and r = ±1 (mod 2n).

7.3. E6 . The group of type E6 is the binary tetrahedral group 2T of order 24. Abstractly it isisomorphic to SL(2, F3 ). It has a centre of order 2 and quotienting by it gives the tetrahedralgroup T ∼= PSL(2, F3 ) ∼= A4 . This last isomorphism can be understood because PSL(2, F3 )acts faithfully on the projective space P1 (F3 ) of lines through the origin in F3 , consisting offour points, so it embeds in S4 as a subgroup of index 2 and there is only one such subgroup.
The full automorphism group of 2T is the symmetric group S4 and a representative outerautomorphism can be obtained by conjugation in GL(2, F3 ) with an element not in SL(2, F3 ).

HALF-BPS QUOTIENTS IN M-THEORY

11

To relate this to the above description, notice that the group GL(2, F3 ) acts faithfully on P1 (F3 )whence it embeds in S4 , but given its order it has to be all of S4 .
We can understand this outer automorphism also in terms of the usual presentation
2T = s, t s3 = t3 = (st)2

,

(21)

where
and
t = 12 (1 + i + j − k).
(22)s = 21 (1 + i + j + k)
The group 2T has seven conjugacy classes, tabulated below in Table 3, along with its size andthe order of any (and hence all) of its elements.
Table 3. Conjugacy classes of 2T
Class
Size
Order

1
1
1

−1
1
2

s
4
6

t
4
6

t2
4
3

s2
4
3

st
6
4

We can see that the only nontrivial outer automorphism is one which sends s to the conjugacy
class of t and viceversa. In fact, since st and ts are conjugate, it is possible to represent the
nontrivial outer automorphisms by the automorphism which exchanges the two generators.
This automorphism consists of conjugation by √12 (i + j) ∈ Sp(1). This means that the twisted
embedding of 2T in SO(8) is conjugate in SO(8) to the diagonal embedding and hence thequotients are isometric. In particular the twisted embedding gives N = 5, which can also bechecked explicitly; although we will refrain from doing so here. Let us nevertheless remarkthat there is a unique spin structure in the quotient, because the commutator subgroup of 2T
has index 3, whence the abelianisation is isomorphic to Z3 , but Hom(Z3 , Z2 ) = {1}.
7.4. E7 . The group of type E7 is the binary octahedral group 2O of order 48. The automorphism group is isomorphic to O × Z2 , where O is the octahedral group, which is the quotient of
2O by its centre, whence it is clear that the factor O corresponds to the inner automorphisms.
The group admits the following presentation
2O = s, t s3 = t4 = (st)2

,

(23)

and we can make this explicit in terms of quaternions by choosing1
s = 21 (1 + i + j + k)(24)
andt = √ (1 + i) .
2
There are 8 conjugacy classes tabulated in Table 4 below, which also contains the computationof the characters, described in more detail below.
Table 4. Conjugacy classes and spinorial characters of 2O
Class
1
−1
s
t
s2
t2
t3
st

Size
1
1
8
6
8
6
6
12

Order
1
2
6
8
3
4
8
4

χ∆+
8
8
5
2σ(2 + τ )5
4
2σ(2 + τ )
4σ

χ∆−
8
−8
√ 4
2 2σ(1 + τ )−4√ 0−2 2σ(1 + τ )012DE MEDEIROS, FIGUEROA-O’FARRILL, GADHIA, AND MÉNDEZ-ESCOBAR

Since automorphisms preserve orders, we see that any automorphism must send s to its
conjugacy class, hence modulo inner automorphisms we can let s remain fixed. The othergenerator t has to be sent to the other conjugacy class of the same size and order to that oft. We see from the table that it must be to the conjugacy class of t3 . It cannot be sent to
t3 , however, because st3 has order 8 and not 4. A little playing around suggest defining theautomorphism by sending t 7→ t5 = −t and fixing s. One checks that the relations are satisfied.
It is not difficult to show that this automorphism cannot be obtained by conjugation in Sp(1).
We now investigate the lift of the generators to Spin(4) ⊂ Cℓ(4) as a preliminary stage
to lifting them via the twisted embedding to Spin(8). First of all, from equation (1) we canalready see that there are two inequivalent spin structures on any quotient. This is because
the commutator subgroup of 2O has index 2, whence the abelianisation is isomorphic to Z2
and Hom(Z2 , Z2 ) ∼= Z2 . So we expect the lift to depend on a sign.
Indeed, in the notation of (2), we find the following lifts
π
π 1
(25)
and
t̂τ = σ exp (2τ − 1)i ,ŝ = exp 3− 2 (i + j + k)34
where σ and τ are signs: σ labels the two inequivalent lifts, whereas τ distinguishes between
the lift t̂+ of t and that t̂− of −t. The diagonal and the twisted embeddings give 2O subgroupsof SO(8) acting freely on the unit sphere. For each of these two embeddings we have two
possible lifts to Spin(8), corresponding to the two spin structures of the quotient. The lifts are
explicitly given by
π 1
πŝ = exp 3− 2 (i + j + k + i′ + j′ + k′ )andt̂ = σ exp (i + (2τ − 1)i′ ) ,(26)
34
where τ = 1 is the diagonal embedding and τ = −1 the twisted embedding and σ labels thetwo spin structures.
It is now a simple matter to use the trace formulae (6) in an explicit representation of the
Clifford algebra in order to compute the characters of 2O in the spinor representations forarbitrary σ and τ . The result of this calculation is found in Table 4. The dimension of theinvariant subspaces is easily computed from the tabulated data using the projection formula(7) and one finds that whereas N− = 0 for both embeddings, we have
N+ = 21 (5 + σ(4 + τ )).

(27)

Therefore for the diagonal embedding (τ = 1), we have N+ = 5 for the positive spin structure(σ = 1) and N+ = 0 for the negative spin structure, providing a check of our calculations.
More interestingly, for the twisted embedding (τ = −1) we find N+ = 4 for the positive spin
structure and N+ = 1 for the negative one.
7.5. E8 . Finally, the group of type E8 is the binary icosahedral group 2I of order 120, which isisomorphic to SL(2, F5 ). The automorphism group is the symmetric group S5 . This is similarin spirit to the case of E6 , in that the representative outer automorphism can be obtained via
conjugation in GL(2, F5 ). A representative for the outer automorphism using this method isgiven in [18, §2.2.5]. We will find another representative here by exploiting the structure of
the group.
The usual presentation is
2I = s, t s3 = t5 = (st)2 ,(28)
and we may take s and t to be the quaternionss = 12 (1 + i + j + k)andt = 21 (ϕ + ϕ−1 i + j) ,
(29)
√
√
1
1
−1
where ϕ = 2 (1 + 5) and ϕ = 2 (−1 + 5) are the Golden Ratio and its reciprocal, respectively. The group has 9 conjugacy classes which are tabulated in Table 5 along with their sizeand order and the spinorial characters in both the diagonal and twisted embeddings.

HALF-BPS QUOTIENTS IN M-THEORY

13

Table 5. Conjugacy classes and spinorial characters of 2I

Class
1
−1
t
t2
t3
t4
s
s4
st

Size
1
1
12
12
12
12
20
20
30

Order
1
2
10
5
10
5
6
3
4

Diagonal
χ∆+
χ∆−
8
8
8
−8
5+ϕ
4ϕ
5 − ϕ−1 4ϕ−1
5 − ϕ−1 −4ϕ−1
5+ϕ
−4ϕ
5
4
5
−4
4
0

Twistedχ∆+ χ∆−
8
8
8
−8
3
2
3
−2
3
2
3
−2
5
4
5
−4
4
0

We see from the description of the conjugacy classes that the generator s belongs to theunique class of size 20 and order 6, whence any automorphism must send it to another element
in its class, which means that we can undo the effect of that automorphism using an innerautomorphism. Hence modulo inner automorphisms, we are free to leave s alone. The generator
t belongs to one of two conjugacy classes of size 12 and order 10. An outer automorphism must
send it to the class of t3 , but alas we cannot send it to t3 itself because st3 is not conjugate
to st. A little experimentation suggests that we send t 7→ t′ := t6 st4 st3 . One checks that thisis an outer automorphism and moreover one checks that it cannot obtained by conjugation inSp(1).
Lifting the elements s, t, t′ to Cℓ(4) one finds a unique lift, whence both the diagonal andtwisted quotients have a unique spin structure. Of course, this is as expected because 2I isperfect, so that the commutator subgroup is all of 2I and hence the abelianisation is trivial.
A little calculation shows that the pairs of elements {ŝ, t̂} and {ŝ, t̂′ } in Spin(4) ⊂ Cℓ(4),defined below in the notation of equation (2), generate separately a group isomorphic to 2I
and surjecting to 2I ⊂ SO(4):π 1ŝ = exp 3− 2 (i + j + k)3
π −1 −1t̂ = exp ϕ 2 5 4 (i + ϕj)(30)
5
3π − 1 − 1
ϕ 2 5 4 (k − ϕi) .
t̂′ = exp
5
The diagonal and the twisted embeddings give 2I subgroups of SO(8) acting freely on theunit sphere. For each of these two embeddings we have a unique lift to Spin(8). The lift forthe diagonal embedding is
π 1ŝ = exp 3− 2 (i + j + k + i′ + j′ + k′ )3(31)
π −1 −1t̂ = exp ϕ 2 5 4 (i + ϕj + i′ + ϕj′ ) ,5
whereas for the twisted embedding we have the same ŝ but nowπ11t̂ = exp ϕ− 2 5− 4 (i + ϕj − 3ϕi′ + 3k′ ) .
(32)
5
It is now a simple matter to use the trace formulae (6) in an explicit representation of the
Clifford algebra in order to compute the characters of 2I in the spinor representations for both
the diagonal (as a check) and the twisted embeddings. The result of this calculation is foundin Table 5. The dimension of the invariant subspaces is easily computed from the tabulated

14

DE MEDEIROS, FIGUEROA-O’FARRILL, GADHIA, AND MÉNDEZ-ESCOBAR

data and one finds that, whereas N− = 0 for both embeddings, one checks that N+ = 5 forthe diagonal embedding and N+ = 4 for the twisted embedding, giving a new N = 4 quotientand concluding their classification.
Notice that the Lie algebra of isometries of an N = 4 quotient is so(4), which is not of big
enough dimension to act (locally) transitively on a 7-dimensional manifold. Therefore N = 4
quotients are not homogeneous. This agrees with the expectation [21, 22] that homogeneity is
only implied by preserving more than one half of the supersymmetry; that is, for N > 4. For
the case of riemannian manifolds admitting Killing spinors, this has recently been proved in[23].
8. Summary
Table 6 summarises the classification of smooth quotients of S 7 of type (N, 0) with N > 3.
We list groups by their ADE labels, but we insist that they are not abstract groups, but the
specific subgroups of SO(8) described above. In each case there is a unique lift to Spin(8) with
the indicated value of N . The notation (Γ, τ ) means a pair consisting of a finite subgroup ofSp(1) of type Γ and an outer automorphism τ . If τ is the identity, we omit it and write only
Γ. The automorphism ν for E7,8 is any automorphism representing the unique nontrivial outerautomorphism.
Table 6. Smooth quotients of S 7 of type (N, 0) with N > 3
NGroups

8
6

A1
An≥2

5
4

Dn≥4 , E6 , E7 , E8
×(An≥4,6=5 , r ∈ Z×n+1 \ {±1}), (Dn≥6 , r ∈ Z2(n−2) \ {±1}), (E7 , ν), (E8 , ν)Acknowledgments

PdM was supported by a Seggie-Brown Postdoctoral Fellowship of the School of Mathematics of the University of Edinburgh. The work of JMF was supported by a World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan. JMF takes pleasurein thanking Hitoshi Murayama for the invitation to visit IPMU and the Leverhulme Trust for
the award of a Research Fellowship freeing him from his duties at the University of Edinburgh.
EME would like to thank the Simons Workshop in Mathematics and Physics 2009 for theirsupport and hospitality during the completion of this work.
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School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Edinburgh EH9 3JZ, UK(JMF also) Institute for the Physics and Mathematics of the Universe, University of Tokyo,
Kashiwa, Chiba 277-8586, Japan
E-mail address: {P.deMedeiros,J.M.Figueroa,E.Mendez}@ed.ac.uk, sunilgadhia@googlemail.com