Marginal Deformations and 3-Algebra Structures
aa r X i v : . [ h e p - t h ] F e b NIKHEF/2009–008TCDMATH 09–15DAMTP 2009–47
Marginal Deformations and 3-Algebra Structures
Nikolas Akerblom a , Christian S¨amann b and Martin Wolf c, ∗† a NIKHEF Theory GroupScience Park 1051098 XG Amsterdam, The Netherlands b Hamilton Mathematics Institute&School of MathematicsTrinity College, Dublin 2, Ireland c Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeWilberforce Road, Cambridge CB3 0WA, United Kingdom
Abstract
We study marginal deformations of superconformal Chern-Simons matter theories thatare based on 3-algebras. For this, we introduce the notion of an associated 3-product,which captures very general gauge invariant deformations of the superpotentials ofthe BLG and ABJM models. We also consider conformal multi-trace deformationspreserving N = 2 supersymmetry. We then use N = 2 supergraph techniques tocompute the two-loop beta functions of these deformations. Besides confirming con-formal invariance of both the BLG and ABJM models, we also verify that the recentlyproposed β -deformations of the ABJM model are indeed marginal to the order we areconsidering.June 09, 2009 ∗ Also at the Wolfson College, Barton Road, Cambridge CB3 9BB, United Kingdom. † E-mail addresses: [email protected] , [email protected] , [email protected] ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. 3-Algebras and associated 3-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1. Real 3-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Matrix representations of real 3-algebras . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Associated 3-products of real 3-algebras . . . . . . . . . . . . . . . . . . . . . . . . 52.4. Hermitian 3-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5. Matrix representations of Hermitian 3-algebras . . . . . . . . . . . . . . . . . . . . 72.6. Associated 3-products for Hermitian 3-algebras . . . . . . . . . . . . . . . . . . . . 83. Deformations of BLG-type actions preserving N = 2 supersymmetry . . . . . . . . . . . 93.1. Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Deformations of the superfield action in the real case . . . . . . . . . . . . . . . . . 93.3. Deformations of the superfield action in the Hermitian case . . . . . . . . . . . . . 114. Marginal deformations of the BLG and ABJM models . . . . . . . . . . . . . . . . . . . 124.1. Quantum action in the real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2. Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3. Powercounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4. Two-loop renormalization in the real case . . . . . . . . . . . . . . . . . . . . . . . 154.5. Two-loop renormalization in the Hermitian case . . . . . . . . . . . . . . . . . . . . 194.6. Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Appendices
A. Casimirs for matrix representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25B. Component form of the actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26C. Feynman rules: Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1. Introduction
In the work of Bagger, Lambert [1] as well as Gustavsson [2], a candidate theory for multipleM2-branes was proposed, which has attracted much attention in the last year. Initially, thistheory was conjectured to be an IR description of stacks of M2-branes in the same sense asmaximally supersymmetric Yang-Mills theory (SYM) provides an effective description of stacks ofD-branes. Soon after its discovery, however, it was realized that this Bagger-Lambert-Gustavsson(BLG) model cannot capture stacks of arbitrarily many M2-branes: Its interactions and the gaugealgebraic structure are based on 3-Lie algebras [4], and there is only one such 3-Lie algebra whichfulfills all reasonable physical requirements [5].One way to circumvent this problem is to generalize the concept of a 3-Lie algebra as donein [6] and [7]. These generalizations yield superconformal field theories which allow for morefreedom but at the cost of a reduced amount of supersymmetry compared to the original BLGmodel. The generalizations discussed in [6], for example, yield the so-called Aharony-Bergman-Jafferis-Maldacena (ABJM) model [8] as a special case, see also [9]. This theory shares many See [3] and references therein for a detailed discussion of algebras with n -ary brackets. N = 4 SYM theory in four dimensions such as planar integrability [10–13] (see [14]for an earlier account). Therefore, it is interesting to ask what phenomena familiar from N = 4SYM theory in four dimensions persist in these generalized BLG-type models.One such phenomenon is the existence of marginal deformations. There is a 3-parameterfamily of such deformations of N = 4 SYM theory, which was found by Leigh and Strassler [15].These include in particular the so-called β -deformations as a subclass. Written in terms of four-dimensional N = 1 superfields, where the field content of N = 4 SYM theory is encoded in threechiral superfields Φ i , i = 1 , ,
3, and a vector superfield, these deformations are given by thesuperpotential terms W = ε ijk tr ([Φ i , Φ j ] β Φ k ) , with [Φ i , Φ j ] β := e i β Φ i Φ j − e − i β Φ j Φ i . (1.1)The theories with such a superpotential are still finite and, as they are written in terms ofsuperfields, they are manifestly N = 1 supersymmetric.In this paper, we make an attempt at the construction of analogous deformations for BLG-type models. For a rough guideline on what structures one expects to arise at 3-algebra level,one can look at the reduction process from M2-branes to D2-branes as described in [16]. For thisreduction, one has to compactify a direction transverse to the M2-branes on a circle. In [16], itwas suggested that in this compactification process, the scalar describing M2-brane fluctuationsin this direction would acquire the vacuum expectation value h X ◦ i = Rℓ / p = g Y M . Here, R isthe radius of the circle, ℓ p the Planck length, and g Y M the Yang-Mills coupling constant. Theinteraction terms of the BLG model are formulated using totally antisymmetric 3-brackets of 3-Lie algebras. In the reduction process, 3-brackets of the form [ X ◦ , X , X ] reduce to commutatorterms g Y M [ X , X ] = [ X ◦ , X , X ], and in a strong coupling expansion, only those 3-bracketexpressions which reduce to a commutator survive. To obtain terms which correspond to β -deformed commutators, one evidently has to relax the total antisymmetry of the 3-bracket. Oneis therefore led to look for marginal deformations amongst models which are built from the 3-Liealgebras introduced in [6] and [7].There are already proposals for β -deformations of both the BLG and the ABJM model in theliterature [17,18] based on considering gravitational duals. Here, we will study such deformationsin more detail from the gauge theory perspective: We will write down the most general gaugeinvariant deformations of BLG-type models based on 3-algebras in N = 2 superspace. Althoughthe generalized 3-Lie algebras of [6] and [7] already allow for certain classes of marginal defor-mations, we find that we should also introduce the notion of an associated 3-product: A newtriple product, which transforms covariantly under gauge transformations. Moreover, we includeall classically conformal multi-trace terms that are compatible with N = 2 supersymmetry. TheLagrangians we find are rather restrictive, but contain the deformations studied in [18]. We thenevaluate the beta functions of the couplings arising from the admissible deformations using super-graph techniques up to two-loop order. We confirm the conformal invariance of the BLG and theABJM model as well as the deformations of [18] at quantum level to this order in perturbationtheory.This paper is structured as follows. In Section 2, we discuss the necessary 3-algebraic struc-tures, the relation between 3-algebras and their associated gauge algebras and introduce asso- Multi-trace terms received attention in this context rather recently in [19].
2. 3-Algebras and associated 3-products
The need for extending the BLG model to higher numbers of M2-branes led to two generalizationsof the notion of a 3-algebra: the generalized 3-Lie algebras [7], which we will refer to as real 3-algebras , and the
Hermitian 3-algebras [6], see also [20] for a summary and a re-interpretation interms of ordinary Lie algebras. In both cases, the underlying 3-bracket is no longer required tobe totally antisymmetric.In the following, we will review these structures as well as their representations using matrixalgebras. We also introduce the notion of an associated 3-product, a generalization of a 3-bracket ,which will allow us to discuss extended superpotential terms yielding marginal deformations ofboth the BLG and ABJM models. A metric real 3-algebra is a real vector space A together with a trilinear bracket [ · , · , · ] : A ×A × A → A and a positive definite bilinear symmetric pairing ( · , · ) : A × A → R satisfying thefollowing properties for all A, B, C, D, E ∈ A :(i) The real fundamental identity:[
A, B, [ C, D, E ]] = [[
A, B, C ] , D, E ] + [ C, [ A, B, D ] , E ] + [ C, D, [ A, B, E ]] , (2.1a)(ii) the real compatibility relation:([ A, B, C ] , D ) + ( C, [ A, B, D ]) = 0 , (2.1b)(iii) and the real symmetry property:( D, [ A, B, C ]) = ( B, [ C, D, A ]) . (2.1c)This is a generalization of the concept of a 3-Lie algebra in the sense of Filippov [4], whichamounts to the special case of a totally antisymmetric 3-bracket.Choosing a basis τ a of A , a = 1 , . . . , dim A , we can introduce the metric h ab and the structureconstants f abcd as h ab := ( τ a , τ b ) and f abcd := ( τ d , [ τ a , τ b , τ c ]) . (2.2)Because of the properties (ii) and (iii), the structure constants obey the following symmetryrelations: f abcd = − f bacd = f cdab = − f abdc . (2.3) A similar generalization has been employed in [17]. Z -graded objects as e.g. bosonic or fermionic fields, we define the3-bracket to be insensitive to the grading:[ A, B, C ] := A a B b C c [ τ a , τ b , τ c ] , with A = A a τ a etc. (2.4)Every real 3-algebra comes with an associated Lie algebra g A , the Lie algebra of inner deriva-tions on A . Choosing a basis τ a of A , we define g A to be the image of the map δ : Λ A →
Der( A )that is given by Λ A ∋ X = X ab τ a ∧ τ b δ X ∈ Der( A ) δ X ( A ) := X ab [ τ a , τ b , A ] (2.5)for A ∈ A . Note that X ab = − X ba . Note also that δ is not an injective map in general and thusthe components X ab in the definition of δ X are usually not uniquely defined. The Lie bracket[[ · , · ]] on g A is defined by the commutator action on A , i.e. [[ δ X , δ Y ]]( A ) := δ X ( δ Y ( A )) − δ Y ( δ X ( A ))for A ∈ A . Closure of this bracket on g A follows from the fundamental identity (2.1a).Additionally, we may endow the Lie algebra g A with a bilinear pairing(( δ X , δ Y )) := X ab Y cd f abcd , (2.6)which is symmetric, non-degenerate and ad -invariant, i.e. (([[ δ X , δ Y ]] , δ Z )) + (( δ Y , [[ δ X , δ Z ]])) = 0.The most prominent example of a 3-Lie algebra is the algebra A , which is the vector space R endowed with the following 3-bracket and bilinear pairing: f abcd = ε abcd and h ab = δ ab . (2.7)The associated Lie algebra is g A ∼ = so (4) ∼ = su (2) ⊕ su (2), and the bilinear pairing induced bythe structure constants on this Lie algebra has split signature : On the first su (2) it is positivedefinite, on the second one negative definite. Further classes of examples of real 3-algebras aregiven in the next section. By a matrix representation ρ ( A ) of a 3-algebra A , we will mean a homomorphism ρ : A → R :=Mat( N, C ), which forms a representation of the 3-algebra A in the following way: The invariantpairing on A is given by the natural scalar product ( A, B ) := tr ( ρ ( A ) † ρ ( B )) for elements A, B ∈ A and the 3-bracket is constructed using the natural operations on the matrix algebra: The productand the Hermitian conjugate. It should be stressed that ρ ( A ) can be a true subset of R ; however,the 3-bracket is certainly required to close on ρ ( A ).In the case of real 3-algebras, the matrix algebra R is restricted to Mat( N, R ) and theHermitian conjugate turns into the transpose. In the sequel, we will often not make a notationaldistinction between an element A ∈ A and its matrix realization ρ ( A ) ∈ R and simply write A in both cases. This property is connected to parity invariance of the Chern-Simons Lagrangian, cf. Section 3.2. One could also choose Hermitian matrices; they, however, can be embedded into the real matrices, so that ourrestriction does not imply any loss of generality. Rα : A, B, C α ([[ A T , B ] , C ] + [[ A, B T ] , C ] + [[ A, B ] , C T ] − [[ A T , B T ] , C T ]) , II Rα : A, B, C α ([[ A, B T ] , C ] + [[ A T , B ] , C ]) , III
Rα,β : A, B, C α ( AB T − BA T ) C + βC ( A T B − B T A ) , IV Rα,β : A, B, C α ([[ A, B ] , C ] + [[ A T , B T ] , C ] + [[ A T , B ] , C T ] + [[ A, B T ] , C T ])+ β ([[ A, B ] , C T ] + [[ A T , B ] , C ] + [[ A, B T ] , C ] + [[ A T , B T ] , C T ]) , (2.8)where α and β are arbitrary (real) parameters. Although α can always be removed from thebracket by a rescaling, we will find it convenient to keep it explicitly.Besides forming representations, these brackets give rise to a real 3-algebra structure onMat( N, R ), and we denote the arising real 3-algebras by M R I α ( N ) , . . . , M R IV α,β ( N ).The case M R III α,β ( N ) is of particular importance: The real 3-algebras C d defined in [7] allowfor representations in the class III Rα,β . The 3-Lie algebra A , which is a sub-3-algebra of C can be identified with a real sub-3-algebra of M R III , − (4). Let us therefore expose the associatedLie algebra structure of M R III α,β ( N ) in the following. A derivation δ X ∈ g A acts on an element C ∈ A = M R III α,β ( N ) according to δ X ( C ) = X ab [ τ a , τ b , C ] = αX ab ( τ a τ Tb − τ b τ Ta ) | {z } =: ˆ X L C + C βX ab ( τ Ta τ b − τ Tb τ a ) | {z } =: ˆ X R = ˆ X L C + C ˆ X R . (2.9)Thus, g A splits into two parts: one acting on A from the left and one acting from the right. Thefact that g A forms a Lie algebra follows from the fundamental identity as mentioned above. Inparticular, [[ δ X , δ Y ]]( C ) = [ ˆ X L , ˆ Y L ] C + C [ ˆ Y R , ˆ X R ] = ˆ Z L C + C ˆ Z R = δ Z ( C ) . (2.10)Note that ˆ X L = − ˆ X TL and ˆ X R = − ˆ X TR , that is, both are antisymmetric matrices and they canbe chosen independently. We therefore conclude that g A ⊆ o ( N ) ⊕ o ( N ) and in particular, if ρ ( A ) = R , we have g A ∼ = o ( N ) ⊕ o ( N ). Moreover, a short calculation reveals that the pairing on g A is given by (( X, Y )) = X ab Y cd f abcd = − α tr ( ˆ X † L ˆ Y L ) − β tr ( ˆ X † R ˆ Y R ) , (2.11)and thus for α = − β , the pairing has split signature. This property is required to render aChern-Simons matter theory based on this gauge algebra parity invariant, see Section 3.2. In gauge theories, the gauge potential (and its superpartners) takes values in a Lie algebra, whilethe matter fields take values in a representation of this Lie algebra. If the matter fields
X, Y sit inthe adjoint matrix representation, there is a product between these fields – the ordinary matrixproduct – which transforms covariantly under gauge transformations δ Λ = [Λ , · ]:[Λ , X · Y ] = [Λ , X ] · Y + X · [Λ , Y ] . (2.12)5oth the matrix product and the commutator are special cases of the more general product α XY − α Y X , with α , ∈ C , (2.13)which also transforms covariantly. An analogous product can be introduced for representationsof 3-algebras: Consider a matrix representation R of a real 3-algebra A . An associated 3-product of A in R is a trilinear map h A, B, C i : R × R × R → R satisfying the following identity:[
A, B, h C, D, E i ] = h [ A, B, C ] , D, E i + h C, [ A, B, D ] , E i + h C, D, [ A, B, E ] i . (2.14)This identity corresponds to the condition that the associated 3-product transforms covariantlyunder gauge transformations governed by the 3-bracket. Later on, this will allow us to replaceordinary 3-brackets in the superpotential by associated 3-products preserving gauge invariance.Evidently, all matrix representations of 3-brackets satisfy this identity and thus they are justspecial cases of associated 3-products. The general associated 3-product, however, allows formore general deformations of the superpotential than the conventional 3-bracket would do. Inthe Hermitian case, this includes in particular the deformations studied in [18], as discussed later.One may now ask for the most general 3-product, which can be written down using nothingbut matrix products and transpositions, analogously to the matrix representations of 3-brackets(2.8). In the representation R of type III Rα,β , the most general such product reads as h A, B, C i = α AB T C + α CB T A + β BC T A + β AC T B + γ CA T B + γ BA T C , (2.15)where α , , β , and γ , are real parameters. A metric Hermitian 3-algebra is a complex vector space A together with a bilinear-antilinear tripleproduct [ · , · ; · ] : A × A × A → A and a positive definite Hermitian pairing ( · , · ) : A × A → C satisfying the following properties for all A, B, C, D, E ∈ A :(i) The Hermitian fundamental identity:[[
C, D ; E ] , A ; B ] = [[ C, A ; B ] , D ; E ] + [ C, [ D, A ; B ]; E ] − [ C, D ; [
E, B ; A ]] , (2.16a)(ii) the Hermitian compatibility relation:( D, [ A, B ; C ]) − ([ D, C ; B ] , A ) = 0 , (2.16b)(iii) and the Hermitian symmetry property:( D, [ A, B ; C ]) = − ( D, [ B, A ; C ]) . (2.16c)With respect to a basis τ a of A , we introduce the metric and the structure constants h ab = ( τ a , τ b ) and f abcd := ( τ d , [ τ a , τ b ; τ c ]) , (2.17) We choose the first slot to be antilinear and the second one to be linear. h ab = ( h ba ) ∗ and f abcd = − f bacd = − f abdc = ( f cdab ) ∗ . (2.18)Analogously to the case of real 3-algebras, a Hermitian 3-algebra comes with an associatedLie algebra, which is naturally a complex Lie algebra g C A . Here, we will merely be interestedin a real form g A of g C A that is defined as follows: Consider a basis τ a of A together with abasis τ ∗ a of the complex conjugate A ∗ of A . An element X = X ab τ a ∧ τ ∗ b of Re ( A ∧ A ∗ ) hascomponents X ab satisfying X ab = − ( X ba ) ∗ , and we then define g A to be the image of the map δ : Re ( A ∧ A ∗ ) → Der( A ), with X δ X and δ X ( A ) := X ab [ A, τ a ; τ b ] , (2.19)for A ∈ A . The Lie bracket [[ · , · ]] on g A is defined as the commutator action of two inner derivations δ X , δ Y ∈ g A on A ∈ A . As in the case of real 3-algebras, closure of this bracket on g A followsfrom the fundamental identity.A pairing on g A can be chosen as [20](( δ X , δ Y )) := X ab Y cd f cabd . (2.20)This pairing is symmetric, bilinear, non-degenerate and ad -invariant. Note that when A is con-sidered as the carrier space for a representation of g A , A ∗ forms the carrier space for the complexconjugate representation. Let us now come to matrix representations of Hermitian 3-algebras as introduced in Section 2.3. Itwas shown in [21] that there is only one such family of representations given by a homomorphism ρ : A →
Mat( N, C ) and the 3-bracketI Hα : A, B, C α ( AC † B − BC † A ) , (2.21)where α is a real parameter. Interestingly, this is also the representation used in [6] to recast theABJM model in 3-algebra language.In the following, we will denote the Hermitian 3-algebra defined by the above bracket onMat( N, C ) by M H I α ( N ). Note that the 3-Lie algebra A introduced above coincides with theHermitian 3-algebra M H I α (2).The associated Lie algebra structure of this Hermitian 3-algebra is easily found to be g A ∼ = su ( N ) ⊕ su ( N ), cf. [6]: Consider an element of δ X = X ab [ · , τ a ; τ b ] ∈ g A , where τ a and τ b arecomplex N × N -matrices and ( X ab ) ∗ = − X ba . With the definition (2.21), we obtain ( α = 1) δ X ( A ) = X ab [ A, τ a ; τ b ] = X ab ( Aτ † b τ a − τ a τ † b A ) . (2.22)Analogously to the case of M R III α,β ( N ), we can associate the following matrices with the innerderivations: ˆ X R = X ab τ † b τ a and ˆ X L = − X ab τ a τ † b , (2.23) The precise definition of A ∗ is irrelevant at this point. Note that our definition differs from that of [20] in that we have introduced an additional factor of 1 / X R ) † = ( X ab τ † b τ a ) † = − X ba τ † a τ b = − ˆ X R .Similar considerations as in the real case show that ˆ X R and ˆ X L can be chosen independently,exhausting the fundamental representation of su ( N ). The trace part is excluded as it would havea trivial action on A . Since left- and right-actions commute, we arrive at the conclusion that g A ∼ = su ( N ) ⊕ su ( N ).The symmetric bilinear pairing of elements δ X , δ Y ∈ g A is then given by(( X, Y )) = X ab Y cd f cabd = tr ( ˆ X † L ˆ Y L ) − tr ( ˆ X † R ˆ Y R ) , (2.24)and this expression shows that the signature on g A is again split, with positive and negativesignature on the left and right acting subalgebra of g A , respectively. Consider again a matrix representation R of a Hermitian 3-algebra A . By an associated 3-productof A in R , we mean a bilinear-antilinear map h A, B ; C i : R × R × R → R satisfying the followingidentity:[ h C, D ; E i , A ; B ] = h [ C, A ; B ] , D ; E i + h C, [ D, A ; B ]; E i − h C, D ; [
E, B ; A ] i . (2.25)We specialize now to the Hermitian 3-algebra M H I α ( N ) with basis τ a for which R = Mat( C , N ).Note that the τ a form a basis for both M H I α ( N ) and R . With respect to this basis, we canintroduce structure constants of the associated 3-product as follows: h τ a , τ b ; τ c i = g abcd τ d and g abcd = g abce h de . (2.26)In the representation R of type I Hα , the most general such product written in terms of matricesand Hermitian conjugation is given by the following expression: h A, B ; C i = α AC † B − α BC † A , (2.27)where α , are complex parameters.Below, we shall solely be interested in the one-parameter family that is given by α = e i β and α = e − i β for β ∈ R . In analogy to the β -deformed commutator given in (1.1), we denote the β -3-bracket by[ τ a , τ b ; τ c ] β := e i β τ a τ † c τ b − e − i β τ b τ † c τ a =: (cid:2) cos βf abcd + i sin β d abcd (cid:3) τ d . (2.28)The f abcd are the structure constants of the Hermitian 3-bracket and d abcd = d abce h de obeys d abcd = d bacd = d abdc = ( d cdab ) ∗ . (2.29)Therefore, g abcd = g badc = − ( g dcab ) ∗ . (2.30)These symmetry properties of the structure constants g abcd can be re-phrased without referringto a particular choice of basis analogously to (2.16b) and (2.16c):( D, [ A, B ; C ] β ) = − ([ D, C ; A ] β , B ) and ( D, [ A, B ; C ] β ) = ( C, [ B, A ; D ] β ) . (2.31)Interestingly, (2.28) will yield precisely the marginal deformations of the ABJM case recentlystudied in [18]. 8 . Deformations of BLG-type actions preserving N = 2 supersymmetry In the following, we present deformations of BLG-type actions which make use of either real3-algebras or Hermitian 3-algebras as their gauge 3-algebra structures. We will refer to these twocases as the real and Hermitian cases, respectively. All deformations will be manifestly N = 2supersymmetric and supergauge invariant. We shall use the usual superfield conventions of [22] dimensionally reduced from four to threedimensions as done in [7]. Our superfields will live on R , | and their expansions are given byΦ i ( y ) = φ i ( y ) + √ θψ i ( y ) + θ F i ( y ) , (3.1a)for the chiral superfield and V ( x ) = − θ α ¯ θ β ( σ µαβ A µ ( x ) + i ε αβ σ ( x )) + i θ (¯ θ ¯ λ ( x )) − i¯ θ ( θλ ( x )) + θ ¯ θ D ( x ) (3.1b)for the vector superfield in Wess-Zumino (WZ) gauge. Here, y are chiral coordinates, i, j, . . . =1 , . . . , N f are flavor indices (counting complex field components) and α, β, . . . = 1 , N f = 4, but keeping N f arbitrary willprove useful as a book-keeping device. Notice that the spin group in 1 + 2 dimensions is SL (2 , R )and hence, we do not need to distinguish between dotted and undotted spinors. In particular,indices of barred spinors can be contracted with those of unbarred ones. Our conventions forspinor contractions are as follows: χψ := χ α ψ α , ¯ χ ¯ ψ := ¯ χ α ¯ ψ α . Furthermore, σ µ are the σ -matricesin three dimensions with σ µαβ = σ µβα and ε αβ = − ε βα with ε αγ ε γβ = δ βα .The superfields Φ i take values in a 3-algebra A , while V takes values in its associated Liealgebra g A . By a bar, we shall mean the appropriate complex conjugation operation (i.e. that ofcomponents and that of the gauge algebra representation).To make our notation more concise, we shall always write X ( A ) or even XA as a shorthandfor the action of an element δ X of the associated Lie algebra g A on A ∈ A . We start from a Wess-Zumino model minimally coupled to a Chern-Simons theory. Correspond-ingly, the superfield action reads as S R = i √ κ Z d | z Z d t (( V, ¯ D α (cid:0) e − √ κ tV D α e √ κ tV (cid:1) )) + Z d | z ( ¯Φ i , e − √ κ V Φ i ) , (3.2)where d | z := d x d θ , cf. [23, 7]. The superfields Φ i are all in the same representation of thegauge algebra g A whose carrier space is A . The coupling constant κ is related to the Chern-Simons level k via κ = k/π . Notice that the vector superfield has been rescaled appropriatelyto ensure that the action (3.2) has a proper free-field limit, 1 / √ κ →
0, needed for perturbationtheory. When discussing the quantum theory, we will not fix WZ gauge; see below. i.e. either a real or a Hermitian 3-algebra g A ∼ = g ⊕ g , where g ∼ = g ,and the bilinear pairing is positive definite on g and negative definite on g . The Chern-SimonsLagrangian then splits into two pieces of Chern-Simons type with a relative sign between thetwo Chern-Simons levels. Parity invariance can now be restored by postulating that under thistransformation, the first Chern-Simons Lagrangian transforms into the second one and vice versa.We also allow for superpotential terms, which we take to be of the following form: S R = Z d | z h R (1) ijkl (Φ l , [Φ i , Φ j , Φ k ]) + R (2) ijkl (Φ i , Φ j )(Φ k , Φ l ) i + Z d | ¯ z h R ijkl (1) ( ¯Φ l , [ ¯Φ i , ¯Φ j , ¯Φ k ]) + R ijkl (2) ( ¯Φ i , ¯Φ j )( ¯Φ k , ¯Φ l ) i , (3.3)where d | z := d x d θ and d | ¯ z := d x d ¯ θ are the (anti)chiral superspace measures. Thesymmetry properties of the 3-bracket and the pairing induces the following symmetry structureson the four-index parameters: R (1) ijkl = − R (1) jikl = − R (1) ijlk = R (1) klij and R (2) ijkl = R (2) jikl = R (2) ijlk = R (2) klij . (3.4)The couplings with upper indices are related to those with lower indices by complex conjugation, R (1) ijkl = ( R ijkl (1) ) ∗ and R (2) ijkl = ( R ijkl (2) ) ∗ . (3.5)The component form of the action S R + S R is given in Appendix B.Note that the double trace term in the superpotential (3.3) corresponds to a double and atriple trace deformation in the potential. Note also that when discussing Feynman rules, thequartic terms R (1) and R (2) may be formally combined into one single vertex, cf. (C.24) togetherwith (C.22d) and (C.22e). Furthermore, the full supergauge transformations are given by δV = £ − i √ κ V (cid:8) Λ − ¯Λ + coth( £ − i √ κ V )(Λ + ¯Λ) (cid:9) = Λ + ¯Λ − i √ κ [ V, Λ − ¯Λ] + O (1 /κ ) ,δ Φ i = √ κ Λ(Φ i ) , (3.6)where £ is the Lie-derivative £ X ( Y ) = [[ X, Y ]], coth( £ − i √ κ V ) is defined via its series expansionand Λ and ¯Λ are the chiral and antichiral gauge parameters.By construction, the above model has at least N = 2 supersymmetry. Higher supersymmetrydepends on the underlying 3-algebra and the choices for the coefficients in the superpotential.For instance, the original BLG model corresponds to A = A , N f = 4 , R (1) ijkl = i4! κ ε ijkl and R (2) ijkl = 0 , (3.7)which yields the maximally supersymmetric theory with N = 8 supersymmetry. We could also have included terms involving the associated 3-product, but in the real case the ordinary 3-bracket already allows for marginal deformations. Additionally, one could introduce mass deformations of the form R d | z R ij (Φ i , Φ j )+c.c. but in this work we shall only be concerned with deformations that do not break conformalinvariance already at the classical level. after performing the integral over t .3. Deformations of the superfield action in the Hermitian case In the Hermitian case which is based on Hermitian 3-algebras, the SU ( N f ) flavor multiplet willnot be chiral, as discussed, e.g., in [10]. Therefore, we have the set of chiral superfields Φ i =(Φ , . . . , Φ N f ) = (Φ m , Φ ˙ m ), but the SU ( N f ) flavor multiplet is formed by (Φ m , ¯Φ ˙ m ). That is, wesplit the flavor index i = 1 , . . . , N f into a pair m, ˙ m = 1 , . . . , N f /
2, where Φ m and ¯Φ ˙ m will nowbe in the same representation of the gauge algebra whose carrier space is A . Accordingly, wehave to adjust the model here to read as S H = i √ κ Z d | z Z d t (( V, ¯ D α (cid:0) e − √ κ tV D α e √ κ tV (cid:1) ))+ Z d | z (cid:2) (Φ m , e − √ κ V Φ m ) + ( ¯Φ ˙ m , e √ κ V ¯Φ ˙ m ) (cid:3) . (3.8)The unusual contraction of the flavor indices is due to the antilinearity of the third slot inthe Hermitian 3-bracket and the first slot in the Hermitian pairing, respectively. The couplingconstant κ is again related to the Chern-Simons level k via κ = k/π . We will allow for thefollowing superpotential deformations, which preserve classical conformal invariance: S H = Z d | z h H (1) mn ˙ m ˙ n ( ¯Φ ˙ n , [Φ m , Φ n ; ¯Φ ˙ m ] β ) + H (2) mn ˙ m ˙ n ( ¯Φ ˙ m , Φ m )( ¯Φ ˙ n , Φ n ) i + Z d | ¯ z h H ˙ m ˙ nmn (1) (Φ n , [ ¯Φ ˙ m , ¯Φ ˙ n ; Φ m ] β ) + H ˙ m ˙ nmn (2) (Φ m , ¯Φ ˙ m )(Φ n , ¯Φ ˙ n ) i , (3.9)where [ · , · ; · ] β was defined in (2.28). The symmetry structure of the couplings here read as H (1) mn ˙ m ˙ n = H (1) nm ˙ n ˙ m and H (2) mn ˙ m ˙ n = H (2) nm ˙ n ˙ m , (3.10)and the relations of couplings with upper indices to the ones with lower indices are H (1) mn ˙ m ˙ n = − ( H ˙ n ˙ mmn (1) ) ∗ and H (2) mn ˙ m ˙ n = ( H ˙ m ˙ nmn (2) ) ∗ . (3.11)For the particular choice β = 0, the β -3-bracket reduces to the Hermitian 3-bracket. In this case,the coupling H (1) mn ˙ m ˙ n has the additional symmetry properties H (1) mn ˙ m ˙ n = − H (1) nm ˙ m ˙ n = − H (1) mn ˙ n ˙ m .Thus, for N f = 4 it is of the form H (1) mn ˙ m ˙ n ∼ ε mn ε ˙ m ˙ n .Moreover, supergauge transformations in this case are given by δV = £ − i √ κ V (cid:8) Λ − ¯Λ + coth( £ − i √ κ V )(Λ + ¯Λ) (cid:9) = Λ + ¯Λ − i √ κ [ V, Λ − ¯Λ] + O (1 /κ ) ,δ Φ m = √ κ Λ(Φ m ) and δ ¯Φ ˙ m = − √ κ ¯Λ( ¯Φ ˙ m ) . (3.12)Note again that the representation formed by Φ ˙ m is the complex conjugate representation of Φ m .We refer to Appendix B. for the component version of the above actions (for β = 0).The ABJM model as formulated in [6] is obtained by choosing A = M H I α ( N ) together withthe couplings N f = 4 , β = 0 , H (1) mn ˙ m ˙ n = κ ε mn ε ˙ m ˙ n and H (2) mn ˙ m ˙ n = 0 , (3.13)and putting α = 1 in (2.21), one obtains exactly the ABJM model as written down, e.g., in [24].11 . Marginal deformations of the BLG and ABJM models All the superpotential terms introduced in the previous section are classically marginal. Recallthat they were captured by parameters R ( ℓ ) ijkl and H ( ℓ ) ijkl for ℓ = 1 ,
2. In the following, we willexamine their behavior under quantization.While the beta function of a pure three-dimensional WZ model is zero due to an argumentanalogous to [25], the situation is different when we couple the model to a Chern-Simons action,cf. [14]: In SYM theories it is possible to argue that the couplings in the superpotential do notrenormalize by promoting the gauge coupling to a chiral superfield. The Chern-Simons level,however, is not a continuous parameter, and therefore this argument does not apply here. Fortu-nately, it is known that the Chern-Simons level itself does not receive any quantum corrections,see e.g. [26, 27], even if the model is coupled to arbitrary renormalizable matter theories. Ittherefore suffices to study the beta function of the superpotential couplings.
To discuss the renormalization of our models, we find it convenient to perform the quantumcomputations directly in superspace. For textbook treatments of the supergraph formalism inthe SYM case in four dimensions, which is very similar to our discussion below, we refer e.g.to [32].Let us start from the action (3.2) in the real setting. The Hermitian case of (3.8) is treatedanalogously, and we will discuss the differences in Section 4.5. We shall suppress the superscript R in the following.First, let us expand (3.2) in powers of V . For our purposes, it will be enough to keep termsonly up to O ( V ), S = Z d | z h i(( V, ¯ D α D α V )) + √ κ (( V, [[ D α V, ¯ D α V ]])) + ( ¯Φ i , Φ i ) + − √ κ ( ¯Φ i , V Φ i )+ (cid:0) − √ κ (cid:1) ( ¯Φ i , V Φ i ) + (cid:0) − √ κ (cid:1) ( ¯Φ i , V Φ i ) + O ( V ) i . (4.1)Here and in the following, the bracket [[ · , · ]] denotes the supercommutator, i.e. an anticommutatorif the Graßmann parity of both arguments is odd and a commutator otherwise.To quantize this action, we adopt a supersymmetric Landau gauge as done e.g. in [33, 26].The corresponding gauge fixing term reads as S gf = Z d | z (( V, { α − ( D + ¯ D ) − i β − ( D − ¯ D ) } V )) , (4.2)where we take the limit αβ →
0. Here, α and β are dimensionless parameters and D := D α D α and ¯ D := ¯ D α ¯ D α . Accordingly, the Faddeev-Popov action is S gh = Z d | z (( b − ¯ b, £ − i √ κ V (cid:8) c − ¯ c + coth (cid:0) £ − i √ κ V (cid:1) ( c + ¯ c ) (cid:9) )) , (4.3) cf. also the discussion in [28–30] and more recently in [31]. Alternatively, we could have introduced the usual gauge fixing Lagrangian L gf ∼ ξ (( V, D ¯ D V + ¯ D D V )) atthe cost of having a dimensionful gauge parameter ξ ; in fact, since V is dimensionless, ξ is of mass-dimension 1.As a consequence, the corresponding gluon propagator has a bad IR behavior for ξ = 0. However, for ξ = 0 thepropagator is the same as the one given in (4.9a) for αβ → c are the ghosts while the b are the antighosts; these are (anti)chiral superfields.As one may check, S gf + S gh is invariant under the following BRST transformation laws: δ BRST V = i √ κ η £ − i √ κ V (cid:8) c − ¯ c + coth (cid:0) £ − i √ κ V (cid:1) ( c + ¯ c ) (cid:9) ,δ BRST c = − η c and δ BRST ¯ c = η ¯ c ,δ BRST b = − i √ κ η ( α − − i β − ) ¯ D V ,δ
BRST ¯ b = i √ κ η ( α − + i β − ) D V , (4.4)where η is some anticommuting parameter.For our purposes, we will need S gh only to O ( V ), S gh = Z d | z h − ((¯ b, c )) − ((¯ c, b )) − i √ κ (( b − ¯ b, [[ V, c − ¯ c ]])) + O ( V ) i . (4.5)The full quantum action is then given by S q = S + S + S gf + S gh . (4.6)In order to have a compact form of the Feynman rules, we use capital Roman letters A, B, . . . =1 , . . . , dim g A to denote gauge algebra indices. For this, it is important to note that there is a priorino bijection between pairs of indices ab denoting elements of Λ A and an index A correspondingto an element of g A . This is due to the fact that δ : Λ A → g A is not injective in general (withan exception being the case of the real 3-algebra A ). This point has to be carefully taken intoaccount in all the calculations in the following.In terms of the gauge algebra indices, the invariant form (( · , · )) on g A is simply given by(( X, Y )) = X ab Y bc f abcd =: X A Y B G AB , with G AB = G BA . (4.7)We assume that G AB has an inverse denoted by G AB with G AC G CB = δ AB . Note that theidentification G AB = f abcd holds only if δ is a bijection (as is the case for A = A ). The structureconstants of g A are denoted by F ABC . In interactions like the 3-gluon vertex, the quantity F ABC := F ABD G DC will appear. Due to ad -invariance of (( · , · )), F ABC is totally antisymmetric in
ABC . Moreover, we will use multi-indices I = ia combining flavor and 3-algebra indices wheneverconvenient. For example, vertices like( ¯Φ i , V (Φ i )) = V ab Φ ic ¯Φ di f abcd = V ab Φ jc ¯Φ id f abcd δ ji , with ¯Φ ia := h ab ¯Φ bi (4.8a)that appear in the expansion of ( ¯Φ i , e − √ κ V Φ i ), will be written as( ¯Φ i , V (Φ i )) = Φ I V A T AI J ¯Φ J , (4.8b)where [ T A , T B ] = F ABC T C . (4.8c)We stress again that the identification T AI J = f abcd δ ji works only if δ : Λ A → g A is a bijection.13 .2. Feynman rules We have now all the necessary ingredients to write down the momentum space Feynman rules forour theory ( ∂ µ
7→ − i p µ ). Propagators:
The propagators are found to be Bθ Aθ −→ p : h V A ( − p, θ ) V B ( p, θ ) i == − i4 p G AB h ¯ D α D α − i4 α β α + β (cid:8) α − ( D + ¯ D ) − i β − ( D − ¯ D ) (cid:9)i δ (4) ( θ − θ ) , (4.9a) Jθ Iθ −→ p : h Φ I ( − p, θ ) ¯Φ J ( p, θ ) i = − i p δ I J δ (4) ( θ − θ ) , (4.9b) Bθ Aθ −→ p : h c A ( − p, θ )¯ b B ( p, θ ) i = i p G AB δ (4) ( θ − θ ) , (4.9c) Bθ Aθ −→ p : h b A ( − p, θ )¯ c B ( p, θ ) i = i p G AB δ (4) ( θ − θ ) , (4.9d)where all derivatives are understood to depend on p and to act on θ . Here, we suppressed theusual i ε -prescription of the poles. As already indicated, in this work we will use Landau gaugewith αβ →
0. We shall also use the convention ¯ D α D α = ¯ DD . Vertices:
Vertices can be read off directly from the action (4.6), and for the reader’s convenience wehave summarized them in Appendix C. As for SYM theory in superspace language, there is oneadditional feature that for each chiral or antichiral line leaving a vertex there is a factor of − ¯ D or − D acting on the corresponding propagator. However, for purely chiral or antichiral verticesthat come from the superpotential, we omit one factor of − ¯ D or − D corresponding to oneinternal line, i.e. a vertex with n internal lines attached carries n − Integration, symmetry factors and regularization:
First, there are the usual loop-momentum integrals R d p (2 π ) for each loop and momentum conserv-ing delta functions. Second, we integrate over d θ at each vertex. Finally, the usual symmetryfactors associated with the diagram have to be taken into account.Our regularization prescription is as follows: We will perform all manipulations of the formulæin D = 3, N = 2 superspace and only compute the final loop-momentum integrals in dimensionalregularization. This prescription corresponds to dimensional reduction [34], a procedure, whichis known to be valid at least up to two loop order [29]. Here and in the sequel, we make no notational distinction between a position space field and its momentumspace version (after Fourier transform). Note that we make no pictorial distinction between h c ¯ b i and h b ¯ c i . .3. Powercounting Before performing the calculation, it is useful to look at the superficial degree of divergence δ (Γ)of some diagram Γ.With the given Feynman rules, the gluon propagator h V V i scales as 1 /p for large momentawhile the propagators for the matter h Φ ¯Φ i and the ghosts h c ¯ b i and h b ¯ c i go like 1 /p . The V n vertexscales as D ¯ D ∼ p , each vertex of type Φ V n ¯Φ goes like D ¯ D ∼ p and the Φ and ¯Φ verticesbehave as ¯ D ∼ p and D ∼ p , respectively. Any ghost/gluon interaction goes like D ¯ D ∼ p .Then each external chiral or antichiral line (matter and ghost lines) goes like 1 / ¯ D ∼ /p or1 /D ∼ /p . Finally, as in SYM theory in four dimensions [35], for each loop one may reduceall the d θ -integrals to just a single one by partially integrating the D - and ¯ D -derivatives, henceleaving each loop-momentum integral to behave as d p/D ¯ D ∼ p .Altogether, the superficial degree of divergence is thus given by δ (Γ) = V g + 2 V cg + 3 V c − I g − I c + L − E c , (4.10)where V g is the number of purely gluonic vertices, V cg the number of matter/gluon and ghost/gluoninteractions and V c is the number of purely chiral vertices of Γ. Then, I g is the number of internalgluon lines, I c is the number of ghost and matter lines and E c is the number of external ghostand matter lines. Finally, L is the number of loops.Using the formulæ L = I − V + 1 = I g + I c − V g − V cg − V c + 1 and E c + 2 I c = 2 V cg + 4 V c , (4.11)we eventually arrive at δ (Γ) = (2 − E c ) . (4.12)Comparing this with the result of SYM theory in four dimensions, [35], we conclude that δ SCS = δ SY M .Equation (4.12) then tells us that all diagrams with more than two external chiral lines aresuperficially convergent. Notice that (4.12) can be refined further. When partially integratingthe supercovariant derivatives some of them will get transferred to external lines (when, e.g.,computing the wave function renormalization of the vector superfield or the renormalization ofthe superpotential). If we let N D be the number of D - and ¯ D -derivatives that are transferred toexternal lines, then the superficial degree of divergence is given by δ (Γ) = (2 − E c − N D ) . (4.13) Let us now come to the computation of the beta functions β ( ℓ ) ijkl for the couplings R ( ℓ ) ijkl with ℓ = 1 ,
2. Upon rescaling Φ i = ( Z / ) ij Φ j , where the subscript ‘0’ refers to the bare quantities,we find R ( ℓ )0 ijkl = ( Z − / ) i ′ i · · · ( Z − / ) l ′ l Z ( ℓ ) i ′ j ′ k ′ l ′ i ′′ j ′′ k ′′ l ′′ R ( ℓ ) i ′′ j ′′ k ′′ l ′′ , (4.14)15nd hence β ( ℓ ) ijkl = ( Z − ) ijkli ′ j ′ k ′ l ′ γ mi ′ Z ( ℓ ) mj ′ k ′ l ′ i ′′ j ′′ k ′′ l ′′ R ( ℓ ) i ′′ j ′′ k ′′ l ′′ + · · · + ( Z − ) ijkli ′ j ′ k ′ l ′ γ ml ′ Z ( ℓ ) i ′ j ′ k ′ mi ′′ j ′′ k ′′ l ′′ R ( ℓ ) i ′′ j ′′ k ′′ l ′′ + (( Z ( ℓ ) ) − ) ijkli ′ j ′ k ′ l ′ d Z ( ℓ ) i ′ j ′ k ′ l ′ i ′′ j ′′ k ′′ l ′′ d log µ R ( ℓ ) i ′′ j ′′ k ′′ l ′′ , (4.15a)where γ ji = ( Z − / ) ki d( Z / ) jk d log µ = 12 d(log Z ) ji d log µ (4.15b)denotes the anomalous dimension of the field Φ i and Z ( ℓ ) ijkli ′ j ′ k ′ l ′ is the renormalization of thequartic vertex ℓ . ¯Φ J ( p, θ )Φ I ( − p, θ ) p k pk + l − pl ( a ) ¯Φ J ( p, θ )Φ I ( − p, θ ) p k p − k − l − pl ( b )¯Φ J ( p, θ )Φ I ( − p, θ ) p pk ( c ) Figure 1: Logarithmically divergent diagrams that contribute to Z ji .To compute β ( ℓ ) ijkl , we emphasize that there is no one-loop renormalization, as there are noFeynman diagrams which could potentially contribute. Note that Lemma 3 of [26] is very helpfulhere, as it immediately rules out contributions from large classes of diagrams. The first non-trivialresult is found at two loops. From the discussion in the previous section, we conclude that allfour-point functions are superficially convergent and indeed, by inspecting all two-loop four-pointdiagrams of types h ΦΦΦΦ i and h ¯Φ ¯Φ ¯Φ ¯Φ i explicitly, one realizes that they all are convergent: Thereis a single such diagram potentially contributing (the two-loop gluon correction to the vertex),which is, however, convergent. We are therefore left with β ( ℓ ) ijkl = γ mi R ( ℓ ) mjkl + · · · + γ ml R ( ℓ ) ijkm . (4.16)Moreover, there are only three diagrams that contribute to γ ji and they are displayed in Fig. 1.All other diagrams either vanish by supersymmetry or by their respective color structure or theyare simply finite. 16urthermore, it will be helpful to introduce the following operators:( O ) JI := G AB T AI K T BK L G CD T CLM T DM J , ( O ) JI := G AC G BD F ABE F CDF T EI K T F K J , ( O ) JI := N f T AK L T BLK G AC G BD T CI M T DM J , (4.17)and one can show that they commute with all T A . However, the T A need not form an irreduciblerepresentation of g A in general, so Schur’s lemma cannot be applied directly. Nevertheless, itturns out that for the 3-algebras we are interested in, i.e. A and the class M R III α,β ( N ) and alsolater for M H I α ( N ), the operators (4.17) are indeed proportional to the identity. In these cases, wedefine ( O ) JI =: k δ JI , ( O ) JI =: k δ JI , ( O ) JI =: k δ JI . (4.18)The explicit values of k , k and k for the various matrix representations are listed in AppendixA. To be concise, we will give all our formulæ using these constants in the following.Let us start from diagram 1a). Using the Feynman rules listed in the previous section and inAppendix C., this diagram is given by the following integral:Σ ( a ) = − i16 · κ (cid:2) k + k (cid:3) Z d p (2 π ) d k (2 π ) d l (2 π ) d θ d θ Φ I ( − p, θ ) ¯Φ I ( p, θ ) × [ D ¯ D ( k, θ ) δ ][ ¯ DD ( k + l − p, θ ) δ ][ ¯ DD ( l, θ ) δ ] k l ( k + l − p ) , (4.19)where δ := δ (4) ( θ − θ ); the 1 / transfer rule D ( p, θ ) δ = − D ( − p, θ ) δ , (4.20)where D represents both, D and ¯ D .Integrating by parts and by employing the D -algebra { D, ¯ D } ∼ p , { D, D } = 0 and { ¯ D, ¯ D } =0, the integral (4.19) simplifies toΣ ( a ) = − κ (cid:2) k + k (cid:3) Z d p (2 π ) d θ Φ I ( − p, θ ) ¯Φ I ( p, θ ) Z d k (2 π ) d l (2 π ) k l ( k + l − p ) | {z } = − log Λ16 π = i4 π κ (cid:2) k + k (cid:3) log Λ Z d p (2 π ) d θ ( ¯Φ i ( p, θ ) , Φ i ( − p, θ )) . (4.21)Thus, the contribution of Σ ( a ) to Z ji is(a) : δZ ji = − log Λ4 π κ (cid:2) k + k (cid:3) δ ji . (4.22)In a very similar manner, we find the contribution coming from diagram 1b) to be(b) : δZ ji = − π h R (1) iklm (cid:0) − c R jklm (1) + 2 c R jmlk (1) + 2 c R jmlk (2) (cid:1) + R (2) iklm (cid:0) d R jklm (2) + 2 R jmlk (2) + 2 c R jmlk (1) (cid:1)i , (4.23)where c , c and c are the three “Casimirs” of A that are given by f accb = c δ ab , f acde f bedc = c δ ab and f acde f bcde = − c δ ba (4.24)17nd d = dim A is the dimension of the 3-algebra. These relations follow from the fundamentalidentity. We refer to Appendix A., where we list c , c and c for the matrix representation M R III α,β ( N ). V B ( p, θ ) V A ( − p, θ ) p pk − pk ( a ) V B ( p, θ ) V A ( − p, θ ) p pk − pk ( b ) V B ( p, θ ) V A ( − p, θ ) p pk − pk ( c ) Figure 2: One-loop diagrams that contribute to the gluon self-energy Π; they are all finite. Theghost diagram (b) represents all four ghost contributions.Finally, we need to find the contribution coming from diagram 1c). To compute this diagram,it is useful to perform the calculation in two steps. Let us first compute the one-loop contributionsto the gluon self-energy Π. For this, we introduce the usual superspin projectors P and P / , P := − p (cid:2) D ¯ D + ¯ D D (cid:3) and P / := 18 p D α ¯ D D α , (4.25)which obey P = P , P / = P / and P + P / = 1 . (4.26)With these, the relevant diagrams displayed in Fig. 2 contribute according toΠ ( a ) = − i8 κ F AC D F BDC Z d p (2 π ) d θ V A ( − p, θ ) p P V B ( p, θ ) , (4.27a)Π ( b ) = i8 κ F AC D F BDC Z d p (2 π ) d θ V A ( − p, θ ) p ( P / + P ) V B ( p, θ ) , (4.27b)Π ( c ) = − i4 κ T AI J T BJ I Z d p (2 π ) d θ V A ( − p, θ ) p P / V B ( p, θ ) , (4.27c)as follows by using the Feynman rules listed in Section 4.1. and in Appendix C. Summing up theterms (4.27), we findΠ = i8 κ (cid:2) F AC D F BDC − T AI J T BJ I (cid:3) Z d p (2 π ) d θ V A ( − p, θ ) p P / V B ( p, θ ) . (4.28)Note that the longitudinal part P does not appear in this expression as required by the Wardidentity for the vector superfield propagator. 18sing the result (4.28), we can now derive the contribution to the anomalous dimension of Φ i coming from diagram 1c). After some algebraic manipulations, we arrive at(c) : δZ ji = − log Λ48 π κ (cid:2) k + N f k (cid:3) δ ji . (4.29)Collecting all the results, (4.22), (4.23) and (4.29), we finally obtain γ ji = 18 π κ n(cid:2) k + k + (2 k + N f k ) (cid:3) δ ji + 8 κ h R (1) iklm (cid:0) − c R jklm (1) + 2 c R jmlk (1) + 2 c R jmlk (2) (cid:1) + R (2) iklm (cid:0) d R jklm (2) + 2 R jmlk (2) + 2 c R jmlk (1) (cid:1)io (4.30)for the anomalous dimension of Φ i . Equation (4.30) may then be substituted into (4.16) to getthe final expressions for the two-loop beta functions β ( ℓ ) ijkl .As a check, let us consider A = A . In this case we have d = 4 and f abcd = ε abcd . Then k = 0, k = − k = 6, c = 0 and c = c = −
6. We also take N f = 4 together with R (1) ijkl = λε ijkl with some constant λ and R (2) ijkl = 0. Using (4.30), the beta functions (4.16) reduce to β (1) ijkl = − π κ (cid:2) − (4! κ ) | λ | (cid:3) R (1) ijkl and β (2) ijkl = 0 , (4.31)and this expression vanishes for either λ = 0 (because R (1) ijkl = λε ijkl ) or | λ | = κ . The lattervalue of λ is precisely the value for the original BLG model (3.7). Furthermore, one might checkthat the phase of λ does not flow (see also Section 4.6.). To characterize the fixed points, itis therefore sufficient to consider the modulus of λ . The value | λ | = 0, the minimally coupledChern-Simons matter theory, is thus a UV stable fixed point, while | λ | = κ , the BLG model,forms an IR stable fixed point. Let us now discuss the Hermitian case with the action given by (3.8), (3.9), (4.2) and (4.3). Thecalculation is essentially the same as in the real case modulo some changes in the color/flavorstructure of the diagrams due to the two different types of matter that transform in oppositerepresentations of the gauge group.We introduce again((
X, Y )) = X ab Y bc f cabd =: X A Y B G AB , with G AB = G BA (4.32)and assume that G AB has an inverse. Due to the ad -invariance of (( · , · )), the structure constants F ABC := F ABD G CD are totally antisymmetric, as in the real case. Here, we have to use multi-indices of two types: I = am and ˙ I = a ˙ m . Correspondingly, the chiral superfields read as Φ I andΦ ˙ I and their conjugates are ¯Φ I and ¯Φ ˙ I ; in writing this, we are implicitly using the metric h ab aswe did in the real setting. With these conventions, the propagators are essentially the same asthose listed in (4.9). The vertices are displayed in Appendix C. Everything else like regularizationand power counting works, of course, as in the real setting.19he beta functions for the two couplings H ( ℓ ) mn ˙ m ˙ n with ℓ = 1 , β ( ℓ ) mn ˙ m ˙ n = ( Z − ) mn ˙ m ˙ nm ′ n ′ ˙ m ′ ˙ n ′ γ km ′ Z ( ℓ ) kn ′ ˙ m ′ ˙ n ′ m ′′ n ′′ ˙ m ′′ ˙ n ′′ H ( ℓ ) m ′′ n ′′ ˙ m ′′ ˙ n ′′ + · · · + ( Z − ) mn ˙ m ˙ nm ′ n ′ ˙ m ′ ˙ n ′ γ ˙ k ˙ n ′ Z ( ℓ ) m ′ n ′ ˙ m ′ ˙ km ′′ n ′′ ˙ m ′′ ˙ n ′′ H ( ℓ ) m ′′ n ′′ ˙ m ′′ ˙ n ′′ + (( Z ( ℓ ) ) − ) mn ˙ m ˙ nm ′ n ′ ˙ m ′ ˙ n ′ d Z ( ℓ ) m ′ n ′ ˙ m ′ ˙ n ′ m ′′ n ′′ ˙ m ′′ ˙ n ′′ d log µ H ( ℓ ) m ′′ n ′′ ˙ m ′′ ˙ n ′′ , (4.33a)where γ nm = 12 d(log Z ) nm d log µ and γ ˙ n ˙ m = 12 d(log Z ) ˙ n ˙ m d log µ (4.33b)denote the anomalous dimensions of the fields Φ m and Φ ˙ m and Z ( ℓ ) mn ˙ m ˙ nm ′ n ′ ˙ m ′ ˙ n ′ is the renormal-ization of the quartic vertex ℓ . As in the real case, there is no renormalization of the vertices totwo-loop order and we are therefore left with the wave function renormalizations β ( ℓ ) mn ˙ m ˙ n = γ km R ( ℓ ) kn ˙ m ˙ n + · · · + γ ˙ k ˙ n R ( ℓ ) mn ˙ m ˙ k . (4.34)Using the conventions introduced above, the diagrams in Fig. 1 yield the following contribu-tions to the wave function renormalization:(a) : δZ nm = − log Λ4 π κ (cid:2) k + k (cid:3) δ nm and δZ ˙ n ˙ m = − log Λ4 π κ (cid:2) k + k (cid:3) δ ˙ n ˙ m , (4.35a)(b) : δZ nm = − log Λ4 π h(cid:0) H (1) mk ˙ m ˙ n H ˙ m ˙ nkn (1) − H (1) mk ˙ m ˙ n H ˙ n ˙ mkn (1) (cid:1) c cos β + (cid:0) H (1) mk ˙ m ˙ n H ˙ m ˙ nkn (1) + H (1) mk ˙ m ˙ n H ˙ n ˙ mkn (1) (cid:1) c ′ sin β + (cid:0) H (1) mk ˙ m ˙ n H ˙ m ˙ nkn (2) + H (2) mk ˙ m ˙ n H ˙ m ˙ nkn (1) (cid:1)(cid:0) c cos β + i c ′ sin β (cid:1) − (cid:0) H (1) mk ˙ m ˙ n H ˙ n ˙ mkn (2) + H (2) mk ˙ m ˙ n H ˙ n ˙ mkn (1) (cid:1)(cid:0) c cos β − i c ′ sin β (cid:1) + (cid:0) H (2) mk ˙ m ˙ n H ˙ m ˙ nkn (2) + d H (2) mk ˙ m ˙ n H ˙ n ˙ mkn (2) (cid:1)i , (4.35b) δZ ˙ n ˙ m = − log Λ4 π h(cid:0) H ˙ n ˙ kmn (1) H (1) mn ˙ k ˙ m − H ˙ k ˙ nmn (1) H (1) mn ˙ k ˙ m (cid:1) c cos β + (cid:0) H ˙ n ˙ kmn (1) H (1) mn ˙ k ˙ m + H ˙ k ˙ nmn (1) H (1) mn ˙ k ˙ m (cid:1) c ′ sin β + (cid:0) H ˙ n ˙ kmn (1) H (2) mn ˙ k ˙ m + H ˙ n ˙ kmn (2) H (1) mn ˙ k ˙ m (cid:1)(cid:0) c cos β + i c ′ sin β (cid:1) − (cid:0) H ˙ k ˙ nmn (1) H (2) mn ˙ k ˙ m + H ˙ k ˙ nmn (2) H (1) nm ˙ k ˙ m (cid:1)(cid:0) c cos β − i c ′ sin β (cid:1) + (cid:0) H ˙ n ˙ kmn (2) H (1) mn ˙ k ˙ m + d H ˙ k ˙ nmn (2) H (2) mn ˙ k ˙ m (cid:1)i , (4.35c)(c) : δZ nm = − log Λ48 π κ (cid:2) k + N f k (cid:3) δ nm and δZ ˙ n ˙ m = − log Λ48 π κ (cid:2) k + N f k (cid:3) δ ˙ n ˙ m , (4.35d)where ( O ) JI := G AB T AI K T BK L G CD T CLM T DM J = k δ JI , ( ˜ O ) ˙ J ˙ I := G AB T A ˙ I ˙ K T B ˙ K ˙ L G CD T C ˙ L ˙ M T D ˙ M ˙ J = k δ ˙ J ˙ I , ( O ) JI := G AC G BD F ABE F CDF T EI K T F K J = k δ JI , ( ˜ O ) ˙ J ˙ I := G AC G BD F ABE F CDF T E ˙ I ˙ K T F ˙ K ˙ J = k δ ˙ J ˙ I , ( O ) JI := N f T AK L T BLK G AC G BD T CI M T DM J = k δ JI , ( ˜ O ) ˙ J ˙ I := N f T A ˙ K ˙ L T B ˙ L ˙ K G AC G BD T C ˙ I ˙ M T D ˙ M ˙ J = k δ ˙ J ˙ I , (4.35e)20nd f accb = c δ ba , f acde f edcb = − c δ ab , d accb = c ′ δ ba , d acde d edcb = − c ′ δ ab (4.35f)with d = dim A . For the explicit values of the Casimirs k i , c i and c ′ i in the matrix representation M H I α ( N ), we refer to Appendix A.Altogether, we obtain the following anomalous dimensions: γ nm = 18 π κ (cid:26)(cid:2) k + k + (2 k + N f k ) (cid:3) δ nm + κ h(cid:0) H (1) mk ˙ m ˙ n H ˙ m ˙ nkn (1) − H (1) mk ˙ m ˙ n H ˙ n ˙ mkn (1) (cid:1) c cos β + (cid:0) H (1) mk ˙ m ˙ n H ˙ m ˙ nkn (1) + H (1) mk ˙ m ˙ n H ˙ n ˙ mkn (1) (cid:1) c ′ sin β + (cid:0) H (1) mk ˙ m ˙ n H ˙ m ˙ nkn (2) + H (2) mk ˙ m ˙ n H ˙ m ˙ nkn (1) (cid:1)(cid:0) c cos β + i c ′ sin β (cid:1) − (cid:0) H (1) mk ˙ m ˙ n H ˙ n ˙ mkn (2) + H (2) mk ˙ m ˙ n H ˙ n ˙ mkn (1) (cid:1)(cid:0) c cos β − i c ′ sin β (cid:1) + (cid:0) H (2) mk ˙ m ˙ n H ˙ m ˙ nkn (2) + d H (2) mk ˙ m ˙ n H ˙ n ˙ mkn (2) (cid:1)i (cid:27) , (4.36a) γ ˙ n ˙ m = 18 π κ (cid:26)(cid:2) k + k + (2 k + N f k ) (cid:3) δ nm + κ h(cid:0) H ˙ n ˙ kmn (1) H (1) mn ˙ k ˙ m − H ˙ k ˙ nmn (1) H (1) mn ˙ k ˙ m (cid:1) c cos β + (cid:0) H ˙ n ˙ kmn (1) H (1) mn ˙ k ˙ m + H ˙ k ˙ nmn (1) H (1) mn ˙ k ˙ m (cid:1) c ′ sin β + (cid:0) H ˙ n ˙ kmn (1) H (2) mn ˙ k ˙ m + H ˙ n ˙ kmn (2) H (1) mn ˙ k ˙ m (cid:1)(cid:0) c cos β + i c ′ sin β (cid:1) − (cid:0) H ˙ k ˙ nmn (1) H (2) mn ˙ k ˙ m + H ˙ k ˙ nmn (2) H (1) nm ˙ k ˙ m (cid:1)(cid:0) c cos β − i c ′ sin β (cid:1) + (cid:0) H ˙ n ˙ kmn (2) H (1) mn ˙ k ˙ m + d H ˙ k ˙ nmn (2) H (2) mn ˙ k ˙ m (cid:1)i (cid:27) . (4.36b)These expressions may be substituted into (4.34) to arrive at the final result for the beta functions.As a check, let us consider the ABJM model. In that case we have, β = 0, N f = 4, H (1) mn ˙ m ˙ n = λε mn ε ˙ m ˙ n for some constant λ and H (2) mn ˙ m ˙ n = 0. Furthermore, we choose M H I α =1 ( N ) and hence k = 0, k = 1 − N , k = − N , c = 0 and c = 2 − N . Therefore, we find γ nm = 116 π κ (1 − N ) (cid:2) − (4 κ ) | λ | (cid:3) δ nm ,γ ˙ n ˙ m = 116 π κ (1 − N ) (cid:2) − (4 κ ) | λ | (cid:3) δ ˙ n ˙ m , (4.37)and thus, we recover precisely the value | λ | = κ for the ABJM model; see equations (3.13). For N = 2, this of course agrees with the result (4.31) as for this particular value of N , the ABJMmodel coincides with the BLG model. As in the real case, the phase of λ does not flow (see alsoSection 4.6.) and so we can restrict ourselves to the modulus | λ | . Therefore, the conformal fixedpoint corresponding to the ABJM model forms an IR fixed point, just like in the case of the BLGmodel. The above expressions for the anomalous dimensions and the resulting expressions for the betafunctions certainly allow for many conformal fixed points depending on the particular choices of21he superpotential couplings and of the underlying 3-algebra structure. For this reason, we shallmerely discuss two examples. We hope to report on a more thorough analysis elsewhere. In oursubsequent discussion, we assume that N f = 4. Real 3-algebras:
Let us consider A = A . We recall that in this case the Casimirs are given by k = 0 , k = − , k = 6 , c = 0 , c = − , c = − . (4.38)Furthermore, we take R (1) ijkl = λ κ ε ijkl and R (2) ijkl = λ κ δ ij δ kl , (4.39)with λ ℓ = r ℓ e i ϕ ℓ . Plugging these values into the expression (4.30) for the two-loop anomalousdimension, one finds that the corresponding beta functions are given by β ( ℓ ) ijkl = f ( r , r ) κ R ( ℓ ) ijkl , with f ( r , r ) := − π (cid:2) − (cid:0) r + r (cid:1)(cid:3) . (4.40)The zero-locus f ( r , r ) = 0 defines an ellipse in R , r = 124 cos t and r = 14 √ t for t ∈ [0 , π ) . (4.41)We thus obtain a one-parameter family of marginal multi-trace deformations (i.e. double-tracein superfields and double- and triple-trace in components) of the BLG model, the latter corre-sponding to t = 0.PSfrag replacements 0.00.00.0 0.05 0.050.1 0.10.10.2 0.15 r f ( r , r ) r Figure 3: The function f ( r , r ) capturing the beta functions of single- and multi-trace deforma-tions.Furthermore, (4.40) implies the following equations for the running couplings ˜ λ ℓ = ˜ r ℓ e i ˜ ϕ ℓ :˙˜ r ℓ = ˜ r ℓ κ f (˜ r , ˜ r ) and ˜ r ℓ ˙˜ ϕ ℓ = 0 , with ˜ λ ℓ ( µ ; λ ℓ ) = λ ℓ , (4.42)22here dot means a total derivative with respect to log( p/µ ). Hence, the phases ˜ ϕ ℓ do not flow.To get a more intuitive picture of the situation, we plotted the function f ( r , r ) in Fig. 3 for r , r >
0. From the figure it is then clear that every point on the fixed point locus of the betafunctions corresponds to an IR fixed point of the renormalization group, as the derivative of thefunction f ( r , r ) in the direction of the outward normal of the curve is positive. Notice furtherthat the minimally coupled Chern-Simons matter theory, r ℓ = 0, is a UV fixed point. Thus, byturning on the above deformation at r ℓ = 0, the theory flows to one of the points on the curve f ( r , r ) = 0 in the IR. Hermitian 3-algebras:
Let us now perform a similar analysis in the Hermitian setting. We take A = M H I α =1 ( N ). In thiscase, we know that k = 0 , k = 1 − N , k = − − N ) , c = 0 , c = 2(1 − N ) ,c ′ = − N , c ′ = − N ) (4.43)and d = N . Let us focus on superpotential couplings of the form H (1) mn ˙ m ˙ n = λ κ (cid:2) ε mn ε ˙ m ˙ n + ρ ( δ ( mn ˙ m ˙ n ) , (1 , , , + δ ( mn ˙ m ˙ n ) , (2 , , , ) (cid:3) ,H (2) mn ˙ m ˙ n = λ κ δ m ˙ m δ n ˙ n . (4.44)Note that λ controls the multi-trace deformations. Substituting these expressions into (4.36)for the two-loop anomalous dimension, we find after some algebraic manipulations that the betafunctions (4.34) vanish if ( λ ℓ = r ℓ e i ϕ ℓ and ρ = r e i ϕ ) a r ( r − r cos ϕ + 4) + b r r + c r + d r r r sin( ϕ − ϕ + ϕ ) = 1 , (4.45a)where a := 4 cos β , b := 2 N + 2 N − β , c := 4 N + 2 N − , d := 8 NN − β . (4.45b)For ρ = − λ = 0, we find the β -deformed ABJM model that was discussed in [18] bystudying the gravitational dual of the theory while for ρ = 0 (implying β = 0 without loss ofgenerality) and λ = 0, we obtain a marginal multi-trace deformation of the ABJM model.
5. Conclusions and outlook
In summary, we have described marginal deformations of Chern-Simons matter theories thatare based on real and Hermitian 3-algebras. In particular, we wrote down the most generalsuperpotentials consisting of single- and multi-trace terms that are i) conformally invariant atthe classical level, ii) compatible with N = 2 supersymmetry and iii) supergauge invariant. Forthese superpotential terms, we computed the two-loop beta functions using N = 2 supergraphtechniques. As familiar from four dimensional SYM theories, supergraphs turned out to be apowerful tool also in the case of supersymmetric Chern-Simons matter theories: The calculationof the two-loop beta functions boiled down to the computation of the three Feynman supergraphs23isplayed in Fig. 1. We expressed our results concisely in terms of certain “Casimir invariants”of the underlying 3-algebra and its associated Lie algebra. Using our expressions for the betafunctions, we confirmed conformality of both the BLG and ABJM models. In addition, wediscussed β -deformations of the ABJM model and certain marginal multi-trace deformations ofboth the BLG and ABJM models, explicitly. We mostly focused on the 3-algebras M R III α,β ( N )and M H I α ( N ), but a similar analysis can easily be carried out for other 3-algebras.Even though real and Hermitian 3-algebras already allow for classes of marginal deformations,we found that not all deformations, and in particular not the β -deformations of [18], are capturedby 3-brackets. Instead, one has to introduce an associated 3-product, i.e. a triple product thattransforms covariantly under gauge transformations. This is in the same spirit to what happens infour-dimensional SYM theory, where one replaces the Lie bracket by some deformed bracket. Todiscuss β -deformations of the ABJM model, for instance, we were led to introduce the β -3-bracket(2.28), which is just a special instance of an associated 3-product. As far as β -deformationsare concerned, we mainly focused on the Hermitian case. Here, we obtained an independentconfirmation of the deformations studied in [18]. Note, however, that more general deformationsthan the β -deformations we focused on can in principle be discussed in both the real and Hermitiancases using associated 3-products.The most interesting open question is certainly to what extent our deformations are exactlymarginal, or at least, to all orders in perturbation theory. Because of the many simplificationswhich arise, e.g., due to Lemma 3 of [26], one might be able to make precise statements usingour superfield formulation. Otherwise, it might be necessary to switch to a different descriptionas, for example, light-cone superspace as done in [36] for β -deformations of N = 4 SYM theory.Another point is certainly to study the ’t Hooft limit of our deformed theories and identifyall geometries which form their gravitational duals, extending the work of [18]. Vice versa, onecould reformulate some of the deformations considered in [18] in terms of 3-algebra language togain more insight into the 3-algebra structures involved.Finally, it would be interesting to extend the analysis of [10, 12] to our deformed BLG-typemodels and to study a possible correspondence of the dilatation operator in these models tothe Hamiltonian of an integrable spin chain, using superspace and 3-algebra language. Thisis possible, because the operators considered in [10] can easily be formulated in terms of theassociated 3-products introduced in this work. Acknowledgements.
We are very grateful to Martin Ammon, Sergey Cherkis, Stefano Kovacsand Riccardo Ricci for discussions, questions and suggestions. N.A. was supported by the DutchFoundation for Fundamental Research on Matter (FOM). C.S. was supported by an IRCSETPostdoctoral Fellowship. M.W. was supported by an STFC Postdoctoral Fellowship and by aSenior Research Fellowship at the Wolfson College, Cambridge, U.K. after appropriate rescalings by factors of ‘ N ’, see e.g. [37] ppendices A. Casimirs for matrix representations
In this appendix, we discuss the Casimirs k i and c i that appear throughout this work for thedifferent matrix representations. Casimirs c i and k i for the real 3-algebra M R III α,β ( N ) : The underlying vector space for this real 3-algebras has dimension N and one easily finds a basiswith elements satisfying the following relationstr ( τ Ta τ b ) = δ ab =: h ab and ( τ a ) ij ( τ a ) kl = δ ik δ jl . (A.1)With the above formulæ, the three Casimirs c , c and c defined by f accb = c δ ab , f acde f bedc = c δ ab and f acde f bcde = − c δ ba (A.2)can be computed straightforwardly and we obtain c = ( N − α − β ) , c = ( N − α − N − αβ + β ) ,c = − N ( N − α + β ) . (A.3)The Casimirs k i can similarly be calculated by using identities for the appearing generatorsof g A ∼ = o ( N ) ⊕ o ( N ) together with formula (2.11) for the bilinear form (( · , · )) on g A . We find herethat k = √ ( α + β ) , k = − ( α + β ) , k = − N ( α + β ) . (A.4)Note that the algebra A is a sub-3-algebra of the 3-algebra M R III , − (4). In this case, one cancompute the Casimirs directly from the structure constants and the fact that g A = su (2) ⊕ su (2)and we obtain c = 0 , c = − , c = − , k = 0 , k = − , k = 6 . (A.5)Analogously, one constructs the Casimirs for the other real 3-algebras M R I α ( N ), M R II α ( N ) and M R IV α,β ( N ), but we refrain from going into more detail at this point. Casimirs c i and k i for the Hermitian 3-algebra M H I α ( N ) : The underlying vector space here is spanned by generators of U ( N ) in the fundamental represen-tation. For simplicity, we fix α = 1, as we did throughout most of the paper. As basis τ a , wehave N anti-Hermitian N × N -matrices and we choose them such that we have the followingidentities: tr ( τ † a τ b ) = δ ab =: h ab , h ab = δ ab and ( τ a ) ij ( τ a ) kl = − δ il δ jk . (A.6)From these, one obtains for the Casimirs c , c and c , which are defined by f accb = c δ ab , f acde f edcb = − c δ ab and f acde f bcde = − c δ ba , (A.7)25he following expressions: c = 0 , c = 2(1 − N ) and c = 2(1 − N ) . (A.8)In addition, we have k = 0 , k = 1 − N and k = − − N ) . (A.9)Recall that M H I α =1 (2) = A , and the above formulæ (A.8) and (A.9) reproduce indeed (A.5) for N = 2. Remarks on the bracket [ A, B ; C ] β : Recall the form of the β -3-bracket[ τ a , τ b ; τ c ] β = g abcd τ d with g abcd = cos βf abcd + i sin β d abcd . (A.10)Therefore, apart from the Casimirs c i we also have the c ′ i d accb = c ′ δ ab and d acde d edcb = − c ′ δ ab . (A.11)Explicitly, we obtain the following values: c ′ = − N and c ′ = − N ) . (A.12) B. Component form of the actions
In this appendix we give the component form of the superspace actions in WZ gauge.
Component action in the real case:
In terms of the component fields (3.1), the action (3.2) reads as S R = Z d x (cid:20) ε µνλ (( A µ , ∂ ν A λ + √ κ [[ A ν , A λ ]])) − i((¯ λ α , λ α )) − i(( λ α , ¯ λ α )) − (( D, σ )) − (( σ, D ))+ ( ¯ F i , F i ) − ( ∇ µ ¯ φ i , ∇ µ φ i ) − i( ¯ ψ αi , ∇ αβ ψ iβ ) − i √ κ ( ¯ φ i , D ( φ i )) + q κ ( ¯ φ i , λ α ( ψ iα ))+ q κ (¯ λ α ( ¯ ψ αi ) , φ i ) + κ ( ¯ φ i , σ ( φ i )) + √ κ ( ¯ ψ iα , σ ( ψ iα )) (cid:21) , (B.13)where ∇ αβ := σ µαβ ∇ µ . Upon performing the integrals over the fermionic directions, the componentform of the superpotential term (3.3) is given by S R = − Z d x (cid:26) R (1) ijkl h(cid:0) φ l , [ ψ iα , ψ jα , φ k ] + 2[ φ i , ψ jα , ψ kα ] (cid:1) − φ i , φ j , φ k ] , F l ) i + R (2) ijkl h ( ψ iα , ψ jα )( φ k , φ l ) + 2( ψ iα , φ j )( ψ kα , φ l ) − F i , φ j )( φ k , φ l ) i (cid:27) + c . c . . (B.14)26he next step is to eliminate the auxiliary fields. After varying S R = S R + S R , we find thefollowing (algebraic) equations of motion for F i , ¯ F i , D , σ , λ and ¯ λ : F i = − R ijkl (1) [ ¯ φ l , ¯ φ k , ¯ φ j ] − R ijkl (2) ( ¯ φ l , ¯ φ k ) ¯ φ j , ¯ F i = − R (1) ijkl [ φ l , φ k , φ j ] − R (2) ijkl ( φ l , φ k ) φ j ,D ( A ) = κ (cid:2) [ φ i , σ ( ¯ φ i ) , A ] − [ σ ( φ i ) , ¯ φ i , A ] (cid:3) + √ κ [ ψ iα , ¯ ψ iα , A ] ,σ ( A ) = − i2 √ κ [ φ i , ¯ φ i , A ] ,λ α ( A ) = − i √ κ [ ¯ ψ iα , φ i , A ] and ¯ λ α ( A ) = i √ κ [ ψ iα , ¯ φ i , A ] , (B.15)and hence S R = R d x L R with L R = ε µνλ (( A µ , ∂ ν A λ + √ κ [[ A ν , A λ ]])) − (cid:12)(cid:12) ∇ µ φ i (cid:12)(cid:12) − i (cid:0) ¯ ψ αi , ∇ αβ ψ iβ (cid:1) + κ (cid:0) [ φ i , ¯ φ i , ¯ φ k ] , [ φ j , ¯ φ j , φ k ] (cid:1) + i2 κ (cid:0) ¯ ψ αj , [ φ i , ¯ φ i , ψ jα ] (cid:1) + i κ (cid:0) [ ¯ ψ αj , φ j , ¯ φ i ] , ψ iα (cid:1) − R (1) ijkl (cid:0) φ l , [ ψ iα , ψ jα , φ k ] + 2[ φ i , ψ jα , ψ kα ] (cid:1) − R ijkl (1) (cid:0) ¯ φ l , [ ¯ ψ iα , ¯ ψ αj , ¯ φ k ] + 2[ ¯ φ i , ¯ ψ jα , ¯ ψ αk ] (cid:1) − R (2) ijkl h ( ψ iα , ψ jα )( φ k , φ l ) + 2( ψ iα , φ j )( ψ kα , φ l ) i − R ijkl (2) h ( ¯ ψ iα , ¯ ψ αj )( ¯ φ k , ¯ φ l ) + 2( ¯ ψ iα , ¯ φ j )( ¯ ψ αk , ¯ φ l ) i − (cid:12)(cid:12)(cid:12) R (1) ijkl [ φ l , φ k , φ j ] + R (2) ijkl ( φ l , φ k ) φ j (cid:12)(cid:12)(cid:12) , (B.16)where | A | := ( ¯ A, A ) for any A ∈ A .For the reader’s convenience, we finally extract the multi-trace terms explicitly: L Rmult = − R (2) ijkl h ( ψ iα , ψ jα )( φ k , φ l ) + 2( ψ iα , φ j )( ψ kα , φ l ) i − R ijkl (2) h ( ¯ ψ iα , ¯ ψ αj )( ¯ φ k , ¯ φ l ) + 2( ¯ ψ iα , ¯ φ j )( ¯ ψ αk , ¯ φ l ) i − h R (1) ijkl R ij ′ k ′ l ′ (2) ([ φ l , φ k , φ j ] , ¯ φ j ′ )( ¯ φ l ′ , ¯ φ k ′ )+ R (2) ijkl R ij ′ k ′ l ′ (1) ([ ¯ φ l ′ , ¯ φ k ′ , ¯ φ j ′ ] , φ j )( φ l , φ k )+ R (2) ijkl R ij ′ k ′ l ′ (2) ( φ l , φ k )( ¯ φ l ′ , ¯ φ k ′ )( φ j , ¯ φ j ′ ) i . (B.17) Component action in the Hermitian case:
Let us now discuss the Hermitian case. Here, we shall assume that β = 0, i.e. we work withthe usual Hermitian 3-bracket in the superpotential. In terms of the component fields (3.1), the27ction (3.8) reads as S H = Z d x (cid:20) ε µνλ (( A µ , ∂ ν A λ + √ κ [[ A ν , A λ ]])) − i(( λ α , ¯ λ α )) − i((¯ λ α , λ α )) − (( D, σ )) − (( σ, D ))+ ( F m , F m ) + ( ¯ F ˙ m , ¯ F ˙ m ) − ( ∇ µ φ m , ∇ µ φ m ) − ( ∇ µ ¯ φ ˙ m , ∇ µ ¯ φ ˙ m ) − i( ψ mα , ∇ αβ ψ mβ )+ i( ¯ ψ mα , ∇ αβ ¯ ψ mβ ) − i √ κ ( φ m , D ( φ m )) + i √ κ ( ¯ φ ˙ m , D ( ¯ φ ˙ m )) + q κ ( φ m , λ α ( ψ mα )) − q κ ( ¯ ψ α ˙ m , λ α ( ¯ φ ˙ m )) + q κ ( λ α ( ψ mα ) , φ m ) − q κ ( λ α ( ¯ φ ˙ m ) , ¯ ψ ˙ mα ) + κ ( φ m , σ ( φ m ))+ κ ( ¯ φ ˙ m , σ ( ¯ φ ˙ m )) + √ κ ( ψ mα , σ ( ψ mα )) − √ κ ( ¯ ψ α ˙ m , σ ( ¯ ψ α ˙ m )) (cid:21) . (B.18)In component form, the superpotential terms (3.9) are given by S H = − Z d x (cid:26) H (1) mn ˙ m ˙ n (cid:20) ( ¯ F ˙ m , [ φ m , φ n ; ¯ φ ˙ n ]) + ([ ¯ φ ˙ m , ¯ φ ˙ n ; φ n ] , F m )+ ( ¯ ψ ˙ nα , [ ψ mα , φ n ; ¯ φ ˙ m ]) + ( ¯ φ ˙ n , [ ψ mα , φ n ; ¯ ψ α ˙ m ]) − ( ¯ ψ ˙ nα , [ φ m , φ n ; ¯ ψ α ˙ m ]) − ( ¯ φ ˙ n , [ ψ mα , ψ nα ; ¯ φ ˙ m ) (cid:21) + H (2) mn ˙ m ˙ n (cid:20) − ( ¯ F ˙ m , φ m )( ¯ φ ˙ n , φ n ) − ( ¯ φ ˙ m , F m )( ¯ φ ˙ n , φ n )+ ( ¯ ψ ˙ mα , ψ mα )( ¯ φ ˙ n , φ n ) + ( ¯ ψ ˙ mα , φ m )( ¯ φ ˙ n , ψ nα ) − ( ¯ ψ ˙ mα , φ m )( ¯ ψ α ˙ n , φ n ) − ( ¯ φ ˙ m , ψ mα )( ¯ φ ˙ n , ψ nα ) (cid:21)(cid:27) + c . c . . (B.19)Varying S H = S H + S H , we find the following (algebraic) equations of motion for the auxiliaryfields F m , ¯ F m , F ˙ m , ¯ F ˙ m , D , σ , λ and ¯ λ : F m = 2 H ˙ m ˙ nmn (1) [ ¯ φ ˙ m , ¯ φ ˙ n ; φ n ] − H ˙ m ˙ nmn (2) ¯ φ ˙ m ( φ n , ¯ φ ˙ n ) , ¯ F m = − H (1) mn ˙ m ˙ n [ φ ˙ m , φ ˙ n ; ¯ φ n ] − H (2) mn ˙ m ˙ n φ ˙ m ( ¯ φ ˙ n , φ n ) ,F ˙ m = − H ˙ m ˙ nmn (1) [ ¯ φ m , ¯ φ n ; φ ˙ m ] − H ˙ m ˙ nmn (2) ¯ φ m ( φ ˙ n , ¯ φ n ) , ¯ F ˙ m = 2 H (1) mn ˙ m ˙ n [ φ m , φ n ; ¯ φ ˙ n ] − H (2) mn ˙ m ˙ n φ m ( ¯ φ n , φ ˙ n ) ,D ( A ) = κ (cid:16) [ A, σ ( φ m ); φ m ] + [ A, σ ( ¯ φ ˙ m ); ¯ φ ˙ m ] − [ A, φ m ; σ ( φ m )] − [ A, ¯ φ ˙ m ; σ ( ¯ φ ˙ m )] (cid:17) − √ κ (cid:16) [ A, ψ mα ; ψ mα ] − [ A, ¯ ψ α ˙ m ; ¯ ψ ˙ mα ] (cid:17) ,σ ( A ) = − i2 √ κ (cid:0) [ A, φ m ; φ m ] − [ A, ¯ φ ˙ m ; ¯ φ ˙ m ] (cid:1) ,λ α ( A ) = i( − ˜ A √ κ (cid:0) [ A, φ m ; ψ mα ] − [ A, ¯ ψ ˙ mα ; ¯ φ ˙ m ] (cid:1) , ¯ λ α ( A ) = i( − ˜ A √ κ (cid:0) [ A, ψ mα ; φ m ] − [ A, ¯ φ ˙ m ; ¯ ψ ˙ mα ] (cid:1) , (B.20)where A is an arbitrary field taking values in A . We may now substitute these expressions intoequations (B.18) and (B.19) to arrive at the final expression for the component action. Since thisis a rather lengthy expression and moreover basically of the same form as (B.16), we shall not28isplay the full action here but only give the multi-trace terms: L Hmult = − H (2) mn ˙ m ˙ n h ( ¯ ψ ˙ mα , ψ mα )( ¯ φ ˙ n , φ n ) + ( ¯ ψ ˙ mα , φ m )( ¯ φ ˙ n , ψ nα ) − ( ¯ ψ ˙ mα , φ m )( ¯ ψ α ˙ n , φ n ) − ( ¯ φ ˙ m , ψ mα )( ¯ φ ˙ n , ψ nα ) i − H ˙ m ˙ nmn (2) h ( ψ mα , ¯ ψ ˙ mα )( φ n , ¯ φ ˙ n ) + ( ψ nα , ¯ φ ˙ n )( φ m , ¯ ψ ˙ mα ) − ( φ n , ¯ ψ α ˙ n )( φ m , ¯ ψ ˙ mα ) − ( ψ nα , ¯ φ ˙ n )( ψ mα , ¯ φ ˙ m ) i + 4 H (1) mn ˙ m ˙ n H ˙ m ′ ˙ n ′ mn ′ (2) ([ ¯ φ ˙ m , ¯ φ ˙ n ; φ n ] , ¯ φ ˙ m ′ )( φ n ′ , ¯ φ ˙ n ′ )+ 4 H (2) mn ˙ m ˙ n H ˙ m ′ ˙ n ′ mn ′ (1) ( ¯ φ ˙ m , [ ¯ φ ˙ m ′ , ¯ φ ˙ n ′ ; φ n ′ ])( ¯ φ ˙ n , φ n ) − H (2) mn ˙ m ˙ n H ˙ m ′ ˙ n ′ mn ′ (2) ( ¯ φ ˙ m , ¯ φ ˙ m ′ )( ¯ φ ˙ n , φ n )( φ n ′ , ¯ φ ˙ n ′ )+ 4 H ˙ m ˙ nmn (1) H (2) m ′ n ′ ˙ m ˙ n ′ ([ φ m , φ n ; ¯ φ ˙ n ] , φ m ′ )( ¯ φ ˙ n ′ , φ n ′ )+ 4 H ˙ m ˙ nmn (2) H (1) m ′ n ′ ˙ m ˙ n ′ ( φ m , [ φ m ′ , φ n ′ ; ¯ φ ˙ n ′ ])( φ n , ¯ φ ˙ n ) − H ˙ m ˙ nmn (2) H (2) m ′ n ′ ˙ m ˙ n ′ ( φ m , φ m ′ )( φ n , ¯ φ ˙ n )( ¯ φ ˙ n ′ , φ n ′ ) . (B.21) C. Feynman rules: Vertices
Vertices for real 3-algebras:
Let us list the Feynman rules for the vertices in Landau gauge αβ →
0. They are: V -vertex : A A A ց k ր k ←−− k − k θ = √ κ V A A A , (C.22a)Φ V ¯Φ-vertex : AJI θ = i − √ κ T AI J , (C.22b)Φ V ¯Φ-vertex : JI ABθ = i (cid:0) − √ κ (cid:1) T ( AI K T B ) K J , (C.22c)Φ -vertex : θLI JK = i4! R IJKL , (C.22d)29Φ -vertex : LI JKθ = i4! R IJKL , (C.22e)ghost/gluon-vertices : ABC θ = i( − √ κ F ABC . (C.22f)Here, ‘ V A A A = X r,s F A A r A s [ ¯ D ( − k r , θ )∆ rθ ( k r )][ D ( − k s , θ )∆ sθ ( k s )] (C.23a)with ∆ ij ( k i ) := − i4 k i ¯ DD ( k i , θ i ) δ ij and δ ij := δ (4) ( θ i − θ j ) . (C.23b)The coefficients appearing in (C.22d) and (C.22e) are R IJKL = (cid:2) R (1) ijkl f abcd + R (2) ijkl h ab h cd (cid:3) s = (cid:0) R (1) ijkl f abcd + R (1) iklj f acdb + R (1) iljk f adbc + R (2) ijkl h ab h cd + R (2) iklj h ac h db + R (2) iljk h ad h bc (cid:1) ,R IJKL = (cid:2) R ijkl (1) f abcd + R ijkl (2) h ab h cd (cid:3) s = (cid:0) R ijkl (1) f abcd + R iklj (1) f acdb + R iljk (1) f adbc + R ijkl (2) h ab h cd + R iklj (2) h ac h db + R iljk (2) h ad h bc (cid:1) . (C.24)The subscript ‘ s ’ refers to total symmetrization in the multi-indices IJ KL . Vertices for Hermitian 3-algebras:
In the Hermitian case, the Feynman rules for the vertices are very similar to the ones for real3-algebras. The purely gluonic and gluon/ghost vertices are the same and we shall again adoptLandau gauge. The only difference is in the gluon/matter and pure matter vertices, since we havetwo different types of matter: Φ I and Φ ˙ I . We haveΦ V ¯Φ-vertex : AJI θ = i − √ κ T AI J , (C.25a)Φ V ¯Φ-vertex : A ˙ J ˙ I θ = i √ κ T A ˙ I ˙ J , (C.25b)30 V ¯Φ-vertex : JI ABθ = i (cid:0) − √ κ (cid:1) T ( AI K T B ) K J , (C.25c)Φ V ¯Φ-vertex : ˙ J ˙ I ABθ = i (cid:0) √ κ (cid:1) T ( A ˙ I ˙ K T B ) ˙ K ˙ J , (C.25d)Φ -vertex : θ ˙ LI J ˙ K = i4 H IJ ˙ K ˙ L , (C.25e)¯Φ -vertex : ˙ LI J ˙ Kθ = i4 H ˙ K ˙ LIJ , (C.25f)where H IJ ˙ K ˙ L = (cid:2) H (1) mn ˙ m ˙ n g abcd + H (2) mn ˙ m ˙ n δ ca δ db (cid:3) s = (cid:2) H (1) mn ˙ m ˙ n g abcd + H (1) mn ˙ n ˙ m g abdc + H (2) mn ˙ m ˙ n δ ca δ db + H (2) mn ˙ n ˙ m δ da δ cb (cid:3) ,H ˙ K ˙ LIJ = (cid:2) H ˙ n ˙ mmn (1) g dcab + H ˙ m ˙ nmn (2) δ ac δ bd (cid:3) s = (cid:2) H ˙ n ˙ mmn (1) g dcab + H ˙ m ˙ nmn (1) g cdab + H ˙ m ˙ nmn (2) δ ac δ bd + H ˙ n ˙ mmn (2) δ ad δ bc (cid:3) , (C.26)where ‘ s ’ refers again to total symmetrization. References [1] J. Bagger and N. Lambert, “Modeling multiple M2’s” , Phys. Rev. D75, 045020 (2007) , hep-th/0611108 . • J. Bagger and N. Lambert, “Gauge symmetry and supersymmetry of multipleM2-branes” , Phys. Rev. D77, 065008 (2008) , arxiv:0711.0955 .[2] A. Gustavsson, “Algebraic structures on parallel M2-branes” , Nucl. Phys. B811, 66 (2009) , arxiv:0709.1260 .[3] C. I. Lazaroiu, D. McNamee, C. Saemann and A. Zejak, “Strong homotopy Lie algebras, generalizedNahm equations and multiple M2-branes” , arxiv:0901.3905 .[4] V. T. Filippov, “ n -Lie algebras” , Sib. Mat. Zh. 26, 126 (1985) .[5] P.-A. Nagy, “Prolongations of Lie algebras and applications” , arxiv:0712.1398 . • G. Papadopoulos, “M2-branes, 3-Lie algebras and Pl¨ucker relations” , JHEP 0805, 054 (2008) , arxiv:0804.2662 . • J. P. Gauntlett and J. B. Gutowski, “Constraining maximally supersymmetricmembrane actions” , JHEP 0806, 053 (2008) , arxiv:0804.3078 .
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