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On 3d N=8 Lorentzian BLG theory as a scaling limit of 3d superconformal N=6 ABJM theory

We elaborate on the suggestion made in arXiv:0806.3498 that the 3d \N=8 superconformal SU(N) Chern-Simons-matter theory of Lorentzian Bagger-Lambert-Gustavson type (L-BLG) can be obtained by a scaling limit (involving sending the level k to infinity and redefining the fields) from the \N=6 superconformal U(N)xU(N) Chern-Simons-matter theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We show that to implement such a limit in a consistent way one is to extend the ABJM theory by an abelian"ghost"multiplet. The corresponding limit at the 3-algebra level also requires extending the non-antisymmetric Bagger-Lambert 3-algebra underlying the ABJM theory by a negative-norm generator. We draw analogy with similar scaling limits discussed previously for bosonic Chern-Simons theory and comment on some implications of this relation between the ABJM and L-BLG theories.


Introduction
An important problem that attracted much recent attention is to understand a low-energy limit of a hypothetical theory that generalizes the worldvolume theory of a single M2-brane [1] to the case of N coincident M2-branes. Using some earlier ideas of [2], Bagger and Lambert, and Gustavson (BLG) succeeded in constructing a three-dimensional N = 8 superconformal Chern-Simons-matter theory based on a 3-algebra [3,4,5]. However, it was soon realized that the original BLG theory can describe only two coincident M2-branes if the 3-algebra is kept finite and having a positive-definite metric [6,7,8].
An interesting suggestion of how to construct a similar theory with right symmetries and degrees of freedom to describe N M2-branes was made in [9,10,11] (see also [12,13,14,15]) where a 3-algebra with a Lorentzian (indefinite) signature metric was used. The resulting Lorentzian-BLG (L-BLG) theory is formally N = 8 superconformal at the classical level, but its interpretation as a quantum theory appears to be non-trivial (in particular, as there is no obvious expansion parameter and its perturbative definition depends on a choice of a vacuum). If expanded near an obvious classical vacuum that spontaneously breaks the superconformal symmetry, it becomes equivalent [10,12,14,16] to the standard low-energy gauge theory of multiple D2-branes (non-conformal N = 8 supersymmetric 3d SYM theory), i.e. it may thus be viewed as a conformal "dressing" of non-conformal 3d N = 8 SYM theory. 1 A different 3d superconformal Chern-Simons-matter theory was proposed by Aharony, Bergman, Jafferis and Maldacena (ABJM) [18]; it has an explicit N = 6 supersymmetry [19] and may be interpreted as describing N coincident M2-branes at the singularity of the orbifold C 4 /Z k . For N = 2 it should be equivalent to the BLG theory in the SU(2) × SU(2) formulation [6].
While the ABJM theory also admits a 3-algebra interpretation [21] (where antisymmetry condition is relaxed, see also [22]) it appears to be very different from the L-BLG one: the two theories have different field content and different symmetries.
In ref. [23] an interesting suggestion was made that the L-BLG theory may be interpreted as a certain limit of the ABJM theory, in which one sends the ABJM coupling k (CS level) to infinity and at the same time rescales some of the fields to zero so that they decouple. However, it was not shown in [23] that such limit of the N = 6 ABJM action does indeed lead to the full N = 8 supersymmetric L-BLG action.
Here we shall refine the suggestion of [23] by pointing out that to be able to relate the two theories by a scaling limit one needs to supplement the ABJM theory with an extra abelian "ghost" multiplet (decoupled from the ABJM fields). We shall demonstrate that there exists a limit (or "contraction") of the 3-algebra underlying the ABJM theory [21] trivially extended by an extra "ghost" generator that leads exactly to the Lorentzian 3-algebra underlying the L-BLG theory [9,10,11]. The BLG construction of the superconformal theory from a 3-algebra then guarantees that the corresponding limit of the (extended) ABJM action is indeed the full 1 There is a close analogy with a conformal plane-wave 2d sigma model [17] having D + 2 dimensional Lorentzian target space ds 2 D+2 = dx + dx − + G ij (x + , x)dx i dx j which may be viewed as a "dressing" of nonconformal 2d sigma model with the Euclidean D dimensional target space metric ds 2 D = G ij (x)dx i dx j that depends on x + according to the RG equation L-BLG action. This relation between 3-algebras reinforces the idea of [21] that 3-algebras may be an essential part of the multiple M2-brane theory.
The scaling limit we shall consider is very similar to the limits considered previously for 2d WZW models [24,25], 3d Chern-Simons and 4d Yang-Mills theories [26] where one goes (via an infinite "boost" in field space and a rescaling of coupling) from a theory defined by a product of a "ghost" (time-like) direction and a simple Lie group to a theory defined by non-semisimple contraction of the corresponding algebra with non-degenerate (but indefinite) metric. 2 In the 3d CS example considered in [26] the starting theory was the U(1) −k × SU(2) k Chern-Simons one and the limiting theory was the CS model based on the centrally extended Euclidean algebra E c 2 in 2 dimensions. Here we shall start with the [U(N) k × U(N) −k ] Chern-Simons-matter N = 6 ABJM theory [18,21] with an extra decoupled "ghost" multiplet and end up with the SU(N) Chern-Simons-matter N = 8 L-BLG theory [9,10,11].
One motivation for considering this limit is that it may be possible to view it as a definition of the L-BLG theory in terms of the ABJM theory and that may shed light on the interpretation of the former. In particular, we shall see how the relation between the L-BLG theory and the 3d N = 8 SYM (or D2-brane) theory can be understood in the ABJM framework: taking the scaling limit and then giving one of the scalars an expectation value is actually equivalent to the Higgs-type procedure of [16,20] for obtaining the D2-brane theory from the ABJM theory.
We shall start in section 2 with defining the relevant 3-algebras and reviewing the ABJM and L-BLG theories. After reviewing in section 3.1 the scaling limit in the bosonic CS theory we shall then explain in sect 3.2 how to relate the kinetic terms in the ABJM action combined with a "ghost" multiplet to those of the L-BLG theory by a similar scaling limit. We will proceed to show in section 3.3 how the scaling limit transforms the ABJM 3-algebra "tensored" with a ghost generator into the Lorentzian BLG 3-algebra. We shall draw some conclusions in section 4, commenting in particular on the relation of the two theories to the D2-brane theory.

Review of ABJM and L-BLG theories
Let us start with a review of basic definition of 3-algebra that can be used [21] to construct the interaction terms in the supersymmetric CS-matter theories like ABJM and L-BLG.
A 3-algebra is a (complex) vector space with a basis T a , a = 1, . . . , M, endowed with a triple product where the structure constants satisfy the following fundamental identity Here we assume a general form of a 3-algebra, for which structure constants are a priori antisymmetric only in the first 2 upper indices [21,27].
It was shown in [4,5] that when f abc d are real and antisymmetric in a, b, c, one can construct a set of 3d equations of motion that are invariant under 16 supersymmetries and SO(8) Rsymmetry. Adding an assumption of existence of an invariant inner product allows one to construct the Chern-Simons-matter N = 8 superconformal BLG action [5]. This applies, in particular, for the SU(N) Lorentzian BLG theory of [9,10,11]. This construction was further generalized in [21] to the case when the structure constants which are no longer real and antisymmetric in all three indices but are required to satisfy (in addition to (2.2)) the following condition It was shown in [21] that such more general algebras lead, in particular, to the N = 6 Chern-Simons-matter theory of ABJM [18].
Let us now review the Lagrangians of the two theories we are interested in. The ABJM theory [18,19] is invariant under 24 supercharge generators (N = 6 superconformal symmetry), SU(4) R-symmetry, and a U(1) internal symmetry. The field content is given by the U (N µ ) and bi-fundamental (N × N matrix-valued) matter fields -4 complex scalar fields Y A (A = 1, 2, 3, 4), and their hermitian conjugates Y † A , as well as the fermions ψ A and their hermitian conjugates ψ A † . Fields with raised A index transform in the 4 of the R-symmetry SU(4) group and those with lowered index transform in the4. The corresponding Lagrangian has the following form Here L CS is a Chern-Simons term and V (Y ) is a sextic scalar potential This action is expected to provide a low-energy description of N M2-branes at an C 4 /Z k orbifold singularity [18].
In [21] the general form for the action of a 3d scale-invariant field theory with N = 6 supersymmetry, SU(4) R-symmetry and U(1) global symmetry was found by starting with a 3-algebra in which the triple product is not antisymmetric. The field content is the same as described above (with the matter fields taking values in the 3-algebra, i.e. Y A = T a Y A a , ψ A = T a ψ Aa , while the gauge field is A µ = A µab T a ⊗ T b ) and the corresponding Lagrangian may be written as where the brackets stand for the inner product in 3-algebra Choosing a particular matrix realization of the 3-algebra such that for any 3 elements X 1 , X 2 , X 3 one has [21] [X 1 , X 2 ; and an inner product given by an ordinary matrix trace one observes that (2.8) becomes the same as the ABJM Lagrangian (2.5). The advantage of this form of the action (2.8) is that its structure is completely determined by the underlying 3-algebra.
In the particular case of the totally antisymmetric 3-algebras the corresponding action [5] has the enhanced N = 8 supersymmetry. While there is only one non-trivial antisymmetric 3-algebra with a Euclidean metric [7], choosing a Lorentzian metric one finds an infinite class of 3-algebras with an underlying Lie algebra structure [8]. For any Lie algebra G with structure constants f ij k and Killing form h ij one can define the corresponding 3-algebra as follows. Let the generators T a of the 3-algebra be denoted by where T i are in one-to-one correspondence with the generators of the Lie algebra. Then the basic 3-algebra relations are chosen to be where a, b take (+, −, i) values and f ijk ≡ f ij l h lk is totally antisymmetric, i.e. this 3-algebra is antisymmetric. The invariant inner product is defined as follows (2.14) Here b is an arbitrary constant; since redefining T + → T + + aT − preserves the 3-algebra structure but shifts b = T + , T + → b − 2a, we can always choose b = 0. Using this Lorentzian 3-algebra in the N = 8 supersymmetric BLG construction (now with real scalar fields X I (I = 1, ..., 8), fermions and gauge fields carrying 3-algebra indices +, − or i, where we may choose i = 1, .., N 2 − 1 to be the index the adjoint representation of the SU(N) Lie algebra) one finds the following Lagrangian [9,10,11] This theory having the right symmetries and an arbitrary-rank SU(N) gauge group was proposed [9, 10] as a candidate for a low-energy limit of a theory on N M2-branes in flat space. 4 Like similar models in two [24] and higher [26] dimensions it has a scale symmetry that allows one to absorb the coefficient in front of the Lagrangian into a field redefinition. It thus has no obvious coupling constant and thus no natural perturbative expansion preserving all the symmetries (if one chooses a particular expansion point in field space one can use the standard loop expansion but a non-zero background would spontaneously break the conformal invariance).
Assuming one is allowed to consider only the observables that do not depend on X − , one may integrate over X − getting a delta-function constraint ∂ 2 X + = 0. Solving it by X + = v=const and integrating out B µ one obtains [9]- [14], via simple path integral duality transformation [16], the 3d SU(N) N = 8 SYM theory with the coupling constant g Y M = v, i.e. one establishes a relation between the L-BLG theory and the low-energy theory of N coincident D2-branes. 5 In principle, this does not necessarily need to imply that the L-BLG theory (2.15) is equivalent to D2-brane theory -this is so only if it is treated in perturbative expansion near the vacuum X + = v=const which spontaneously breaks conformal symmetry and introduces v as a coupling constant. To preserve the conformal symmetry one may insist on summing over all solutions of ∂ 2 X + = 0, and, in particular, on integrating over v (cf. [13,15,23]). It remains to be seen if one can give a precise meaning to such "summation" [15] or "integration over coupling constant" (which obviously depends on a definition of a measure of integration, etc.).
Our aim below will be to make precise, following the suggestion of [23], in which sense the L-BLG theory (2.15) can be interpreted as a scaling limit of the ABJM theory (2.5). One may 3 Here we set A i µ = A µi+ , B i µ = 1 2 f ijk A µjk and used that A µ+− and A µ−i decouple. We then replaced the fields with adjoint index i with matrices in the fundamental representation of SU (N ) (i.e. X I i , etc, by N × N hermitian matrices) and thus replaced summation over i (with h ij = δ ij ) by the trace. 4 In contrast to similar plane-wave sigma models where the existence of a negative norm direction in field space is natural and does not represent a problem due to reparametrization invariance allowing to fix a lightcone gauge, here there is an apparent non-unitarity issue. It is possible that it can be avoided by a consistent truncation of the spectrum or by gauging the shifts in X − direction [12,13]. 5 In a sense, the L-BLG theory can be interpreted as a "conformal dressing" of the 3d N = 8 SYM theory where one effectively integrates over its coupling constant. This does not, however, mean that the two theories are completely equivalent, cf. footnote 1 and a discussion below.
hope that understanding the L-BLG theory via this scaling limit may possibly shed light on its proper interpretation and suggest an alternative way of defining the quantum L-BLG theory without explicit breaking of its conformal invariance. As we shall see, to implement such a scaling limit in a systematic fashion one will need to supplement the ABJM theory by an extra "ghost" supermultiplet, and thus it may not help with clarifying the unitarity issue of L-BLG theory.

Scaling limit
Here we shall first review the scaling limit of Chern-Simons theory and then show how a similar limit can be used to relate the ABJM action to L-BLG action by first adding an extra singlet ghost multiplet to the ABJM theory.
Next, we shall show that there exists a scaling limit or contraction of the non-antisymmetric 3-algebra (2.10) associated to the U(N) × U(N) gauge group [21] supplemented by an extra negative-norm generator that reduces it to the antisymmetric Lorentzian 3-algebra associated to the SU(N) gauge group. This will provide a rigorous confirmation of the relation between the full non-linear ABJM and L-BLG actions via the scaling limit.

Examples of scaling limits of Chern-Simons theory
Let us start with recalling a simple example of a relevant type of scaling limit -the one that leads from the U(1)×SU(2) algebra to the centrally extended Euclidean algebra in 2 dimensions E c 2 . The idea [25] will be to mix together the U(1) generator J 0 with a Cartan U(1) generator J 3 of SU(2). Denoting the remaining two generators of SU(2) by J n , n = 1, 2 we have (3.1) Let us introduce J + , J − , J n by setting and take the limit ε → 0. Then we end up with the algebra of E c 2 : 6 If we consider the corresponding quadratic form (or Casimir) on the original algebra taking the U(1) part with negative sign we get −J 0 J 0 + J 3 J 3 + J n J n = ε −2 (2J − J + + J n J n ) which is non-degenerate after an overall rescaling. While the standard Killing form on E c 2 is degenerate, it admits a non-degenerate invariant bilinear form with the signature (−1, 1, 1, 1) with entries [24,28] where b can be set to 0 by a redefinition of generators. This may be compared to (2.13), (2.14) where T − corresponds to the central element J − and T + -to the redefined rotation generator J + .
As discussed in [26], the CS theory for E c 2 with the above non-degenerate invariant form can be obtained by a scaling limit from CS theory for U(1) −k ×SU(2) k : one should do a redefinition of gauge field components and take k → ∞. Explicitly, starting with and setting and then taking the limit ε → 0 (with A a andk fixed) we end up with which is the CS action for E c 2 with the metric (3.4) with b = 0. 7 The redefinition in (3.6) is consistent with the one (3.2) used in taking the scaling limit in the algebra, namely, (3.8) Note that the action (3.7) does not actually depend onk as it can be redefined away by rescaling the fields. This feature is shared by the L-BLG action -the overall coefficient in (2.15) can be set equal to 1 by a field redefinition [13]. 8 Let us mention also another example of a limit of CS theory that leads to a BF theory (cf. [29]). In this case we start with CS theory for the group G −k × G k , e.g., G = U(N). Let us denote the two gauge fields as A (L,R) µ and define their combinations A µ and B µ as Then taking the limit ε → 0 (i.e. k → ∞ for fixedk) in the corresponding action we end up with (cf. (3.7)) Herek can be set to 1 by a redefinition of B µ . 7 In general, the CS Lagrangian written in terms of an invariant metric Ω ab and structure constants of the algebra f a bc is L = Ω ad ǫ µνλ A a µ ∂ ν A d λ + 1 3 f a bc A d µ A b ν A c λ . 8 One is to rescale the fields in (2.15) as follows: In the limit ε → 0 we will then get a semidirect sum of the algebra of G and translations, i.e.
This relation of the BF structure of the CS part of the L-BLG action (2.15) to such Inonu-Wigner contraction-type scaling limit was mentioned earlier in [9,10,23].

From ABJM + "ghost" action to L-BLG action via a scaling limit
We are now ready to discuss the scaling limit of the ABJM action suggested in [23]. We shall concentrate on the bosonic part of the action (the fermions are readily included after we describe the corresponding scaling limit at the level of the 3-algebras below). In addition to the U(N) −k × U(N) k CS action the bosonic part of the ABJM action (2.5) contains also the "matter" part -complex N × N scalar field matrix where we used the same field redefinition as in (3.9). If we now separate the trace part in Y A by setting we get 4 complex singlet fields Y A + and 4(N 2 − 1) complex scalar components ofỸ A . We have introduce the factor of ε −1 in the singlet part to define the scaling limit leading to the L-BLG action. Then we get Taking the limit ε → 0 the second term in (3.15) gives As was observed in [23], the first term in (3.15) and the second term in (3.17) are as in the L-BLG action (2.15) with the 4 complex singlet scalars Y A + corresponding to the 8 real scalar fields X I + ; the same was found to be true also for the bosonic interaction terms [23]. 9 One is still left with a singular first term in (3.17), ∼ ε −2 |∂ µ Y A + | 2 . In [23] it was suggested that to make the action finite one is to require that ∂ 2 Y A + = 0 which was then concluded to be the same as the equation in L-BLG theory (2.15) obtained by variation over X I − . This suggestion, however, appears to be hardly satisfactory for several reasons: (i) The condition for the vanishing of that singular term is, in general, ambiguous (depending, e.g., on boundary conditions) and, in fact, appears to require that ∂ µ Y A + = 0. Then Y A + =const but in this case the L-BLG action becomes equivalent to 3d SYM or D2-brane action. The relation to the ABJM theory via a scaling limit with k → ∞ is then not too surprising as the D2-brane theory is also a limit of the ABJM theory [18,30]. The limit considered in [23] then does not represent a consistent derivation of the L-BLG theory from the ABJM one but rather of a D2-brane theory from the ABJM theory.
(ii) The non-trivial difference of the L-BLG theory from D2-brane theory is in the presence of extra 8 scalar fields X I − on which "observables" (composite conformal operators representing states) may, in general, depend and which may enter the external sources (cf. [15]). However, there is no place for such fields in the above limit of the ABJM theory considered in [23]: the matter fields Y A or Y A + ,Ỹ A correspond only to X I + , X I . (iii) The above limit missing X I − fields is also not consistent with an expectation that the scaling limit may be may be carried out directly at the level of the corresponding 3-algebras: the Lorentzian 3-algebra contains the generator T − corresponding to X − (in addition to the generators corresponding to the SU(N) algebra and the X + fields) but the 3-algebra for the ABJM theory lacks the corresponding generator. A related point is that while the scalar product on the Lorentzian 3-algebra vector space is indefinite, the scalar product on the ABJM 3-algebra vector space is hermitian (reflecting the positivity of the scalar kinetic term in the ABJM action).
Here we are going to improve the suggestion of [23] by proposing that in order to obtain the L-BLG theory by a consistent scaling limit similar to the one that relates the corresponding CS parts of the actions one is to extend the original ABJM theory by an extra "ghost" U(1) multiplet containing 4 complex bosonic fields U A and the corresponding singlet fermions. To implement such scaling limit at the 3-algebra level one will then need to extend the ABJM 3-algebra by an extra negative-norm generator.
Namely, we shall start with the (bosonic) ABJM Lagrangian supplemented by an extra term N|∂ µ U A | 2 having "wrong" sign of its kinetic term. Then setting where Y A + is defined in (3.14) and Y A − is a new variable, and taking the limit ε → 0 we shall then get instead of (3.17) As a result, we recover the full kinetic term of the L-BLG theory (2.15) (with X I ± being the real parts of Y A ± ). This limiting procedure is obviously similar to the one discussed above on the example of the U(1) −k × SU(2) k CS model or the one used in the string sigma model context in getting a pp-wave model via a Penrose-type limit [25] (with U A playing the role of the "target space time" direction).
With a hindsight, the need to extend the ABJM action by an extra "ghost" U A field is hardly unexpected: it would be strange to obtain the L-BLG action which has an indefinite kineticterm signature from a manifestly definite ABJM action by a regular scaling limit. The existence of this scaling limit does not seem to shed extra light on the unitarity issue of the L-BLG theory: 10 while at the level of the ABJM theory one may assume that the extra ghost multiplet is completely decoupled (and does not enter the observables), it gets effectively coupled via the redefinition (3.18) in the process of taking the scaling limit.
3.3 From ABJM + "ghost" 3-algebra to L-BLG 3-algebra via a scaling limit Let us now show that starting with the 3-algebra (2.10) for the ABJM theory extended by an extra "ghost" generator one can get the Lorentzian 3-algebra (2.13) associated to L-BLG theory by a scaling limit. Since the two supersymmetric actions can be directly constructed from the corresponding 3-algebras [9,10,21], that will imply, in particular, that the scaling limit will go through at the level of the full actions including the fermions and all interaction terms. The Lorentzian 3-algebra vector space has an indefinite scalar product (2.14) while that of the ABJM algebra (2.11) is positive definite. Counting the degrees of freedom in the U(N) × U(N) ABJM theory we get 4 × 2N 2 = 8N 2 scalars, while in the SU(N) L-BLG theory we get 8 × (1 + 1 + (N 2 − 1)) = 8N 2 + 8 scalars. To match the degrees of freedom we thus need to add an extra generator to the ABJM 3-algebra; it should have a negative norm and should thus correspond to the "ghost" multiplet introduced in the previous subsection.
Let us start by considering the scaling limit for the 3-algebra corresponding to the simplest case of the U(2) × U(2) gauge group. We decompose the 2 × 2 complex matrices X in (2.10) (which may be identified with maps from C 2 to C * 2 , or elements of bi-fundamental representation of U(2) × U(2) [21]) in the following basis (with complex coefficients): {E, σ i }. Here E = iI 2 with I 2 as the 2 × 2 identity matrix and σ i are the Pauli matrices. The 3-algebra relations (2.10) for these basic elements then become Note that this U(2) × U(2) (or N = 2) case is special in that the 3-algebra is actually fully antisymmetric. This implies that in this case the corresponding action will have the extended N = 8 supersymmetry [18]. 11 By analogy with the discussion in the previous two subsections let us now extend the 3algebra by adding an extra generator e in a trivial way, i.e. without modifying the existing relations and with [e, T a ; T b ] = 0. We shall also assume that the scalar product on the extended algebra is defined so that e is perpendicular to the {E, σ i } generators and has negative norm: e, e = −2 . (3.21) Similarly to the U(1) × SU(2) CS example discussed above in section 3.1, let us now rescale k and rename the generators to T + , T − , T i as follows i.e. T − = 1 2 εe, T + = ε −1 (E − e). Next, let us take the limit ε → 0 for fixed T ± , T i ,k. This leads to the following (totally antisymmetric) 3-algebra (a, b = +, −, i; i = 1, 2, 3) Hereκ can be set equal to 1 by further rescaling of the generators. This is just the Lorentzian 3-algebra (2.13) associated to the gauge group SU(2), with (3.21) leading to the appropriate scalar product (2.14): In the general case of U(N) × U(N) Lie algebra, the generators T i of SU(N) satisfy so that the generalization of the N = 2 3-algebra relations (3.20) is where the coefficients A ijk m are determined by f ij k and d ij k : Unlike the N = 2 case (3.20) here there is an additional term in the last relation in (3.26) which is not antisymmetric in i, j, k. However, after adding an extra "ghost" generator e that commutes with all other generators and taking a similar scaling limit with the second term in the last line of (3.26) will vanish and we will get the antisymmetric SU(N) L-BLG algebra (2.13). Note that the definition of the generators in (3.28) is in direct correspondence with the definition of fields in (3.14), (3.18):

Concluding remarks
To conclude, the 3-algebra [21] corresponding to the ABJM theory is related to the Lorentzian antisymmetric 3-algebra corresponding to the L-BLG theory after one extends the former by a "ghost" generator and takes the scaling limit defined above. This implies that the corresponding interacting actions are also related by this scaling limit. One may wonder if this scaling-limit relation may have some implications for the physical interpretation of the L-BLG theory, and the unitarity issue in particular. The ABJM theory extended by an abelian "ghost" supermultiplet (U A in and the corresponding fermions) may formally be unitary assuming we define the corresponding observables (conformal operators and their correlation functions) so that they do not depend on the "ghost" fields. The observables of the L-BLG theory obtained by taking the scaling limit will then be a certain subset of all possible composite operators one could consider by starting directly with the full L-BLG action.
The relation via the scaling limit does not necessarily mean that the correlation functions of the two theories will also be directly related: while the scaling limit is smooth in the action, it need not be so at the level of the correlation functions. One may view the L-BLG theory as a certain "truncation" of the ABJM theory (similar to the case of 2d sigma models related via Penrose-type limit).
One consistency test of this scaling limit relation between the ABJM and the L-BLG theories is found by considering their known connection to 3d N = 8 SU(N) SYM theory describing N coincident D2 branes. Starting from the ABJM theory one can get the theory on D2-branes by assuming that one of the scalars Y A develops an expectation value Y = √ k v and then taking the limit k → ∞, v → ∞ with g 2 YM = v 2 k kept fixed [18,20,30]. Since the new "ghost" field U that we introduced is completely decoupled it can be integrated out without changing this relation to 3d SYM theory. Given that the L-BLG also reduces to the D2-brane theory when one of the 8 scalars X I + gets an expectation value, one would naturally expect that taking the above scaling limit and then setting X + =ṽ would be equivalent to the D2-brane reduction procedure of [20,30].
Indeed, the scaling limit translates the D2-brane limit of assuming a scalar expectation value and taking it to infinity to first scaling the relevant fields to infinity and then giving one of them a finite expectation value. Comparing the resulting coupling constant g YM for the two ways of getting the N = 8 SYM theory -from the ABJM theory and from the L-BLG theory -we see that it is the same Thus L-BLG theory may be interpreted as a background-independent intermediate step that can be considered when reducing from the M2-branes to the D2-branes. The limit from the ABJM theory to D2-brane theory may be interpreted [20] as the cone C 4 /Z k becoming locally a cylinder for large k, with the radius of the cylinder being related to the distance of the branes from the singularity, given by the expectation value of the scalar Y . The scaling limit leading to the L-BLG theory also converts the cone into a cylinder, but in this case the radius of the cylinder is a dynamical variable and only when we give it an expectation value do we recover D2-brane theory.