SUPERPOTENTIALS FOR SUPERCONFORMAL CHERN–SIMONSTHEORIES FROM REPRESENTATION THEORYPAUL DE MEDEIROS, JOSÉ FIGUEROA-O’FARRILL, AND ELENA MÉNDEZ-ESCOBAR
Abstract. These notes provide a detailed account of the universal structure of superpotentials defining a large class of superconformal Chern–Simons theories with matter, many of whichappear as the low-energy descriptions of multiple M2-brane configurations. The amount of superconformal symmetry in the Chern–Simons-matter theory determines the minimum amountof global symmetry that the associated quartic superpotential must realise, which in turn restricts the matter superfield representations. Our analysis clarifies the necessary representationtheoretic data which guarantees a particular amount of superconformal symmetry. Thereby weshall recover all the examples of M2-brane effective field theories that have appeared in the
recent literature. The results are based on a refinement of the unitary representation theory of
Lie algebras to the case when the Lie algebra admits an ad-invariant inner product. The typesof representation singled out by the superconformal symmetry turn out to be intimately associated with triple systems admitting embedding Lie (super)algebras and we obtain a number of
new results about these triple systems which might be of independent interest. In particular,
we prove that any metric 3-Lie algebra embeds into a real metric 3-graded Lie superalgebra insuch a way that the 3-bracket is given by a nested Lie bracket.

Contents
1. Introduction and summary
Matters of notation
1.1. Matter content of three-dimensional superconformal theories
1.2. Constraints from supersymmetry
1.3. Indecomposability and irreducibility
1.4. Superpotentials
2. Generic superconformal Chern–Simons-matter theories
2.1. N=1 supersymmetry in three dimensions
2.2. N=1 supersymmetric lagrangians in three dimensions
2.2.1. Supersymmetric Chern-Simons term
2.2.2. Supersymmetric matter term
2.2.3. Superpotential
2.2.4. On-shell N=1 supersymmetric lagrangians
2.3. N=2 supersymmetry for M complex
2.3.1. N=2 F-term superpotential
2.3.2. On-shell N=2 supersymmetric lagrangians
2.4. N=3 supersymmetry for M quaternionic2.4.1. On-shell N=3 supersymmetric lagrangian
2.4.2. R-symmetry
3. N>3 superconformal Chern–Simons-matter theories
3.1. N=4 supersymmetry
3.1.1. W ∈ Rep(g, H)aLTS
Date: 15th August 2009.
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3.1.2. W = W1 ⊕ W2 with W1 , W2 ∈ Rep(g, H)aLTS
3.2. N=5 supersymmetry for W1 ∼= W2 ∈ Rep(g, H)aLTS
3.3. N=6 supersymmetry for W = V ⊕ V with V ∈ Irr(g, C)aJTS
3.4. N=8 supersymmetry for V = UC with U ∈ Irr(g, R)3LA
3.4.1. so(8) R-symmetry
Acknowledgments
Appendix A. Unitary representations of metric Lie algebras
A.1. Notation and basic notions
A.1.1. Real representations
A.1.2. Complex representations
A.1.3. Quaternionic representations as complex representations
A.1.4. Real representations as complex representations
A.1.5. Complex and quaternionic representations as real representations
A.1.6. Reality conditions
A.1.7. Relations between real, complex and quaternionic representations
A.2. Lie-embeddable representations of metric Lie algebras
A.2.1. Real orthogonal representations
A.2.2. Complex unitary representations
A.2.3. Complex representations in real terms
A.2.4. Quaternionic unitary representations
A.2.5. Some relations between these representations
A.2.6. Embedding Lie (super)algebras
A.2.7. Simplicity
Appendix B. Deriving the superpotentials
B.1. N =2 superpotential
B.2. N =3 superpotential
References

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1. Introduction and summary
The past couple of years have witnessed a remarkable amount of progress in our ability
to understand configurations of coincident M2-branes in M-theory preserving large amountsof supersymmetry. This progress followed the seminal work of Bagger and Lambert [1, 2]
and Gustavsson [3] who managed to construct a lagrangian field theory in three-dimensional
Minkowski spacetime that is invariant under the same maximal N =8 superconformal algebraas that of the near-horizon geometry of the M2-brane solution of eleven-dimensional supergravity. This theory is now understood to arise as a special case of a more general type of N =6
superconformal field theory in three dimensions discovered by Aharony, Bergman, Jafferis andMaldacena in [4] (see also [5]). These N =6 theories are thought to describe the low-energy
dynamics of multiple coincident M2-brane configurations whose near-horizon geometries areof the form AdS4 × S 7 /Zk , for some positive integer k, with maximal N =8 supersymmetry
recovered only for k = 1, 2. New N =5 superconformal field theories were obtained in [6, 7] andthe regular ones which cannot enhance to N = 6 are thought [8] to describe near-horizon geometries of the form AdS4 × S 7 /D̂k (D̂k being the binary dihedral group of order 4k). Moreover
they can be considered as special cases of the N =4 superconformal field theories found in [9]generalising the N =4 theories obtained first by Gaiotto and Witten in [10].

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

3

The lagrangian for all these new superconformal field theories can be written in terms ofa Chern–Simons term for a non-dynamical gauge field which is coupled to matter fields ina particular unitary representation of the Lie algebra which describes the gauge symmetry.
In addition to the standard kinetic terms for the bosonic scalar and fermionic spinor matterfields, there are quartic scalar-fermion Yukawa couplings and a sextic scalar potential. The
order of these matter couplings is precisely as expected for a generic on-shell superconformal
Chern–Simons-matter theory and is just a consequence of superconformal symmetry. What is
novel, at least relative to known superconformal gauge theories in other dimensions, is that the
representation and indeed the Lie algebra describing the gauge-matter couplings is restricted by
the amount of superconformal symmetry to be realised. As noted in [11], for the N =8 theoryof [1–3] the unitary representation for the matter fields corresponds to the bifundamental ofsu(2) ⊕ su(2) ∼= so(4). The fact that this is the only possibility follows from the classificationsin [12–14] of euclidean 3-Lie algebras, in terms of which the original description in [1–3] wasbased. A classification of the N =6 theories of [4] has been provided by [15] (see also [16,17]) andthe unitary representation for the matter fields must correspond to the bifundamental of eitheru(m) ⊕ u(n) or sp(m) ⊕ u(1). The N =5 theories obtained in [6,7] can have either so(m) ⊕ sp(n),spin(7) ⊕ sp(1), g2 ⊕ sp(1) or so(4) ⊕ sp(1) (the second and fourth being distinguished from thefirst of these possibilities due to the different representation inhabited by the matter fields inthese cases). The N =4 theories in [9, 10] can have matter fields in any of the representations
noted above.
The goal of this paper, which is to be viewed as a companion to [16], is to survey thisemerging landscape of superconformal Chern–Simons theories and establish precisely what
representation-theoretic conditions must be obeyed by the matter couplings in order to realisea particular amount of superconformal symmetry. Our starting point will be the generic off-shell
N =1 superconformal Chern–Simons-matter theory. To define this (at least as a classical field
theory) requires two sets of ingredients: a unitary representation M (which the matter fields will
inhabit) of a metric Lie algebra g and a real, quartic, g-invariant function on M (that defines asuperpotential W from which the quartic Yukawa couplings and sextic scalar potential will beobtained in the on-shell theory after integrating out all the auxiliary fields). The construction of
N =2 superconformal Chern–Simons-matter theories is somewhat canonical given the existenceof an off-shell N =2 superspace formalism in three dimensions and the generic N =2 theory canbe obtained from the N =1 construction whenever M is complex unitary (provided one fixes
certain components of the N =1 superpotential appropriately so as to reproduce the correct N =2
gauge-matter couplings). Prior to the recent progress described above, the maximum amount ofsuperconformal symmetry that could be realised in the lagrangian for a Chern–Simons-mattertheory was suspected to be N =3 [18]. The construction of the N =3 superconformal Chern–
Simons-matter theory not only requires M to be quaternionic unitary but also fixes uniquely
the superpotential and so indeed one should expect no generic possibilities for obtaining N > 3
superconformal symmetry. Recalling that N =3 theories should describe M2-brane near-horizon
geometries of the form AdS4 ×X7 , where X7 is a 3-Sasakian 7-manifold, a nice consistency checkis that there exists a corresponding infinitesimal rigidity theorem [19, Theorem 13.3.24] for suchgeometries.
The way we shall proceed to N > 3 is by examining certain special classes of unitary representations for which this same superpotential realises an enhanced so(N − 1) global symmetry.
For a given value of N > 3, this property, albeit non-trivial, need not guarantee that the resulting on-shell Chern–Simons-matter theory realises an enhanced N -extended superconformal
symmetry. Clearly though for such an on-shell theory to be derivable from an N =1 superpotential requires that superpotential to be invariant under those R-symmetries preserving the choice

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of N =1 superspace parameter, which is an so(N − 1) subalgebra of the so(N ) R-symmetry.
Nevertheless in all the cases we consider we will find that this property does give rise to superconformal symmetry enhancement and thereby this construction will recover all the known
examples of N ≥ 4 superconformal Chern–Simons-matter theories described above.
These notes are organised as follows. In the rest of this introductory section we outline
the representation-theoretic structure of three-dimensional superconformal Chern–Simons theories with matter. We begin with a cursory glance at some of the essential notation to be
used for the generic types of unitary representations we shall encounter throughout the paper. The representation theory of N -extended superconformal algebras in three dimensions
is then reviewed focusing on the specific types of unitary representations of so(N ) ⊕ g which
the matter fields must inhabit for each value of N ≤ 8. We then summarise for N ≥ 4 therepresentation-theoretic criteria which determine the indecomposable N -extended superconformal Chern–Simons theories. Section 1 is concluded with a summary of the functional formsof the quartic superpotentials that we will find give rise to all the N -extended superconformal
theories in the rest of the paper.
In Section 2 we will review the generic N ≤ 3 superconformal Chern–Simons-matter theories, describing the lagrangians and supersymmetry transformations both off- and on-shell.
Starting from the generic N = 1 theory, we will describe in detail how one obtains the genericN =2, 3 theories by taking the ground field for the matter representation to be respectively
K = C, H and then choosing an appropriate superpotential WC,H in terms of this data. The
realisation of the non-abelian usp(2) R-symmetry in the N =3 theory is also detailed. In Section 3, for incremental values of N > 3, we proceed to describe how one obtains N -extended
superconformal Chern–Simons-matter theories from the generic N =1 superspace formalism bytaking the matter representations to be of the types established in Section 1 and using thecorresponding so(N − 1)-invariant forms of the rigid superpotential WH describing the N =3
theory. For N =4 we will thereby recover the theories of [9,10] and determine the conditions forindecomposability of these theories. For the theories of [10] this condition is just irreducibilityof the matter representation, while for the more general theories in [9] indecomposability is
the connectedness of the corresponding quiver. Similarly for N =5, 6, 8 we recover the theoriesof [6, 7], [4] and [1–3] respectively. We find that indecomposability in all the N > 4 theories
is equivalent to irreducibility of the associated matter representations, thus allowing their classification in terms of certain simple Lie superalgebras into which the matter fields embed. In
the process of obtaining the N =8 theory we rule out the possibility of a Chern–Simons-mattertheory with precisely N =7 superconformal symmetry in the sense that we find such a theory
must automatically enhance to N =8.
In the interest of accessibility we have collected many crucial technical results concerning
unitary representations of metric Lie algebras into a comprehensive Appendix A. This appendix contains precise definitions of all the basic representation-theoretic notions employed in thispaper. We begin by reviewing the notions of real, complex and quaternionic unitary representations of a Lie algebra g. We then describe the canonical functors mapping between thesethree categories of representations as we change the ground field. We also describe the various
useful identities which these functors satisfy under composition and finally the effect of thesefunctors on the irreducible representations. When the Lie algebra g admits an ad-invariant
inner product, it is possible to refine the standard representation theory and this allows us
to distinguish among the generic unitary representation some which a posteriori can be characterised by the existence of a certain embedding Lie (super)algebra structure. We discuss
how to each of these “Lie-embeddable” representations one can attach a triple system of thetype which appears naturally in the 3-algebraic description [1–3, 20] of the N ≥ 4 theories

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

5

related to M2-branes. In particular, a new characterisation (Theorem 7) is established of themetric 3-Lie algebras of the N =8 theory in terms of certain real metric 3-graded Lie superalgebras. We then determine the action of the aforementioned functors on the Lie-embeddable
representations and in this way shed some light on the representation-theoretic underpinningof supersymmetry enhancement in these theories. Finally, for completeness, we include a derivation of the superpotentials describing the generic N =2, 3 theories in Appendix B.
Matters of notation. In this paper we will make much use of certain basic concepts in thetheory of unitary representations of a Lie algebra. There is a great deal of bookkeeping forwhich perhaps there is no universally agreed notation. Although the notation is presented as
the definitions are introduced, it may benefit the reader to see the notation at glance in one
place. This small section aims to be that place.
Following [21, 22], we have used the letters U , V and W , also with decorations, to refer toreal, complex and quaternionic representations, respectively. The notation Rep(g, K) (resp.
Irr(g, K)), where g is a Lie algebra, denotes the (resp. irreducible) unitary representations over
K = R, C, H. We will also let Rep(g, K)C (resp. Irr(g, K)C ) denote the subset of those (resp.
irreducible) K-representations of g of class C, where these are defined in Appendix A.2. There
are natural operations on representations which act on the ground field K:• extending from R to C: U 7→ V = UC (complexification)• extending from C to H: V 7→ W = VH (quaternionification)• conjugation: V 7→ V
• restricting from H to C: W 7→ V = ((W ))• restricting from C to R: V 7→ U = [[V ]]These operations obey relations which are reviewed in Proposition 1 in Appendix A.1.7 andwhose effect on the irreducibles is described by Proposition 2 in the same appendix, and on thespecial classes of representations by Proposition 5 in Appendix A.2.5.
1.1. Matter content of three-dimensional superconformal theories. Superconformal
field theories in three-dimensional Minkowski spacetime R1,2 are invariant under a conformal
superalgebra. These algebras are denoted type VII in [23, Proposition 2.2] and are indexed bya positive integer N . The even Lie algebra is isomorphic to so(N ) ⊕ so(2, 3), where so(N ) is the
R-symmetry of the field theory and so(2, 3) is the three-dimensional conformal algebra. The
odd subspace is in the tensor product representation of the vector of so(N ) and the spinor ofso(2, 3). The spinor representation of so(2, 3) defines an isomorphism so(2, 3) ∼= sp(4, R), whichmeans that the spin representation is real, four-dimensional and symplectic. In other words,
the conformal superalgebra is isomorphic to the orthosymplectic Lie superalgebra osp(N |4).
This Lie superalgebra can be understood as endomorphisms of the vector superspace RN |4 of
N even and 4 odd dimensions preserving a euclidean structure on the even subspace and a
symplectic structure on the odd subspace. The supercharges in the superalgebra are the odd
endomorphisms which map the even and odd subspaces to each other. Standard arguments
restricting unitary interacting field theories to admit at most 32 supercharges impose an upperbound N ≤ 8 for the interesting theories. Having said that, such arguments must be takenwith a pinch of salt for theories in less than 4 dimensions which do not admit a dimensionaloxidation to four dimensions, as is the case with the three-dimensional superconformal Chern–
Simons theories which are the subject of this paper. Nevertheless we will restrict attention toN ≤ 8; although see [24] for a topologically massive Chern–Simons theory admitting N > 8.
As we will discuss in more detail in Section 2, the lagrangian for superconformal Chern–
Simons theories with matter consists of two types of terms: a Chern–Simons lagrangian for agauge field taking values in a metric Lie algebra g and a lagrangian involving matter fields in

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

a faithful unitary representation M of g. Unitarity of the theory requires the inner producton M to be positive-definite, but since the Chern–Simons gauge fields are non-propagating,the inner product on g can, and in some cases must, have indefinite signature. Indeed, one of
the remarkable features of the theories with N ≥ 4 supersymmetry is that the Lie algebra gcannot be taken to be simple. The assumption that the representation be faithful is done forconvenience and we believe that there is minimal (if any) loss in generality. We recall that aunitary representation ρ : g → u(M) is faithful if ρ has trivial kernel. In most other contexts,
this assumption can be made without any loss of generality: the kernel k of ρ is an ideal andthe quotient g/k is a Lie algebra which does act faithfully. However here we have gauge fields
taking values in g, hence in particular also in k and one would like to show that these fields
somehow decouple. The gauge fields A enter the matter lagrangian via covariant derivatives ofthe form d + ρ(A), whence any gauge field in k appears only in the Chern–Simons lagrangian,which depends not just on g as a Lie algebra but as a metric Lie algebra. In other words, it
depends on a choice of ad-invariant inner product on g. If k is a nondegenerate ideal, so thatg = k ⊕ k⊥ , then it is not hard to see that since both k and k⊥ are ideals and are orthogonal, thetheory decouples into a Chern–Simons term for k and a Chern–Simons-matter term for k⊥ . In
the case when k is not nondegenerate, preliminary results with abelian quiver theories suggestthat one ends up with a Chern–Simons-matter theory for k⊥ /(k ∩ k⊥ ) which is a metric Liealgebra which does act faithfully.
The matter fields in three dimensions arrange themselves into supermultiplets which, ignoring
auxiliary fields, consist of a bosonic scalar X and a fermionic Majorana spinor Ψ on R1,2 inrepresentations which we will denote schematically by B ⊗ M1 and F ⊗ M2 , respectively,
where B and F are the bosonic and fermionic R-symmetry representations and M1 , M2 arerepresentations of g. The supersymmetry transformations take the generic formδ X = Ψandδ Ψ = dX ·  + · · · ,(1)where  is the supersymmetry parameter which is a vector under the R-symmetry, but inert
under g, reflecting the fact that for a rigidly supersymmetric theory, supersymmetry and gaugetransformations commute. It then follows that M1 = M2 , whence we will drop the subscript,
and focusing on the R-symmetry, we see that letting V denote the vector representation of the
R-symmetry,
B⊂V⊗F

and

F ⊂ V ⊗ B.

This suggests taking B and F to be spinor representations in such a way that the above
inclusions are induced from the Clifford actions V ⊗ F → B and V ⊗ B → F, respectively.
We will do so. This means that when N is odd, bosons and fermions will be in the same
representation, whereas if N is even, since Clifford multiplication by vectors reverses chirality,
the fermionic representation will be obtained from the bosonic one by changing the chirality ofthe spinor representations.
Table 1 summarises the spinor representations for N ≤ 8. It lists the exceptional lowdimensional isomorphisms which are induced by the spinor representations and lists the typesof representation with their dimension. For N odd there is a unique irreducible spinor representation (up to isomorphism) which is real for N ≡ ±1 (mod 8) and quaternionic for N ≡ ±3
(mod 8). For N even there are two, distinguished by chirality. They are complex for N ≡ ±2
(mod 8), with opposite chiralities being related by complex conjugation, real for N ≡ 0 (mod 8)
and quaternionic for N ≡ 4 (mod 8). It will be convenient to introduce the following notation for the spinor representations: for N odd, we let ∆(N ) denote the unique irreducible(N )spinor representation of so(N ), whereas for N even, we let ∆± denote the unique irreducible

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

N

so(N )

spinor irreps

2

u(1)

C

3

sp(1)

H

4

sp(1) ⊕ sp(1)

H⊕H

5

sp(2)

H2

6

su(4)

C4

7

so(7)

R8

8

so(8)

R8 ⊕ R8

7

Table 1. Spinor representations of so(N ) for N ≤ 8spinor representation of so(N ) with positive/negative chirality, with the understanding that for
(N )

(N )

N =2, 6, ∆± = ∆∓ .
The degrees of freedom described by the matter fields are fundamentally real and hence thisfact determines the type of the representation M in terms of the type of the relevant spinor
representation. This means that if the spinor representation is real then so must M, whereas if
the spinor representation is quaternionic then so must M, but we are then supposed to take thefields to be in the underlying real representation of the tensor product of the two quaternionic
representations. In practical terms, this means imposing a reality condition on the fields which
involves the symplectic structure of both the spinor representation and M, as described forexample in Appendix A.1.6. Finally, if the spinor representation is complex, we can take M
to be complex without loss of generality, with the understanding that we may think of both
reals and quaternionic representations as special types of complex representations. In this case,the matter fields take values in the real representation given by their real and imaginary parts.
In conclusion, for N =1, 7, 8 the representations M are real, for N =3, 4, 5 quaternionic and for
N =2, 6 complex.
Summarising thus far, for odd N the bosonic and fermionic matter fields both take values in
the representation ∆(N ) ⊗M, with the proviso that for N =3, 5, when ∆(N ) is quaternionic, fields
must obey the natural reality condition. For even N the bosonic matter fields can take values
(N )
(N )
in the representation ∆+ ⊗ M1 ⊕ ∆− ⊗ M2 , whereas the fermionic matter fields take values
(N )
(N )
in ∆− ⊗ M1 ⊕ ∆+ ⊗ M2 , where a priori both representations M1 and M2 can be different.
Again, if N =4, then all representations are quaternionic, so that we must impose the natural
symplectic reality condition on the fields. If N =2, 6 then all representations are complex and
we must consider the representation made up of by the real and imaginary parts of the fields
or, said differently, to consider both the fields and their complex conjugates. In this case one
may ignore the distinction between M1 and M2 because taking real and imaginary parts of
(N )
(N )
(N )
∆+ ⊗ M1 ⊕ ∆− ⊗ M2 is the same as taking real and imaginary parts of ∆+ ⊗ (M1 ⊕ M2 ),so that we can always take the matter fields to be in (the underlying real form of) a particularchiral spinor representation of so(N ). Of course, for N =4 the symplectic reality condition
for the matter fields does not eliminate the distinction and we will see in later sections how
this can lead to the notion of “twisted” and “untwisted” N =4 hypermultiplets according to
the relative chiralities of the spinor representation of so(4) they transform under. Similarly
for N =8 the matter representations are real and one might expect to be able to distinguish

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DE MEDEIROS, FIGUEROA-O’FARRILL, AND MÉNDEZ-ESCOBAR(8)

between different types of matter in ∆± . However, this case of maximal supersymmetry willturn out to be rather special in that we will find one can obtain any N =8 supermultiplet from(8)
an N =7 one wherein both ∆± are identified with ∆(7) under the embedding so(7) ,→ so(8)thus eliminating the apparent distinction between the two possible types of matter.
1.2. Constraints from supersymmetry. We will see that for the N =1, 2, 3 theories we cantake the matter to be in any real, complex or quaternionic unitary representations, respectively, whilst for N > 3 the allowed representations are subject to further restrictions. In Appendix A.2 we introduce several classes of “Lie-embeddable” unitary representations of a metric Liealgebra, which are summarised in Table 2 below. The notation is explained in Appendix A.2.
For now suffice it to say that R is a natural fourth-rank tensor constructed canonically from therepresentation data. The names of the classes stand for the corresponding triple system naturally attached to such representations: Lie (LTS), anti-Lie (aLTS), Jordan (JTS), anti-Jordan(aJTS), 3-Lie (3LA) and quaternionic (QTS), this last one being a non-standard nomenclature
introduced in this paper.
Class type

R

LTS

R

U

JTS

C

S 2V ⊗ S 2V

3-graded complex metric Lie algebra

QTS

H

S 4W

3-graded complex metric Lie algebra

aLTS

H

W

complex metric Lie superalgebra

aJTS

C

Λ2 V ⊗ Λ2 V

3LA

R

Λ4 U

Embedding Lie (super)algebra
2-graded metric Lie algebra

3-graded complex metric Lie superalgebra
3-graded metric Lie superalgebra

Table 2. Lie-embeddable unitary representations of a metric Lie algebra

As discussed in Section 3, the matter representations for N > 3 supersymmetry are forced tobelong to some of these special classes. Table 3 summarises the situation, where the notation
Rep(g, K)C denotes the unitary representations of g of type K and class C, where K = R, C, H
and C can be either 3LA, aJTS or aLTS, and similarly Irr(g, K)C for the irreducibles. If the class
C is absent, then we mean any representation of type K. The irreducibility of the representation
for N ≥ 5 theories is imposed by the requirement that the theory should not decouple into two
or more nontrivial theories. For N even, the bosonic and fermionic matter representations aredifferent, related by changing the chirality of the R-symmetry spinor representations. In thosecases we list them both in one line, with the top reading for bosons and the bottom reading
for fermions. In the case of representations which are not manifestly real, one is instructed totake the underlying real representation by imposing the appropriate reality conditions on thefields, which follow from the discussion in Appendix A.1.6. This applies to 2 ≤ N ≤ 6.
The representation theory also helps to explain the conditions for supersymmetry enhancement. Table 4 summarises how the spinor representations decompose as a result of the embedding of the R-symmetry Lie algebras so(N − 1) ,→ so(N ). The notation [[V ]], introduced inAppendix A.1.7, means the real representation obtained from the complex representation V byrestricting scalars to R.

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

N

Matter representation

Remarks

1

U

U ∈ Rep(g, R)

2∆± ⊗ V ⊕ ∆∓ ⊗ V(2)

V ∈ Rep(g, C)3∆(3) ⊗ W

W ∈ Rep(g, H)

4

(2)

(4)

9

(4)

∆± ⊗ W1 ⊕ ∆∓ ⊗ W2 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∆± ⊗ U(8)U ∈ Irr(g, R)3LA

Table 3. Matter representations for N -extended supersymmetry

N so(N ) ⊃ so(N − 1)
(8)

8

∆±

7

∆(7)

6

∆±

(6)

∼
= ∆(7)(6)∼= [[∆+ ]]∼= ∆(5)

4(4)(4)∆(5) ∼= ∆+ ⊕ ∆−(4)∆± ∼= ∆(3)

3(2)(2)∆(3) ∼= ∆+ ⊕ ∆−5

Table 4. Spinor representations under so(N − 1) ,→ so(N )This then implies the decomposition of the matter representations from N - to (N − 1)extended supersymmetry which is summarised in Table 5. In that table we use notation introduced in Appendix A.1.7. In particular, 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 the underlying 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 practice one finds the N -extended
matter representation in the second column of Table 5, then moves over to the third column
which shows this representation in terms of (N − 1)-extended supersymmetry and then moves
back to the second column but one row below to compare with the generic (N − 1)-extended
representations. This allows us to understand the enhancements N =4 → N =5, N =5 → N =6
and N =6 → N > 6, as follows.

10

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

N N − matter representation (N − 1) − matter representation(8)8∆+ ⊗ U∆(7) ⊗ U

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

4∆+ ⊗ W1 ⊕ ∆− ⊗ W2∆(3) ⊗ (W1 ⊕ W2 )3∆(3) ⊗ W(∆+ ⊕ ∆− ) ⊗ ((W ))

(6)

(6)

(6)

(4)

(6)

(4)

(4)

(4)

(2)

(2)

Table 5. Decomposition of matter representations• In N =4, W1 , W2 ∈ Rep(g, H)aLTS and the enhancement to N =5 occurs precisely when
W1 = W2 :/ ∆(4) ⊗ W ⊕ ∆(4) ⊗ W∆(5) ⊗ W

−

+

(2)

t
(4)
(4)
∆+ ⊗ W1 ⊕ ∆− ⊗ W2

• In N =5, W ∈ Irr(g, H)aLTS and the enhancement to N =6 occurs when W = VH , forV ∈ Irr(g, C)aJTS :(6)/ ∆(5) ⊗ V(6)∆+ ⊗ V ⊕ ∆− ⊗ V

H

(3)

v

∆(5) ⊗ W• Finally, in N =6, V ∈ Irr(g, C)aJTS and enhancement to N =7 occurs when V = UC forU ∈ Irr(g, R)3LA :(7)∆+ ⊗ U/ [[∆(6) ⊗ U ]]
+

C

(4)

x
(6)
[[∆+ ⊗ V ]]We also see from Table 5 that enhancement from N =7 to N =8 does not constrain the
representation further. This suggests that N =7 implies N =8 and we will show in Section 3.4
that this is indeed the case.
1.3. Indecomposability and irreducibility. Given two N -extended superconformal Chern–
Simons theories with matter with data (g1 , M1 ) and (g2 , M2 ) one can add their lagrangians to
obtain a theory with the same amount of supersymmetry and with data (g1 ⊕g2 , (M1 ⊗K)⊕(K⊗
M2 )), where K = R, C denotes the relevant trivial one-dimensional representation. In otherwords, superconformal Chern–Simons theories admit direct sums and hence there is a notionof indecomposability; namely, an indecomposable theory is one which cannot be decoupled asa direct sum of two nontrivial theories.
For N < 4 indecomposability places very weak constraints on the allowed representations.
For example, if the Chern–Simons Lie algebra g is simple, then any direct sum of nontrivial

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

11

irreducible unitary representations of the right type will give rise to an indecomposable theory,
the Chern–Simons terms acting as the “glue” binding the matter together.
For the N =4 theories of the type discussed by Gaiotto and Witten [10], where the bosonic(4)
matter lives in ∆+ ⊗ W , for W ∈ Rep(g, H)aLTS , indecomposability forces W to be irreducible.
For the general N =4 theories with twisted matter, indecomposability implies the connectednessof the corresponding quiver [9], which imposes conditions — albeit not irreducibility — on theallowed representations.
Finally for N > 4 indecomposability coincides with irreducibility of the matter representation. The matter representations for N > 4 are Lie-embeddable, which means that one canattach a Lie superalgebra to them and hence a triple system by nesting the Lie bracket: the Liebracket of two odd elements is even and its Lie bracket with a third odd element will again be
odd. As discussed in Appendix A.2.7, which is based on [17], the notions of irreducibility of therepresentation agrees with the simplicity of the embedding Lie superalgebra and with that of
the triple system, provided the representation is positive-definite, which is a tacit assumption
in this paper. This allows a classification of positive-definite irreducible matter representations
for N > 4 in terms of Lie superalgebras, which is summarised in Table 6 below, where u(1)charges are denoted by subscripts where appropriate. It should be noted that simplicity ofthe embedding Lie superalgebra does not imply the simplicity of the gauge Lie algebra g andindeed in none of the cases in the table is g allowed to be simple. This fact may explain why
these theories took a relatively long time to be discovered.
N

Class

Representationg

Lie superalgebra

83LA(2, 2)

su(2) ⊕ su(2)A(1, 1)5, 6 aLTS, aJTS (m + 1, n + 1)m−n su(m + 1) ⊕ su(n + 1) ⊕ u(1)A(m, n), m 6= n

5, 6 aLTS, aJTS

(n + 1, n + 1)

su(n + 1) ⊕ su(n + 1)

A(n, n)5, 6 aLTS, aJTS(2n)+1

usp(2n) ⊕ u(1)C(n + 1)5aLTS

(2m + 1, 2n)so(2m + 1) ⊕ usp(2n)B(m, n)5aLTS(2m, 2n)so(2m) ⊕ usp(2n)D(m, n)5

aLTS(2, 2, 2)

su(2) ⊕ su(2) ⊕ su(2)D(2, 1; α)

5

aLTS

(2, 8)

su(2) ⊕ spin(7)

F (4)

5

aLTS

(2, 7)

su(2) ⊕ g2

G(3)Table 6. Irreducible, positive-definite Lie-embeddable representations
It must be remarked that all the matter representations for N > 4 superconformal Chern–
Simons theories in our Table 6 have been found already in [7] via a certain global limit ofconformally gauged supergravities in three dimensions. The gauging can be most conveniently
described in terms of a so-called embedding tensor and it is from the linear constraint imposedon this object by supersymmetry that allows one to identify the different classes of representations in Table 6 with those in Table 3 of [7]. In each case, it is the tensor R defined inAppendix A that corresponds to an R-symmetry-singlet of the embedding tensor in the aforementioned global limit. Thus our classification from first principles establishes that there exist
no other indecomposable N > 4 theories.

12

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

1.4. Superpotentials. The on-shell superconformal Chern–Simons-matter theories we willconsider can all be derived in terms of an off-shell N =1 superspace formalism from a particular
choice of quartic, gauge-invariant superpotential W. For an N -extended superconformal symmetry to be realised in the on-shell theory requires W to be invariant under a global so(N − 1)symmetry. This can be understood as the subalgebra of the so(N ) R-symmetry in the on-shelltheory which preserves the choice of N =1 superspace parameter in the off-shell theory.
The off-shell superfield that will describe the matter content in the on-shell N -extended
superconformal Chern–Simons-matter theory can always be assembled into the representationof so(N − 1) ⊕ g appearing in the third column of Table 5. The g-modules M for 2 ≤ N ≤ 6
are always of complex or quaternionic type while only for N =8 is M of real type. As describedin Appendix A.2.2, associated with any complex or quaternionic unitary representation M
there is canonical complex-sesquilinear map T : M × M → gC defined by the transpose of
the action of g with respect to the hermitian inner product. When M is real, the canonicalreal skewsymmetric map T : M × M → g is defined in Appendix A.2.1 and, as explained inAppendix A.2.3, can be thought of as the real part of T by thinking of the real M as a special
kind of complex representation. Using the inner product (−, −) on g one can thus define thecanonical quartic, g-invariant tensor R = (T(−, −), T(−, −)) on M when it is of complex orquaternionic type and R = (T (−, −), T (−, −)) when M is real. All the superpotentials weshall consider can be expressed in terms of these tensors. However, the fact that we are dealing
with superfields in representations of so(N − 1) ⊕ g rather than just g means that for each2 ≤ N ≤ 8, so(N − 1)-invariance of the superpotential is achieved only after using a particularquartic tensor on the appropriate spinor representation of so(N − 1) appearing in the thirdcolumn of Table 5 to create a singlet.
This structure will be made much more explicit in the rest of the paper but a schematic
picture of what these superpotentials look like can be achieved by writing the superfield Ξ =Ξa ea in terms of the basis {ea } for the spinor representation ∆(N −1) of so(N − 1). (For N(N −1)(N −1)
odd, ∆(N −1) = ∆+⊕ ∆−and this basis is further decomposed in terms of the bases(N −1)(N −1){eα } on ∆+and {eα̇ } on ∆−.) For N =4, 5, 6, 8, the component superfields Ξa arevalued respectively in unitary representations M = W1 ⊕ W2 , W, V, U where W1 , W2 , W ∈Rep(g, H)aLTS , V ∈ Rep(g, C)aJTS and U ∈ Rep(g, R)3LA . For N =2 the single component
superfield Ξ is valued in a generic complex unitary representation whereas for N =3 the single
component superfield Ξ is valued in a generic quaternionic unitary representation and taken to(2)∼have charge 21 corresponding to chiral representation ∆+ of so(2) = u(1).
R
12
The superpotentials can all be expressed as W = 16 d θ W(Ξ), where the measure is for the
N =1 superspace parameter θ, in terms of a real, quartic, so(N − 1) ⊕ g-invariant function W.
Table 7 defines what this function is for all N > 1. The tensor Ω 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 theso(7)-invariant self-dual Cayley 4-form on ∆(7) . Repeated indices are contracted with respect
to the hermitian inner product on ∆(N −1) . The function WF in the N =2 row denotes anarbitrary quartic, g-invariant, holomorphic F-term superpotential that is compatible with N =2
supersymmetry. The superpotential giving rise to N =3 superconformal symmetry is rigid. In
the course of the paper, we will show how one ascends Table 7 realising more global symmetry
for this same superpotential purely as a consequence of restricting attention to increasingly
specialised types of unitary representations for the matter fields.

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

N

W(Ξ)

8

1
Ω
(T (Ξa , Ξb ), T (Ξc , Ξd ))3 abcd6(T(Ξa , Ξb ), T(Ξb , Ξa )) + Ωab Ωcd (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(T1 (Ξa , Ξb ), T1 (Ξb , Ξa )) + 16 (T2 (Ξa , Ξb ), T2 (Ξb , Ξa )) − (T1 (Ξa , Ξb ), T2 (Ξb , Ξa ))6

3(T(Ξ, Ξ), T(Ξ, Ξ)) + Re (T(Ξ, JΞ), T(Ξ, JΞ))

2

(T(Ξ, Ξ), T(Ξ, Ξ)) + Re WF (Ξ)

13Table 7. Superpotentials

2. Generic superconformal Chern–Simons-matter theories
Having summarised the necessary data associated with matter representations of the superconformal algebra in three dimensions, we are now ready to describe some of the physicalproperties of the associated superconformal Chern–Simons-matter theories. This section will
focus on the structure of the generic theories with N ≤ 3 superconformal symmetry. This
will pave the way for our consideration of further supersymmetry enhancement in subsequent
sections. We will begin by describing the generic N =1 superconformal Chern–Simons-mattertheory that is built from any faithful unitary representation of a metric Lie algebra. We thendescribe how supersymmetry is enhanced to N =2 and N =3 when the representation is assumed to be respectively of a generic complex and quaternionic type. Some useful backgroundreferences on this material are [18, 25].
2.1. N=1 supersymmetry in three dimensions. N =1 supermultiplets in three dimensions
are classed as either gauge or matter .
A gauge supermultiplet consists of a bosonic gauge field Aµ and a fermionic Majorana spinor
χ. We will assume that both fields take values in a Lie algebra g that is equipped with an
ad-invariant inner product (−, −). It is sometimes assumed that these fields are valued in
the adjoint representation of a semisimple Lie algebra. This ensures that one can construct asupersymmetric lagrangian for the gauge supermultiplet in terms of the Killing form on the
Lie algebra. This is sufficient but not necessary for the construction of a lagrangian and inseveral important examples we will see that it is necessary for the inner product (−, −) we haveassumed on g to not be the Killing form.
A matter supermultiplet consists of a bosonic scalar field X and an auxiliary field C plus a
fermionic Majorana spinor Ψ. We will assume that all the matter fields take values in a faithfulunitary representation M of g and can be collected into a superfield Ξ = X + θ̄Ψ + 21 θ̄θC where
the superspace coordinate θ is a fermionic Majorana spinor.
Our spinor conventions in three dimensions are as follows. We take the Minkowski metricηµν on R1,2 to have mostly plus signature and the orientation tensor εµνρ such that ε012 = 1.
The Clifford algebra C`(1, 2) has two inequivalent representations, both of which are real andtwo-dimensional. Having chosen one of these representations, the Clifford algebra acts via
2 × 2 real matrices γµ which obey γµ γν + γν γµ = 2ηµν 1 — a suitable choice being γ0 = iσ2 ,γ1 = σ1 and γ2 = σ3 . A Majorana spinor ξ has two real components and we define ξ¯ :=¯ and χ̄γµ ξ = −ξγ¯ µ χ for any fermionic Majorana spinors χ andξ t γ 0 . This implies χ̄ξ = ξχ14

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

ξ. Finally, it will be useful to recall that γµν = εµνρ γ ρ , γµνρ = εµνρ 1 and the Fierz identityξ χ̄ = − 21 [(χ̄ξ)1 + (χ̄γ µ ξ)γµ ].
The coupling of gauge and matter supermultiplets is achieved using the action · of the gmodule on M. The N =1 supersymmetry transformations for the matter and gauge fields areδX = ¯ΨδΨ = −(Dµ X)γ µ  + CδC = −¯γ µ (Dµ Ψ) − ¯χ · X
δAµ = ¯γµ χ(5)δχ = 21 Fµν γ µν ,
where the parameter  is a fermionic Majorana spinor and Dµ φ = ∂µ φ + Aµ · φ for any field
φ valued in M. The derivative Dµ is covariant with respect to the gauge transformations
δφ = −Λ · φ and δAµ = ∂µ Λ + [Aµ , Λ], for any gauge parameter Λ valued in g and where [−, −]denotes the Lie bracket on g. The curvature of this covariant derivative is g-valued and definedby Fµν = [Dµ , Dν ] = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ]. It is easy to check that the commutator of two
supersymmetry transformations in (5) closes off-shell giving a translation on R1,2 plus a gaugetransformation in g. It is worth emphasising that up to this point we need not have assumed
that g is metric nor that the representation M be unitary since neither of the inner products
appear in (5) nor are they required for closure of the supersymmetry algebra. It is in order
to construct a lagrangian that is invariant under these supersymmetry transformations thatnecessitates this extra data.
2.2. N=1 supersymmetric lagrangians in three dimensions. There are three distinct
contributions making up the most general Chern–Simons-matter lagrangian that is invariant
under (5). They will be referred to as the supersymmetric Chern-Simons term LCS , the supersymmetric matter term LM and the superpotential W and will now be discussed in turn.
2.2.1. Supersymmetric Chern-Simons term. Given a gauge supermultiplet (Aµ , χ) valued in the
Lie algebra g with ad-invariant inner product (−, −), the integral of
LCS = −εµνρ Aµ , ∂ν Aρ + 13 [Aν , Aρ ] − (χ̄, χ),(6)
is invariant under the last two supersymmetry transformations in (5). It is, of course, also
manifestly gauge-invariant as a consequence of the ad-invariance of (−, −).
2.2.2. Supersymmetric matter term. Given a matter supermultiplet (X, Ψ, C) valued in a real
representation M of g with an invariant positive-definite symmetric inner product h−, −i, thatis coupled to the aforementioned gauge supermultiplet (Aµ , χ), the supersymmetric matter termis
LM = − 21 hDµ X, Dµ Xi + 21 Ψ̄, γ µ Dµ Ψ + 12 hC, Ci − hX, χ̄ · Ψi .

(7)

Replacing covariant with partial derivatives in the first two terms and dropping the fourth term
would describe the supersymmetric lagrangian for the matter fields in the ungauged theory. As
it is, the integral of (7) is invariant under (5) with the fourth term describing an additional
gauge-matter coupling that is required to cancel the supersymmetry variation of the first threeterms in the gauged theory.

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

15

2.2.3. Superpotential. The integral of LCS + LM from (6) and (7) is classically scale-invariant
with respect to the fields (X,weights ( 12 , 1, 32 , 1, 23 ). One could add
R 2Ψ, C, Aµ , χ) being assigned
1
1
a mass term of the form 2 d θ hΞ, Ξi = hX, Ci − 2 Ψ̄, Ψ which is manifestly supersymmetricbut of course breaks the classical scale invariance.
More generally, one could consider a superpotential
Z
∂
∂2
1 a b
W = d2 θ W(Ξ) = C a
W(X)
−
Ψ̄
Ψ
W(X),
(8)
2
∂X a
∂X a ∂X b
where W is an arbitrary polynomial function on M and the matter superfield components havebeen written relative to a basis {ea } for M. Scale-invariance requires that W must in fact bea quartic function. Notice that the expression (8) does not require an inner product on M.
However, given the gauge couplings to the matter fields in (5), invariance of the integral of thesuperpotential under these supersymmetry transformations requires W to be an g-invariant
function.
Given a g-invariant inner product on M, as was required for the existenceof LM in (7), oneR
has a generic type of quartic g-invariant superpotential proportional to d2 θ hΞ, Ξi2 . However,this can be thought of as arising as a marginal deformation by the square of the g-invariant
operator hΞ, Ξi of an existing classically superconformal N =1 Chern–Simons-matter theory.
Such operators are of course generically unprotected from quantum corrections given only N =1
supersymmetry. We shall not consider superpotentials which take the form of such deformations
in our subsequent analysis.
2.2.4. On-shell N=1 supersymmetric lagrangians. Before discussing the superpotentials leading
to increased amounts of supersymmetry, let us conclude this subsection by noting the on-shellform of the generic N =1 supersymmetric Chern-Simons-matter lagrangian LCS + LM + W,
after integrating out the auxiliary fields χ and C. Their respective equations of motion areχ = 21 T (X, Ψ) and hC, −i = −dW(X) (the terms in the equation for C being thought of asM∗ -valued). Substituting these expressions into the lagrangian gives
LCS + LM + W = − εµνρ Aµ , ∂ν Aρ + 13 [Aν , Aρ ] − 12 hDµ X, Dµ Xi − 12 hdW(X), dW(X)i(9)+ 21 Ψ̄, γ µ Dµ Ψ − 12 Ψ̄a Ψb ∂a ∂b W(X) + 14 (T (X, Ψ̄), T (X, Ψ)).
In a slight abuse of notation, the expression − 21 hdW(X), dW(X)i for the scalar potential isshorthand for − 12 g ab ∂a W(X)∂b W(X) with g ab denoting components of the matrix inverse ofhea , eb i on M. It is straightforward to check that (9) is invariant under the N =1 supersymmetrytransformations (5), upon substituting into their expressions the field equations for the auxiliary
fields χ and C. These supersymmetry transformations areδX = ¯ΨδΨ = −(Dµ X)γ µ  − dW(X)

(10)

δAµ = 21 T (X, ¯γµ Ψ) ,and close up to a translation on R1,2 plus a gauge transformation, using the equations of motion
from (9). Notice the effect of integrating out the auxiliary fields has been to generate a sextic
potential for the scalar fields and various scalar-fermion Yukawa couplings that appear in thesecond line of (9).

16

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

2.3. N=2 supersymmetry for M complex. When the matter representation M = V ∈Rep(g, C), there exists a superpotential
Z
1
WC = 16 d2 θ (T(Ξ, Ξ), T(Ξ, Ξ)).
(11)
Proposition 4 implies that T(Ξ, Ξ) is pure imaginary (indeed T(Ξ, Ξ) = iT (Ξ, IΞ) from Lemma 3),
whence (11) is real. Notice that T (Ξ, Ξ) ≡ 0 and so there is no possibility to build an alternative
superpotential based on the real part T of T here. The value of the coefficient is fixed uniquelyby the requirement that this superpotential gives rise to an enhanced N =2 supersymmetry,when added to LCS + LM . That is, it provides precisely the additional gauge-matter couplings
that are required for N =2 supersymmetry.
To understand why this superpotential gives rise to an enhancement in supersymmetry, it will
be useful to note that one can obtain (11) via integrating out an auxiliary matter supermultiplet.
The auxiliary matter superfield Π = σ − θ̄χ̂ + 12 θ̄θD here is just an N =1 matter superfield thatis valued in g instead of V (the supersymmetry transformations for these fields just follow from(5) by taking the action · of g to be the adjoint action of g on itself). The real superpotentialZ
d2 θ (Π, Π) + 2i (Π, T(Ξ, Ξ)),(12)then gives precisely (11) after integrating out Π. Classical scale invariance here follows fromthe auxiliary components (σ, χ̂, D) being assigned weights (1, 23 , 2).
However, before integrating out Π, notice that adding the first term in (12) to the supersymmetric Chern-Simons term (6) givesZ¯ χ̂) + 2 (σ, D),
LCS + d2 θ (Π, Π) = −εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − (χ̄, χ) − (χ̂,
(13)
N =2
for N =2 supersymmetric Chern-Simons theory. The
which is precisely the lagrangian LCS
supersymmetry enhancement here can be seen to arise from the choice of taking either χ orχ̂ to describe the superpartner of the gauge field Aµ in the N =1 gauge supermultiplet. It
will be convenient to assemble the fermions into a complex spinor χC = χ + iχ̂, whereby¯ χ̂) = (χ̄∗ , χC ), with χ∗ = χ − iχ̂ here denoting the complex conjugate of χC . The(χ̄, χ) + (χ̂,CC
N =2
integral of LCS
in (13) is invariant under the N =2 supersymmetry transformations

δAµ = Re (¯∗C γµ χC )
δχC = 12 Fµν γ µν C + i(Dµ σ)γ µ C − iD Cδσ = −Im (¯∗C χC )(14)

δD = Im (¯∗C (γ µ Dµ χC + i[σ, χC ])) ,where the parameter C is a complex spinor on R1,2 . Associated with this enhanced N =2
supersymmetry there is a u(1) R-symmetry under which χC and C have charge -1 (their complex
conjugates having charge +1) while Aµ , σ and D are uncharged.
Combining the remaining term in (12) with the supersymmetric matter term (7) givesZ
1
LM + 2 d2 θ hΞ, iΠ · Ξi = − 12 hDµ X, Dµ Xi + 12 Ψ̄, γ µ Dµ Ψ + 21 hC, Ci − hX, χ̄∗C · Ψi
(15)
1
1
+ 2 hX, iD · Xi − 2 Ψ̄, iσ · Ψ + hX, iσ · Ci .

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

17

In terms of the redefined auxiliary matter field F := C + iσ · X one finally recovers from (15)the standard off-shell gauged N =2 supersymmetric matter lagrangian
LMN =2 = − 21 hDµ X, Dµ Xi + 12 Ψ̄, γ µ Dµ Ψ + 21 hF, F i − hX, χ̄∗C · Ψi
+ 12 hX, iD · Xi − 21 Ψ̄, iσ · Ψ − 21 hσ · X, σ · Xi ,(16)

whose integral is invariant under the N =2 supersymmetry transformations
δX = ¯∗C ΨδΨ = −(Dµ X)γ µ C + F ∗C − iσ · X C

(17)

µ

δF = −¯C (γ Dµ Ψ + χC · X − iσ · Ψ) ,for the matter supermultiplet, combined with the transformations (14) for the gauge supermultiplet. The matter fields (X, Ψ, F ) have charges ( 21 , − 21 , − 32 ) under the u(1) R-symmetry.
The transformations (17) with C =  real generate an N =1 subalgebra whose transformations
are recovered from (5) in terms of the matter superfields Π ∈ g and Ξ ∈ V defined above
precisely as a consequence of the identification F = C + iσ · X. The off-shell closure of the
N =2 supersymmetry algebra from (14), (17) is straightforward to check.
A rather more economical expression for LMN =2 can be obtained by collecting the fields into
N =2 superfields involving a complex superspace coordinate θC . The matter fields (X, Ψ, F )can be assembled into the chiral N =2 superfield ΞC = X + θ̄C∗ Ψ + 12 θ̄C∗ θC∗ F . The N =2 gauge
supermultiplet fields (Aµ , χC , σ, D) can be assembled into the N =2 superfield operator V =θ̄C∗ γ µ θC Dµ + iθ̄C∗ θC σ − 4i (θ̄C∗ θC∗ )(θ̄C θC )D + 12 (θ̄C∗ θC∗ )θ̄C χC − 21 (θ̄C θC )θ̄C∗ χ∗C . Thus V is pure imaginary.
The superspace coordinate θC has u(1) R-charge -1 and so ΞC has R-charge 21 while V is neutral.
In terms of these quantities, one can writeZ
N =2
LM = d4 θC 12 ΞC , e−V · ΞC ,(18)
up to total derivatives and where d4 θC = d2 θC d2 θC∗ is shorthand for the full N =2 superspacemeasure.
2.3.1. N=2 F-term superpotential. To the N =2 supersymmetric Chern-Simons-matter lagrangian
N =2
LCS+ LMN =2 , one can add a so-called F-term superpotential of the form
Z
Z
2
WF = d θC WF (ΞC ) + d2 θC∗ WF (ΞC )∗ ,(19)
which is also N =2 supersymmetric provided that WF is a g-invariant holomorphic function of
the matter fields. Scale-invariance again requires WF to be a quartic function. Notice that(19) does not require an inner product on V but demands it is complex. Notice also that thechiral superspace measure appearing in (19) guarantees that WF is invariant under the u(1)R-symmetry of the N =2 superalgebra.
One can generally obtain the NR=2 F-term superpotential in (19) off-shell from a particular
N =1 superpotential of the form d2 θ 2 Re WF (Ξ) when M is of complex type, where Ξ =X + θ̄Ψ + 21 θ̄θF is the N =1 superfield on which the the chiral N =2 superfield ΞC is constructed.
The extra data here being precisely the quartic g-invariant holomorphic function WF . It
is worth pointing out that one recovers precisely the same F-term superpotential from theaforementioned N =1 superpotential based on the N =1 superfield Ξ = X + θ̄Ψ + 12 θ̄θC we
had been using before. This follows from the fact that F − C = iσ · X and so the potential
discrepancy between the resulting superpotentials is proportional to hiσ · X, ∂WF (X)i which
vanishes identically as a consequence of WF being g-invariant.

18

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

In summary, we have seen that, when M ∈ Rep(g, C), one can obtain the general N =2
N =2
Chern-Simons-matter lagrangian LCS+ LMN =2 + WF from the choice of N =1 superpotentialZ
1(T(Ξ, Ξ), T(Ξ, Ξ)) + 2 Re WF (Ξ).
(20)
WC + WF = d2 θ 16
Whenever M ∈ Rep(g, C), one can decompose a generic quartic N =1 superpotential into its(4, 0) + (3, 1) + (2, 2) + (1, 3) + (0, 4) components, with respect to the complex structure on M.
Thus we have found that enhancement to N =2 supersymmetry for the N =1 Chern-Simonsmatter lagrangian is guaranteed provided the (3, 1) + (1, 3) component is absent and the (2, 2)
component is WC . Furthermore, as proven in Appendix B.1, the expression in (20) is the uniquechoice of N =1 superpotential giving rise to an on-shell lagrangian which is invariant under theu(1) R-symmetry that is necessary for N =2 supersymmetry.
2.3.2. On-shell N=2 supersymmetric lagrangians. Before going on to look at further types ofsupersymmetry enhancing superpotentials which exist when M is quaternionic, let us conclude
this subsection by noting the on-shell form of the generic N =2 supersymmetric Chern-SimonsN =2
matter lagrangian LCS+ LMN =2 + WF , after integrating out the auxiliary fields χC , D and
F . Their equations of motion are respectively χ∗C = 21 T(X, Ψ), σ = − 4i T(X, X) and hF, −i =−∂WF (X). Substituting these expressions into the lagrangian gives

N =2
LCS
+ LMN =2 + WF = − εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 12 hDµ X, Dµ Xi − VD (X) − VF (X)+ 12 Ψ̄, γ µ Dµ Ψ − 12 Ψ̄a Ψb ∂a ∂b WF (X) − 12 Ψ̄ā Ψb̄ ∂ā ∂b̄ WF (X)∗− 14 (T(X, Ψ̄), T(Ψ, X)) − 18 (T(X, X), T(Ψ̄, Ψ)).
(21)
where we have introduced the positive-definite D-term and F-term sextic scalar potentials
1
VD (X) = 32
hT(X, X) · X, T(X, X) · Xi and VF (X) = 21 h∂WF (X), ∂WF (X)i and the indices
are with respect to a complex basis {ea } on V .
Invariance of (21) under the N =2 supersymmetry transformations (14) and (17) can beestablished after substituting the equations of motion for χC , D and F . These supersymmetrytransformations areδX = ¯∗C ΨδΨ = −(Dµ X)γ µ C − ∂WF (X) ∗C − 41 T(X, X) · X C

(22)

δAµ = 21 T (X, ¯∗C γµ Ψ) ,and close up to a translation on R1,2 plus a gauge transformation, using the equations of motion
from (21).
2.4. N=3 supersymmetry for M quaternionic. Let M = W ∈ Rep(g, H). Recall that
we view quaternionic representations as complex representations with a quaternionic structure
map J. In the language of Appendix A.1.7, we work not with W but with ((W )); although weshall not enforce this notational distinction and talk of fields taking values in W when in fact
they take values in ((W )). In the case of a quaternionic representation W , we have in addition
to T(−, −) based on just the hermitian structure on W , now also the map T(−, J−) in termsof which we will write down a superpotential which provides a further N =3 supersymmetry
enhancement.
The map T(−, J−) is symmetric and complex bilinear in its arguments. Thus one can define
the following quartic, g-invariant, holomorphic N =2 F-term superpotential
1(T(ΞC , JΞC ), T(ΞC , JΞC )),WF (ΞC ) = 32

(23)

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

19

where ΞC is a chiral N =2 matter superfield, just as in the previous section but here taking values
in W . The value of the coefficient is again fixed by the requirement that the F-term superpotenN =2
tial WF for (23) gives rise to an enhanced N =3 supersymmetry, when added to LCS+ LMN =2 .
Equivalently, following the results of the previous section, this F-term superpotential also arises
from the choice of N =1 superpotentialZ
1
WH = 16 d2 θ [(T(Ξ, Ξ), T(Ξ, Ξ)) + Re (T(Ξ, JΞ), T(Ξ, JΞ))]Z
1= 16 d2 θ [−(T (Ξ, IΞ), T (Ξ, IΞ)) + (T (Ξ, JΞ), T (Ξ, JΞ)) − (T (Ξ, IJΞ), T (Ξ, IJΞ))] ,(24)
where Ξ is an N =1 matter superfield, again taking values in W . The superpotential above
WH = WC + WF just describes a special case of the generic expression in (20) for the choice of
F-term superpotential (23) based on the quaternionic structure here.
The enhanced N =3 supersymmetry can be seen using a similar method to that which was employed in the previous subsection for understanding enhancement from N =1 to N =2 supersymmetry. In this case however we note that one can obtain the N =2 F-term superpotential basedon (23) via integrating out an auxiliary chiral N =2 matter superfield ΠC = τC + θ̄C∗ ζC∗ + 21 θ̄C∗ θC∗ EC
that is valued in gC rather than W (the N =2 supersymmetry transformations for these fields
just follow from (17) by taking the action · of g to be the adjoint action of gC on itself). The
F-term superpotential resulting from− 12 (ΠC , ΠC ) − 41 (ΠC , T(ΞC , JΞC )),(25)

then gives precisely (23) after integrating out ΠC . Classical scale invariance here follows fromthe auxiliary components (τC , ζC , EC ) being assigned weights (1, 23 , 2) while their u(1) R-charges
are (1, 0, −1).
Before integrating out ΠC though, notice that adding the F-term superpotential associatedwith the first term in (25) to the N =2 supersymmetric Chern-Simons term (13) gives
Z
Z
N =2
2
1
1
LCS − 2 d θC (ΠC , ΠC ) − 2 d2 θC∗ (Π∗C , Π∗C )
(26)= − εµνρ Aµ , ∂ν Aρ + 13 [Aν , Aρ ] + 2 (σ, D) − (τC , EC ) − (τC∗ , EC∗ )− (χ̄∗C , χC ) + 21 (ζ̄C , ζC ) + 21 (ζ̄C∗ , ζC∗ ),N =3
which is in fact precisely the lagrangian LCS
for N =3 supersymmetric Chern–Simons theory.
This can be seen by recalling that the off-shell N =4 supersymmetry transformations for the
N =2 superfield components (Aµ , χC , σ, D) and (τC , ζC , EC ) above, which collectively comprisean N =4 vector supermultiplet in three dimensions, areδAµ = Re (¯∗C γµ χC + η̄C∗ γµ ζC )

δχC = 21 Fµν γ µν C + i(Dµ σ)γ µ C − iD C+ (Dµ τC∗ )γ µ ηC∗ − i[σ, τC∗ ]ηC∗ − EC ηCδζC = 21 Fµν γ µν ηC + i(Dµ σ)γ µ ηC + (iD − [τC , τC∗ ]) ηC− (Dµ τC∗ )γ µ ∗C + i[σ, τC∗ ]∗C + EC∗ C
δσ = −Im (¯∗C χC + η̄C∗ ζC )
δτC∗ = ¯C ζC − η̄C χC
δD = Im (¯∗C (γ µ Dµ χC + i[σ, χC ]) − η̄C∗ (γ µ Dµ ζC + i[σ, ζC ]) − 2 [η̄C∗ χ∗C , τC∗ ])δEC = −¯C (γ µ Dµ ζC∗ − i[σ, ζC∗ ] + [χC , τC ]) + η̄C∗ (γ µ Dµ χC + i[σ, χC ] − [ζC∗ , τC∗ ]) ,

(27)

20

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

where the parameters C and ηC are complex spinors on R1,2 , with u(1) R-charges −1 and 0
respectively. Upon setting ηC = 0 in (27) one recovers the N =2 supersymmetry transformationsin (14) for (Aµ , χC , σ, D) and those in (17) for the components of the auxiliary superfield ΠC =
τC + θ̄C∗ ζC∗ + 21 θ̄C∗ θC∗ EC . It is not too difficult to check that the supersymmetry transformations in(27) close off-shell for any C and ηC , thus generating an N =4 superconformal algebra. However,invariance of the integral of the Chern–Simons lagrangian in (26) under (27) is only possible
provided Re ηC = 0. Thus it is indeed only invariant under the subset of N =3 supersymmetrytransformations generated by the parameters C and Im ηC in (27).
Let us now consider the remaining contribution coming from adding the F-term superpotential associated with the second term in (25) to the N =2 supersymmetric matter lagrangian(16). This gives
Z
Z
N =2
2
1
1
LM − 2 d θC (ΠC , T(ΞC , JΞC )) + 2 d2 θC∗ (Π∗C , T(JΞC , ΞC ))= − 12 hDµ X, Dµ Xi + 21 hX, iD · Xi − 12 hσ · X, σ · Xi + 12 hF, F i
+ 21
− 21

µ

Ψ̄, γ Dµ Ψ

− 12

(28)

Ψ̄, iσ · Ψ + Ψ̄, χC · X

Ψ̄, τC∗ · JΨ − Ψ̄, ζC · JX − 12 hEC · X, JXi + hF, τC∗ · JXi ,and upon integrating out the auxiliary matter field F , thus fixing F = −τC∗ · JX, one obtains
the N =4 supersymmetric matter lagrangian
LMN =4 = − 12 hDµ X, Dµ Xi + 21 hX, iD · Xi − 12 hσ · X, σ · Xi − 12 hEC · X, JXi − 21 hτC · X, τC · Xi
+ 12 Ψ̄, γ µ Dµ Ψ − 12 Ψ̄, iσ · Ψ + Ψ̄, χC · X − 21 Ψ̄, τC∗ · JΨ − Ψ̄, ζC · JX .
(29)
The integral of LMN =4 is indeed invariant under the on-shell N =4 supersymmetry transformations

δX = (¯∗C + η̄C J)Ψ
δΨ = −γ µ (C − ηC J)Dµ X − (τC∗ · JX)∗C − (τC∗ · X)ηC∗ − i(C − ηC J)σ · X,

(30)

for the matter fields X and Ψ, which comprise an on-shell N =4 hypermultiplet in three dimensions. The transformations in (30) are gauged with respect to the off-shell N =4 vector
supermultiplet described above. It is straightforward to check that the N =4 supersymmetrytransformations (30) combined with (27) close precisely up to the equation of motion
γ µ Dµ Ψ − iσ · Ψ + χC · X − τC∗ · JΨ − ζC · JX = 0,(31)

for the fermionic field Ψ resulting from (29). (We are forced to work on-shell here for the matterfields to realise more than N =2 supersymmetry since in order to do so off-shell would require
the use of rather elaborate harmonic or projective superspace techniques that will be unnecessary for our present analysis.) Notice that one recovers precisely the N =2 supersymmetrytransformations in (17) for the matter fields upon setting ηC = 0 and imposing the equation of
motion F = −τC∗ · JX for the auxiliary matter field F . Similarly one can obtain the full N =4transformations in (30) from two sets of N =2 transformations in (17), one with matter fields(X, Ψ) and parameter C and the other with matter fields (−JX, Ψ) and parameter ηC .
In summary, we have shown that for M = W ∈ Rep(g, H), the N =1 superpotential WH in(24) added to the N =1 Chern–Simons-matter lagrangian LCS +LM (or equivalently the F-term
N =2
superpotential in (23) added to the N =2 Chern–Simons-matter lagrangian LCS+LMN =2 ) gives
N =3
rise to precisely the N =3 Chern–Simons-matter lagrangian LCS+ LMN =4 . A proof is given
in Appendix B.2 that WH in (24) is the unique choice of N =1 superpotential giving rise to an
on-shell lagrangian which is invariant under the usp(2) R-symmetry that is necessary for N =3

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

21

supersymmetry. Thus one establishes that the class of N =3 superconformal Chern–Simonsmatter theories is rigid in the sense that the superpotential function WH is fixed uniquely by
the requirement of N =3 supersymmetry.
It is worth stressing that, were it not for the Chern-Simons term, the quaternionic structure of
W would have allowed an even greater enhancement to N =4 supersymmetry here. The obvious
question is therefore whether there are special kinds of quaternionic unitary representations forwhich the realisation of (at least) N =4 superconformal symmetry is possible? This is indeed thecase and the resulting theories will be detailed in the next section. For N =4 supersymmetry,
W is found to necessarily involve a special class of quaternionic unitary representations related
to anti-Lie triple systems and defined in Appendix A.2.4.
Before moving on to this however, let us first describe the on-shell form of the N =3 Chern–
Simons-matter lagrangian and supersymmetry transformations. The N =3 discussion will thenbe concluded with a description of how the R-symmetry is realised in the Chern–Simons-mattertheory above. This will be useful later when we come to investigate how further supersymmetry
enhancement can occur via embedding this in a larger R-symmetry algebra.
2.4.1. On-shell N=3 supersymmetric lagrangian. Having already integrated out F in order toobtain the matter lagrangian (29), it remains to impose the equations of motion χ∗C = 12 T(X, Ψ),ζC∗ = − 21 T(X, JΨ), σ = − 4i T(X, X) and τC = − 41 T(X, JX) for the respective auxiliary fields
χC , ζC , D and EC in the N =4 vector supermultiplet. Substituting these expressions into thelagrangian gives

N =3
LCS
+ LMN =4 = − εµνρ Aµ , ∂ν Aρ + 13 [Aν , Aρ ] − 12 hDµ X, Dµ Xi − VD (X) − VF (X)+ 21 Ψ̄, γ µ Dµ Ψ − 14 (T(X, Ψ̄), T(Ψ, X)) − 81 (T(X, X), T(Ψ̄, Ψ))
1(T(X, JX), T(Ψ̄, JΨ))
− 81 (T(X, J Ψ̄), T(X, JΨ)) − 16

(32)

1
− 18 (T(JX, Ψ̄), T(JX, Ψ)) − 16
(T(JX, X), T(J Ψ̄, Ψ)),1
hT(X, JX) · X, T(X, JX) · Xi while
where the F-term scalar potential here is VF (X) = 32
VD (X) is just as in (21).
Invariance of (32) under the N =3 supersymmetry transformations (27) and (30) can beestablished after substituting the aforementioned equations of motion for χC , ζC , D, EC and
F . These supersymmetry transformations areδX = (¯∗C + η̄C J)Ψ

δΨ = −γ µ (C − ηC J)Dµ X − 41 T(JX, X) · (∗C J + ηC∗ )X − 14 (C + ηC J)T(X, X) · X
δAµ = 21 T (X, (¯∗C − η̄C∗ J)γµ Ψ) ,1,2

and close up to a translation on R
from (32).

(33)

plus a gauge transformation, using the equations of motion

2.4.2. R-symmetry. Relative to a basis {eα } on C2 , let us denote by v α the components of acomplex vector v which transforms in the defining representation of u(2) acting on C2 . Identifying the complex conjugate with the dual of this vector, the components of the complex
conjugate vector u∗ are written uα with the index downstairs (whereby uα v α is u(2)-invariant
with repeated indices summed). With respect to this basis, εαβ = −εβα denotes the componentof the sp(2, C)-invariant holomorphic 2-form ε = e1 ∧ e2 on C2 . A tensor w in the adjoint representation 3 of usp(2) = u(2) ∩ sp(2, C) can be taken to have complex components wαβ = wβαobeying the reality condition wαβ = εαγ εβδ wγδ . Consequently, the tensor wα β := εβγ wαγ canbe thought of as a skew-hermitian 2 × 2 matrix (in the sense that wα β = −wβ α ) as befits theadjoint representation of usp(2) = su(2). A vector v ∈ W in the defining representation of

22

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

u(2) is in the fundamental representation 2 of usp(2) if it obeys the pseudo-reality condition
Jv α = εαβ v β . Consequently h(uα , v α ) = −εαβ ω(uα , v β ) for any u and v in the fundamental
representation.
Let us now collect the N =4 vector supermultiplet auxiliary fields (χC , ζC , σ, τC , D, EC ) into
the usp(2) tensors






1
∗
∗
∗
∗
iD − 2 [τC , τC ]
χC ζC
iσ
τC
EC
 , ∆α β = 
 , Σα β = ,χαβ = −ζC χ∗C−τC −iσ−EC−iD + 21 [τC , τC∗ ](34)

and the N =4 supersymmetry parameter αβ = 

C

ηC∗
∗C

 which obeys αβ = εαγ εβδ γδ with−ηC
αβ defined as the complex conjugate of αβ . Since they are skew-hermitian, Σ and ∆ are taken
to transform in the 3 of usp(2) while the fermionic matrices χ and  transform in the reducible2 ⊗ 2 = 3 ⊕ 1 representation. Of course, being an N =4 supermultiplet, one can more naturally
denote the matrices above as χαβ̇ , Σα̇ β̇ , ∆α β and αβ̇ in the appropriate representations (2, 2),(1, 3), (3, 1) and (2, 2) of the N =4 R-symmetry algebra so(4) = su(2) ⊕ su(2) (with indices
α and α̇ denoting the fundamental representations of the left and right su(2) = usp(2) factors
in so(4)). The N =3 structure is then recovered by embedding the R-symmetry usp(2) as
the diagonal subalgebra of so(4). Notice that this selects precisely the one supersymmetry
parameter Re ηC which could not preserve the supersymmetric Chern–Simons term in (26) tobe the singlet in the decomposition 2 ⊗ 2 = 3 ⊕ 1 for the N =4 parameter αβ̇ .
In this notation, the N =4 supersymmetry transformations in (27) can be written more
succinctly asδAµ = 21 Re ¯αβ̇ γµ χαβ̇
δχαβ̇ = 21 Fµν γ µν αβ̇ − (γ µ αγ̇ )Dµ Σγ̇ β̇ + 21 αγ̇ [Σγ̇ δ̇ , Σδ̇ β̇ ] + ∆α γ γ β̇δΣα̇ β̇ = ¯γ α̇ χγ β̇ − 12 δβ̇α̇ ¯γ δ̇ χγ δ̇




γ̇ µ
γ̇ µ
β
γ̇
β δ̇
γ̇
1 β
δ∆α = ¯αγ̇ δδ̇ γ Dµ + Σ δ̇ χ − 2 δα ¯γ γ̇ δδ̇ γ Dµ + Σ δ̇ χγ δ̇ .

(35)

The N =3 supersymmetric Chern–Simons term (26) is

N =3
LCS
= −εµνρ Aµ , ∂ν Aρ + 13 [Aν , Aρ ] + (Σα β , ∆β α ) − 61 (Σα β , [Σβ γ , Σγ α ]) − 12 (χ̄βα , χαβ ). (36)
It is worth emphasising that the − 61 Σ3 term is purely to account for the fact the diagonal
elements of ∆ in (34) involve the shifted auxiliary field D + 2i [τC , τC∗ ] rather than just D. The
reason that this shifted definition is useful is that it allows one to obtain from (35) precisely
the N =2 supersymmetry structure with parameter C we have found already by simply setting
the other parameter ηC = 0. Thus, just as in (26), there are no cubic terms involving any of theauxiliary fields in (36). Notice that (36) cannot be expressed invariantly in terms of the vector
supermultiplet fields in the representations of so(4) described above. This is another signal of
only N =3 supersymmetry for the Chern–Simons term.
The N =4 hypermultiplet matter fields X and Ψ can be assembled into the vectors


 
X
Ψ
 , Ψα̇ =   ,Xα = (37)
JX
JΨ

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

23

corresponding respectively to the representations (2, 1) and (1, 2) of so(4). Thus we satisfy
the pseudo-reality conditions JX α = εαβ X β and JΨα̇ = εα̇β̇ Ψβ̇ identically. They can both be
thought of as inhabiting the graph of J in W ⊕ W .
The on-shell N =4 supersymmetry transformations (30) for (37) can now be more compactly
expressed as
δX α = ¯αβ̇ Ψβ̇(38)δΨα̇ = −(γ µ β α̇ )Dµ X β − Σα̇ β̇ · X γ γ β̇ ,and the N =4 supersymmetric matter lagrangian (29) becomes
LMN =4 = − 14 hDµ X α , Dµ X α i + 41
+ 14 Ψ̄α̇ , γ µ Dµ Ψα̇ − 14

D

α

α

β

− 81

D

β̇

γ

β̇

X ,∆ β · X
Σα̇ · X , Σα̇ · XEΨ̄α̇ , Σα̇ β̇ · Ψβ̇ + 21 Ψ̄α̇ , χβ α̇ · X β .

γ

E
(39)

Notice that, for example, huα , v α i = 2 hu, vi = −εαβ Re ω(uα , v β ) for any u, v ∈ W , withuα = (u, Ju) and v α = (v, Jv), and the contraction of α indices here is usp(2)-invariant as aconsequence of h(−, −) being complex sesquilinear. The same applies for contracted α̇ indiceswith respect to the other usp(2) factor in so(4). Thus the matter lagrangian (39) is manifestly
so(4)-invariant as befits the fact that it is N =4 supersymmetric.
The lack of off-shell N =4 supersymmetry and so(4)-invariance for the Chern–Simons term
propagates into the form of the equations of motion for some of the auxiliary fields. In particular,
the equations of motion for χC , ζC , D and EC collect into the following usp(2) representations
χαβ = − 21 T(Ψα , X β ),Σα β = 14 T(X β , X α ),(40)with the indices matching as a consequence of T being a complex sesquilinear map. Theseusp(2)-invariant equations are clearly not so(4)-invariant since they would not make sense after
sprinkling dots commensurate with the so(4) representations that the fields were declared toinhabit above. Notice though that the first equation in (40) would have been so(4)-invariant,with χαβ̇ = − 21 T(Ψβ̇ , X α ), if it had been the transposed vector appearing on the right handside. This seemingly innocuous statement will turn out to be a key feature of realising N =4
supersymmetry to be described in the next section.
Let us now close by noting the manifestly usp(2)-invariant form of the on-shell N =3 lagrangian (32) given by
L N =3 = − εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 14 hDµ X α , Dµ X α i + 14 Ψ̄α , γ µ Dµ Ψα
1− 81 (T(X α , Ψ̄β ), T(Ψα , X β )) − 16
(T(X α , X β ), T(Ψ̄α , Ψβ ))

1
1
+ 384
[T(X α , X β ), T(X β , X γ )], T(X γ , X α ) − 128
T(X α , X β ) · X γ , T(X α , X β ) · X γ ,(41)and the N =3 supersymmetry transformations in (33) which becomeδX α = ¯αβ ΨβδΨα = −(γ µ αβ )Dµ X β − 41 T(X β , X α ) · X γ βγδAµ = 41 T X α , ¯αβ γµ Ψβ .

(42)

3. N>3 superconformal Chern–Simons-matter theories
Having described the structure of N ≤ 3 superconformal Chern–Simons-matter theories
with matter in unitary representations of generic type, we are now in a position to examine in

24

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

more detail the special kinds of matter representations which guarantee further supersymmetry
enhancement.
3.1. N=4 supersymmetry. We shall begin by considering matter representations M = W ∈Rep(g, H)aLTS in the notation of Appendix A.2.4. This class of representation was shown by
Gaiotto and Witten in [10] to give rise to an enhanced N =4 superconformal symmetry. By
substituting this data in the N =3 theory based on the superpotential WH in (24) described in
the previous section, we recover the known examples of N =4 superconformal Chern–Simonsmatter theories. When W ∈ Irr(g, H)aLTS , the resulting class of indecomposable N =4 theories
will be shown in Section 3.1.1 to coincide with those found by Gaiotto and Witten in [10].
When W = W1 ⊕ W2 with W1 , W2 ∈ Rep(g, H)aLTS (but W not necessarily of aLTS class), it
will be shown in Section 3.1.2 that the resulting class of N =4 theories coincide with those foundby Hosomichi, Lee, Lee, Lee and Park in [9] by coupling to so-called twisted hypermultiplets.
We will see how this special type of coupling, which exists only for N = 4 theories, is really
what distinguishes the theories of [10] and [9] and, contrary to what the notation above mightsuggest, it is not merely a question of whether the representations are irreducible or not. For
instance, in terms of the notation in Section 1.1, we will find that the bosonic matter in the
N = 4 theories of [10] and [9] transform under (the appropriate real form of) the representations
(4)
(4)
(4)
∆+ ⊗ W and ∆+ ⊗ W1 ⊕ ∆− ⊗ W2 respectively of so(4) ⊕ g (i.e. only for the theories of [9](4)do both so(4) chiralities ∆± appear).
3.1.1. W ∈ Rep(g, H)aLTS . Such representations W are characterised by the existence of a Lie
superalgebra structure on gC ⊕ W — a fact which was first appreciated and utilised in the context of N =4 superconformal Chern–Simons-matter theories in [10]. Recall that a similar (butdistinct) embedding Lie superalgebra structure on V ⊕ gC ⊕ V characterises a complex unitary
g-module V being in the aJTS class. However, this extreme case is too severe to have meritedconsideration in Section 2.3 simply because it would imply that the minimal superpotential WC
in (11) (that is required to produce the N =2 gauge-matter couplings) must vanish identically.
In the quaternionic case, there is still not too much room for manoeuvre. In particular, noticethat the aLTS cyclicity condition (130) means that T(X, JX) · X = 0 identically for any W valued field X in this special case. Consequently the F-term superpotential in (23) vanishes
identically. Thus the superpotential WH in (24) is equal to the minimal superpotential WC in(11) for W ∈ Rep(g, H)aLTS .
Some crucial identities implied by the aLTS cyclicity condition (130) are(T (X, IX), T (X, IX)) = (T (X, JX), T (X, JX)) = (T (X, IJX), T (X, IJX)),(43)(T (X, IX), T (X, JX)) = (T (X, IX), T (X, IJX)) = (T (X, JX), T (X, IJX)) = 0,

(44)

and

for any W -valued field X. Remarkably these algebraic conditions imply that the minimal
superpotential WH = WC for W is precisely that which was used by Gaiotto and Witten[10], thus allowing the realisation of N =4 superconformal symmetry! Initially it will be moreconvenient to express this superpotential asZ1
WH = − 16 d2 θ (T (Ξ, JΞ), T (Ξ, JΞ)).
(45)

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

25

Writing the W -valued N =1 superfield Ξ appearing above as Ξα = (Ξ, JΞ) in terms of a usp(2)doublet then the superpotential (45) can be reexpressed asZ1
WH = − 48 d2 θ 21 εαβ εγδ (T(Ξα , JΞγ ), T(Ξβ , JΞδ )).
(46)
Of course, on its own, this is not so(4)-invariant because the constituent matter fields X α andΨα̇ naturally transform in the fundamental representations of the two different usp(2) factorsof so(4).
Since we are now beyond the realms of a wieldy off-shell supersymmetric framework, we shallproceed to describe the on-shell N =4 supersymmetric structure of the theory associated with
N =4
the superpotential in (45). The on-shell form (9) of the lagrangian LGW= LCS + LM + WH
is given by

N =4
LGW
= − εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 21 hDµ X, Dµ Xi + 21 Ψ̄, γ µ Dµ Ψ
+ 81 (T (Ψ̄, JΨ), T (X, JX)) + 41 (T (X, J Ψ̄), T (X, JΨ)) + 41 (T (X, Ψ̄), T (X, Ψ))

(47)

1
− 32
hT (X, JX) · X, T (X, JX) · Xi .

This lagrangian can be given a manifestly so(4)-invariant expression as

N =4
LGW
= − εµνρ Aµ , ∂ν Aρ + 13 [Aν , Aρ ] − 14 hDµ X α , Dµ X α i + 41 Ψ̄α̇ , γ µ Dµ Ψα̇1+ 16εαβ εγ̇ δ̇ (T(X α , J Ψ̄γ̇ ), T(X β , JΨδ̇ ))

(48)

1
− 768
([T(X α , X β ), T(X β , X γ )], T(X γ , X α )),which, mutatis mutandis, is indeed the N =4 Gaiotto–Witten lagrangian in [10]. It is worth
stressing that the precise value of the coefficient in the superpotential (45) is what has allowed
the various Yukawa couplings to assemble themselves in an so(4)-invariant manner in the secondline above and has fixed the overall coefficient for this term.
The on-shell N =4 supersymmetry transformations, under which the integral of the lagrangian
N =4
is invariant, are given by
LGWδX = (¯∗C + η̄C J)Ψ
δΨ = −γ µ (C − ηC J)Dµ X − 41 (C − ηC J)T(X, X) · X(49)

δAµ = 21 T (X, (¯∗C + η̄C J)γµ Ψ) ,1,2

and close up to a translation on R plus a gauge transformation, using the equations of motion
from (47). Their so(4)-covariant expressions areδX α = ¯αβ̇ Ψβ̇1δΨα̇ = − γ µ Dµ X β + 12T(X β , X γ ) · X γ β α̇δAµ = 14 T X α , ¯αβ̇ γµ Ψβ̇ .

(50)Comparing first the on-shell N =4 transformations in (49) with the N =3 ones in (33), wenote the following differences. First, the second term in the transformation of Ψ in (33) (thatarose from imposing the equation of motion τC = − 14 T(X, JX)) is absent in (49) which isconsistent with the fact that we have no F-term superpotential here. Second, the sign of theparameter ηC has changed in the second term in the variation of Ψ in (49) relative to (33).
Finally, in the on-shell transformation of Aµ above we have effectively replaced the parameterηC∗ in (33) with −ηC in (49). These changes are all necessary for the realisation of N =4
supersymmetry on-shell and the final change has a natural interpretation based on the remark at
the end of the penultimate paragraph in Section 2.4.2. Namely, it is precisely the on-shell N =4

26

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

transformation for Aµ in (50) that would have resulted from substituting the so(4)-covariant
equation χαβ̇ = − 21 T(Ψβ̇ , X α ) for the auxiliary field into the first line of (35). Moreover, thisequation of motion could be obtained by replacing the usp(2)-invariant expression − 12 (χ̄βα , χαβ )for the auxiliary fermions in the off-shell N =3 Chern–Simons lagrangian (36) with the so(4)invariant term − 12 (χ̄αβ̇ , χαβ̇ ). One way to think of this change is from Wick rotating one of
the four auxiliary fermions changing the so(1, 3)-invariant inner product in the N =3 case to anso(4)-invariant one in the N =4 case. We should stress though that is just a formal conceit inthat there still exists no off-shell formulation of the N =4 theory above that does not involve
more elaborate harmonic or projective superspace methods.
Before moving on, it will be convenient to introduce some notation that will allow us toencapsulate the structure above more compactly. We will define the tensor µα β := 41 T(X α , X β )which transforms in the adjoint of one of the usp(2) factors of so(4). This tensor is precisely the
moment map associated with the action of g on the flat hyperkähler manifold W . It will also be
useful define a superpartner of sorts for the hyperkähler moment map to be ν αβ̇ := 41 T(X α , Ψβ̇ )(whose complex conjugate is ναβ̇ = − 41 T(Ψβ̇ , X α )). These two expressions can be collected into
a useful N =1 superfield moment map M α β := 41 T(Ξα , Ξβ ), albeit only usp(2)-covariant. In
terms of these tensors, the usp(2)-invariant form of the superpotential in (46) is simplyZ
1
WH = 6 d2 θ (M α β , M β α ),(51)
leading to the Gaiotto–Witten lagrangian (48) given by

N =4
LGW
= − εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 14 hDµ X α , Dµ X α i + 41 Ψ̄α̇ , γ µ Dµ Ψα̇1+ (ν̄ αβ̇ , ναβ̇ ) − 12([µα β , µβ γ ], µγ α ),(52)

and the N =4 supersymmetry transformations (50) areδX α = ¯αβ̇ Ψβ̇δΨα̇ = − γ µ Dµ X β + 13 µβ γ · X γ β α̇(53)

δAµ = ¯αβ̇ γµ ν αβ̇ .
3.1.2. W = W1 ⊕ W2 with W1 , W2 ∈ Rep(g, H)aLTS . The quaternionic hermitian structure
on W is defined in the obvious way such that (W, h, J) = (W1 ⊕ W2 , h1 ⊕ h2 , J1 ⊕ J2 ) interms of the corresponding structures on W1 and W2 . The action of g on W is defined by
X · (v1 , v2 ) = (X · v1 , X · v2 ) for any X ∈ g. Consequently the map T = T1 ⊕ T2 decomposes
orthogonally in terms of its restrictions to W1 and W2 ; in other words, T(w1 , w2 ) = 0 for all
w1 ∈ W1 and w2 ∈ W2 .
Demanding that W ∈ Rep(g, H)aLTS (and faithful) is too strong and the resulting theory
decouples into a Gaiotto–Witten theory for W1 and another for W2 . Indeed, consider the aLTS
cyclicity condition (130)T(u, Jv) · w + T(v, Jw) · u + T(w, Ju) · v = 0(54)

for all u, v, w ∈ W . Decomposing this equation on W1 ⊕ W2 shows that it is identicallysatisfied on the individual components W1 and W2 , since they are both aLTS. However, thecontributions from the mixed components imply that T1 (u1 , v1 ) · w2 = 0 and T2 (u2 , v2 ) · w1 = 0,
for all u1 , v1 , w1 ∈ W1 and u2 , v2 , w2 ∈ W2 . Let g1 = T(W1 , W1 ) and g2 = T(W2 , W2 ). Then
g1 acts trivially on W2 and g2 acts trivially on W1 . Since W is faithful, g1 ∩ g2 = 0 and sinceT(W1 , W2 ) = 0, we see that gC = g1 ⊕ g2 . Furthermore the direct sum is orthogonal withrespect to the inner product (−, −). Consequently the superconformal Chern–Simons-matter

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

27

theories based on such data would effectively decouple in terms of distinct Gaiotto–Witten
N =4 theories on W1 and W2 . We will therefore exclude the possibility that W itself be aLTS.
Let us now examine the superpotential WH in (24) for an N =1 matter superfield Ξ valued
in W = W1 ⊕ W2 . Of course, this is no longer identical to the expression in (45) when W is
not aLTS. That said, if one decomposes WH on W into its component parts on W1 and W2 one
obtains a sum of three distinct contributions. Two of these are simply the decoupled superpotentials on the individual components W1 and W2 . Since we have assumed that both thesecomponents are aLTS then clearly these contributions do each agree with the expression (45)for the Gaiotto–Witten superpotentials associated with W1 and W2 individually. In addition
one has a contribution from the mixture of components on W1 and W2 . It is this mixed term
which can be thought of as providing a non-trivial F-term superpotential contribution in the
resulting theory (which is of course absent from the individual W1 and W2 contributions).
The next question is whether this theory can realise N =4 supersymmetry. Recall that thesuperpotential WH in (24) gives rise to an on-shell N =3 superconformal theory with manifest
usp(2) R-symmetry. However, on its own, WH is generically only invariant under the u(1)R-symmetry subalgebra arising from the the N =2 supersymmetric framework from whence
it came. A crucial indicator of the enhanced N =4 supersymmetry in the Gaiotto–Witten
theory is that its superpotential WH enjoys the larger global symmetry usp(2) > u(1) (as wasmade explicit in (46) and (51)). The reason for this is simple. To see why, assume moregenerally that there exists an on-shell N =n superconformal Chern–Simons-matter theory with
so(n) R-symmetry that one is trying to obtain from an off-shell N =1 superspace formalism.
Relative to the resulting N =n theory, clearly ones choice of N =1 superspace breaks the so(n) Rsymmetry down to the so(n − 1) subalgebra preserving the chosen N =1 superspace parameter.
Hence, assuming that there exists an off-shell N =1 superpotential that gives rise to this N =nsuperconformal theory, it must be invariant under precisely this so(n − 1) subalgebra. Thisis exactly what has been described above in the Gaiotto–Witten theory when n = 4. The
realisation of this isotropy subalgebra generating a symmetry of the N =1 superpotential will
be a useful guiding principle for us in our search here and in later sections for theories withenhanced superconformal symmetry.
With this in mind we note that if one expresses the N =1 matter superfield Ξ = (Ξ1 , Ξ2 )valued in W = W1 ⊕ W2 as a usp(2)-doublet in the obvious way, such that Ξα1 = (Ξ1 , J1 Ξ1 ) andΞα2 = (Ξ2 , J2 Ξ2 ), then the superpotential WH is not usp(2)-invariant. The trick is to instead
write the N =1 matter superfield as Ξ = (Ξ1 , J2 Ξ2 ) when evaluating WH on W = W1 ⊕ W2 . Of
course, this may seem rather trivial since it is nothing but a field redefinition of the W2 superfieldcomponent. However, this seemingly innocuous modification allows the superpotential to be
expressed in a manifestly usp(2)-invariant way as
WH |W1 ⊕W2 = WH |W1 +WH |W2 +Wmixed ,

(55)

where
1
WH |W1 = − 16

1
WH |W2 = − 16

Z

Z

2

d θ (T1 (Ξ1 , J1 Ξ1 ), T1 (Ξ1 , J1 Ξ1 )) = 612d θ (T2 (Ξ2 , J2 Ξ2 ), T2 (Ξ2 , J2 Ξ2 )) = 61ZZ

d2 θ ((M1 )α β , (M1 )β α ),(56)

d2 θ ((M2 )α β , (M2 )β α ),

(57)

28

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

and1
Wmixed = − 16Z

d2 θ [2 (T1 (Ξ1 , Ξ1 ), T2 (Ξ2 , Ξ2 ))+(T1 (Ξ1 , J1 Ξ1 ), T2 (J2 Ξ2 , Ξ2 )) + (T1 (J1 Ξ1 , Ξ1 ), T2 (Ξ2 , J2 Ξ2 ))]

Z
=−

(58)

d2 θ ((M1 )α β , (M2 )β α ),with the usp(2)-doublet superfields Ξα1 = (Ξ1 , J1 Ξ1 ), Ξα2 = (Ξ2 , J2 Ξ2 ) as before and (M1 )α β =1T (Ξα1 , Ξβ1 ), (M2 )α β = 41 T2 (Ξα2 , Ξβ2 ). The unmixed term WH |W1 is insensitive to the dis4 1
tinction between choosing the superfield to be Ξ1 or J1 Ξ1 on W1 (since T1 (J1 Ξ1 , J12 Ξ1 ) =−T1 (J1 Ξ1 , Ξ1 ) = T1 (Ξ1 , J1 Ξ1 )) and likewise for the unmixed W2 contribution. By the same
reasoning, the mixed term Wmixed would be the same were we to choose either (Ξ1 , J2 Ξ2 ) orJ(Ξ1 , J2 Ξ2 ) = (J1 Ξ1 , −Ξ2 ) as the matter superfield. The significant point here is that there is arelative factor of J between the W1 and W2 superfield contributions. This is necessary so thatthe terms in the second line of (58) take the form z1 z2∗ + z1∗ z2 , rather than the generic formz1 z2 + z1∗ z2∗ predicated on (24), thereby combining with the first term in a way that is invariant
under usp(2) = so(3), under which the superfield moment maps M1 and M2 both transformin the adjoint.
N =4= LCS + LM + WH |W1 ⊕W2 based on the
The on-shell form (9) of the lagrangian LH3LP
superpotential (55) can, after some manipulations, be expressed as− εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 21 hDµ X1 , Dµ X1 i1 − 21 hDµ X2 , Dµ X2 i2+ 21 Ψ̄1 , γ µ Dµ Ψ1 1 + 12 Ψ̄2 , γ µ Dµ Ψ2 2
+ 18 (T1 (Ψ̄1 , J1 Ψ1 ), T1 (X1 , J1 X1 )) + 14 (T1 (X1 , J1 Ψ̄1 ), T1 (X1 , J1 Ψ1 )) + 41 (T1 (X1 , Ψ̄1 ), T1 (X1 , Ψ1 ))+ 81 (T2 (Ψ̄2 , J2 Ψ2 ), T2 (X2 , J2 X2 )) + 14 (T2 (X2 , J2 Ψ̄2 ), T2 (X2 , J2 Ψ2 )) + 41 (T2 (X2 , Ψ̄2 ), T2 (X2 , Ψ2 ))
+ 81 (T1 (Ψ̄1 , Ψ1 ), T2 (X2 , X2 )) + Re (T1 (Ψ̄1 , J1 Ψ1 ), T2 (J2 X2 , X2 ))+ 81 (T2 (Ψ̄2 , Ψ2 ), T1 (X1 , X1 )) + Re (T2 (Ψ̄2 , J2 Ψ2 ), T1 (J1 X1 , X1 ))+ 21 Re (T1 (X1 , Ψ̄1 ), T2 (X2 , Ψ2 )) + (T1 (X1 , J1 Ψ̄1 ), T2 (J2 X2 , Ψ2 ))11− 32
hT1 (X1 , J1 X1 ) · X1 , T1 (X1 , J1 X1 ) · X1 i1 − 32
hT2 (X2 , J2 X2 ) · X2 , T2 (X2 , J2 X2 ) · X2 i211
hT1 (X1 , X1 ) · X2 , T1 (X1 , X1 ) · X2 i2 − 32
hT2 (X2 , X2 ) · X1 , T2 (X2 , X2 ) · X1 i1
− 32
1
1
− 64
hT1 (X1 , J1 X1 ) · X2 , T1 (X1 , J1 X1 ) · X2 i2 − 64
hT2 (X2 , J2 X2 ) · X1 , T2 (X2 , J2 X2 ) · X1 i111
hT1 (J1 X1 , X1 ) · X2 , T1 (J1 X1 , X1 ) · X2 i2 − 64
hT2 (J2 X2 , X2 ) · X1 , T2 (J2 X2 , X2 ) · X1 i1 .
− 64
(59)
Despite appearances, this lagrangian has some neat and familiar structure. Notice first that thefirst four lines together with the eighth line in (59) gives just the decoupled matter contributions
from the individual components W1 and W2 to the on-shell Gaiotto–Witten lagrangian (47).
Thus, as we saw in the previous section, these terms can certainly be given a manifestly so(4)invariant expression on their own.
In order to write the remaining mixed terms in an so(4)-invariant manner, let us declare thematter fields X1 , X2 , Ψ1 and Ψ2 to transform in the following representations of so(4):








X1
Ψ
X
Ψ
 , Ψα̇ =  1  , X̃α̇ =  2  , Ψ̃α =  2  ,Xα = 
(60)
J1 X1
J1 Ψ1
J2 X2
J2 Ψ2

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

29

whereby J1 X α = εαβ X β , J1 Ψα̇ = εα̇β̇ Ψβ̇ , J2 X̃α̇ = εα̇β̇ X̃β̇ and J2 Ψ̃α = εαβ Ψ̃β identically. The
representations (2, 1) and (1, 2) for the bosonic and fermionic matter fields in W1 is just as wewould expect from (37) if they are to comprise an N =4 hypermultiplet. The fact that we havedefined the matter fields in W2 to transform in the opposite representations of so(4) stems fromthe fact that we required the matter superfield in W2 to be J2 Ξ2 rather than Ξ2 in order toobtain the usp(2)-invariant superpotential (55). Of course, this is still isomorphic to an N =4
hypermultiplet representation on W2 though it is clearly useful to distinguish between these
two types of so(4) representations and the latter is often referred to as a twisted N =4 hypermultiplet in the literature. The N =4 supersymmetry transformations for a twisted hypermultiplet follow by acting with the quaternionic structure J on the untwisted N =4 hypermultiplet
transformations in (49) and then absorbing the factor of J into the definition of the twisted
hypermultiplet matter fields X̃ and Ψ̃. In terms of the subsequent so(4)-covariant forms of the
N =4 supersymmetry transformations in (50) and (53), the corresponding prescription for goingfrom an untwisted to a twisted hypermultiplet consists of switching all upstairs/downstairs anddotted/undotted indices followed by relabelling all the fields with tildes.
Given the so(4) representations in (60), the lagrangian (59) can indeed be given the more
succinct and manifestly so(4)-invariant expression
D
E

N =4
LH3LP
= − εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 41 hDµ X α , Dµ X α i − 14 Dµ X̃α̇ , Dµ X̃α̇
D
E
¯
µ
α
µ
α
1
1
+ 4 Ψ̄α̇ , γ Dµ Ψα̇ + 4 Ψ̃ , γ Dµ Ψ̃¯ β , Ψ̃α )) + 1 (µ̃ β̇ , T(Ψ̄ , Ψ ))+ (ν̄ αβ̇ , ναβ̇ ) + (ν̃¯α̇β , ν̃ α̇β ) + 4 (ν̄ αβ̇ , ν̃β̇α ) + 41 (µα β , T(Ψ̃
α̇
α̇
β̇
4
1
1
− 12
([µα β , µβ γ ], µγ α ) − 12([µ̃α̇ β̇ , µ̃β̇ γ̇ ], µ̃γ̇ α̇ )DE
DE
− 81 µα β · X̃γ̇ , µα β · X̃γ̇ − 81 µ̃α̇ β̇ · X γ , µ̃α̇ β̇ · X γ ,

(61)
β̇

:= 14 T(X̃α̇ , X̃β̇ ) and ν̃α̇β := 14 T(X̃α̇ , Ψ̃β ) in keeping with the notation

where we have defined µ̃α̇
introduced at the end of the last section. (To avoid cluttering the expression unnecessarily, we
have omitted the subscripts denoting operations involving W1 and W2 since this should be
obvious from whether the term involves tildes or not.) Mutatis mutandis, the lagrangian (61)is precisely that found already in [9] via the coupling of a twisted and untwisted hypermultiplet
in a way that is compatible with N =4 superconformal symmetry.
N =4
The integral of LH3LP
is indeed invariant under the N =4 supersymmetry transformations
δX α = ¯αβ̇ Ψβ̇δ X̃α̇ = ¯β α̇ Ψ̃βδΨα̇ = − γ µ Dµ X β + 31 µβ γ · X γ β α̇ + µ̃α̇ β̇ · X α αβ̇hiδ Ψ̃α = − γ µ Dµ X̃β̇ + 13 µ̃β̇ γ̇ · X̃γ̇ αβ̇ + µα β · X̃α̇ β α̇δAµ = ¯αβ̇ γµ ν αβ̇ + ν̃ β̇α ,(62)which close up to a translation on R1,2 plus a gauge transformation, using the equations of
motion from (61).
In summary, we have established that the superpotential WH in (24), which genericallyguarantees N =3 supersymmetry when M = W is quaternionic unitary, also describes the onshell theories found in [9, 10] with enhanced N =4 superconformal symmetry for the specific30

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

types of W involving the aLTS class assumed above. Of course, this is to be expected fromthe point of view that a theory with N =4 superconformal symmetry is a special kind of N =3
theory, whose generic superpotential WH is rigid. On the other hand, for example, in the
Gaiotto–Witten theory we have seen that the associated N =4 supersymmetry transformationsin (49) do not simply follow from the those in (33) for the generic N =3 theory by restricting
W to be aLTS. Hence the aforementioned deduction is perhaps not so trivial as it may appear.
Moreover, it is worth stressing that the guiding principle that has led us to the N =4 theories
in [9, 10] has simply been to look for special cases of quaternionic unitary g-modules for which
the superpotential WH can be written in a usp(2)-invariant way.
Let us conclude by detailing the possible gauge-theoretic structures on which the N =4 theories of [9, 10] described above can be based. In the case of the N =4 Gaiotto–Witten theory described in Section 3.1.1, the fundamental ingredients describing indecomposable N =4
Gaiotto–Witten theories are W ∈ Irr(g; H)aLTS . By Theorem 11, W ∈ Irr(g; H)aLTS are in
one-to-one correspondence with metric complex simple Lie superalgebras gC ⊕ W . These havebeen classified and thus one has an indecomposable N =4 Gaiotto–Witten theory for each of
the classical complex simple Lie superalgebras whose odd component admits a quaternionic
structure: namely, A(m, n), B(m, n), C(n + 1), D(m, n), F (4), G(3) and D(2, 1; α).
For the more general N =4 superconformal Chern–Simons-matter theories of [9] described in
Section 3.1.2, let us begin by considering the simplest nontrivial setup wherein W = W1 ⊕ W2
with W1 , W2 ∈ Irr(g, H)aLTS . Theorem 11 allows us to attach to W1 and W2 complex metricsimple Lie superalgebras G1 and G2 , respectively. Despite W = W1 ⊕ W2 being a faithful representation of g, the irreducible aLTS constituents W1 and W2 need not be. Thus
the real forms g1 and g2 of the even parts of G1 and G2 need not be isomorphic to g itself (even though g1 and g2 do collectively span g). The special case of W1 ∼= W2 where∼∼∼
G1 = G2 and thus g1 = g2 = g will be the topic of the next section where it will beshown to give rise to an enhanced N =5 superconformal symmetry. In order that the coupling terms between the untwisted and twisted hypermultiplets in the N =4 lagrangian (61)are non-vanishing, such that we obtain an indecomposable theory, it is necessary for thesemisimple Lie algebras g1 and g2 to have at least one common simple factor. Since both
G1 and G2 are simple, they must be one of the classical Lie superalgebras listed at the end of
the last paragraph. Thus, from the regular simple Lie superalgebras, one may choose any ofthe pairs (G1 , G2 ) = (A(m, p), A(n, p)), (B(m, p), B(n, p)), (B(p, m), B(p, n)), (B(m, n), C(n +1)), (B(m, p), D(n, p)), (C(m+1), C(m+1)), (D(m, n), C(n+1)), (D(m, p), D(n, p)), (D(p, m), D(p, n))and in each case identify the simple Lie algebra factor they have in common. Clearly this technique can be continued to incorporate all the exceptional Lie superalgebras too as well as the
additional possibilities which follow from utilising the various low-dimensional Lie algebra isomorphisms. However, it will not be useful for us to elaborate further on these other possibilities.
The generalisation of this construction when either of the aLTS constituents W1 or W2 isreducible is straightforward. For example, assume that the untwisted hypermultiplet matteris in an irreducible aLTS W1 but the twisted hypermultiplet matter is taken to be in a reducible aLTS of the form W2 ⊕ W3 where both W2 and W3 are irreducible aLTS representations.
Associated with W1 , W2 and W3 we have three simple Lie superalgebras G1 , G2 and G3 andthe construction above can be employed for the two pairs (G1 , G2 ) and (G1 , G3 ) such that theordered triple (G3 , G1 , G2 ) is constrained only by the requirement that adjacent simple Lie superalgebras must have identified at least one simple Lie algebra factor in their even components(e.g. (G3 , G1 , G2 ) could be (A(m, p), A(p, q), A(q, n)) or (B(m, p), D(q, p), D(q, n)) to name but
two of many possibilities). The most general situation can be described such that the faithful

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

31

L
reducible representation takes the form W = ni=1 Wi in terms n irreducible aLTS representations {Wi | i = 1, ..., n} (with n associated simple Lie superalgebras {Gi | i = 1, ..., n}) where,
for convenience, one can assume there is a relative twist between the hypermultiplet matter inadjacent Wi and Wi+1 for i = 1, ..., n−1. Thus one has an indecomposable N =4 superconformal
Chern–Simons-matter theory for each ordered n-tuple of simple Lie superalgebras (G1 , ..., Gn )such that each adjacent pair have identified at least one simple Lie algebra in their even parts
and where there is a relative twist between the hypermultiplet matter in adjacent pairs. (Thus,
when n is even, there is also the possibility of identifying a simple Lie algebra factor in theeven parts of G1 and Gn at the extremities.) Of course, there are many possibilities but mostof the generic ones, described first in [9], involve chains of simple Lie superalgebras of the sameclassical type. To the best of our knowledge, these possibilities comprise all the known examplesof N =4 superconformal Chern–Simons-matter theories.
3.2. N=5 supersymmetry for W1 ∼= W2 ∈ Rep(g, H)aLTS . As discussed at the end of Section 1.2 and particularly in the diagram (2), to obtain a theory with enhanced N =5 superconformal symmetry, we reconsider the N =4 theory described in Section 3.1.2 but now we willassume that W1 , W2 ∈ Rep(g, H)aLTS are isomorphic or, in other words, that the matter content
is now taken to transform under two copies of W ∈ Rep(g, H)aLTS . This prescription follows [6]wherein a new class of N =5 superconformal Chern–Simons-matter theories is constructed inprecisely this way, as a special case of the class of N =4 theories they had found previously
in [9] by coupling to a twisted hypermultiplet.
We shall therefore employ the notation of the previous section and identify W1 ∼=W
= W2 ∼as quaternionic unitary g-modules. The untwisted and twisted hypermultiplet matter fields
are X = X1 , Ψ = Ψ1 , X̃ = X2 and Ψ̃ = Ψ2 , relative to their counterparts in the previous
section, and each field here takes values in W . They can be assembled into a pair of W -valued
N =1 matter superfields Ξ and Ξ̃ in the usual way. It is important to stress that, in identifying
W1 ∼= W2 ∼= W , one is no longer obliged to take h(W1 , W2 ) = 0 which followed from defining
W1 ⊕W2 as an orthogonal direct sum with respect to the hermitian inner product. Indeed, whatis needed to describe the N =5 theory when identifying W1 ∼= W2 ∼= W is to simply evaluate
all the inner products involving the matter fields appearing in Section 3.1.2 using the single
hermitian inner product h on W .
With respect to this structure, the expression in (55) for the superpotential WH readsZh1
WH = 16 d2 θ (T(Ξ, Ξ) − T(Ξ̃, Ξ̃), T(Ξ, Ξ) − T(Ξ̃, Ξ̃))
(63)i
−2 (T(Ξ, Ξ̃), T(Ξ, Ξ̃)) − 2 (T(Ξ̃, Ξ), T(Ξ̃, Ξ)) ,where T is the map associated with W . The second line contains the contribution fromthe F-term superpotential and has been simplified using the identity (T(u, Ju), T(Jv, v)) =2 (T(u, v), T(u, v)), for any u, v ∈ W , which follows using the aLTS cyclicity condition (130) forW.
Whereas the superpotential WH in (55) for W1 ⊕ W2 could only be given a usp(2)-invariant
expression, we will now show that the superpotential in (63) is actually so(4)-invariant. This
is to be expected of course if the theory is to have an N =5 superconformal symmetry since
by writing the on-shell theory in terms of N =1 superfields one necessarily breaks the so(5)R-symmetry down to an so(4) isotropy subalgebra preserving the choice of N =1 superspaceparameter. However, since the enhanced so(4)-invariance of (63) (relative to the manifestly
usp(2)-invariant expression in (55)) is not immediately apparent, it will be enlightening to see
explicitly how this works. The trick is to not immediately try to write the hypermultiplet

32

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

matter fields in terms of the representations of so(4) that appeared in (60) for the N =4 theory.
Instead, one must first define the linear combinations Ξ± := √12 Ξ ± Ξ̃ of the W -valued
N =1 superfields above (such combinations do not exist on W1 ⊕ W2 ). One then defines Ξα+ =(Ξ+ , JΞ+ ) to transform in the (2, 1) representation of so(4) while Ξ− α̇ = (Ξ− , JΞ− ) is definedto transform in the opposite (1, 2) representation. In terms of these combinations of N =1
matter superfields, the superpotential (63) takes a manifestly so(4)-invariant form given byZ
1
WH = 16 d2 θ (T(Ξα+ , Ξ− β̇ ), T(Ξ− β̇ , Ξα+ ))
(64)
i
β
β
α
α
1
1
− 6 (T(Ξ+ , Ξ+ ), T(Ξ+ , Ξ+ )) − 6 (T(Ξ− α̇ , Ξ− β̇ ), T(Ξ− β̇ , Ξ− α̇ )) .
Let us now consider the on-shell form (9) of the lagrangian L N =5 = LCS + LM + WH
based on the superpotential in (63). Not surprisingly, this gives precisely the lagrangian in(59) after identifying W1 ∼= W2 ∼= W . Thus it can subsequently also be written just asin (61) in terms of the so(4) representations defined in (60). As befits the expected N =5
superconformal symmetry of this on-shell lagrangian, one can assemble the constituent matterfields into representations of the expected so(5) = usp(4) R-symmetry algebra. We shall define
representations of usp(4) via straightforward extension of the way we defined representations ofusp(2). That is, relative to a basis {eA } on C4 , we denote by v A the components of a complex
vector v transforming in the defining representation of u(4) (while components of the complex
conjugate vector v ∗ have a downstairs index). With respect to this basis, we take the sp(4, C)invariant complex symplectic form to be Ω = e1 ∧ e2 + e3 ∧ e4 . A vector v ∈ W in the defining
representation of u(4) is in the fundamental representation 4 of usp(4) = u(4) ∩ sp(4, C) if itobeys Jv A = ΩAB v B . The embedding of so(4) in so(5) we shall require corresponds to fixing
a subalgebra usp(2) ⊕ usp(2) < usp(4) which defines a decomposition of C4 = C2 ⊕ C2 withthe two usp(2) factors in so(4) acting on the respective C2 components (for convenience wewill identify {eα } with e1 and e2 and {eα̇ } with e3 and e4 ). In terms of this embedding, we
assemble the matter fields into
 
 
α
Ψ̃α
X
(65)
X A =   , ΨA =   ,X̃α̇Ψα̇with both bosons and fermions transforming in the fundamental representation of usp(4) since
the pseudo-reality conditions JX A = ΩAB X B and JΨA = ΩAB ΨB are identically satisfied.
In terms of the representations in (65), the on-shell lagrangian takes the manifestly usp(4)invariant expression

L N =5 = −εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 41 Dµ X A , Dµ X A + 14 Ψ̄A , γ µ Dµ ΨA1
3+ (ν̄ A B , νA B ) + 2 (ν̄ A B , ν B A ) + 15([µA B , µB C ], µC A ) − 40
µA B · X C , µA B · X C , (66)where we have defined µA B := 14 T(X A , X B ) and ν A B := 14 T(X A , ΨB ). Mutatis mutandis, thelagrangian (66) is indeed precisely the one for the N =5 superconformal Chern–Simons-mattertheory found in [6] and its integral is invariant under the N =5 supersymmetry transformations
δX A = ¯A B ΨBδΨA = − γ µ Dµ X B + 31 µB C · X C B A + 23 µA B · X C C B
B

A

δAµ = ¯A γµ ν B ,(67)

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

33

where the complex N =5 supersymmetry parameter AB := ΩBC A C is skewsymmetric AB =−BA , symplectic traceless ΩAB AB = 0 and obeys the reality condition AB = ΩAC ΩBD CD ,thus describing five linearly independent Majorana spinors on R1,2 .
We conclude this section by summarising the consequences of the (ir)reducibility of W for the
N =5 theory. If W ∈ Rep(g, H)aLTS is reducible, so that W = W1 ⊕ W2 , then as discussed in thesecond paragraph of Section 3.1.2 the N =5 lagrangian in (66) decouples into the sum of N =5
lagrangians on the individual irreducible components. (Recall that W1 , W2 ∈ Rep(g, H)aLTS
since so is W .) The potential mixed terms in the lagrangian vanish identically as a consequenceof the mixed components for the different irreducible factors in the aLTS cyclicity condition
for W . It is therefore enough to consider W ∈ Irr(g, H)aLTS . As stated in Proposition 2,
if W ∈ Irr(g, H)aLTS , the underlying complex representation ((W )) ∈ Rep(g, C) is irreducible
unless W = VH , so that ((W )) ∼= V ⊕V for some V ∈ Irr(g, C). This case will be examined in thenext section where it will be found to give rise to an enhanced N =6 superconformal symmetry.
Thus we deduce that the indecomposable Chern–Simons-matter theories with precisely N =5
superconformal symmetry are in one-to-one correspondence with W ∈ Irr(g, H)aLTS for which((W )) ∈ Irr(g, C), which according to Theorem 11, are in turn in one-to-one correspondence withcomplex simple Lie superalgebras gC ⊕ W . Such Lie superalgebras have been classified and are
given by the classical simple Lie superalgebras A(m, n), B(m, n), C(n+1), D(m, n), F (4), G(3)and D(2, 1; α). This list exhausts the examples of N =5 superconformal Chern–Simons-mattertheories that have been obtained already in [6, 7].
3.3. N=6 supersymmetry for W = V ⊕ V with V ∈ Irr(g, C)aJTS . Following again theprescription in [6], and as discussed in Section 1.2 and particularly diagram (3), one can enhance
the supersymmetry of an N =5 theory to N =6 by considering matter representations W ∈Irr(g, H)aLTS which are quaternionifications W = VH of some V ∈ Irr(g, C), so that ((W )) ∼=V ⊕ V . As shown in Proposition 5(i), this means that in fact V ∈ Irr(g, C)aJTS .
Let us now investigate the structure of the superpotential WH in (63) for W = V ⊕ V . (Thequaternionic unitary structure associated with this representation is defined in the proof of
Proposition 5(i) in Appendix A.2.5.) The constituent N =1 matter superfields will be writtenΞ = (Ξ1 , Ξ̄2 ) and Ξ̃ = (Ξ3 , Ξ̄4 ) in terms of four N =1 superfields Ξ1 , Ξ2 , Ξ3 , Ξ4 ∈ V . In terms ofthese superfields, the superpotential (63) becomesZ1
WH = − 8 d2 θ (T(Ξ1 , Ξ1 ), T(Ξ2 , Ξ2 )) + (T(Ξ1 , Ξ1 ), T(Ξ3 , Ξ3 )) − (T(Ξ1 , Ξ1 ), T(Ξ4 , Ξ4 ))−(T(Ξ2 , Ξ2 ), T(Ξ3 , Ξ3 )) + (T(Ξ2 , Ξ2 ), T(Ξ4 , Ξ4 )) + (T(Ξ3 , Ξ3 ), T(Ξ4 , Ξ4 ))
+2 (T(Ξ1 , Ξ2 ), T(Ξ4 , Ξ3 )) + 2 (T(Ξ2 , Ξ1 ), T(Ξ3 , Ξ4 )) ,(68)
where T is the map associated with V and the aJTS skewsymmetry condition T(x, y) · z =−T(z, y) · x has been used.
The third line in (68) represents the contribution from the F-term superpotential which
was shown to admit a manifestly so(4)-invariant expression for the N =6 theory in [4]. To
demonstrate how this works, and mimicking the nomenclature in [4], let us collect the four
V -valued N =1 superfields into the two pairs A = (Ξ1 , Ξ4 ) and B = (Ξ2 , −Ξ3 ). It will be
convenient to take the pairs A and B to transform separately in the fundamental representationof two different copies of sp(2, C). With respect to the orthonormal bases {eα } and {eα̇ }associated with these two fundamental representations, we take the respective sp(2, C)-invariant
symplectic forms to be ε = e1 ∧ e2 and ε̃ = e1̇ ∧ e2̇ . In terms of this structure, the third line

34

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

in (68) can be written as
18Z

d2 θ εαβ ε̃α̇β̇ Re (T(Aα , B α̇ ), T(Aβ , B β̇ )).

(69)

which is manifestly invariant under sp(2, C) ⊕ sp(2, C). The addition of the kinetic terms forthe matter fields to this F-term superpotential however breaks each sp(2, C) down to usp(2)(since both h(Aα , Aα ) and h(B α̇ , B α̇ ) must also be invariant). Hence the resulting symmetry
realised by (69) is indeed usp(2) ⊕ usp(2) ∼= so(4).
Of course, for an N =6 superconformal Chern–Simons-matter theory written in terms of N =1
superfields we expect that the full superpotential should be invariant under the so(5) isotropy
subalgebra of the so(6) R-symmetry preserving our choice of N =1 superspace parameter. This
is indeed the case and follows by assembling the superfields into the array (Ξ1 , Ξ2 , Ξ3 , Ξ4 )which is to be thought of as a V -valued element of C4 whose components we denote by ΞA withrespect to a basis {eA } on C4 . This C4 is to be equipped with an action of sp(4, C) such that ΞAtransforms in the fundamental representation. We will define the sp(4, C) subalgebra as thosecomplex linear transformations which preserve the complex symplectic form Ω = e1 ∧e3 +e2 ∧e4 .
The full superpotential in (68) then takes the explicitly sp(4, C)-invariant formZ
1
WH = − 16 d2 θ δAC δBD − ΩAB ΩCD (T(ΞA , ΞC ), T(ΞB , ΞD )).
(70)
Again, the addition of the kinetic terms for the matter fields breaks this sp(4, C) symmetry
down to the expected usp(4) ∼= so(5) (since h(ΞA , ΞA ) must also be invariant).
Let us now consider the on-shell form (9) of the lagrangian L N =6 = LCS + LM + WH basedon the superpotential in (68). This simply amounts to rewriting (66) for the special case of
W = V ⊕ V in a form which is explicitly invariant under the so(6) ∼= su(4) R-symmetry of theN =6 superconformal algebra. To this end, let us begin by writing the untwisted and twisted
hypermultiplet matter fields from the original N =4 theory as X = (X1 , X2 ), X̃ = (X3 , X4 ),Ψ = (Ψ4 , −Ψ3 ) and Ψ̃ = (Ψ2 , −Ψ1 ) on W = V ⊕ V in terms of the four V -valued bosons(X1 , X2 , X3 , X4 ) and fermions (Ψ1 , Ψ2 , Ψ3 , Ψ4 ) whose components we will denote by XA andΨA respectively. This is a convenient parametrisation inasmuch as it allows one to assemble
the components into the usp(4)-covariant expressions
X A = (XA , ΩAB XB ),ΨA = (ΩAB ΨB , −ΨA ),(71)

with the components on the left hand sides being defined just as in (65). The pseudo-reality
conditions JX A = ΩAB X B and JΨA = ΩAB ΨB are thus identically satisfied, with no constraint on XA and ΨA , from the definition of J acting on V ⊕ V . The expressions in (71)
can be understood as describing the canonical embedding of the fundamental representationof usp(4) into the real form of the fundamental representation of su(4), wherein XA and ΨArespectively transform in the complex representations corresponding to the fundamental 4 andantifundamental 4̄ of su(4).
In terms of these representations, the on-shell lagrangian (66) can be given the following
su(4)-invariant expression

L N =6 = − εµνρ Aµ , ∂ν Aρ + 31 [Aν , Aρ ] − 12 Dµ XA , Dµ XA + 12 Ψ̄A , γ µ Dµ ΨA+ 2 (ν̄ AB , ν AB − 2 ν BA ) + εABCD (ν̄ AC , ν BD ) + εABCD (ν̄ AC , ν BD )
A

B

C

A

C

A

+ 32 ([µ B , µ C ], µ A ) − 21 µ B · X , µ B · X

C

(72)

,

where we have defined µA B := 41 T(XA , XB ) and ν AB := 41 T(XA , ΨB ) in terms of the map
T associated with V and ε = e1 ∧ e2 ∧ e3 ∧ e4 is the su(4)-invariant 4-form with respect an

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

35

orthonormal basis {eA } on the dual of C4 . Under the embedding of usp(4) in su(4) describedabove, the moment maps here are related to their N =5 counterparts defined below (66) suchthat µA B = µA B −ΩAC ΩBD µD C and ν A B = ΩBC ν AC −ΩAC ν CB (where ν AB = − 41 T(ΨB , XA ) is
the complex conjugate of ν AB ). The components of the su(4)-invariant 4-form can be expressedas εABCD = ΩAB ΩCD + ΩAC ΩDB + ΩAD ΩBC in terms of the usp(4)-invariant symplectic form
under the aforementioned embedding.
Notice that the N =6 lagrangian above has a global u(1) symmetry under which the gauge
field is uncharged while the bosonic and fermionic matter fields XA and ΨA both have thesame charge. It is to be distinguished from the u(1) < su(4) R-symmetry subalgebra that isrealised in the description of this theory in terms of N =2 superfields under which the bosons
and fermions have opposite charges 21 and − 12 . Indeed, this global u(1) is a flavour symmetry
since it commutes with the N =6 superconformal algebra.
The lagrangian (72) describes precisely the N =6 theory in [4,6,20] and its integral is invariant
under the N =6 supersymmetry transformations
δXA = ¯AB ΨBδΨA = − γ µ Dµ XB + µB C · XC AB − µB A · XC BCδAµ = −¯AB γµ ν

AB

AB

− ¯

(73)

γµ ν AB ,where the complex N =6 supersymmetry parameter AB here is skewsymmetric AB = −BA
and obeys the reality condition AB = 21 εABCD CD , thus describing six linearly independent
Majorana spinors on R1,2 . The N =5 supersymmetry transformations in (67) are recovered
following the embedding of usp(4) in su(4) described above and then imposing the symplectic
tracelessness condition on AB .
Given the construction of the N =6 theory as an enhanced N =5 theory, indecomposabilityof the N =6 theory follows from that of the N =5 theory, which required W ∈ Irr(g, H)aLTS .
Enhancement further requires W = VH and hence V ∈ Irr(g, C)aJTS . As stated in Proposition 2,
the underlying real representation [[V ]] ∈ Rep(g, R) is still irreducible unless V = UC is the complexification of U ∈ Irr(g, R). As we will see in the next section, in this case supersymmetry
will be enhanced to N =8. Thus we deduce that the Chern–Simons-matter theories with precisely N =6 superconformal symmetry are in one-to-one correspondence with V ∈ Irr(g, C)aJTS
which are not the complexification of a real representation. According to Theorem 10, such
V ∈ Irr(g, C)aJTS are in turn in one-to-one correspondence with complex simple 3-graded Lie
superalgebras V ⊕ gC ⊕ V . These have been classified [26, Theorem 4] and are given by the twoclassical simple Lie superalgebras A(m, n) and C(n + 1). Again this conclusion is in accordancewith the earlier classification of N =6 superconformal Chern–Simons-matter theories in [15] andremarks in [6, 10].
3.4. N=8 supersymmetry for V = UC with U ∈ Irr(g, R)3LA . We will now describe how
to obtain the theory that was first discovered by Bagger and Lambert [1, 2] and Gustavsson [3]with maximal N =8 superconformal symmetry as a special case of the N =6 theory encounteredin the previous section by taking the representation V ∈ Irr(g, C)aJTS on which that theory
is based to be the complexification V = UC of U ∈ Irr(g, R)3LA . That is V here will betaken to be irreducible as a complex representation but reducible as a real representation. The
canonical complex unitary structure on UC is defined in Appendix A.1.4. Proposition 5(f) in
Appendix A.2.5 establishes the crucial relation that V = UC ∈ Rep(g, C)aJTS if and only if
U ∈ Rep(g, R)3LA .
Given the incremental amounts of supersymmetry enhancement we have been observing,based on assuming increasingly specialised types of unitary representations, one might wonder

36

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

why we have omitted the possibility of realising N =7 supersymmetry. Indeed, the on-shelltheory we shall initially find has a manifest global so(7) symmetry and is invariant under an
N =7 superconformal algebra. However, it will be shown that, without further representationtheoretic assumptions, the lagrangian for this theory can be rewritten in a manifestly so(8)invariant way that is precisely the maximal N =8 superconformal Chern–Simons-matter theoryin [1–3].
Let us now investigate the structure of the superpotential WH in (68) for the special caseof V = UC . The four V -valued N =1 matter superfields will be written Ξ1 = ξ 1 + iξˆ1 ,Ξ2 = ξ 2 + iξˆ2 , Ξ3 = ξ 3 + iξˆ3 and Ξ4 = ξ 4 + iξˆ4 in terms of eight U -valued N =1 superfields
ξ 1 , ξ 2 , ξ 3 , ξ 4 , ξˆ1 , ξˆ2 , ξˆ3 , ξˆ4 . In terms of these superfields, the superpotential (68) becomesZh2
1
WH = 2 d θ (T (ξ 1 , ξˆ1 ), T (ξ 2 , ξˆ2 )) + (T (ξ 1 , ξˆ1 ), T (ξ 3 , ξˆ3 )) − (T (ξ 2 , ξˆ2 ), T (ξ 3 , ξˆ3 ))+(T (ξ 2 , ξˆ2 ), T (ξ 4 , ξˆ4 )) + (T (ξ 3 , ξˆ3 ), T (ξ 4 , ξˆ4 )) − (T (ξ 1 , ξˆ1 ), T (ξ 4 , ξˆ4 ))+(T (ξ 1 , ξ 2 ), T (ξˆ3 , ξˆ4 )) + (T (ξ 3 , ξ 4 ), T (ξˆ1 , ξˆ2 )) − (T (ξ 1 , ξ 4 ), T (ξˆ2 , ξˆ3 ))(74)−(T (ξ 1 , ξ 3 ), T (ξˆ2 , ξˆ4 )) − (T (ξ 2 , ξ 4 ), T (ξˆ1 , ξˆ3 )) − (T (ξ 2 , ξ 3 ), T (ξˆ1 , ξˆ4 ))i+(T (ξ 1 , ξ 2 ), T (ξ 3 , ξ 4 )) + (T (ξˆ1 , ξˆ2 ), T (ξˆ3 , ξˆ4 )) ,where T is the map associated with U and the total skewsymmetry of T (x, y) · z has beenused. For an N =8 superconformal Chern–Simons-matter theory written in terms of N =1
superfields we expect that the superpotential should be invariant under the so(7) subalgebraof the so(8) R-symmetry which preserves the N =1 superspace parameter. This is indeed
the case and follows by assembling the eight real constituent N =1 superfields into the array(ξ 3 , ξ 4 , ξ 1 , ξ 2 , ξˆ1 , ξˆ2 , ξˆ3 , ξˆ4 ) which is to be thought of as a U -valued element of R8 . Let us denoteits components by ΞI with respect to a basis {eI } on R8 . This R8 is to be equipped withthe action of so(7) that defines the real spinor representation in seven dimensions. A natural
quartic tensor on R8 that is preserved by this action of so(7) is the Cayley 4-form Ω. If we take{eI } to define an orthonormal basis on the dual of R8 then the Cayley form can be taken to beΩ = e1234 + (e12 − e34 ) ∧ (e56 − e78 ) + (e13 + e24 ) ∧ (e57 + e68 ) + (e14 − e23 ) ∧ (e58 − e67 ) + e5678 , (75)where multiple indices denote wedge products of the corresponding basis elements. In termsof the Cayley form, the superpotential in (74) can be written in the manifestly so(7)-invariant
form
Z
1
WH = 48d2 θ ΩIJKL (T (ΞI , ΞJ ), T (ΞK , ΞL )).

(76)

This so(7)-invariant expression for the superpotential that gives rise to the N =8 Bagger–
Lambert lagrangian has appeared already in [27]. What we have shown above is that it isprecisely this superpotential which follows from evaluating WH in (68) for the special case of
V = UC .
We will now examine the on-shell form (9) of the lagrangian L N =8 = LCS + LM + WH
based on the superpotential (74). There are two ways to evaluate this. The simplest is tointegrate out the auxiliary fields in the generic N =1 Chern–Simons-matter lagrangian with theform of the superpotential in (76) based on an N =1 matter superfield ΞI valued in R8 ⊗ U (theterms in the matter part of the lagrangian being evaluated with respect to the obvious tensor
product inner product involving the unit inner product on R8 , with components δIJ , and theinner product h−, −i on U ). The other way is to just evaluate the on-shell N =6 lagrangian

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY(72) for V = UC . Of course, as consistency dictates, both methods give the same answer

L N =8 = − εµνρ Aµ , ∂ν Aρ + 13 [Aν , Aρ ] − 21 Dµ X I , Dµ X I + 12 Ψ̄I , γ µ Dµ ΨI+ 2 (2 δIK δJL + ΩIJKL ) (ν̄ IK , ν JL ) − 13 µIJ · X K , µIJ · X K ,

37

(77)

where µIJ := 14 T (X I , X J ) and ν IJ := 14 T (X I , ΨJ ) in terms of the map T associated with U .
Of course, this form of the lagrangian is only manifestly so(7)-invariant and we shall detailits so(8)-invariant expression in a moment. First though it will be useful to elaborate a little
on how the matter fields appearing in (77) are realised in the two different derivations noted
above. In the first and simpler derivation, X I and ΨI are just the obvious bosonic and fermioniccomponents of the N =1 superfield ΞI . To obtain the scalar potential in (77) from this approachrequires the identity[M
PΩIJKL ΩM N P L = 6 δ[IM δJN δK]− 9 δ[I ΩJK] N P ] ,(78)
for the components of the Cayley form to be employed and the contribution of the second termon the right hand side above to the scalar potential can be shown to vanish identically as aconsequence of the fundamental identity (104) for the 3-Lie bracket associated with T on U .
In the second derivation, after taking V = UC , in order that the kinetic terms for the matterfields in the lagrangians (72) and (77) agree, the identification of C4 with R8 must be isometric,corresponding to the canonical embedding of su(4) in so(8). A convenient way to achieve thisfor the bosonic fields is to identify the first and last four of the eight real scalars X I respectively
with the real and imaginary parts of the four complex scalars XA in the N =6 theory. For thefermionic fields though one identifies the first and last four fermions ΨI respectively with theimaginary and and real parts of the four complex fermions ΨA in the N =6 theory. (Thisdistinction between the way the bosons and fermions are identified is necessary in order tothen rewrite the Yukawa couplings for the N =6 theory in an so(7)-invariant way.) In deriving
the expression for the Yukawa couplings in the second line of (77) from (72), use has been
made of the identity Ω = Re ε − 12 k ∧ k for the Cayley form (75) on R8 in terms of (the realpart of) the holomorphic 4-form ε = e1 ∧ e2 ∧ e3 ∧ e4 and the Kähler form k = 2i eA ∧ eA
on C4 (the latter being a real 2-form and can be expressed as e15 + e26 + e37 + e48 on R8
under the aforesaid identification). Finally, in deriving the scalar potential in (77), it was
useful to first note that the N =6 scalar potential in the third line in (72) can be rewritten as− 13 µA B · XC , µA B · XC − 21 µA B · XB , µA C · XC .
The integral of the lagrangian in (77) is invariant under the N =7 supersymmetry transformations
δX I = ¯IJ ΨJδΨI = −γ µ Dµ X J IJ + 32 µIJ · X K JK + 13 ΩIJKL µJK · X M LM(79)

δAµ = −2 ¯IJ γµ ν IJ ,where the real N =7 supersymmetry parameter IJ is skewsymmetric IJ = −JI and obeys
IJ = − 16 ΩIJKL KL which defines the projection onto the seven-dimensional so(7)-invariant
subspace of Λ2 R8 . Thus it describes seven linearly independent Majorana spinors on R1,2 .
The N =6 supersymmetry transformations in (73) can be recovered on V = UC under theidentification of R8 with C4 described above after imposing the the extra condition k IJ IJ = 0
using the Kähler form to eliminate the seventh supersymmetry parameter.
3.4.1. so(8) R-symmetry. To rewrite the lagrangian (77) in a manifestly so(8)-invariant way
first requires a choice of embedding for so(7) in so(8). Recall that there are three real eightdimensional irreducible representations of so(8): the vector representation 8v and the positive
and negative chirality spinor representations 8s and 8c . Consequently there are three distinct

38

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

embeddings of so(7) in so(8) that can be understood as the subalgebras preserving a fixednonzero element in either 8v , 8s or 8c . Of course, ’distinct’ only up to the triality symmetry
which relates these three representations. In each case, whichever of the three representations
the fixed element resides in breaks into the vector and singlet representations 7⊕1 of so(7) while
the remaining two representations both reduce to the spinor representation 8. Both the bosonicand fermionic matter fields X I and ΨI above transform in the 8 while the supersymmetry
parameter IJ transforms in the 7 of so(7). We are therefore free to choose any of the threeembeddings so long as it is the supersymmetry parameter which embeds into whichever of
the three representations of so(8) that contains a singlet under the so(7) subalgebra. As
in [23], it has been convenient for us so far to assume that the supercharges in an N -extended
superconformal algebra transform in the vector representation of the so(N ) R-symmetry. In
the case at hand of N =8 this would suggest we lift the supercharges into the 8v of so(8) withthe matter fields lifting to the 8s and 8c . However, it will prove more convenient here to lift
the supercharges into the 8s of so(8) with the matter fields X I and ΨI lifting respectively to
the 8v and 8c representations. The main reason for this is technical inasmuch as it allows one
to rewrite things more neatly in terms of elements of the Clifford algebra in eight dimensions
and whence recover the well-known form of the N =8 lagrangian originally presented in [2]. Of
course, one can always apply triality to obtain whichever of the three representations of so(8)one wants for the supercharges (for example, this has been done explicitly in [28] in the context
of so-called ‘trial BLG’ theories).
We shall adopt the conventions of [29] for the Clifford algebra C`(8). An so(8)-covariant
basis for C`(8) can be constructed in terms of products of the 8 real 16 × 16 skewsymmetricmatrices ΓI obeying ΓI ΓJ + ΓJ ΓI = −2δIJ 1. (The index I will be used here to denote thevector representation of so(8) and δIJ denotes the components of the unit so(8)-invariant innerproduct.) The real 16-dimensional vector space acted upon by the matrices ΓI corresponds
to the spinor representation of so(8). The chirality matrix is defined by Γ := Γ1 ...Γ8 , which
is idempotent and anticommutes with each ΓI . The chiral and antichiral representations 8s
and 8c correspond respectively to the positive and negative chirality eigenspaces of Γ. The
supersymmetry parameter IJ is to be lifted to  = Γ in 8s while the fermions ΨI are to belifted to Ψ = −ΓΨ in 8c of so(8) (of course, both are also Majorana spinors on R1,2 ). The liftof the bosons X I is rather more trivial requiring only the reinterpretation of the index I fromthe 8 of so(7) to the 8v of so(8).
Let us now assume the existence of a fixed (commuting) chiral spinor ϑ ∈ 8s which we taketo be unit normalised such that ϑt ϑ = 1 (Majorana conjugation is just transposition here since
the charge conjugation matrix for C`(8) can be taken to be the identity). This defines thedesired embedding of so(7) in so(8) as the stabiliser of ϑ. In terms of this fixed chiral spinor,
one can deduce the precise identifications for the supersymmetry parameter and fermions to be

IJ = ϑt ΓIJ ,ΨI = ϑt ΓI Ψ,(80)

where ΓIJ := Γ[I ΓJ] and the Cayley
form ΩIJKL = ϑt ΓIJKL ϑ. Making use of the Fierz identity1ϑϑt = 16
1 + Γ + 4!1 ΩIJKL ΓIJKL , one can check that the right-hand side of the first equationin (80) obeys the same projection condition IJ = − 16 ΩIJKL KL satisfied by the left-handside. Hence the eighth supersymmetry parameter in  is automatically projected out on theright-hand side.

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

39

Substituting this into the lagrangian (77) then gives the sought after so(8)-invariant expression
D
E
 1
t
N =8
µνρ
I
µ I
µ
1
1
L
=−ε
Aµ , ∂ν Aρ + 3 [Aν , Aρ ] − 2 Dµ X , D X + 2 Ψ̄ , γ Dµ Ψ
(81)
t
IJ
K
IJ
K
IJ
1
1
+ 2 (µ , T (Ψ̄ , ΓIJ Ψ)) − 3 µ · X , µ · X ,whose integral is indeed invariant under the N =8 supersymmetry transformations
δX I = ¯t ΓI ΨδΨ = −γ µ Dµ X I ΓI  − 13 µIJ · X K ΓIJK (82)

δAµ = − 12 T (X I , ¯t γµ ΓI Ψ).
The N =8 supersymmetry parameter above can be decomposed as  = − 81 IJ ΓIJ ϑ + ηϑ withrespect to the so(7) subalgebra, where η is a single fermionic Majorana spinor on R1,2 . Setting η = 0 one recovers precisely the N =7 supersymmetry transformations in (79). Mutatis
mutandis, the lagrangian (81) and supersymmetry transformations (82) indeed agree with thosein [2]. To be precise, equations (45) and (42) in [2] match up with (81) and (82) by identifying
their 3-bracket [−, −, −] with − 21 T (−, −) · − on U and their Lie algebra inner product with2(−, −) on g here. Of course, the form presented in [2] is in terms of projected Majorana
spinors of C`(1, 10) broken to C`(1, 2) ⊗ C`(0, 8) and one identifies with our expressions above
such that the gamma matrices of C`(1, 10) take the form γµ ⊗ Γ and 1 ⊗ iΓI .
Acknowledgments
PdM would like to thank Jan de Boer, Amihay Hanany, Chris Hull and Neil Lambert for
useful discussions related to this work. PdM is 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 Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan. JMF takes pleasure in 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.
Appendix A. Unitary representations of metric Lie algebras
The organisational principle advocated in this paper for the classification of superconformal
Chern–Simons theory is the representation theory of metric Lie algebras. This is a refinementof the usual representation theory of Lie algebras, on which it is based. Therefore in thisappendix we collect the basic facts about unitary representations of a metric Lie algebra. We
will fix once and for all a real finite-dimensional metric Lie algebra g with ad-invariant innerproduct (−, −). After reviewing the useful yoga of real, complex and quaternionic unitary
representations, we discuss special types of these representations which can only be definedwhen the Lie algebra is metric. They turn out to be related to certain triple systems whichembed in Lie (super)algebras.
A.1. Notation and basic notions. Unitary representations of real Lie algebras come in three
types, depending on the ground field: real, complex or quaternionic. It is possible to view all
three as special classes of real or complex representations. Let us review this briefly only to set
the notation. All our representations will be assumed finite-dimensional.

40

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

A.1.1. Real representations. We shall usually denote real representations by U and denote the(symmetric) inner product by h−, −i. This defines a Lie algebra so(U ) consisting of linear
transformations of U which are skewsymmetric relative to the inner product. An orthogonal
representation of g on U is a Lie algebra homomorphism g → so(U ). This means that for eachX ∈ g and u, v ∈ U ,hX · u, vi = − hu, X · vi ,(83)
where the · denotes the action of g.
A.1.2. Complex representations. We shall usually denote complex representations by V anddenote the hermitian inner product by h. This is a sesquilinear form h : V × V → C, which iscomplex linear in the first entry and complex antilinear in the second. It defines a Lie algebrau(V ) consisting of complex linear transformations of V which are skewhermitian relative to h.
A unitary representation of g on the hermitian vector space V, h is a Lie algebra homomorphismg → u(V ), which translates toh(X · u, v) = −h(u, X · v).

(84)

Given a complex representation V we shall denote by V the complex conjugate representation.
This shares the same underlying set as V , but the complex number z ∈ C acts on V in the waythat its complex conjugate z̄ acts on V . It is therefore convenient notationally to parametrise
the vectors in V by the vectors in V , but denoting them as v, where v ∈ V . In this way, zv = z̄v
and the action of g is such that X · v = X · v. If V is a complex unitary representation, thenso is V , with hermitian structureh(u, v) = h(v, u) = h(u, v).

(85)

A.1.3. Quaternionic representations as complex representations. For reasons which will hopefully become clearer below, the nonexistence of quaternionic Lie algebras forbid us from working
over the quaternions. Instead we will consider a quaternionic representation to be given by acomplex hermitian vector space V, h as in Section A.1.2 above and a quaternionic structure,that is, a complex antilinear map J : V → V obeying J 2 = −1 that is both g-invariant andcompatible with the hermitian inner product in the sense thath(Ju, Jv) = h(v, u).

(86)

In particular, we have a complex symplectic structureω(u, v) := h(u, Jv),

(87)

whence V must have even complex dimension. The Lie subalgebra of u(V ) consisting of endomorphisms which commute with J is called usp(V ). The notation stems from the fact thatusp(V ) = u(V ) ∩ sp(V ), where sp(V ) is the Lie subalgebra of gl(V ) which preserves ω. For
us in this paper, a quaternionic representation of g on V is then a Lie algebra homomorphismg → usp(V ).
A.1.4. Real representations as complex representations. It is possible to view real orthogonal
representations as a special kind of complex unitary representations. If U is a real orthogonal
representation of g, then let UC = C ⊗R U denote its complexification. This becomes a complex
representation of g by declaring X · (z ⊗ u) = z ⊗ (X · u) for all z ∈ C and u ∈ U and lettingz 0 ∈ C act on UC by z 0 (z ⊗ u) = (z 0 z) ⊗ u. It is clear that this action of g is complex linear.
We define a hermitian structure on UC by extending the following additively:h(z1 ⊗ u, z2 ⊗ v) = z1 z2 hu, vi ,(88)

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

41

which makes it clear that g preserves h as well, whence UC , h is a complex unitary representationof g.
Conversely, complex unitary representations V which are of this type are characterised by
the existence of a real structure, that is, a complex antilinear map R : V → V satisfying R2 = 1
that both commutes with the action of g and is compatible with the hermitian structure inthath(Ru, Rv) = h(v, u).
(89)
In the case of UC , R is simply complex conjugation: R(z ⊗ u) = z ⊗ u. In the abstract
case, if V, h, R is such a representation, then we may define U to be the eigenspace of R witheigenvalue 1. Since R is complex antilinear, this is only a real subspace of V . The eigenspacewith eigenvalue −1 is given by iU , whence V = U ⊕ iU ∼= UC . On V we can define a g-invariant
complex bilinear inner product byb(u, v) = h(u, Rv),

(90)

whose restriction to U , by condition (89), is real and hence makes U into a real orthogonal
representation. We will often employ the notation U = [V ] for this sort of real representations.
A.1.5. Complex and quaternionic representations as real representations. Finally, it is possible
to discuss both complex and quaternionic unitary representations as particular cases of real
orthogonal representations. A complex unitary representation V, h gives rise to a real representation by simply restricting scalars to the real numbers. Relative to a basis, one is simply
taking real and imaginary parts of the vectors in V . This real representation is usually denoted[[V ]] to remind us that its real dimension is twice the complex dimension of V . Multiplication
by i then defines a real linear endomorphism I of [[V ]], satisfying I 2 = −1; that is, a complex
structure. Since g acts complex linearly on V , it commutes with the action of I. The real partof h defines a symmetric inner product h−, −i on [[V ]], relative to which I is orthogonal.
Conversely if U is a real orthogonal representation with a g-invariant orthogonal complex
structure I, then we can define on U the structure of a complex vector space by having thecomplex number a + ib ∈ C act on u ∈ U by (a + ib)u = au + bIu. We can also define ahermitian structure h on U byh(u, v) = hu, vi + i hu, Ivi .

(91)

Similarly, a quaternionic unitary representation is a special type of real orthogonal representation U where we have two invariant orthogonal complex structures I and J which anticommute,in that IJ = −JI. This allows us to define a complex symplectic structure by
ω(u, v) = hu, Jvi + i hu, IJvi .

(92)

If so inclined, we could now define on U the structure of a quaternionic vector space with all
the trimmings, but we will not do so in this paper.
A.1.6. Reality conditions. If W1 and W2 are two quaternionic unitary representations in thesense of Appendix A.1.3, then their tensor product W1 ⊗C W2 is a complex representation witha real structure. Indeed, if J1 , J2 are the quaternionic structure maps of W1 , W2 , respectively,
then R = J1 ⊗ J2 is a real structure map. The underlying real representation [W1 ⊗C W2 ] is thereal subspace given by the eigenspace of R with eigenvalue 1.
A common device to obtain a real representation out of a quaternionic representation W is
to tensor with the quaternions, understood as a trivial quaternionic representation. Indeed,
give H the structure of a two-dimensional complex representation by having C act on the rightand consider V = H ⊗C W . Let J denote the quaternionic structure on W and j that on H

42

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

and let R = j ⊗ J. As a representation of g, V ∼= W ⊕ W , with the isomorphism being given
explicitly by w1
1 ⊗ w1 + j ⊗ w2 ↔   .
w2

(93)

In this picture, the conjugation c acts by extending the following expression complex antilinearly:
  

w1
−Jw2
.
R  = 
w2
Jw1

(94)

The real subspace U is the eigenspace of c with eigenvalue 1, which means that w2 = Jw1 , so
that

 

 w
(95)
U =  w∈W
 Jw
is the graph of J as a subspace of W ⊕ W . Since J is antilinear, U is only a real subspace.
A.1.7. Relations between real, complex and quaternionic representations. As seen above, there
are natural maps between representations obtained by altering the ground field. We can extend,
restrict and, in the case of the complex field, also conjugate. These maps are summarised
succinctly in the following (noncommutative!) diagram, borrowed from [21] via [22]:Rep(g, C)O

p8
c ppppppp
ppp

Rep(g, R)

fNNN
NNN
N
r NNNN

t

fNNN
NNNr0
NNN
NN

(96)

Rep(g, H)

pp8
ppp
p
p
p q
ppp

Rep(g, C)
where Rep(g, K) means the category of representations of the Lie algebra g over the field K,
and the arrows denote the following functors:
t: if V ∈ Rep(g, C), then t(V ) = V denote the complex conjugate representation;
q: if V ∈ Rep(g, C), then q(V ) = VH = H ⊗C V ∈ Rep(g, H), where H is a right C-module;
c: if U ∈ Rep(g, R), then c(U ) = UC = C ⊗R U is its complexification;
r: if V ∈ Rep(g, C), then r(V ) = [[V ]] ∈ Rep(g, R) is obtained by restricting scalars; andr0 : if W ∈ Rep(g, H), then r0 (W ) = ((W )) ∈ Rep(g, C) is obtained by restricting scalars.
The map t does not change the dimension, and neither do c or q in the sense that dimC UC =dimR U and dimH VH = dimC V . However r and r0 double the dimension: dimR [[V ]] = 2 dimC V
and dimC ((W )) = 2 dimH W . In this paper we are not working with quaternionic representations
themselves but with their image under r0 ; although this will not always be reflected in ournotation. In other words, we will often write simply W for ((W )) if in so doing the possibilityof confusion is minimal.
The above maps obey some relations, which are more or less obvious (see [21, Proposition 3.6]or [22, Proposition (6.1)]).
Proposition 1. The following relations hold:(a) t2 = 1 or V ∼= V for all V ∈ Rep(g, C);(b) tc = c or UC ∼= UC for all U ∈ Rep(g, R);

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

43(c) qt = q or V H ∼= VH for all V ∈ Rep(g, C);(d) rc = 2 or [[UC ]] ∼= U ⊕ U for all U ∈ Rep(g, R);(e) cr = 1 + t or [[V ]]C ∼= V ⊕ V for all V ∈ Rep(g, C);(f) rt = r or [[V ]] ∼= [[V ]] for all V ∈ Rep(g, C);(g) tr0 = r0 or ((W )) ∼= ((W )) for all W ∈ Rep(g, H);(h) qr0 = 2 or ((W ))H ∼= W ⊕ W for all W ∈ Rep(g, H); and0(i) r q = 1 + t or ((VH )) ∼= V ⊕ V for all V ∈ Rep(g, C).
Recall that a real (resp. complex, quaternionic) representation is irreducible if it admits
no proper real (resp. complex, quaternionic) subrepresentations. We shall denote by Irr(g, K)
the irreducible representations of g of type K. The following proposition states what happens
to irreducible representations under the above maps. It is not always the case that irreducibles
go to irreducibles, but their images are under control in any case.
Proposition 2. The following hold:(a) V ∈ Irr(g, C) ⇐⇒ V ∈ Irr(g, C);(b) if U ∈ Irr(g, R) then UC ∈ Irr(g, C), unless U = [[V ]] for some V ∈ Irr(g, C), in whichcase UC ∼= V ⊕V;(c) if V ∈ Irr(g, C) then [[V ]] ∈ Irr(g, R), unless V = UC for some U ∈ Irr(g, R), in whichcase [[V ]] ∼= U ⊕ U;(d) if W ∈ Irr(g, H) then ((W )) ∈ Irr(g, C), unless W = VH for some V ∈ Irr(g, C), in whichcase ((W )) ∼= V ⊕ V ; and(e) if V ∈ Irr(g, C) then VH ∈ Irr(g, H), unless V = ((W )) for some W ∈ Irr(g, H), in whichcase VH ∼= V ⊕V.
See [22, Proposition (6.6)] for a partial proof.
A.2. Lie-embeddable representations of metric Lie algebras. We will now use that the
Lie algebra g admits an ad-invariant inner product (−, −). This allows us to distinguish certain
privileged types of unitary representations, which are summarised in Table 2 in the Introduction. For reasons that will become clear in Appendix A.2.6, we call them Lie-embeddable
representations in this paper; although we are not claiming that these are the only representations which could be given this name.
A.2.1. Real orthogonal representations. Let U ∈ Rep(g, R) with corresponding homomorphism
g → so(U ). Using the inner products on U and on g we may dualise this map to arrive at ag-equivariant bilinear map T : U × U → g. Explicitly, we have that for all u, v ∈ U and X ∈ g,
(T (u, v), X) = hX · u, vi ,(97)

from where it follows at once that it is alternating:T (v, w) = −T (w, v).

(98)(T (u, v), T (w, x)) = hT (w, x) · u, vi = hT (u, v) · w, xi ,(99)

whence the fourth-rank tensor R(u, v, w, x) := hT (u, v) · w, xi belongs toS 2 Λ2 U ∼= Λ4 U ⊕ U .

(100)

It also follows thatFor generic real orthogonal representations U , the tensor R ∈ S 2 Λ2 U will have components inboth representations, but for some special representations one of the components will vanish.
There are thus two classes of special real orthogonal representations to be described below. In

44

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

order to explain the names we will give to these special classes, it will be convenient to review
the construction in [16].
The map T above allows us to define a trilinear product on U by[u, v, w] := T (u, v) · w(101)

for all u, v, w ∈ U , defining a triple system on U . The resulting triple system, which appeared
originally in [30] but more recently in [31] in the context of superconformal Chern–Simonsmatter theories, satisfies the following axioms for all x, y, z, v, w ∈ U :(a) the unitarity condition
h[x, y, z], wi = − hz, [x, y, w]i ;(102)h[x, y, z], wi = h[z, w, x], yi ;(103)(b) the symmetry condition(c) and the fundamental identity[x, y, [v, w, z]] − [v, w, [x, y, z]] = [[x, y, v], w, z] + [v, [x, y, w], z].

(104)

The fundamental identity is simply the g-equivariance of the map T , whereas the symmetry
condition is a rewriting of equation (99) and the unitarity condition is just the fact that g
preserves the inner product.
The first class of special representations consists of those where R ∈ U , which is equivalent
to the Bianchi-like identity
T (u, v) · w + T (v, w) · u + T (w, u) · v = 0,(105)

or, in terms of the 3-bracket,[u, v, w] + [v, w, u] + [w, u, v] = 0.

(106)

Such a 3-bracket defines on U the structure of a Lie triple system (LTS) and hence we saythat the unitary representation U is LTS, written U ∈ Rep(g, R)LTS . Lie triple systems arelinear approximations to riemannian symmetric spaces and indeed the tensor R is nothing but
the Riemann curvature tensor.
The other special class consists of representations U where R ∈ Λ4 U or, equivalently, forwhich the 3-bracket if totally skewsymmetric. Such triple systems are known as 3-Lie algebras(3LA) [32] and hence such representations are said to be 3LA, written U ∈ Rep(g, R)3LA . As
conjectured in [33] and shown in [12–14], there is a unique positive-definite representation
U ∈ Irr(g, R)3LA , corresponding to g = so(4) = so(3) ⊕ so(3) with the inner product being
equal to the Killing form on one of the so(3) and the negative of the Killing form in the other,and U = R4 being the vector representation. Dropping the positive-definite condition, one
obtains many such representations, of which there are a number of partial classifications [34–36].
They are the relevant representations for N =8 supersymmetry and the totally skewsymmetric
3-bracket is precisely the same as that which figured in the original description of the N =8
theory of Bagger and Lambert [1, 2] and Gustavsson [3].
A.2.2. Complex unitary representations. Let V ∈ Rep(g, C) with corresponding homomorphism g → u(V ), whose transpose is now a sesquilinear map T : V × V → gC to the complexification gC = C ⊗R g of g. We extend the bracket and the inner product on g complex bilinearly insuch a way as to make gC into a complex metric Lie algebra. In this way, the map T is defined
explicitly by(T(u, v), X) = h(X · u, v),(107)

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

45

for all u, v ∈ V and X ∈ gC . In particular,(T(u, v), T(w, x)) = h(T(w, x) · u, v) = h(T(u, v) · w, x),(108)whence the fourth-rank tensor R(u, v, w, x) := h(T(u, v) · w, x) belongs to
S 2 (V ⊗ V ) ∼= S 2 V ⊗ S 2 V ⊕ Λ2 V ⊗ Λ2 V

(109)

For generic V ∈ Rep(g, C), the tensor R ∈ S 2 (V ⊗ V ) will have both components, but for
special representations one of the components will vanish. There are two such types of special
complex unitary representations which we describe below.
The map T above allows us to define a complex sesquibilinear product on V by[[u, v, w]] := T(u, v) · w(110)for all u, v, w ∈ V . This means that it is complex linear in u, w and complex antilinear inv. In the notation of [16], which was chosen to ease the comparison with [20], the bracket[[u, v, w]] = [w, u; v]. The bracket (110) satisfies a number of identities. For all x, y, z, v, w ∈ V ,we have(a) the unitarity conditionh([[v, w, x]], y) = h(x, [[w, v, y]]);(111)h([[v, w, x]], y) = h([[x, y, v]], w);(112)(b) the symmetry condition(c) and the fundamental identity[[x, y, [[v, w, z]]]] − [[v, w, [[x, y, z]]]] = [[[[x, y, v]], w, z]] − [[v, [[y, x, w]], z]].

(113)

Again the fundamental identity is basically the g-equivariance of T, whereas the other two
identities follow as before from the fact that T(u, v) = −T(v, u) and that for all X ∈ gC , wehave thath(X · u, v) = −h(u, X · v),(114)
and from equation (108).
The first special class is when R ∈ S 2 V ⊗ S 2 V , which corresponds to those V whereT(u, v) · w = T(w, v) · u(115)

or equivalently where [[u, v, w]] = [[w, v, u]]. Such a bracket defines on V the structure of a
Jordan triple system (JTS) [37] and we will say such a representation V is JTS, written
V ∈ Rep(g, C)JTS . These representations are linear approximations to hermitian symmetricspaces and, as in case of Lie triple systems, the tensor R coincides with the Riemann curvaturetensor, this time on the complexified tangent bundle of the symmetric space.
The other special class is when R ∈ Λ2 V ⊗ Λ2 V , which corresponds to those V whereT(u, v) · w = −T(w, v) · u(116)

or, equivalently, where [[u, v, w]] = −[[w, v, u]]. Such a 3-bracket defines on V the structureof an anti-Jordan triple system (aJTS) (see, e.g., [38, Remark 4.3]) and hence we will
say that these representations are aJTS, written V ∈ Rep(g, C)JTS . They are the relevant
representations for N =6 supersymmetry and the skewsymmetry condition (116) identifies thisclass of triple systems with those used by Bagger and Lambert in [20] to recover the N =6
theories discovered by Aharony, Bergman, Jafferis and Maldacena in [4].

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A.2.3. Complex representations in real terms. It is useful to rewrite the complex unitary construction above in terms of real representations. Let V ∈ Rep(g, C) and let U = [[V ]] ∈Rep(g, R) with inner product given by the real part of the hermitian inner product on V .
Complex multiplication in V is implemented via an orthogonal g-invariant complex structure
I : U → U . This means that I 2 = −1, hIv, Iwi = hv, wi and X · (Iv) = I(X · v) for allv, w ∈ U and X ∈ g. In particular, hIv, wi = − hv, Iwi, whence I is both orthogonal andskewsymmetric. In terms of h−, −i, the hermitian structure is given by equation (91).
Let gC denote the complexification of g, which we make into a complex metric Lie algebra byextending both the Lie bracket and the ad-invariant inner product (−, −) complex bilinearly.
There is a natural action of gC on U as follows. If X = X + iY ∈ gC , with X, Y ∈ g, then for
all v ∈ U ,(X + iY ) · v = X · v + I(Y · v) = X · v + Y · (Iv),the second equation following from the g-invariance of I. The real bilinear map T : U × U → gC
defined in equation (107) is written in reals terms as(T(u, v), X) = hX · u, vi + i hX · u, Ivi ,(117)

for all u, v ∈ U , X ∈ gC .
Lemma 3. For all u, v ∈ U , we have T(u, v) = T (u, v) + iT (u, Iv).
Proof. Write T(u, v) = A(u, v) + iB(u, v) and expand equation (107) for X = X + iY complex
bilinearly. The left-hand side becomes(T(u, v), X) = (A(u, v) + iB(u, v), X + iY )= (A(u, v), X) + i (A(u, v), Y ) + i (B(u, v), X) − (B(u, v), Y ) ,whereas the right-hand side becomesh(X · u, v) = h(X + iY ) · u, vi + i h(X + iY ) · u, Ivi= hX · u, vi + hI(Y · u), vi + i hX · u, Ivi + i hI(Y · u), Ivi= hX · u, vi − hY · u, Ivi + i hX · u, Ivi + i hY · u, vi .
Comparing real and imaginary parts and the terms depending on X and Y , we arrive at thefollowing two equations:(A(u, v), X) = hX · u, viand(B(u, v), X) = hX · u, Ivi .

Now using equation (97), we see that the first of the above equations says that A(u, v) = T (u, v),
whereas the second equation says that B(u, v) = T (u, Iv).

Proposition 4. The map T : V × V → gC satisfies the following properties
T(u, v) = −T(v, u)T(u, Iv) = −iT(u, v).

T(Iu, v) = iT(u, v)

Proof. Again it is sufficient to prove the first two. Notice first of all that
(T (u, Iv), X) = hX · u, Ivi

by equation (97)

= − hI(X · u), vi

by skewsymmetry of I

= − hX · Iu, vi

by g-invariance of I

= − (T (Iu, v), X)

again by equation (97),whenceT (u, Iv) = −T (Iu, v).

(118)

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47

Together with equation (98), we have in addition that
T (u, Iv) = T (v, Iu).

(119)

To show the first identity, we calculate
T(u, v) = T (u, v) + iT (u, Iv)= T (u, v) − iT (u, Iv)= −T (v, u) − iT (v, Iu)

using equations (98) and (119)= −T(v, u);and to show the second, we calculate
T(Iu, v) = T (Iu, v) + iT (Iu, Iv)= −T (u, Iv) + iT (u, v)

using equation (118)= i(T (u, v) + iT (u, Iv))
= iT(u, v).
We again define a 3-bracket on V by equation (110), which, by Lemma 3, can be rewritten
as[[u, v, w]] = [u, v, w] + [u, Iv, Iw],
(120)
in terms of the 3-bracket [u, v, w] in (101). The bracket (110) of course satisfies the unitarity(111), symmetry (112) and fundamental (113) identities, but in addition, as an easy consequenceof Proposition 4, the following:[[Iu, v, w]] = I[[u, v, w]][[u, Iv, w]] = −I[[u, v, w]][[u, v, Iw]] = I[[u, v, w]],

which is simply the sesquibilinearity of the 3-bracket.
A.2.4. Quaternionic unitary representations. Let W, h, J be a quaternionic unitary representation. Again we recall that for us in this paper this means that W, h is a complex hermitian
vector space and J a quaternionic structure map. The reason we work with complex representations with a quaternionic structure instead of a quaternionic representation should now become
clear. In the case of an honest quaternionic representation, the dualising procedure employed
here would map to a quaternionification of the Lie algebra, but such an object does not exist;
although see [39] for a possibly related concept. Let T be the sesquilinear map defined in (107).
By suitably inserting the quaternionic structure map, one can turn T into a complex bilinear
map: (u, v) 7→ T(u, Jv). This map can be understood as the transpose of the representation
homomorphism g → usp(W ) with respect to the complex symplectic structure ω defined in(87):(T(u, Jv), X) = ω(X · u, v).
(121)
Since ω is symplectic, it follows that
T(u, Jv) = T(v, Ju),(122)

whence it defines a symmetric map V × V → gC . Compatibility with the quaternionic structuresays that J ◦ T(u, Jv) = T(u, Jv) ◦ J, where T(u, Jv) = −T(Jv, u). Again we have that
(T(u, Jv), T(w, Jx)) = ω(T(w, Jx) · u, v) = ω(T(u, Jv) · w, x),
whence the fourth-rank tensor R(u, v, w, x) := ω(T(u, Jv) · w, x) belongs to
S 2S 2W ∼= W ⊕ S 4 W.

(123)
(124)

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For generic quaternionic unitary representations W , the tensor R ∈ S 2 S 2 W will have components in both representations, but for the special representations one of the components will
vanish. This defines two special classes: one where R ∈ S 4 W and one where R ∈ W .
Again inserting J in the complex sesquilinear 3-bracket (110), defines a complex complex
trilinear 3-bracket on W by (u, v, w) 7→ [[u, Jv, w]], for all u, v, w ∈ W . The resulting triple
system satisfies the axioms inherited from (111), (112) and (113) and the compatibility with
the quaternionic structure — namely,(a) the symplectic condition
ω([[x, Jy, z]], w) = ω([[x, Jy, w]], z);(125)(b) the symmetry condition
ω([[x, Jy, z]], w) = ω([[z, Jw, x]], y);(126)(c) the fundamental identity[[x, Jy, [[v, Jw, z]]]] − [[v, Jw, [[x, Jy, z]]]] = [[[[x, Jy, v]], Jw, z]] + [[v, J[[x, Jy, w]], z]],(127)(d) and the quaternionic condition
J[[x, Jy, z]] = −[[Jx, y, Jz]].

(128)

It follows from the symplectic and symmetry conditions that for all x, y, z ∈ W ,[[x, Jy, z]] = [[y, Jx, z]],(129)

which is precisely the symmetry property (122) noted above.
The first special class is when R ∈ S 4 W , so that the 3-bracket is totally symmetric. We
will see that they correspond to hyperkähler symmetric spaces and hence this class is trivial in
positive-definite signature [40]. Nevertheless we will refer to the triple systems that they giverise to as quaternionic triple systems (QTS), for lack of a more appropriate name, and therepresentations will be said to be QTS, written W ∈ Rep(g, H)QTS . The tensor R is related
to the Riemann curvature tensor of the symmetric space.
The other special class corresponds to the case R ∈ W , or equivalently,
T(u, Jv) · w + T(v, Jw) · u + T(w, Ju) · v = 0,(130)

or in terms of the 3-bracket[[u, Jv, w]] + [[v, Jw, u]] + [[w, Ju, v]] = 0.

(131)

Equivalently, taking the symmetry conditions (122) or (129) into account, we may write theabove two conditions as asT(u, Ju) · u = 0

and[[u, Ju, u]] = 0,(132)

respectively, for all u ∈ W . Either of these conditions defines a (quaternionic) anti-Lie
triple system (aLTS) and hence we wll say that W is an aLTS representation, written W ∈Rep(g, H)aLTS . They are the relevant representations for the N =4, 5 theories.
A.2.5. Some relations between these representations. Some of the special cases are related toeach other via the maps in Appendix A.1.7 consistent with the requirements of supersymmetryof the corresponding Chern–Simons theory. Let us formalise the notation we have been using
until now and let U , V and W stand for a real, complex or quaternionic representation, respectively. Remember, however, that for us quaternionic representations are always complex
representations in the image of r0 . Let Rep(g, K)C denote the unitary representations of g oftype K and class C, where K = R, C, H and C can be either 3LA, LTS, aJTS, JTS, aLTS or
QTS.

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49

Proposition 5. The following relations hold:(a) V ∈ Rep(g, C)aJTS ⇐⇒ V ∈ Rep(g, C)aJTS(b) V ∈ Rep(g, C)JTS ⇐⇒ V ∈ Rep(g, C)JTS(c) V ∈ Rep(g, C)aJTS ⇐ [[V ]] ∈ Rep(g, R)3LA(d) V ∈ Rep(g, C)JTS ⇐⇒ [[V ]] ∈ Rep(g, R)LTS(e) UC ∈ Rep(g, C)JTS =⇒ U is trivial(f) U ∈ Rep(g, R)3LA ⇐⇒ UC ∈ Rep(g, C)aJTS(g) W ∈ Rep(g, H)QTS ⇐⇒ ((W )) ∈ Rep(g, C)JTS(h) ((W )) ∈ Rep(g, C)aJTS =⇒ W is trivial(i) V ∈ Rep(g, C)aJTS ⇐⇒ VH ∈ Rep(g, H)aLTS
Proof. First of all (a) and (b) are obvious because in both cases the fourth-rank tensor R lives
in a self-conjugate representation: Λ2 V ⊗ Λ2 V or S 2 V ⊗ S 2 V , respectively.
To prove (c), let V ∈ Rep(g, C) and let U = [[V ]] ∈ Rep(g, R)3LA . The relation between T on
V and T on [[V ]] is given by Lemma 3, whenceT(u, v) · w = T (u, v) · w + T (u, Iv) · Iw
= T (u, v) · w + IT (u, Iv) · w

since I is g-invariant= −T (w, v) · u − IT (w, Iv) · u

since r(V ) ∈ Rep(g, R)3LA= −T (w, v) · u − T (w, Iv) · Iu

since I is g-invariant= −T(w, v) · u,
whence V ∈ Rep(g, C)aJTS .
We prove (d) along similar lines. In one direction, let [[V ]] ∈ Rep(g, R)LTS and use Lemma 3
and the g-invariance of I to calculate
T(u, v) · w − T(w, v) · u = T (u, v) · w + IT (u, Iv) · w − T (w, v) · u − IT (w, Iv) · u.
The first and third terms and the second and fourth terms combine, using equation (105), to
produceT(u, v) · w − T(w, v) · u = −T (w, u) · v − IT (w, u) · Iv,
which vanishes due to the g-invariance of the complex structure I. In the other direction, from
Lemma 3 we see that T (u, v) is the real part of T(u, v):
T (u, v) = 21 (T(u, v) − T(v, u)) ,whence writing down the LTS condition (105) in full, we find
T (u, v) · w + T (v, w) · u + T (w, u) · v = 21 (T(u, v) − T(v, u)) · w + 21 (T(v, w) − T(w, v)) · u+ 21 (T(w, u) − T(u, w)) · v,
which cancels pairwise using the JTS condition (115).
To prove (e), let V = UC ∈ Rep(g, C). Then V admits a real structure R satisfying equation(89) from where it follows that the map T : V × V → gC obeys T(Ru, Rv) = T(u, v). Under
the action of R, V decomposes as V = U ⊕ iU , where U and iU are the real eigenspaces of
R with eigenvalues ±1, respectively. It follows that if u, v ∈ U then T(u, v) is real and, sinceT(u, v) = −T(v, u), it is skewsymmetric. Hence it defines a real alternating map U × U → g.
This map is seen to be the map T in (97), since equation (89) says that h on U is real, so thatit agrees with the inner product h−, −i on U . Now let w ∈ U and consider T(u, v) · w. Now, if
V ∈ Rep(g, C)JTS , then in particular T(u, v)·w = +T(w, v)·u, but since T(u, v)·w = −T(v, u)·w
we see that T(u, v) · w = 0.

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If, on the contrary, V = UC ∈ Rep(g, C)aJTS , then T(u, v) · w = −T(w, v) · u, whence it istotally skewsymmetric and U ∈ Rep(g, R)3LA , proving the reverse implication in (f). To finish
proving (f), notice that if U ∈ Rep(g, R)3LA , then T(u, v) · w = −T(w, v) · u for all u, v, w ∈ U .
Now use the sesquibilinearity of T(u, v) · w to show that this is satisfied for all u, v, w ∈ V .
To prove (g) simply notice that if ((W )) ∈ Rep(g, C)JTS , so that the map T satisfies the JTS
condition (115), then in particularT(u, Jv) · w = +T(w, Jv) · u,
which together with the symmetry condition T(u, Jv) = T(v, Ju) says that T(u, Jv) · w is
totally symmetric, whence W ∈ Rep(g, H)QTS . The argument is clearly reversible, so we getboth implications.
If instead ((W )) ∈ Rep(g, C)aJTS , then T(u, Jv) · w is symmetric in u ↔ v but skewsymmetricin u ↔ w, whence it has to vanish, which says that ((W )) and hence W is a trivial representation.
This proves (h).
Finally, let us prove (i). Let W = VH . Recall that we do not work with W but with its
image ((W )) under r0 , which from Proposition 1(i) is given by ((VH )) ∼= V ⊕ V . As explained inAppendix A.1.2, we will denote vectors in V by v, for v ∈ V . Then the quaternionic structure
J on V ⊕ V is defined by
Jv = vandJv = −v

for all v ∈ V .

The hermitian structure on V ⊕V is given by the hermitian structures on V and V and declaring
the direct sum to be orthogonal. The complex symplectic structure on V ⊕ V is such that V
and V are lagrangian submodules andω(u, v) = −h(u, v).
The only nonzero components of the map T are
T(u, Jv) = −T(u, v).

(133)

The aLTS condition (130) is satisfied if and only if
T(u, Jv) · w + T(v, Jw) · u + T(w, Ju) · v = 0.
The last term vanishes since V is a lagrangian submodule, hence the aLTS condition is equivalent toT(u, Jv) · w + T(v, Jw) · u = 0,
which using equation (133) is equivalent toT(u, v) · w + T(w, v) · u = 0,
which is equivalent to the aJTS condition (116) on V .

It follows form this proposition (and some geometrical results) that if W ∈ Rep(g, H)QTS thenV = ((W )) ∈ Rep(g, C)JTS and U = [[V ]] ∈ Rep(g, R)LTS . This means that the action of g on U is
the holonomy representation of a symmetric space M . Since U admits an invariant quaternionic
structure, the holonomy representation maps g to sp(n) ⊂ so(4n), with dimR U = 4n. Thismeans that M is a hyperkähler symmetric space and hence, in particular, Ricci-flat. However
any homogeneous Ricci-flat riemannian manifold is actually flat [41, Theorem 7.61], a result
due originally to Alekseevsky and Kimelfeld [42]. This allows us to conclude the following.
Corollary 6. If W ∈ Rep(g, H)QTS is positive-definite, then W is the trivial representation.
There do exist indefinite QTS representations, associated with indefinite hyperkähler manifolds [40].

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51

A.2.6. Embedding Lie (super)algebras. The special classes of unitary representations defined
above all share one characteristic: namely, the fact that together with g, they define a Lie(super)algebra. This is well-known for the case of the representations associated with thesymmetric spaces, as we now briefly recall.
Indeed, if U ∈ Rep(g, R)LTS , then on the 2-graded vector space g ⊕ U , with g having degree
0 and U having degree 1, one can define the structure of a graded metric Lie algebra in thefollowing way. The Lie bracket is given by the Lie bracket of g, the action of g on U and themap T defined by equation (97) above. Then identity (105) is the one component of the Jacobi
identity which is not implicit in the construction. The inner product consisting of the one on
g and the one on U , with both spaces being mutually perpendicular, is invariant under theadjoint action. Conversely, given any 2-graded metric Lie algebra, the degree-1 subspace as arepresentation of the degree-0 Lie subalgebra is an LTS representation.
Now let V ∈ Rep(g, C)JTS . In this case we can define on the 3-graded vector space V ⊕gC ⊕V— with degrees −1, 0, 1, respectively — the structure of a Lie algebra by adding to the Liebracket on gC and the action of gC on V and V , the map T defined in equation (107), butviewed here as a complex bilinear map V × V → gC . Then the defining condition (115) for an
JTS representation implies the two components of the Jacobi identity which are not alreadytrivially satisfied. The 3-graded Lie algebra V ⊕ gC ⊕ V is metric relative to the complex innerproduct defined by the one on gC and by h, thought of as a complex bilinear inner product
V × V → C. Relative to this inner product, the subspaces V and V are isotropic abelian Liesubalgebras. These representations are in one-to-one correspondence with hermitian symmetricspaces. Indeed, the 3-graded Lie algebra V ⊕ gC ⊕ V is the complexification of a 2-graded real
metric Lie algebra g ⊕ [[V ]], and the inner product is given by the one on g together with thereal part of the hermitian inner product on V . We remark that if W is a QTS representationof g, then by Proposition 5(g), ((W )) gives rise to a JTS representation, hence it admits anembedding Lie algebra with a 3-grading, but with both the subspaces of degree ±1 isomorphic
to ((W )).
It turns out that similar results hold for the classes of representations relevant to superconformal Chern–Simons theories, except that now the result of the construction will be a metric
Lie superalgebra, suggesting a larger rôle in the superconformal theory for these Lie superalgebras. The embedding Lie superalgebras were constructed in detail in [16] for the case of
aJTS and aLTS representations: see Theorem 22 for the aJTS case and the discussion aroundequation (45) for the aLTS case. For completeness we review these constructions here.
Let V ∈ Rep(g, C)aJTS . Then on the 3-graded vector space V ⊕ gC ⊕ V we define thestructure of a complex 3-graded Lie superalgebra using the Lie algebra structure on gC , theaction of gC on V and V and the map T defined in equation (107), but again thought of as
a complex bilinear map T : V × V → gC . The identity (116) corresponds now to the one
component of the Jacobi identity which is not already trivially satisfied by the construction.
The resulting complex 3-graded Lie superalgebra is metric relative to the inner product on
gC and the symplectic structure on V ⊕ V defined by declaring V and V to be lagrangian
subspaces and (u, v̄) = h(u, v). This complex Lie superalgebra is the complexification of a
metric Lie superalgebra with underlying vector space g ⊕ [[V ]] and with inner product definedby the one on g together with the imaginary part of the hermitian inner product on V , which isa symplectic structure on [[V ]]. This construction appeared already in [16] and was consideredfurther in [43] and [17].
Similarly, let W ∈ Rep(g, H)aLTS . Consider the 2-graded complex vector space gC ⊕ W , with
gC in degree 0 and W in degree 1. We define the Lie bracket by extending the one on gC andthe action of gC on W by [u, v] = T(u, Jv) for u, v ∈ W . Then the identity (130) is the one

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component of the Jacobi identity for a complex Lie superalgebra which is not automaticallysatisfied in the case of any W ∈ Rep(g, H). This Lie superalgebra is metric relative to the innerproduct on gC and to the complex symplectic form ω on W . This construction appeared in [16]already and was considered further in [17].
Finally, we discuss the case of 3LA representations. It follows from Proposition 5(f) that if
U ∈ Rep(g, R)3LA , then its complexification V = UC ∈ Rep(g, C)aJTS . By Theorem 22 in [16],recalled briefly above, and using that V ∼= V in this case, we may define a 3-graded metric
Lie superalgebra structure on V ⊕ gC ⊕ V . Furthermore, since V here is the complexificationof a real representation, this complex Lie superalgebra is the complexification of a metric Lie
superalgebra which, unlike in the general case of aJTS representations, is also 3-graded. We
can see this explicitly as follows. Consider the 3-graded real vector space U−1 ⊕ g0 ⊕ U1 , where
the subscripts reflect the degree. For every u ∈ U , we will write u1 and u2 , respectively, thecorresponding vectors in U1 and U−1 . We write ua generically, where a = 1, 2. Then we define
the following Lie brackets in addition to the ones of g:[X, ua ] = (X · u)aand[ua , vb ] = ab T (u, v),(134)

with ab the Levi-Civita symbol with 12 = 1, say. The inner product is defined to be the one
on g extended by(ua , vb ) = ab hu, vi .
(135)
It is a simple exercise to verify that the Jacobi identity is satisfied and that the resulting innerproduct is ad-invariant.
Conversely, given any 3-graded metric Lie superalgebra U−1 ⊕ g0 ⊕ U1 with U1 and U−1 both
isomorphic to an orthogonal representation U of g, then the 3-bracket [u, v, w] on U defined by[[ua , vb ], wc ] = ab [u, v, w]c ,(136)

defines a metric 3-Lie algebra structure on U .
In summary, we have the following characterisation of metric 3-Lie algebras, which gives one
answer to an open question stated in [16] and allows us to paraphrase Kantor, as quoted in [44],and suggest that there are no 3-Lie algebras, only Lie superalgebras.
Theorem 7. Metric 3-Lie algebras (U, [u, v, w], hu, vi) are in one-to-one correspondence withmetric 3-graded Lie superalgebras U−1 ⊕ g0 ⊕ U1 , where U1 and U−1 are both isomorphic to U ,a faithful orthogonal representation of g.
Example 8. As shown by Ling [45] there is a unique complex simple 3-Lie algebra. Thereis a unique real form of this 3-Lie algebra which is metric relative to a positive-definite innerproduct. The corresponding vector space is R4 with the standard euclidean inner product andg = so(4) the Lie algebra of skewsymmetric endomorphisms, with inner product given under
the isomorphism so(4) = su(2)⊕su(2) by the Killing form on on the first so(3) and the negativeof the Killing form on the second. The corresponding 3-graded Lie superalgebra is a “compact”
real form of A(1, 1) in the Kac classification [26]. Notice that the Killing form of A(1, 1) vanishes
identically, but here we see that it does nevertheless have a non-degenerate inner product.
A.2.7. Simplicity. We have seen above that to every Lie-embeddable representation of a metric
Lie algebra we can attach a triple system and a Lie (super)algebra. In principle, there are three
separate notions of simplicity or irreducibility we can consider: irreducibility of the representation, simplicity of the embedding Lie (super)algebra and simplicity of the triple system — thislatter one being defined as the nonexistence of proper ideals in the triple system, ideals being
defined as kernels of homomorphisms. For the case of positive-definite aJTS representations,this has been discussed recently in [43] and from the present point of view in [17], where LTS

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

53

and aLTS representations are also treated. The following theorems are proved in [17]; although
the first is of course classical. In that paper we had not yet identified anti-Jordan triple systems
for what they were and referred to them as N =6 triple systems instead.
Theorem 9. Let g be a metric Lie algebra, U ∈ Rep(g, R)LTS faithful and positive-definite andlet k = g ⊕ U denote its embedding 2-graded Lie algebra. The following are equivalent:(a) U ∈ Irr(g, R)LTS ,(b) U is a simple Lie triple system,(c) k is a simple Lie algebra or else U ∼= g (as g-modules) and g is a simple Lie algebra.
Theorem 10. Let g be a metric Lie algebra, V ∈ Rep(g, C)aJTS faithful and positive-definite
and let k = V ⊕ gC ⊕ V denote its embedding 3-graded Lie superalgebra. The following are
equivalent:(a) V ∈ Irr(g, C)aJTS ,(b) V is a simple anti-Jordan triple system,(c) k is a simple Lie superalgebra.
Theorem 11. Let g be a metric Lie algebra, W ∈ Rep(g, H)aLTS positive-definite and let
k = gC ⊕ W denote its embedding Lie superalgebra. The following are equivalent:(a) W ∈ Irr(g, H)aLTS ,(b) W is a simple quaternionic anti-Lie triple system,(c) k is a simple Lie superalgebra.
Similar results can be proved also for the Jordan triple systems, but as they do not playsuch an important role in the study of superconformal Chern–Simons theories, we will not
mention them in this paper. Also since by Corollary 6 there are no positive-definite QTS
representations, this question does not arise in this case. There is, however, a classificationof hyperkähler symmetric spaces [40]. Finally, example 8 shows that the same result holds
for the unique positive-definite U ∈ Irr(g, R)3LA , whose associated triple system is the uniquepositive-definite nonabelian simple 3-Lie algebra in [32] and which embeds in the simple Liesuperalgebra A(1, 1).
The above results allow a classification of positive-definite irreducible Lie-embeddable representations which is summarised in Table 6, where we only the ones of classes 3LA, aJTS andaLTS, as these are the ones relevant for the study of three-dimensional superconformal Chern–
Simons-matter theories. The other positive-definite Lie-embeddable classes are associated withthe riemannian and hermitian symmetric spaces and that classification is classical and can befound, for example, in [46].
Appendix B. Deriving the superpotentials
This appendix provides a derivation of the N =1 superpotentials WC + WF in (20) and WH
in (24) which respectively provide enhanced N =2 and N =3 superconformal symmetry whenthe matter representation M is of generic complex and quaternionic type. The proof follows
from the requirement of obtaining the appropriate R-symmetry commensurate with the degreeof supersymmetry enhancement in the on-shell lagrangian.
B.1. N =2 superpotential. Let us begin by assuming M = V ∈ Rep(g, C). If we wish toobtain an enhanced N =2 superconformal symmetry in the on-shell N =1 lagrangian in (9) thenit is necessary (but not sufficient) for it to be invariant under the associated u(1) R-symmetry.
With respect to this desired enhancement we can ascribe the u(1) R-charge 12 to the bosonicmatter field X and − 12 to the fermionic matter field Ψ (with their complex conjugates having

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the opposite R-charges). Notice that this assignment implies that all the terms in (9) areautomatically u(1)-invariant (as a consequence of the hermitian inner product h on V being
complex-sesquilinear) except for the scalar-fermion Yukawa couplings.
It is convenient to break up the Yukawa couplings into the contributions transforming withdifferent overall u(1) charges. To this end, let us first decompose the quartic superpotential
W = W4,0 + W3,1 + W2,2 + W1,3 + W0,4 with respect to the complex structure on V , where
Wp,4−p = W4−p,p for all p = 0, 1, 2, 3, 4 since W is real. Each component Wp,4−p is a quartic
tensor which is taken to be complex-linear in its first p arguments and complex-antilinear in
its remaining 4 − p arguments. Thus W2,2 (X) is real and uncharged, W3,1 (X) has charge 1
and W4,0 (X) has charge 2. The other contribution to the Yukawa couplings involves T(X, Ψ)which has charge 1 (and its complex conjugate −T(Ψ, X) with charge −1). Assembling thesecontributions to the Yukawa couplings, we see that there are separate contributions from termswith overall charges 0, ±1 and ±2. The uncharged contributions are therefore unconstrained
and the u(1) R-symmetry can only be realised if the complex terms with charges −1 and −2(and their complex conjugates with charges 1 and 2) vanish identically.
The component W4,0 (and its complex conjugate) only appear in a term with no overall u(1)charge. This unconstrained component is to be identified with the F-term superpotential in the
N =2 theory. There is only one contribution to the charge −1 term involving the component
W3,1 and it is straightforward to check that this term can vanish only if W3,1 = 0. The
remaining contributions to the charge −2 terms involve W2,2 and it is easily checked that they1
vanish only if W2,2 (X) = 16
(T(X, X), T(X, X)). Thus we have established that WC + WF
in (20) is the most general N =1 superpotential which can realise the aforementioned u(1) Rsymmetry in the on-shell lagrangian. The fact that we have already established that this theory
is invariant under the N =2 superconformal algebra thus means that the realisation of this u(1)R-symmetry is in fact necessary and sufficient in this instance for N =2 enhancement.
B.2. N =3 superpotential. Let us now take M = W ∈ Rep(g, H) but as usual understood as
the complex representation ((W )) with a quaternionic structure. Since a theory with enhanced
N =3 superconformal symmetry can be thought of as a special kind of N =2 superconformal
Chern–Simons-matter theory, we can utilise the result of the previous appendix to deduce theconstraints which an enhanced usp(2) > u(1) R-symmetry in the on-shell N =2 lagrangian(2.3.2) puts on the generic holomorphic N =2 F-term superpotential WF .
With respect to this desired enhancement we can collect the W -valued bosonic and fermionicmatter fields X and Ψ into the fields X α = (X, JX) and Ψα = (Ψ, JΨ), obeying identically thepseudo-reality conditions JX α = εαβ X β and JΨα = εαβ Ψβ , corresponding to the fundamental
representation of usp(2). Consequently, the kinetic terms hDµ X, Dµ Xi = 21 hDµ X α , Dµ X α iand Ψ̄, γ µ Dµ Ψ = 12 Ψ̄α , γ µ Dµ Ψα in (2.3.2) are automatically usp(2)-invariant, but the samecannot be said of the Yukawa couplings and scalar potential. The extra conditions for usp(2)R-symmetry in these terms can be deduced most easily by focusing initially on the Yukawa
couplings. We will see that the condition that they be usp(2)-invariant then guarantees thatthe scalar potential is too.
Consider the Yukawa couplings in the third line of (2.3.2) which do not involve the Fterm superpotential. Since T(X α , X α ) vanishes identically, it turns out that there are notmany options for writing down manifestly usp(2)-invariant terms which recover these Yukawa
1
couplings. In particular, one finds that only from − 16
(T(X α , X β ), T(Ψ̄α , Ψβ )) can one recover
the second term while the first term must come from − 81 (T(X α , Ψ̄β ), T(Ψα , X β )) (other possible
usp(2)-invariant permutations of indices can be rewritten in terms of these using the pseudoreality condition for the matter fields). Of course, these contributions do not give just the third

CHERN–SIMONS SUPERPOTENTIALS FROM REPRESENTATION THEORY

55

line of (2.3.2) or else it would already be usp(2)-invariant. Indeed, the point of doing this is
to isolate the terms responsible for obstructing the enhanced usp(2) R-symmetry. Combining
these extra terms with the contribution from the F-term superpotential (which one finds also
cannot be usp(2)-invariant on its own) to the Yukawa couplings gives an overall obstruction
1
term which vanishes only if WF (X) = 32
(T(X, JX), T(X, JX)).
This is precisely the F-term superpotential in (23) which gives rise to the N =1 superpotential
WH in (24). The fact that we have already established this theory to be invariant under the
N =3 superconformal algebra means that the realisation of this usp(2) R-symmetry in the
Yukawa couplings is in fact necessary and sufficient for N =3 enhancement here — the resulting
usp(2)-invariant form of the on-shell N =3 lagrangian being given by (41).

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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