M2-BRANES, EINSTEIN MANIFOLDS AND TRIPLE SYSTEMSJOSÉ MIGUEL FIGUEROA-O’FARRILLPoświęcone pamięci Mojego Przyjaciela Krzysztofa Galickiego

Abstract. This is the written version of a talk given on 1 July 2009 at the
XXV Max Born Symposium: the Planck Scale, held in Wrocław, Poland. I
review the possible transverse geometries to supersymmetric M2-brane configurations and discuss the representation-theoretic description of their conjectured dual superconformal Chern–Simons theories.

1. Introduction
It is a pleasure to speak at this Max Born Symposium in Wrocław, not just forthe obvious reason, but at least on two other accounts: firstly, because I share with
Max Born the odd fate of having ended up at the University of Edinburgh (albeit
in different departments); and secondly, because my friend Krzysztof Galicki camefrom Wrocław and studied here before we met as graduate students doing our PhD
with Martin Roček in Stony Brook a quarter of a century ago. It was from himthat I learnt, among other things, the proper Polish pronunciation of the city I used
to refer to as Breslavia and it is to his memory that I dedicate the written versionof this talk.
It is not clear whether the subject of my talk is appropriate for a conference on
the Planck scale. It is not a talk about quantum gravity, even though it derives its
motivation from an attempt to understand M-theory [1], the strong coupling limitof a candidate theory of quantum gravity: namely, type IIA superstring theory.
We know relatively little about M-theory away from its low-energy limit: elevendimensional supergravity [2, 3]. The absence of string solutions suggests that thisis not a theory of strings, whereas the existence of membrane solutions (so-called
M2-branes) suggests that it might be a theory of membranes. However it is difficult to make this more precise because membranes have resisted every attempt at
quantisation.
One way to try to understand at least some aspects of M-theory is via the
AdS/CFT correspondence (see, e.g., [4]). In that context, the M2-branes play asimilar role to the D3-branes of type IIB string theory: namely they give rise to a
conjectural correspondence between a gravitational theory and a conformal gauge
field theory. In the case of the D3-brane, this is the duality between type IIB string
1
theory on AdS5 × S 5 , with equal radii of curvature proportional to N 4 , where N
is the number of coincident D3-branes, and the maximally supersymmetric fourdimensional SU(N ) Yang–Mills theory. In the case of the M2-brane, this duality
should relate M-theory on AdS4 × S 7 , with commensurate radii of curvature now
1
proportional to N 6 , and a maximally supersymmetric three-dimensional conformal
field theory, which at the time of the original conjecture [5] and for almost a decade
thence proved elusive. In fact, even as recently as 2004, doubts were cast as to theexistence of a lagrangian description of such a theory [6].
1

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JOSÉ MIGUEL FIGUEROA-O’FARRILL

This all changed with the pioneering work of Bagger and Lambert [7, 8] and Gustavsson [9] who constructed a maximally supersymmetric three-dimensional conformal field theory. As shown in [10] (see also [11]) the BLG model can be recast as
a superconformal Chern–Simons theory with group SU(2) × SU(2) with opposite
levels coupled to matter in the bifundamental representation of the group. The
BLG model is the only known (indecomposable) unitary maximally superconformal three-dimensional field theory involving a finite number of fields and it has beenargued [12] to describe two M2-branes at an orbifold R8 /Z2 . The theory dual to
any number of M2-branes in R8 /Zk was constructed by Aharony, Bergman, Jafferis
and Maldacena [13] as an N =6 superconformal Chern–Simons theory with gauge
group U(N )k ×U(N )−k . One expects that for k = 1, 2 that there ought to be supersymmetry enhancement to N =8 and this has recently been demonstrated [14] byconsidering monopole operators. Similarly the dual theories to M2-branes on someother geometries have been constructed as superconformal Chern–Simons theories
with the appropriate amount of supersymmetry the geometry dictates; although
the dual theory to by far most of the possible transverse geometries (especiallythose with little supersymmetry) has not been identified.
This state of affairs motivates the desire to establish a more precise dictionary
between the possible transverse geometries to supersymmetric M2-brane configurations and superconformal Chern–Simons theories. To this end one needs to first
determine the possible geometries and also the possible theories and this is what
I will discuss in this talk. The task of determining the possible superconformal
Chern–Simons theories is best accomplished using the language and tools of representation theory, particularly the theory of unitary representations of a Lie algebra
admitting an ad-invariant inner product. This “metric” property of the Lie algebra
allows to relate its unitary representations to certain triple systems, which explainsa posteriori the important rôle played by ternary algebras in the original BLG
model.
The talk will thus consist of two parts. In the first part, departing from the wellknown elementary M2-brane solution of eleven-dimensional supergravity, I will review other supersymmetric solutions which are obtained by replacing the euclidean
space transverse to the brane by a Ricci-flat cone admitting parallel spinors. In
dimension 8 there is a wealth of such geometries which were reviewed in [15, 16]
and will be recalled here briefly. In the second part of the talk I will describe therepresentation-theoretic underpinnings of superconformal Chern–Simons theories
along the lines of [17], which is the companion paper to [18], and discuss their classification. This sets the stage for a more thorough investigation of the dictionary,
which is the subject of an ongoing investigation to be reported on elsewhere. In
particular it suggests the tantalising prospect of associating triple systems to a certain Einstein manifolds. In the meantime we can view the results described here as
one more example of how supersymmetry shines its light into mathematical objects
such as triple systems and gives us a fresh reason to investigate them further.

2. Supersymmetric M2-brane geometries
For the purposes of this talk, eleven-dimensional supergravity is a system ofgeometric partial differential equations for an eleven-dimensional lorentzian metricg and a closed 4-form F . We will not write these equations down, but simply
mention that they admit a two-parameter family of half-supersymmetric solutions
describing a stack of N coincident M2-branes [19]. Explicitly, we have the following

M2-BRANES, EINSTEIN MANIFOLDS AND TRIPLE SYSTEMS

3

expressions for g, F :21

g = H − 3 ds2 (R2,1 ) + H 3 ds2 (R8 )F = dvol(R2,1 ) ∧ dH −1 ,(1)where H is a harmonic function on R8 which we will take to be maximally symmetric:β(2)
H =α+ 6 ,rfor α, β ∈ R not both equal to zero. The parameter β depends linearly on thenumber N of M2-branes. For generic values of α, β, which means αβ 6= 0, thissolution preserves one-half of the supersymmetry, but if either α or β vanish, supersymmetry is enhanced to maximal. If β = 0 there are no M2-branes and thesolution is isometric to the Minkowski vacuum R10,1 with zero F , whereas if wetake α = 0, the solution is isometric to AdS4 × S 7 with the radii of curvature ina ratio of 1 : 2 and with F proportional to the volume form on the AdS4 [20, 21].
This latter solution is known as the near-horizon geometry of the M2-branes, since
taking α to zero is formally the same as taking r to zero.
Writing the metric on the euclidean transverse space in spherical polar coordinates, suggests a way to generalise this solution. We replaceds2 (R8 ) = dr2 + r2 ds2 (S 7 )by

dr2 + r2 ds2 (M 7 ),(3)

where M is a 7-dimensional riemannian manifold. For the new g (and the old F )to be a solution of eleven-dimensional supergravity it is necessary and sufficient for
M to be Einstein with unit radius of curvature. However, if we want the solution
to be supersymmetric, then M should be a spin manifold admitting real Killingspinors; namely, there should exist nonzero spinor fields ψ obeying the equation∇X ψ = 21 X · ψ

for all vector fields X,(4)where X · ψ is the action of the Clifford bundle C`(T M ) on the spinor bundle.
As observed by Bär [22], this is equivalent to the cone C(M ) = R+ × M , withmetric dr2 + r2 ds2 (M ), admitting parallel spinors. If M is assumed to be complete,then a theorem of Gallot [23] says that the cone is either flat, in which case M
is locally isometric to S 7 , or else the holonomy of the cone is irreducible. Wang[24] determined the holonomy representations of irreducible riemannian manifolds
admitting parallel spinors and in dimension 8, which is the dimension of the coneover M , these are Spin(7) acting on the spinor representation, SU(4) ⊂ SO(8) andSp(2) ⊂ SO(8). The last two correspond to Calabi–Yau 4-folds and hyperkähler
8-manifolds, respectively. Those 7-manifolds whose cones have Spin(7), SU(4) andSp(2) holonomy belong to well-known classes: weak G2 holonomy, Sasaki-Einsteinand 3-Sasakian, respectively. The corresponding solutions have the interpretation
as the near-horizon geometry of M2-branes at a conical singularity in a special
holonomy 8-manifold, which near the singularity is described as the cone over oneof the 7-manifolds just mentioned.
Let N denote the dimension of the space of Killing spinors on M , so thatthe corresponding M2-brane solution will preserve 2N supercharges. The positive
values that N may take are 1, 2, 3, 4, 5, 6, 8. For N > 3 the manifolds M are allsmooth quotients of S 7 by a finite subgroup Γ of SO(8) which leaves invariant an
N -dimensional subspace of chiral spinors. There are two smooth manifolds with
N =8: S 7 itself and RP7 , which is the quotient of S 7 by the order two subgroup
generated by the antipodal map. For every finite subgroup of SU(2), there is anembedding in SO(8) in such a way that the resulting quotient is smooth and has
N =5, unless the subgroup is cyclic in which case N =6, if the order is > 2, andN =8 if the order is 2. There are cyclic and binary dihedral subgroups of SO(8) for4

JOSÉ MIGUEL FIGUEROA-O’FARRILL

which the corresponding quotient has N =4. The values N =1, 2, 3 correspond to
weak G2 holonomy, Sasaki-Einstein and 3-Sasakian manifolds, respectively, someof which can of course be sphere quotients. Many classes of such manifolds are
discussed in [15, 16] and references therein. The book [25] by Boyer and Galicki
contains a wealth of information on Sasakian and 3-Sasakian geometry. Table 1
summarises the results reviewed so far.
Table 1. Supersymmetric M2-brane geometries
NCone

7-dimensional geometry

8

R8 , R8 /Z2

S 7 , RP74, 5, 6R8 /ΓS 7 /Γ

3

hyperkähler

3-Sasaki2

Calabi–Yau

Sasaki-Einstein

1

Spin(7) holonomy

weak G2 holonomy

To every supersymmetric solution of eleven-dimensional supergravity, one canattach a Lie superalgebra which is generated by the Killing spinors. This is described in complete generality in [26]; although for the near-horizon geometries ofthe supersymmetric M2-brane configurations which are the focus of this talk, they
were calculated in [27] and shown to be isomorphic to the orthosymplectic Lie superalgebra osp(N |4), as expected from the AdS/CFT correspondence. Indeed, thiscorrespondence posits that to every M2-brane configuration with a near-horizon
geometry of the form AdS4 × M with M admitting an N -dimensional space of
Killing spinors, there corresponds a three-dimensional N -extended superconformal field theory. The three-dimensional conformal superalgebras were classified by
Nahm [2] and shown to be isomorphic to osp(N |4), although it is realised differently. Focusing on the even Lie algebra so(N ) ⊕ sp(4, R), we recognise so(N ) as
the generic isometry algebra of the 7-manifold M and also the R-symmetry of thesuperconformal field theory, whereas sp(4, R) ∼= so(3, 2) is the isometry algebra of
AdS4 and also the conformal algebra of R2,1 . Examples of field theories exhibiting
this symmetry are the superconformal Chern–Simons theories with matter, which
we now begin to describe.
3. Superconformal Chern–Simons theories with matter
The degrees of freedom of a theory dual to a supersymmetric brane configuration
usually include some scalars which parametrise the normal bundle to the brane in
the spacetime and its fermionic partners which supersymmetry demands. Unlike inthe case of a D3-brane, for an M2-brane the degrees of freedom of the scalars andthe fermions already match, whence any gauge fields present in the theory should
contribute no new dynamical degrees of freedom. This forbids a Yang–Mills-likeaction, since in three dimensions this has propagating degrees of freedom, but it
does not forbid a Chern–Simons term whose gauge fields are non-propagating. The
sort of theories we will be considering thus contain the supersymmetric completionof a Chern-Simons term(A, dA) + 31 ([A, A], A) ,(5)
where A is a one-form on R2,1 with values in a Lie algebra g and (−, −) is an
ad-invariant inner product on g. This turns g into a metric Lie algebra. If g is
simple, then (−, −) is a multiple of the Killing form. Quantum consistency of

M2-BRANES, EINSTEIN MANIFOLDS AND TRIPLE SYSTEMS

5

the Chern–Simons term quantises this multiple, which is then called the level of
the Chern–Simons theory. If g is not simple, then there will be a larger space of
ad-invariant inner products and quantum consistency now selects a lattice in it.
Matter fields live in supermultiplets which contain a scalar, a Majorana fermion
and an auxiliary field we shall ignore in this talk. The action consists of the standard gauge-covariant kinetic terms, Yukawa couplings and a sextic scalar potential,
consistent with conformal invariance. In a manifestly N =1 superspace formulation, there are two kinds of terms in the matter lagrangian: the kinetic terms anda quartic superpotential which, upon integrating out the auxiliary fields, gives rise
to a sextic scalar potential and the Yukawa couplings. The matter fields belong toa unitary representation of g. Unitarity of the theory requires the inner product ofthe matter fields to be positive-definite, but since the Chern–Simons gauge fields
do not propagate their inner product (−, −) need not be positive-definite and inmany cases it will be forced to be indefinite.
Matter fields will also transform in a unitary representation of the so(N ) Rsymmetry. It follows from [2] that the supercharges in the conformal superalgebratransform in the fundamental representation of so(N ). Since the supercharges relate bosons to fermions, it follows that the bosonic R-symmetry representation B
must appear in the tensor product decomposition V ⊗ F of the vector representation V and the fermionic R-symmetry representation F, and similarly for B and
F interchanged. The simplest way to achieve this is for B and F to be spinorial
representations of so(N ), with the intertwiners V ⊗ B → F and V ⊗ F → B given
by Clifford action. This means that when N is odd, bosons and fermions will bein the same representation whereas if N is even the fermionic representation will
be obtained from the bosonic one by changing the chirality of the spinor representations. Table 2 summarises the spinor representations for N ≤ 8. The notation
is such that only the types (real, complex or quaternionic) and their dimension are(N )explicitly written down. I will use the notation ∆(N ) or ∆± for the irreducible
spinor representations of so(N ) for N odd and even, respectively. The subscripts
refer to the chirality, of course.
Table 2. Spinor representations of so(N ) for N ≤ 8N1so(N )
Spinors

R

2

3

4

5

6

7

8

u(1)

sp(1)

sp(1) ⊕ sp(1)

sp(2)

su(4)

so(7)

so(8)

C

H

H⊕H

H2

C4

R8

R8 ⊕ R8

Since gauge transformations and supersymmetry commute (we are not talking
about a supergravity theory), both bosons and fermions transform under the same
representation of the gauge Lie algebra g: let’s call it M generically. Since matter
degrees of freedom are fundamentally real, the type of the R-symmetry representation determines the type of M, which can read off from the table: real if N =1, 7, 8,
complex if N = 2, 6 and quaternionic if N = 3, 4, 5.
For the N =1, 2, 3 theories we can take the matter to be in any real, complex orquaternionic unitary representations, respectively, with the usual proviso that for
N =2 we must take fields and their complex conjugates, in effect working with theunderlying real representation obtained by restricting scalars from C to R, and thatfor N =3 fields are subject to the usual symplectic reality condition on the tensor
product of two quaternionic representations. For a good review of these theories
see [28].

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JOSÉ MIGUEL FIGUEROA-O’FARRILL

Things get more interesting for N ≥ 4 in that the type of representation andunexpectedly perhaps also the gauge symmetry g are constrained. In order towrite down these constraints we must first discuss a refinement of the usual unitary
representation theory of Lie algebras which is possible when the Lie algebra ismetric. As we now discuss, this leads quite naturally to the subject of threealgebras or triple systems and may explain a posteriori the rôle played by such
algebraic objects in the early literature on this topic.
4. Lie-embeddable unitary representations and triple systems
We now discuss special kinds of unitary representations of a metric Lie algebra.
This summarises the results in [18], which derived its inspiration from [29].
4.1. Real unitary representations. Let g be a metric Lie algebra as above andlet U denote a real unitary representation. The inner products on g and U will bedenoted (−, −) and h−, −i, respectively. Since we have g-invariant inner products
on both g and U , we can take the transpose of the representation map g → so(U )to obtain a g-equivariant linear map T : Λ2 U → g, which is surjective if U is afaithful representation. Explicitly, given u, v ∈ U , we define T (u, v) ∈ g by
(T (u, v), X) = hX · u, vi

for all X ∈ g.

(6)j
(In indices, if Xa is a basis for g and ui is a basis for U , then Xa · ui = Taiuj andjaa
the map T (ui , uj ) = Tij Xa , where Tij is obtained from Tai by raising and lowering

indices with the relevant inner product.) Now consider the tensor Ω defined byΩ(u, v, x, y) = (T (u, v), T (x, y)) .

(7)

It follows that Ω is a g-invariant tensor in the representation
S 2 Λ2 U = Λ4 U ⊕ U .

(8)

For general U , Ω will have components in both representations, but for some special
representations one or the other component will vanish. Such representations canbe described in the language of triple systems or 3-algebras, more precisely metric
3-Leibniz algebras, as described, for example in [30].
Define a trilinear product U × U × U → U by[u, v, w] := T (u, v) · w,

(9)in terms of which the tensor Ω can be rewritten as Ω(u, v, x, y) = h[u, v, x], yi. The
g-equivariance of T translates into a fundamental identity for the triple product:[x, y, [v, w, z]] = [[x, y, v], w, z] + [v, [x, y, w], z] + [v, w, [x, y, z]],(10)whereas the product obeys symmetry properties which follow from the tensorial
decomposition of Ω. These 3-algebras can trace their origin to [29] and to [31] inthe present context.
If Ω ∈ Λ4 U , then [u, v, w] is totally skew-symmetric and defines on U the structure of a 3-Lie algebra, a structure formalised by Filippov [32] but going backto the work of Nambu [33]. As conjectured (in a separate context and using aslightly different language) in [34], there is a unique positive-definite irreducible
representation U : it is four-dimensional and g = su(2) ⊕ su(2) with invariant innerproduct given by the Killing form on one of the simple ideals and the negative of theKilling form on the other. There exist at least four independent proofs of this fact:a geometric proof based on prolongations of Lie algebras [35], two similar proofs
based on a combinatorial analysis of the equation [36, 37] and a structural proof[38] based on the classification theorem for simple 3-Lie algebras [39]. This 3-Liealgebra is precisely the one in the original work of Bagger–Lambert and Gustavsson

M2-BRANES, EINSTEIN MANIFOLDS AND TRIPLE SYSTEMS

7

and the corresponding Chern–Simons theory is the unique interacting, manifestly
unitary, maximally superconformal such theory.
The other extreme case is where Ω ∈ U is an algebraic curvature tensor. The
triple product now satisfies[u, v, w] + [v, w, u] + [w, u, v] = 0,

(11)

making U into a Lie triple system. Lie triple systems are linear approximations
to symmetric spaces, in the same way that Lie algebras are linear approximations
to Lie groups, and in fact the tensor Ω in this case is the curvature tensor of
the symmetric space. The classification of positive-definite Lie triple systems isclassical and goes back to Cartan’s classification of riemannian symmetric spaces.
Such representations U can be used to construct N =1 superconformal theories, but
as far as I know these theories are not any more special than the ones constructed
out of a generic real representation U .
4.2. Complex unitary representations. Now let V be a complex unitary representation of g and let h−, −i now denote a hermitian inner product on V , complex
antilinear in the second slot in my conventions. The transpose of the g action on
V now defines a bilinear map T : V × V → gC to the complexification of g, where
V is the conjugate representation to V . We make gC into a complex metric Lie algebra by extending both the Lie bracket and the inner product complex-bilinearly.
If u, v ∈ V we define T (u, v) ∈ gC by (6), but where X ∈ gC now. The tensor Ω,
defined in (7), now belongs to the g-invariants in
S 2 (V ⊗ V ) = (S 2 V ⊗ S 2 V ) ⊕ (Λ2 V ⊗ Λ2 V ).

(12)

The sesquibilinear triple product V × V × V → V defined by (9) obeys thefollowing version of the fundamental identity[x, y, [v, w, z]] = [[x, y, v], w, z] − [v, [y, x, w], z] + [v, w, [x, y, z]],(13)

where the change in the middle term in due to T (x, y) = −T (y, x). The two
extremes, when one or the other component of Ω vanishes, correspond to representations V where[u, v, w] = ±[w, v, u].
(14)
The positive sign gives rise to Jordan triple systems and the negative to anti
Jordan triple systems [40]. Jordan triple systems are in bijective correspondencewith hermitian symmetric spaces, and again in this case Ω is the curvature tensor
of the relevant Kähler metric. Their classification is therefore again classical. The
classification of positive-definite anti Jordan triple system reduces, as we will see,to the classification of certain complex Lie superalgebras [41, 42]. Anti Jordan
triple systems are precisely the 3-algebras put forward in [43] to reformulate theN =6 theories of [13]. Again one can use Jordan triple systems to construct N =2
theories, but to my knowledge they are not more special than the N =2 theories
built out of generic V .
4.3. Quaternionic unitary representations. Finally we come to the case ofquaternionic unitary representations. The nonexistence of quaternionic Lie algebras means that it is more convenient to think of these representations as complex
unitary representations with an invariant quaternionic structure map, denoted J.
Hence let W be a complex unitary representation with hermitian structure h−, −iand let J : W → W be a g-equivariant complex antilinear map, obeying J 2 = −1
and compatible with the hermitian structure in thatω(u, v) = hu, Jvi

(15)

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JOSÉ MIGUEL FIGUEROA-O’FARRILL

is a g-invariant complex symplectic structure. Defining the transpose of the g-action
but relative to ω instead, we obtain a map T : S 2 W → gC and a tensor Ω defined
as in (7) which is g-invariant in the representation
S 2 (S 2 W ) = S 4 W ⊕ W .

(16)Ω belongs to W , the triple product defined by (9) satisfies equation (11), exceptthat now [u, v, w] = [v, u, w] so we have an anti Lie triple system. These are
the representations for the N =5 theories of [44, 45] and the N =4 theories of [46]and also appear as building blocks for the N =4 theories of [47]. The classificationof positive-definite representations again reduces to the classification of certain Lie
superalgebras [42]. The representations where Ω is totally symmetric correspondto hyperkähler symmetric spaces, but there are no nontrivial representations inpositive-definite signature and hence no unitary theories based on them.
This discussion is summarised in Table 3, which employs the following notation.
For N odd the scalars and fermions live in the same representation, whereas for
N even they live in representations with opposite chirality for the R-symmetry
spinors, and both representations are written, with the top sign corresponding to
the scalars and the bottom sign to the fermions, in some conventions. The notation
Rep(g, K) denotes the category of positive-definite unitary representations of g oftype K = R, C, H, and Rep(g, K)C the set ot those which are of class C , where
C can be either aLTS, aJTS or 3LA for anti Lie triple systems, anti Jordan triple
systems or 3-Lie algebras, respectively. The notation Irr(g, K)C denotes the subsets
of irreducible objects.
Table 3. Matter representations for N -extended supersymmetry
N

Matter representation

Remarks

1

U

U ∈ Rep(g, R)

2

(2)
(2)
∆± ⊗ V ⊕ ∆∓ ⊗ V

V ∈ Rep(g, C)3∆(3) ⊗ W

W ∈ Rep(g, H)

4

(4)
(4)
∆ ± ⊗ W 1 ⊕ ∆∓ ⊗ W 2

W1,2 ∈ Rep(g, H)aLTS

5∆(5) ⊗ W

W ∈ Irr(g, H)aLTS

6

(6)
(6)
∆± ⊗ V ⊕ ∆∓ ⊗ V

V ∈ Irr(g, C)aJTS

7∆(7) ⊗ U

U ∈ Irr(g, R)3LA

8(8)∆± ⊗ UU ∈ Irr(g, R)3LA

The irreducibility conditions correspond to the notion of an indecomposable
theory; namely, one which does not decouple into two or more nontrivial noninteracting theories. For N < 4 indecomposability does not imply irreducibility(e.g., take g simple), whereas for N > 4 indecomposability does imply irreducibility. For N =4 if W1 = 0 then W2 has to be irreducible and viceversa; otherwise,
indecomposability imposes connectedness of the corresponding quiver.
4.4. Embedding Lie (super)algebras. One peculiar property of the special representations described above is that they embed in Lie (super)algebras, which meansthat the triple product in the corresponding triple system is given by nesting two
Lie brackets. For the Lie and Jordan triple systems this is of course a classicalresult.

M2-BRANES, EINSTEIN MANIFOLDS AND TRIPLE SYSTEMS

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Indeed, if U is a Lie triple system, then on the Z2 -graded vector space k = g ⊕ U ,with g in degree 0 and U in degree 1, we may define a Lie algebra structure extendingthat of g and the action of g on U by declaring [u, v] = T (u, v), for u, v ∈ U . (Recall
that T is skewsymmetric, so this bracket is trying to define a Lie algebra.) Mostof the components of the Jacobi identity are immediate except for one, which isequivalent to (11). The resulting Z2 -graded Lie algebra k is metric by using theinner products on g and U . The symmetric space associated to this representationis K/G, where K is a Lie group with Lie algebra k and G is the closed subgroupwith Lie algebra g. Similarly, if V is a Jordan triple system, then we consider thecomplex graded vector space k = V ⊕ gC ⊕ V , in degrees −1, 0, 1. Then we define a
Lie algebra structure on k in such a way that gC is a Lie subalgebra by extending
the actions of gC on V and V by [u, v] = T (u, v) for u ∈ V and v ∈ V . Again
the only nontrivial component of the Jacobi identity is equation (14) with the plus
sign.
There are similar results for the three classes of representations demanded byN ≥ 4 supersymmetry. An anti Lie triple systems embeds in a complex Lie
superalgebra gC ⊕ W , whereas an anti Jordan triple system embeds in a complex
3-graded Lie superalgebra V ⊕ gC ⊕ V . The situation mimics the case of Lie andJordan triple systems, except that the symmetry of the anti Lie triple system saysthat gC ⊕W is a Lie super algebra, while the negative sign in (14) is (one componentof) the Jacobi identity only for a Lie super algebra. Finally, metric 3-Lie algebras
embed in a real 3-graded Lie superalgebra U ⊕ g ⊕ U .
For positive-definite unitary representations, irreducibility implies (with a minor
exception in the case of Lie triple systems) the simplicity of the embedding Lie
superalgebra. Hence this allows one to reduce the classification of positive-definite
irreducible representations to extant classifications of simple Lie (super)algebras.
In this way one recovers the classifications of N ≥ 5 superconformal Chern–Simons
theories in [44, 45, 48] from conceptually clear representation-theoretic results.
4.5. Supersymmetry enhancement. The representation theory also explains
the conditions for supersymmetry enhancement. By studying the decompositionof the R-symmetry spinor representations as a result of the embedding of the Rsymmetry Lie algebras so(N − 1) ,→ so(N ), we can read off the conditions which
are required for supersymmetry enhancement. Table 4 summarises the decomposition of the matter representations from N - to (N − 1)-extended supersymmetry.
The notation in the table is such that UC is the complexification of a real representation U , whereas VH is the quaternionification of a complex representation V
and ((W )) is a complex representation obtained from a quaternionic representation
W by forgetting the quaternionic structure. As usual, square brackets denote theunderlying real representation, so that if V is a complex representation with a real
structure, then [V ]C ∼=V.
We may understand the following supersymmetry enhancements, by looking at
the N -extended matter representation in terms of the (N − 1)-extended representation and then comparing with the generic (N − 1)-extended representation:• in N =4, W1 , W2 ∈ Rep(g, H)aLTS and the enhancement to N =5 occurs
precisely when W1 = W2 ;• in N =5, W ∈ Irr(g, H)aLTS and the enhancement to N =6 occurs when
W = VH , for V ∈ Irr(g, C)aJTS ; and• finally, in N =6, V ∈ Irr(g, C)aJTS and enhancement to N =7 occurs when
V = UC for U ∈ Irr(g, R)3LA .
These enhancements are consistent with relations between the different triple
systems; namely, a complex representation V is an anti Jordan triple system if and10

JOSÉ MIGUEL FIGUEROA-O’FARRILL

Table 4. Decomposition of matter representations
N

N − matter representation

8∆+ ⊗ U∆(7) ⊗ U

7∆(7) ⊗ U[(∆+ ⊕ ∆− ) ⊗ UC ]6∆+ ⊗ V ⊕ ∆− ⊗ V∆(5) ⊗ VH5∆(5) ⊗ W∆+ ⊗ W ⊕ ∆ − ⊗ W

4∆+ ⊗ W 1 ⊕ ∆− ⊗ W 2∆(3) ⊗ (W1 ⊕ W2 )3∆(3) ⊗ W(∆+ ⊕ ∆− ) ⊗ ((W ))(N − 1) − matter representation

(8)

(6)

(4)

(6)

(6)

(6)

(4)

(4)

(2)

(4)

(2)

only if its quaternionification is an anti Lie triple system; and a real representation
is a 3-Lie algebra if and only if its complexification is an anti Jordan triple system,
whereas if the underlying real representation [[V ]] of a complex representation V is
a 3-Lie algebra, then V is an anti Jordan triple system. Some of these results seemto be new and are described in [17, Appendix A].
Finally we remark that it follows form Table 4 that enhancement from N =7
to N =8 does not constrain the representation further, suggesting that N =7 implies N =8. This is indeed the case, as proved in [17] by a detailed study of thesuperpotentials.
4.6. Superpotentials. It is often convenient to write the superconformal Chern–
Simons theories in an off-shell formalism in which one of the supersymmetries ismanifest. This is done by working in an N =1 superspace where one of the supercharges acts as a supertranslation. The choice of supercharge breaks the so(N )R-symmetry to the so(N − 1) stabilizer of the supercharge. In this formalism thetheory is determined by a quartic, gauge-invariant superpotential which is inert
under the global so(N − 1) symmetry. The off-shell superfield Ξ that describes thematter content can always be assembled into the representation of so(N −1)⊕g appearing in the third columnR of Table 4. The superpotentials can all be expressed as1
the superspace integral 16
d2 θ W(Ξ), where W is a real, quartic, (so(N − 1) ⊕ g)invariant function. For all N ≥ 4 the expression for this function is given in
Table 5. In [17] one can find the expression also for N =2, 3. In the table, thetensor Θ appearing in the N =6 row is the so(5) ∼= usp(4)-invariant symplectic
form on ∆(5) while in the N =8 row it denotes the so(7)-invariant self-dual Cayley
4-form on ∆(7) . Repeated indices are contracted with respect to the hermitian innerproduct on ∆(N −1) .
Table 5. Superpotentials
N

W(Ξ)

8

1
a
b
c
d
3 Θabcd (T (Ξ , Ξ ), T (Ξ , Ξ ))6

(T (Ξa , Ξb ), T (Ξb , Ξa )) + Θab Θcd (T (Ξa , Ξc ), T (Ξb , Ξd ))5− 16 (T (Ξα , Ξβ ), T (Ξβ , Ξα )) − 16 (T (Ξα̇ , Ξβ̇ ), T (Ξβ̇ , Ξα̇ )) + (T (Ξα , Ξβ̇ ), T (Ξβ̇ , Ξα ))

4

1
1
a
b
b
a
a
b
b
a
a
b
b
a
6 (T1 (Ξ , Ξ ), T1 (Ξ , Ξ )) + 6 (T2 (Ξ , Ξ ), T2 (Ξ , Ξ )) − (T1 (Ξ , Ξ ), T2 (Ξ , Ξ ))Finally let me remark that the rigidity of the N ≥ 3 theories translates, using the
AdS/CFT correspondence, to a rigidity of 3-Sasakian manifolds and it is has indeed

M2-BRANES, EINSTEIN MANIFOLDS AND TRIPLE SYSTEMS

11

been shown by Pedersen and Poon [49] (see [25, Theorem 13.3.24]) that complete 3Sasakian manifolds are infinitesimally rigid, a result which came after the AdS/CFT
correspondence. Had the dual theories to M2-branes been understood earlier, thiswould have provided a nice mathematical conjecture which I’m sure Krzysztof would
have appreciated.
Acknowledgments
It is a pleasure to thank Remigiusz Durka and Jerzy Kowalski-Glikman for theinvitation to participate in this meeting and the IPMU for support. This work
was supported by a World Premier International Research Center Initiative (WPI
Initiative), MEXT, Japan. It is a pleasure to thank Hitoshi Murayama for theinvitation to visit IPMU and the Leverhulme Trust for the award of a Research
Fellowship relieving me of my duties at the University of Edinburgh. Last, but by
no means least, I would like to thank Paul de Medeiros, Elena Méndez-Escobar
and Patricia Ritter for a most enjoyable collaboration leading up to the results
described in this talk, and Roger Bielawski for the Polish lesson.
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Institute for the Physics and Mathematics of the Universe, University of Tokyo,
Kashiwa, Chiba 277-8586, Japan
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3JZ, UK