A New Duality Between $\mathcal{N}=8$ Superconformal Field Theories in Three Dimensions

We propose a new duality between two 3d $\mathcal{N}=8$ superconformal Chern-Simons-matter theories: the $U(3)_1 \times U(3)_{-1}$ ABJM theory and a theory consisting of the product between the $\left(SU(2)_3\times SU(2)_{-3}\right)/\mathbb{Z}_2$ BLG theory and a free ${\cal N} = 8$ theory of eight real scalars and eight Majorana fermions. As evidence supporting this duality, we show that the moduli spaces, superconformal indices, $S^3$ partition functions, and certain OPE coefficients of BPS operators in the two theories agree.


Introduction and Summary
Maximally supersymmetric (N = 8) superconformal field theories in three dimensions have received quite a bit of attention due to their interpretation as M2-brane theories and due to the many new exact supersymmetric localization results that have allowed for several precision tests of AdS/CFT. While some of these theories have several distinct microscopic descriptions, they can all be described by a few infinite families of Chern-Simons (CS) theories with a product gauge group coupled to two pairs of matter chiral multiplets transforming in the bifundamental representation of the gauge group-see Figure 1. These families are: • BLG theories: These are SU (2) k ×SU (2) −k (denoted BLG k ) and (SU (2) k ×SU (2) −k )/Z 2 (denoted BLG k ) gauge theories, which preserve manifest N = 8 supersymmetry for any integer Chern-Simons level k. This description of the BLG theories is a reformulation [1,2] of the original work of Bagger, Lambert, [3][4][5] and Gustavsson [6] (BLG). Figure 1: The field content of the two-gauge group description of N = 8 SCFTs. The gauge group is G 1 × G 2 with opposite Chern-Simons levels for the two factors. The matter content consists of two pairs of bifundamental chiral multiplets whose bottom components are denoted by A 1 , A 2 and B 1 , B 2 . As explained in the main text, such theories have N = 8 SUSY at the IR fixed point only for special values of k and/or for special gauge groups G 1 and G 2 .
While these theories were originally believed to have an interpretation as effective theories on 2 coincident M2-branes, and this is indeed true for certain small values of k, their M-theory interpretation, if any, is still unknown for arbitrary k. The theories with M = N were first introduced by Aharony, Bergman, Jafferis, and Maldacena (ABJM) in [7], and those with M = N by Aharony, Bergman, and Jafferis (ABJ) in [8]. These theories can be interpreted as effective theories on N coincident M2-branes placed at a C 4 /Z k singularity in the transverse directions. Due to the dualities [7,14] ABJ N +1,N,1 ∼ = ABJM N,1 , the only independent theories in this family are the ABJM N,1 , ABJM N,2 , and ABJ N +1,N,2 theories.
The case of the ABJM 1,1 theory is worth noting: this theory is equivalent to a free theory of 8 massless real scalars and 8 massless Majorana fermions. This is the theory on one M2-brane in flat space, where the 8 scalars parameterize the location of the brane in the transverse space. The case ABJM N,1 for N > 1, which corresponds to a stack of N M2-branes in flat space, flows to a product of two decoupled CFTs in the infrared (see for instance [9]).
One of these CFTs is free (and equivalent to the ABJM 1,1 theory), and corresponds to the center-of-mass motion of the stack of branes. The other CFT in the product is interacting and strongly coupled.
In addition to the dualities between ABJM / ABJ theories already mentioned, there are further dualities that relate the BLG and ABJM theories at certain small values of k. For instance [14,22]: Furthermore, it is possible to conjecture other dualities that come from the fact that the k = 1, 2 ABJM and the k = 2 ABJ theories can be thought of as the IR limits of the maximally supersymmetric Yang-Mills theory with gauge algebra u(N ), so(2N ), and so(2N + 1), respectively [10][11][12][13]. At small N , there are various coincidental isomorphisms between these Lie algebras, which themselves induce isomorphisms between the corresponding N = 8 SCFTs. For instance, since u(2) ∼ = u(1) ⊕ so (3), one expects that the ABJM 2,1 theory should be isomorphic to the product between the ABJM 1,1 theory and the ABJ 2,1,1 theory.
The purpose of this paper is to present yet another duality between the ABJ(M) and BLG theories that is not included in the list above. It is: Recalling that the ABJM 3,1 theory has a decoupled free sector isomorphic to ABJM 1,1 theory as well as an interacting one, this duality can be rephrased as Thus, our new duality (3)-(4) provides an interpretation for the BLG theory at level k = 3: it is the interacting sector of the theory on three coincident M2-branes. Quite curiously, this duality casts the k = 3 BLG theory as a theory on three coincident M2-branes, unlike the original intuition that BLG theories should be related to theories on two M2-branes.
hypermultiplet and an adjoint hypermultiplet [9,14]. In fact, it is this latter description that we will use in some of our computations in the ABJM 3,1 theory that we perform in order to check (3)-(4).
As was checked in previous dualities, for our proposed duality we match the moduli spaces and superconformal indices on each side of the duality. We also match the values of the S 3 partition functions of the two theories. In addition, we provide a new check using the recently proposed supersymmetric localization of 3d N = 4 theories to a topological 1d sector [15]. Using this method, we calculate the two-and three-point functions of low-lying half and quarter BPS operators, which we use to extract their OPE coefficients, listed in (49), (50), and (70). For the OPE coefficients in the four point function of the stress tensor, we compare these values to the conformal bootstrap bounds of [16] in Figure 2. We find that the exact values come close to saturating the lower bounds.
The paper is organized as follows. In Section 2 we review ABJM 3,1 and BLG 3 theories and demonstrate the explicit operator matching for low-lying BPS operators, including matching the superconformal index. In Section 3 we match the moduli spaces. In Section 4 we compute and match the values of the S 3 partition functions. In Section 5 we study certain 1d topological sectors of each theory, and extract the OPE coefficients of low-lying BPS operators. In Appendix A we discuss the four point function, and in Appendix B we list more OPE coefficients.

Low-lying BPS Operator Spectrum
Let us start by introducing the two N = 8 theories we argue are dual in more detail and compare their operator spectra. These theories belong to the family of N = 6 Chern-Simons-matter theories [7] that have gauge group U (N ) × U (N ) or (SU (N ) × SU (N ))/Z N with Chern-Simons coefficients k and −k for the two gauge groups. In N = 2 notation, the matter content consists of chiral multiplets with scalar components A 1 , A 2 and B 1 , B 2 that transform under the product gauge group as (N, N) and (N, N), respectively. The theories have a quartic superpotential which preserves an SU (2) × SU (2) flavor symmetry under which A a transforms as (2,1) and Bȧ transforms as (1,2). These theories also have a manifest SU (2) R symmetry (corre- Bȧ have charge −1), which combines with the SU (2) R symmetry mentioned above to form an SO(8) R R-symmetry. Such an enhancement occurs for any k, but we will focus on the case k = 3.
We would like to compare the operator spectra of the U We will exhibit a few operators that belong to these representations. In order to construct operators, we can use the fields in the Lagrangian, as well as monopole operators.
The monopole operators M n 1 ,...,n Ñ n 1 ,...,ñ N create diag{n 1 , . . . , n N } and diag{ñ 1 , . . . ,ñ N } units of magnetic flux through the two gauge groups, respectively. Here, we take both the n r and n r to be in descending order. If the gauge groups are U (N ), then the equations of motion imply that where Q T is the charge under the U (1) T symmetry mentioned above, quantized in halfinteger units. 1 If the gauge groups are SU (N ), then the n r andñ r must each sum to zero.
We will only be considering BPS monopole operators, with zero R-charge. In general, the R-charge is as was first proposed in [18] and derived in [9,10,19]. It is easy to see from (7) that E = 0 only for n r =ñ r . In order to avoid clutter, we denote such operators simply by M n 1 ,...,n N . For k = 0 these monopole operators transform nontrivially under the gauge group in a way to be described shortly. To form gauge-invariant operators, the monopole operators M n 1 ,...,n N need to be dressed with the matter fields. Let us show this explicitly for the lowest few multiplets.

ABJM 3,1
For the ABJM 3,1 theory, the monopole operators M n 1 ,n 2 , gauge group as where we have denoted a U We can construct gauge invariant BPS states with nonzero Q T by dressing M n 1 ,n 2 ,n 3 with appropriate products of C I = (A 1 , A 2 , B † 1 , B † 2 ) and C † I , where upper/lower I = 1, 2, 3, 4 is a fundamental/anti-fundamental index for SU (4) R . In the notation above, the C I transform in the gauge irrep (Υ ν , Υ −ν ), with Υ = and ν = 1. Including explicit gauge indices, we would write (C I ) α α and (C † I )α α . Using a single matter field, we find that C † I M 1,0,0 and C I M 0,0,−1 (with the gauge indices contracted in the only possible way, namely ( respectively. These operators have scaling dimension ∆ = 1 2 , and are thus free. They are part of the free sector of ABJM theory, which also contains all operators that appear in the OPE of C † I M 1,0,0 and C I M 0,0,−1 . The lowest few scalar operators in this free sector are given schematically in Table 1. 2 The hallmark of the free sector is the OSp(8|4) irrep (B, +) [0010] whose scalar operators were mentioned above. Another feature is the presence of a stress tensor multiplet (B, +) [0020] . Table 1: BPS operators with ∆ ≤ 3 2 in the free sector of the ABJM 3,1 theory.
The interacting sector, whose lowest few scalar operators are given schematically in Table 2, consists of all operators that decouple from the free sector. For instance, the first operator in Table 2, C † (I C † J) M 1,1,0 , can be written more explicitly as: Note that the flavor indices are symmetrized, because the gauge indices for both gauge groups are simultaneously anti-symmetrized, and thus this operator transforms in the 10 of SU (4) R and has U (1) T charge −1. Also note the presence of another stress tensor multiplet (B, +) [0020] , which is different from the one appearing in the free sector. Thus, this ABJM theory has two N = 8 stress tensor multiplets, each corresponding to a decoupled sector.
Lastly, there is a mixed sector whose lowest few scalar operators are given in Table 3, 3 which consists of all operators built using both free and interacting sector operators. Note that there are no free or stress tensor multiplets in the mixed sector, as expected, but there are now both (B, +) and (B, 2) operators with dimension 3 2 .
mixing between operators in the same representations. 3 The appearance of C I C † (J C † K) M 1,0,0 in both the mixed and interacting sector is because there are two singlets in the product 3 ⊗ 3 ⊗3 ⊗3 of gauge irreps, and thus two inequivalent ways of contracting the gauge indices.  Table 3: BPS operators with ∆ ≤ 3 2 in the mixed sector of ABJM 3,1 theory.

BLG 3
A similar construction holds for the BLG 3 theory. One difference between this theory and the ABJM 3,1 example we studied above is that the BLG 3 theory has a different set of monopole operators with E = 0, labeled by only a single positive half-integer GNO charge n. They (For the BLG k theory with arbitrary k, the gauge irrep is (2kn + 1, 2kn + 1).) These monopole operators must be combined with the matter fields C I and C † I , each of which transform as (2,2) under the gauge group.
The lowest dimension gauge invariant operators are quadratic in C I and C † I and do not require monopole operators. The next lowest are cubic in the C I and C † I and require the monopole operator with n = 1/2. See Table 4. These operators are in one-to-one correspondence with operators from the interacting sector of the ABJM 3,1 theory given in Table 2.
We take this match to be the first piece of evidence for the duality (3)-(4) between the two theories.

Superconformal Index
As an alternative to the explicit construction given in the previous section, one can use the superconformal index. The superconformal index, to be defined more precisely shortly, captures information about protected representations of the superconformal algebra. Its advantage over the explicit construction of the previous section is that it can be rigorously computed using supersymmetric localization. Its disadvantage is that the information it encodes does not unambiguously identify all the osp(8|4) representations.
In order to define the superconformal index, it is convenient to view an N = 8 SCFT as an N = 2 SCFT with SU (4) flavor symmetry. One can then consider a supercharge Q within the N = 2 superconformal algebra such that {Q, where ∆ is the scaling dimension, j 3 is the third component of the angular momentum, and R is the U (1) R charge. (There is a unique such supercharge, and it has ∆ = 1/2, R = 1, and The superconformal index with respect to Q is defined as the trace over the S 2 × R Hilbert space where F = (−1) 2j 3 is the fermion number and F f are the charges under the Cartan of the SU (4) flavor symmetry. Standard arguments imply that the only states contributing to the trace in (11) obey ∆ = R + j 3 ; all others cancel pairwise.
The indices for the theories we are interested in have been computed using supersymmetric localization in [20], following the general computation in [10]. It can be shown that is the index of the ABJM 3,1 theory, I BLG 3 is that of the BLG 3 theory, and I free is that of the ABJM 1,1 theory, which is free. For instance, keeping only one fugacity z corresponding to the Cartan element of SU (4) given by either U (1) T or One can indeed check that these expressions obey I ABJM 3,1 = I BLG 3 I free up to the order given. We regard this match of the indices as the second piece of evidence supporting our conjectured duality (3)-(4).

Moduli Space
We now show how to relate the (classical) moduli space of vacua of the ABJM 3,1 theory to that of the BLG 3 theory. The moduli space can be found by modding out the zero locus of the scalar potential by the gauge transformations. For both theories, one can check that the 4 We fix a typo in [20] for the coefficient of z 2 x 3 in the expression for I ABJM3,1 .
scalar potential vanishes provided that [1,7] Aα aβ = a βa δα β , where a βa , b β a are complex numbers, and where we used part of the gauge symmetry to put Aα aβ and B β aα in diagonal form. For a gauge group of rank N , the moduli space is thus parameterized by 4N complex numbers z r = {a r1 , a r2 , b r 1 , b r 2 } for r = 1, . . . , N , modulo residual gauge transformations.
The residual gauge symmetry gives further relations on z r . For the ABJM 3,1 theory, these relations are [7] where r = 1, 2, 3 and S 3 is the symmetric group of order six. The moduli space is thus From the M-theory perspective, this is the moduli space of three M2-branes in flat space, where the S 3 corresponds to permuting the indistinguishable branes.
For the BLG 3 theory, for which we denote the corresponding coordinates by z r instead of z r , the relations are [21][22][23] The first relation comes from permuting the identical gauge groups, while the last two come from identifications that depend on the Chern-Simons coupling. These relations define the moduli space (C 4 ) 2 /D 3 , where D 3 is the dihedral group of order six. We wish to identify this with the interacting sector of ABJM 3,1 . To distinguish between the free and interacting sector of the latter, consider the reparameterization The parameter w 3 is invariant under S 3 and thus parameterizes the moduli space of the free theory. The interacting sector is parameterized by w 1 , w 2 , which transform under the permutations (12), (123) ∈ S 3 as (12) : : where (12) permutes z 1 ↔ z 2 and (123) permutes z 1 → z 2 , z 2 → z 3 , z 3 → z 1 . These relations are the same as (15), which establishes the isomorphism where C 4 corresponds to the free sector of the ABJM 3,1 theory, and (C 4 ) 2 /D 3 corresponds to the interacting sector as well as to the BLG 3 theory. We regard the match between the moduli spaces (18) as the third piece of evidence supporting our conjectured duality (3)-(4).

The S 3 Partition Function
We will now compare the S 3 partition functions of the two theories. The partition function for the ABJM N,k theory can be written as the following finite dimensional integral [24]: where σ α , σ α are integration variables that can be interpreted as the eigenvalues of the scalars in the vector multiplets associated with the two U (N ) gauge groups. For k = 1 and N = 1, 3 we find where recall that the ABJM 1,1 theory is free.
The partition function of the BLG k theory can be derived from the ABJM N,k partition function (19) by setting N = 2, imposing the constraints σ 1 + σ 2 = σ 1 + σ 2 = 0, and multiplying by 2 to take into account the Z 2 quotient in the (SU (2) × SU (2))/Z 2 gauge group. The result is where we have made the change of variables σ ± = π(σ 1 ± σ 1 ). For k = 3, we find that as we expect from our duality. We regard (22) as our fourth piece of evidence supporting the conjectured duality (3)-(4).

One-Dimensional Topological Sector
Lastly, let us attempt to make a more detailed check of the duality (3) where O i 1 ...i 2j H (ϕ) is a 3d operator with ∆ = j H and j C = 0, written as a symmetric, rank-2j H tensor of SU (2) H . For more details, see [15] as well as [25][26][27].
For the particular case of N = 8 SCFTs, the Higgs and Coulomb topological sectors are isomorphic, so without loss of generality we will study the Higgs one. In [15], it was shown that for N = 4 SCFTs described by a Lagrangian with a vector multiplet with gauge algebra g and a hypermultiplet in representation R of g, it is possible to use supersymmetric localization to obtain an explicit description of the 1d sector associated with the Higgs branch. When the 1d topological sector is defined on a great circle within S 3 parameterized by ϕ, as above, its explicit description takes the form of a Gaussian 1d theory coupled to a matrix model:
Here, σ is the matrix degree of freedom that has its origin in the 3d vector multiplet and was diagonalized to lie within the Cartan of the gauge algebra. The 1d fields Q(ϕ) and Q(ϕ) have their origin in the 3d hypermultiplet and transform in the representations R and R, respectively. Their definition in terms of the hypermultiplet scalars is as in (23) Fortunately, there is a way around this difficulty. The right-hand side of (3)-(4), or more generally the ABJM N,1 theory, has a dual description as an N = 4 U (N ) gauge theory coupled to an adjoint hypermultiplet and a fundamental hypermultiplet [7,14,28]. So if we worked with this dual description we could use (24) to compute correlation functions in the Higgs branch topological sector, and we will do so in the case of interest N = 3. For the BLG theories no such dual description is available, but we will conjecture that a modification of (24) will allow us to compute some of the correlation functions in the Higgs branch sector.
Our conjecture is that to the integrand of (24) we should insert e iπk tr σ 2 (26) for every gauge group factor that has a Chern-Simons level k, where the trace is taken in the fundamental representation of that gauge group factor and in the trivial representation of the rest. This conjecture is motivated by the fact that this is the correct prescription in the matrix model (25). Importantly, it allows us to compute correlation functions of gauge-invariant operators built from Q and Q. However, unlike when k = 0, these operators are not the most general operators in the 1d theory; some of the operators in the 1d theory descend from 3d monopole operators, and these are not captured by (24) supplemented by (26). Nevertheless, we will still be able to compute correlation functions of non-monopole operators in the BLG 3 theory and compare them with the analogous correlators in the ABJM 3,1 theory. As we will see, the results of these computations are consistent with our proposed duality in (3) A careful analysis [25] shows that the only operators in the 1d theory come from the superconformal primaries of N = 8 multiplets that are at least 1/4-BPS-in our case, these will be the (B, +) where we have denoted the 1d operators as O (∆,j F ) a 1 ...a 2j F (ϕ), writing them explicitly as rank-2j F symmetric tensors of the SU (2) F . This SU (2) F is thus a global symmetry of the 1d topological theory.
As in [25], in order keep track of the SU (2) F indices more efficiently, we introduce polar-ization variables y a , a = 1, 2, and denote the operators in the 1d theory by where in order to avoid clutter we simply denote j F = j. We consider a basis of 1d operators with diagonal two-point functions, normalized such that where ϕ 21 ≡ ϕ 2 − ϕ 1 , and the product between SU (2) F polarizations is defined as The form of the three point functions is fixed by the SU (2) F symmetry up to an overall coefficient that we denote by λ (∆ 1 ,j 1 ),(∆ 2 ,j 2 ),(∆ 3 ,j 3 ) : × y 1 , y 2 j 123 y 2 , y 3 j 231 y 3 , y 1 j 312 (sgn ϕ 21 ) ∆ 123 (sgn ϕ 32 ) ∆ 231 (sgn ϕ 13 ) ∆ 312 , where j k 1 k 2 k 3 ≡ j k 1 + j k 2 − j k 3 . Eq. (32) is correct as long as j 1 , j 2 , and j 3 obey the triangle inequality. If this requirement is not fulfilled, the RHS of (32) vanishes.

ABJM 3,1
Let us now apply the formalism introduced above to the U (3) 1 × U (3) −1 ABJM theory in its dual description as a U (3) gauge theory with both an adjoint and fundamental N = 4 hypermultiplet. The result (24) reads in this case with where α, β = 1, 2, 3. The 1d fields X α β and X β α correspond to the adjoint hypermultiplet, Q α and Q α correspond to the fundamental hypermultiplet, and σ α are the matrix degrees of freedom in the Cartan of the U (3).
The D-term relations of the 3d theory allow us to rewrite the Q's in terms of the X's, so we will only use the latter to construct operators. Correlation functions of such operators can be computed by performing Wick contractions at fixed σ with the propagator and then integrating over σ: where · · · σ is the correlation function for the Gaussian theory in (34) at fixed σ computed using (36).
Being a 1d sector of an N = 8 SCFT, the theory (33) must have a flavor SU (2) F symmetry. Indeed, it is not hard to see that the fields ( X, X T ) transform as a doublet under SU (2) F . It is thus convenient to define X (ϕ, y) = y 1 X(ϕ, y) + y 2 X T (ϕ, y) , where the y a are the same polarization variables introduced earlier in (29).

Free Sector
As explained above, the ABJM 3,1 theory has a decoupled free sector. Consequently, the 1d theory (33) also has a decoupled free sector. It is generated by the gauge invariant operator which has its origin in the free multiplet (B, +) [0010] , whose superconformal primaries are scalars of scaling dimension ∆ = 1/2.
Since tr X and tr X only appear in the kinetic term of (34), we can simply read off the free (ϕ 2 , y 2 ) = 3 8πr y 1 , y 2 sgn ϕ 21 .
All other 1d operators belonging to the free sector are powers of O It follows that all free theory correlations functions can be computed using Wick contractions with the propagator (39). For the two and three point functions, we find and, when j 1 , j 2 , j 3 obey the triangle inequality, sgn ϕ 32 y 1 , y 2 Rescaling the O (j,j) free by a positive factor in order to match (30) and comparing with (32), we extract the OPE coefficients

Interacting Sector
Let us now discuss operators in the interacting sector in increasing order of the number of X 's they are built from. The interacting sector cannot have any operators linear in X , because such operators would have originated from ∆ = 1/2 operators in 3d, which are free.
So, the first non-trivial operator in the interacting sector must involve two X 's. It must also be orthogonal to the free theory operator that is quadratic in X , namely O (tr X ) 2 (ϕ, y) .
Next, we can construct operators with four X 's. It can be shown that the interacting sector contains two such operators. One of them has j = 2 and is The other has j = 0, and is given by: where here we have used explicit SU (2) F indices. The second term in the above expression ensures that this operator is orthogonal to the unit operator. It is straightforward to continue and construct operators with more than four X 's.
We also computed the OPE coefficients for Higgs branch operators in the O OPEs. These expressions are more complicated, so we relegate them to Appendix B.

BLG 3
As explained above, the 1d theory corresponding to the BLG theory requires a generalization of [15]. If we are not interested in correlation functions of operators arising from monopole operators in 3d, we conjecture that we can simply insert (26) into (24) and compute correlation functions of gauge-invariant operators built from Q and Q. For the BLG 3 theory, this conjecture produces the 1d theory where α, β andα,β are fundamental indices for each gauge group, Q α β and Qβ α correspond to the bifundamental hypermultiplets, and σ ± are the same integration variables as in (21).
(Eq. (21) is obtained after integrating out Q and Q in (53).) We can rewrite the action in terms of the mass matrix-like quantity to read off the propagator where there is no sum over the gauge indices. We then compute correlation functions as where O 1 (ϕ 1 , y 1 ) · · · O n (ϕ n , y n ) σ is the correlation function for the Gaussian theory (54) at fixed σ, given in (56).
Since the 1d theory (54) arises from an N = 8 SCFT, it must have a flavor SU (2) F symmetry. Indeed, it can be checked that such a symmetry is present and that (Qβ α , αγ βδ Q γδ ) form a doublet. It is thus convenient to combine the 2 × 2 matrices Q and Q into the matrix where y a are our usual SU (2) F polarization variables.
BLG 3 ) 2 , and one with j = 0, namely where here we have again used explicit SU (2) F indices and have included a second term to ensure that it is orthogonal to the unit operator. It is straightforward to proceed further using five Q's and higher.
The operators constructed so far, namely O
We can also write the four point function in the t-channel by exchanging (1 ↔ 3) in (64).