Studying the supersymmetry enhancement mechanism of Aharony, Bergman, Jafferis and Maldacena, we find a simple condition on the gauge group generators for the matter fields. We analyze all possible compact Lie groups and their representations. The only allowed gauge groups leading to the manifest N=6 supersymmetry are, up to discrete quotients, SU(n) x U(1), Sp(n) x U(1), SU(n) x SU(n), and SU(n) x SU(m) x U(1) with possibly additional U(1)'s. Matter representations are restricted to be the (bi)fundamentals. As a byproduct we obtain another proof of the complete classification of the three algebras considered by Bagger and Lambert.
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-a…
We study operator insertions into 1/2 BPS Wilson loops in N = 6 ABJM theory and investigate their two-point correlators. In this framework, the energy emitted by a heavy moving probe can be exactly obtained from some two-point coefficients of bosonic and fermionic insertions. This allows us to confirm an early proposal for computing the Bremsstrahlung function in terms of certain supersymmetric circular Wilson loops, whose value might be accessible to localization techniques. In the derivation of this result we also elucidate the structure of protected multiplets in the relevant superconformal…
We propose a recursion relation for tree-level scattering amplitudes in three-dimensional Chern-Simons-matter theories. The recursion relation involves a complex deformation of momenta which generalizes the BCFW-deformation used in higher dimensions. Using background field methods, we show that all tree-level superamplitudes of the ABJM theory vanish for large deformations, establishing the validity of the recursion formula. Furthermore, we use the recursion relation to compute six-point and eight-point component amplitudes and match them with independent computations based on Feynman diagrams…
We study the partition function of three-dimensional ${\mathcal N}=4$ superconformal Chern-Simons theories of the circular quiver type, which are natural generalizations of the ABJM theory, the worldvolume theory of M2-branes. In the ABJM case, it was known that the perturbative part of the partition function sums up to the Airy function as $Z(N)=e^{A}C^{-1/3}\mathrm{Ai}[C^{-1/3}(N-B)]$ with coefficients $C$, $B$ and $A$ and that for the non-perturbative part the divergences coming from the coefficients of worldsheet instantons and membrane instantons cancel among themselves. We find that many…
This review covers a number of applications of conformal field theory methods for perturbative calculations in N=4 Super Yang-Mills and ABJM theory. After motivating the role of superconformal symmetry for elementary particle physics research, we review some basic facts about conformal symmetry and conformal transformations of fields in a pedagogical manner. The implications for perturbative quantum field theory calculations are discussed in terms of Ward identities. Recent developments and relations between amplitudes, correlators and Wilson loops are reviewed for N=4 Super Yang-Mills as well…
We examine the three-dimensional N = 6 superconformal Chern--Simons theory with U(N) X U(N) gauge symmetry, which was recently constructed by Aharony, Bergman, Jafferis, and Maldacena (ABJM). Using a formulation with manifest SU(4) R-symmetry and no auxiliary fields, we verify in complete detail both the Poincare supersymmetry and the conformal supersymmetry of the action. Together, these imply the complete OSp(6|4) superconformal symmetry of the theory. The potential, which is sixth order in scalar fields, is recast as a sum of squares.
It was known that the ABJM matrix model is dual to the topological string theory on a Calabi-Yau manifold. Using this relation it was possible to write down the exact instanton expansion of the partition function of the ABJM matrix model. The expression consists of a universal function constructed from the free energy of the refined topological string theory with an overall topological invariant characterizing the Calabi-Yau manifold. In this paper we explore two other superconformal Chern-Simons theories of the circular quiver type. We find that the partition function of one theory enjoys the…
For three dimensional N=6 superconformal field theories we compute one-loop scattering amplitudes for any number of external particles. We focus on a particular subsector of N=2 invariant superamplitudes for which the ordinary perturbative evaluation becomes very easy. The result we obtain is in general non-vanishing. For six external particles our findings are sufficient for determining the complete expression of the N=6 superamplitude at this order. We discuss the symmetries of the result and its anomalous variation under superconformal generators.
We study the large N expansion of the partition function of the quiver superconformal Chern-Simons theories deformed by two continuous parameters which correspond to the general R-charge assignment to the matter fields. Though the deformation breaks the conformal symmetry, we find that the partition function shares various structures with the superconformal cases, such as the Airy function expression of the perturbative expansion in 1/N with the overall constant A(k) related to the constant map in the ABJM case through a simple rescaling of k. We also identify five kinds of the non-perturbativ…
We study the $\mathcal{N}=6$ superconformal Chern-Simons matter field theory (the ABJM theory) conformally coupled to a Lorentzian, curved background spacetime. To support rigid supersymmetry, such backgrounds have to admit twistor spinors. At the classical level, the symmetry of the theory can be described by a conformal symmetry superalgebra. We show that the full $\mathcal{N}=6$ superconformal algebra persists at the quantum level using the BV-BRST method.
We examine the string dual of the recently constructed $\mathcal{N}=6$ superconformal Chern-Simons theory of Aharony, Bergman, Jafferis and Maldacena (ABJM theory). We focus in particular on the $SU(2)\times SU(2)$ sector. We find a sigma-model limit in which the resulting sigma-model is two Landau-Lifshitz models added together. We consider a Penrose limit for which we can approach the $SU(2)\times SU(2)$ sector. Finally, we find a new Giant Magnon solution in the $SU(2)\times SU(2)$ sector corresponding to one magnon in each $SU(2)$. We put these results together to find the full magnon disp…
We explore further our recent generalization of the $\mathcal{N}=4$ superconformal Chern-Simons theories of Gaiotto and Witten. We find and construct explicitly theories of enhanced $\mathcal{N}=5$ or 6 supersymmetry, especially $\mathcal{N}=5$, $Sp(2M)\times O(N)$ and $\mathcal{N}=6$, $Sp(2M)\times O(2)$ theories. The $U(M)\times U(N)$ theory coincides with the $\mathcal{N}=6$ theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We argue that the $\mathcal{N}=5$ theory with $Sp(2N)\times O(2N)$ gauge group can be understood as an orientifolding of the ABJM model with $U(2N)\times U(2N)$…
We review recent progress in the study of line defects in three-dimensional Chern-Simons-matter superconformal field theories, notably the ABJM theory. The first part is focused on kinematical defects supporting a topological sector of the theory. After reviewing the construction of this sector, we concentrate on the evaluation of topological correlators from the partition function of the mass-deformed ABJM theory and provide evidence on the existence of a topological quantum mechanics living on the line. In the second part, we consider dynamical defects realized as latitude BPS Wilson loops f…
We revisit the N=6 superconformal Chern-Simons-matter theories and their supergravity duals in the context of generalized symmetries. This allows us to finally clarify how the $SU(N)\times SU(N)$ and $(SU(N)\times SU(N))/\mathbb{Z}_N$ theories, as well as other quotient theories that have recently been discussed, fit into the holographic framework. It also resolves a long standing puzzle regarding the di-baryon operator in the $U(N)\times U(N)$ theory.
We discuss higher-order corrections to superconformal invariance for a class of N=2 supersymmetric Chern-Simons theories including the ABJM model. We argue that corrections are inevitable for general theories in this class; but that it is probable that any corrections are of a particular "maximally transcendental" form.
The coefficients of the membrane instantons in the ABJM theory are known to be quadratic polynomials of the chemical potential. We show that, after deforming the ABJM theory into more general superconformal Chern-Simons theories labelled by (q,p) where the original ABJM theory corresponds to (q,p)=(1,1), we can decompose the membrane instanton into three types of non-perturbative effects with constant coefficients independent of the chemical potential. We find that, although these constants contain poles at certain values of q and p including the ABJM case, all of the poles cancel among themse…
In $\mathcal N \geq 2$ superconformal Chern-Simons-matter theories we construct the infinite family of Bogomol'nyi-Prasad-Sommerfield (BPS) Wilson loops featured by constant parametric couplings to scalar and fermion matter, including both line Wilson loops in Minkowski spacetime and circle Wilson loops in Euclidean space. We find that the connection of the most general BPS Wilson loop cannot be decomposed in terms of double-node connections. Moreover, if the quiver contains triangles, it cannot be interpreted as a supermatrix inside a superalgebra. However, for particular choices of the param…
Some remarks on Gribov mechanism on N=1 Supersymmetric 3D theories are presented. The two point correlation function is analysed and the possibility of obtaining the confining Gribov regime is discussed. Also the possibility of obtaining Gribov behaviour in ABJM due to a symmetry breaking is presented.
We present the higgsing of three-dimensional N=6 superconformal ABJM type theories coupled to conformal supergravity, so called topologically gauged ABJM theory, thus providing a gravitational extension of previous work on the relation between N M2 and N D2-branes. The resulting N=6 supergravity theory appears at a chiral point similar to that of three-dimensional chiral gravity introduced recently by Li, Song and Strominger, but with the opposite sign for the Ricci scalar term in the lagrangian. We identify the supersymmetry in the broken phase as a particular linear combination of the supers…
We calculate the superconformal indices of recently discovered three-dimensional N=4,5 Chern-Simons-matter theories and compare them with the corresponding indices of supergravity on AdS4 times orbifolds of S7. We find perfect agreement in the large N and large k limit, provided that the twisted sector contributions at the fixed loci of the orbifolds are properly taken into account. We also discuss the index for the so-called "dual ABJM" proposal.
We use supertwistor space to construct scattering amplitudes of maximal superconformal theories in three and six dimensions. In both cases, the constraints of superconformal invariance and rationality imply that the three-point amplitude vanishes on-shell, which constrains the four-point amplitude to have vanishing residues in all channels. In three dimensions, we find a unique solution for the four-point amplitude and demonstrate that it agrees with the component result in the BLG theory. This suggests that BLG is the unique three-dimensional theory with classical OSp(8|4) symmetry that admit…
We test several expected isomorphisms between the U(N)xU(N) ABJM theory and (SU(N)xSU(N))/Z_N theory including the BLG theory by comparing their superconformal indices. From moduli space analysis, it is expected that this equivalence can hold if and only if the rank N and Chern-Simons level k are coprime. We also calculate the index of the ABJ theory and investigate whether some theories with identical moduli spaces are isomorphic or not.
The ABJM theory refers to superconformal Chern-Simons-matter theory with product gauge group U(L)xU(R) and level +k, -k, respectively. The theory is a candidate for worldvolume dynamics of M2-branes sitting at C(4)/Z(k)k. By utilizing monopole operators, we prove that ABJM theory gets enhanced N=8 supersymmetry and SO(8) R-symmetry at Chern-Simons levels k=1,2. We first show that the ABJM Lagrangian can be written in a manifestly SO(8) invariant form up to certain extra terms. We then show that upon integrating out Chern-Simons gauge fields these extra terms vanish precisely at levels k=1,2. U…
We extend the hermitian three-algebra formulation of ABJM theory to include $U(1)$ factors. With attention payed to extra $U(1)$ factors, we refine the classification of $\N=6$ ABJM theories. We argue that essentially the only allowed gauge groups are $SU(N)\times SU(N)$, $U(N)\times U(M)$ and $Sp(N)\times U(1)$ and that we have only one independent Chern-Simons level in all these cases. Our argument is based on integrality of the $U(1)$ Chern-Simons levels and supersymmetry. A relation between monopole operators and Wilson lines in Chern-Simons theory suggests certain gauge representations of…
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.
We construct the classical action of the Aharony-Bergman-Jafferis-Maldacena (ABJM) model in the N=3, d=3 harmonic superspace. In such a formulation three out of six supersymmetries are realized off shell while the other three mix the superfields and close on shell. The superfield action involves two hypermultiplet superfields in the bifundamental representation of the gauge group and two Chern-Simons gauge superfields corresponding to the left and right gauge groups. The N=3 superconformal invariance allows only for a minimal gauge interaction of the hypermultiplets. Amazingly, the correct sex…
Starting with generic Wess-Zumino type coupling to constant four-form and the dual seven-form field strengths in the ABJM theory, we obtain mass-deformed theories with ${\cal N}=2,4$ supersymmetries. These theories contain massless scalar fields and allow the implementation of the Mukhi-Papageorgakis Higgsing procedure. Using this procedure, we connect the Higgsed theories to three-dimensional mass-deformed SYM theories. These are also connected by the four-dimensional ${\cal N}=1^*,2^*$ mass-deformed SYM theories through dimensional reduction. We classify the three-dimensional mass-deformed S…
Integrable structure has played a very important role in the study of various non-perturbative aspects of planar ABJM theories. In this paper we showed that this remarkable structure survive after orbifold operation with discrete group $Γ(\simeq\mathbb{Z}_n)<SU(4)_R\times U(1)_b$. For general $Γ$, we prove the integrability in the scalar sector at the planar two-loop order and get the Bethe ansatz equations. The eigenvalues of the anomalous dimension matrix are also obtained. For $Γ<SU(4)$, two-loop all-sector and all-loop BAEs are proposed. Supersymmetric orbifolds are discussed in this frame…
We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N=2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Ga…
For N=6 superconformal Chern-Simons-matter theories in three dimensions, by a direct superspace Feynman diagram approach, we compute the two-loop four-point scattering amplitude with external chiral matter fields. We find that the result is in perfect agreement with the two-loop result for a light-like four-polygon Wilson loop. This is a nontrivial evidence of the scattering amplitudes/Wilson loop duality in three dimensions. Moreover, both the IR divergent and the finite parts of our two-loop result agree with a BDS-like ansatz for all-loop amplitudes where the scaling function is given in te…
We investigate several aspects of BPS latitude Wilson loops in gauge theories in three dimensions with $\mathcal{N}\ge 4$ supersymmetry. We derive a matrix model for the bosonic latitude Wilson loop in ABJM using supersymmetric localization, and show how to extend the computation to more general Chern-Simons-matter theories. We then define latitude type Wilson and vortex loop operators in theories without Chern-Simons terms, and explore a connection to the recently derived superalgebra defining local Higgs and Coulomb branch operators in these theories. Finally, we identify a BPS loop operator…
We study marginal deformations of superconformal Chern-Simons matter theories that are 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 of the BLG and ABJM models. We also consider conformal multi-trace deformations preserving N=2 supersymmetry. We then use N=2 supergraph techniques to compute the two-loop beta functions of these deformations. Besides confirming conformal invariance of both the BLG and ABJM models, we also verify that the recently proposed beta-deformations of the …
We study the large-$N$ limit of superconformal index for two strongly interacting Chern-Simons theories in three dimensions with $\mathcal{N}=2$ supersymmetry, and compare the result against the AdS/CFT dual, namely the data of full Kaluza-Klein reduction spectra obtained using the exceptional field theory technique. The two theories of interest are mABJM and GJV theories, which are obtained as IR fixed point of RG whose UV description is ABJM and maximally supersymmetric Yang-Mills theory respectively. We confirm agreement and the duality persists. For the case of mABJM it turns out that we n…
We clarify how mirror symmetry acts on 3d theories with N=2,3 or 4 supersymmetries and non-abelian Chern-Simons terms and then construct many new examples. We identify a new duality, geometric duality, that allows us to generate large families of gauge theories, with and without Chern-Simons term, that all flow to the same conformal field theory in the infrared. In particular, we find an interesting duality of dualities: a pair of theories related via mirror symmetry can be mapped, via geometric duality, into a pair of gauge theories related by Seiberg duality. This network of dualities can be…
We explore the idea that large $N$, non-supersymmetric conformal field theories with a parametrically large gap to higher spin single-trace operators may be obtained as infrared fixed points of relevant double-trace deformations of superconformal field theories. After recalling the AdS interpretation and some potential pathologies of such flows, we introduce a concrete example that appears to avoid them: the ABJM theory at finite $k$, deformed by $\int\!{\cal O}^2$, where ${\cal O}$ is the superconformal primary in the stress-tensor multiplet. We address its relation to recent conjectures base…
Using the Eguchi-Kawai equivalence, we provide regularizations of supersymmetric quiver and Chern-Simons gauge theories which leave the supersymmetries unbroken. This allow us to study many interesting theories on a computer. As examples we construct large-$N$ reduced models of supersymmetric QCD with flavor and the ABJM model of multiple M2 branes.
We show that baryon-like operators exist in the N=6 superconformal Chern-Simons-matter theory constructed by Aharony, Bergman, Jafferis and Maldacena (ABJM). This involves the introduction of Wilson(or 't Hooft) lines ending on baryon-like operators and we show that the presence of such lines cannot be detected by the fields in the theory. The same construction can be used to make magnetic monopoles charged under the diagonal U(1) gauge group with arbitrary non-integral charge. If we do not include such monopole configurations in the path integral, the moduli space is known to be given by the …
We analyse a light-like cusped Wilson loop in N=6 superconformal Chern-Simons theory at both weak and strong coupling in light-cone gauge. At the second order in the 't Hooft coupling $λ$ the correct cusp anomalous dimension $Γ_{\rm cusp}=-ϕ/ 2 λ^2$ is recovered through a deformation of the contour that takes both rays of the cusp slightly off of the light-cone. The strong coupling behaviour is addressed by means of the Bethe-Salpeter equation for ladders of tree-level gauge propagators and ladders of one-loop corrected gauge propagators. It turns out that, as might be expected, the contributi…
We consider the partition function of the superconformal Chern-Simons theories with the quiver diagram being the affine D-type Dynkin diagram. Rewriting the partition function into that of a Fermi gas system, we show that the perturbative expansions in 1/N are summed up to an Airy function, as in the ABJM theory or more generally the theories of the affine A-type quiver. As a corollary, this provides a proof for the previous proposal in the large N limit. For special values of the Chern-Simons levels, we further identify three species of the membrane instantons and also conjecture an exact exp…
We analyse the fluctuations of the ground-state/funnel solutions proposed to describe M2-M5 systems in the level-k mass-deformed/pure Chern-Simons-matter ABJM theory of multiple membranes. We show that in the large N limit the fluctuations approach the space of functions on the 2-sphere rather than the naively expected 3-sphere. This is a novel realisation of the fuzzy 2-sphere in the context of Matrix Theories, which uses bifundamental instead of adjoint scalars. Starting from the multiple M2-brane action, a U(1) Yang-Mills theory on R^{2,1} x S^2 is recovered at large N, which is consistent …
We investigate the large $N$ instanton effects of partition functions in a class of $\mathcal{N}=4$ circular quiver Chern-Simons theories on a three-sphere. Our analysis is based on the supersymmetry localization and the Fermi-gas formalism. The resulting matrix model can be regarded as a two-parameter deformation of the ABJM matrix model, and has richer non-perturbative structures. Based on a systematic semi-classical analysis, we find analytic expressions of membrane instanton corrections. We also exactly compute the partition function for various cases and find some exact forms of worldshee…
We construct monopole operators for 3d Yang-Mills-matter theories and Chern-Simons-matter theories in canonical formalism. In this framework, monopole operators, although as the disorder operators, could be written in terms of the fundamental fields of the theory, and thus could be treated in the same way as the ordinary operators. We study the properties of the constructed monopole operators. In Chern-Simons-matter theories, monopole operators transform as the local operators with the classical conformal dimension $ 0 $ under the action of the dilation and are also covariantly constant. In su…
In this paper we construct the $\mathcal N=6$ conformal supergravity in three dimensions from a set of Chern-Simons-like terms one for each of the graviton, gravitino, and R-symmetry gauge field and then couple this theory to the $\mathcal N=6$ superconformal ABJM theory. In a first step part of the coupled Lagrangian for this topologically gauged ABJM theory is derived by demanding that all terms of third and second order in covariant derivatives cancel in the supersymmtry variation of the Lagrangian. To achieve this the transformation rules of the two separate sectors must be augmented by ne…
We discuss the N=2 superspace formulation of the N=8 superconformal Bagger-Lambert-Gustavsson theory, and of the N=6 superconformal Aharony-Bergman-Jafferis-Maldacena U(N)xU(N) Chern-Simons theory. In particular, we prove the full SU(4) R-symmetry of the ABJM theory. We then consider orbifold projections of this theory that give non-chiral and chiral (U(N)xU(N))^n superconformal quiver gauge theories. We argue that these theories are dual to certain AdS_4 x S^7/(Z_n x Z_k) backgrounds of M-theory. We also study a SU(3) invariant mass term in the superpotential that makes the N=8 theory flow to…
By integrating out the U(1)_B gauge field, we show that the U(n)xU(n) ABJM theory at level k is equivalent to a Z_k identification of the (SU(n)xSU(n))/Z_n Chern-Simons theory, but only when n and k are coprime. As a consequence, the k=1 ABJM model for two M2-branes in R^8 can be identified with the N=8 (SU(2)xSU(2))/Z_2 theory. We also conjecture that the U(2)xU(2) ABJM model at k=2 is equivalent to the N=8 SU(2)xSU(2)-theory.
We consider two-loop anomalous dimensions for fermionic operators in the ABJM model and the ABJ model. We find the appropriate Hamiltonian and show that it is consistent with a previously predicted Bethe ansatz for the ABJM model. The difference between the ABJ and ABJM models is invisible at the two-loop level by cancelation of parity violating diagrams. We then construct a Hamiltonian for the full two-loop OSp(6|4) spin chain by first constructing the Hamiltonian for an SL(2|1) subgroup, and then lifting to OSp(6|4). We show that this Hamiltonian is consistent with the Hamiltonian found for …
We define a 3-algebra with structure constants being symmetric in the first two indices. We also introduce an invariant anti-symmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The general N=5 superconformal Chern-Simons-matter (CSM) theory with SO(5) R-symmetry in three dimensions is constructed by using this algebraic structure. We demonstrate that the supersymmetry can be enhanced to N=6 if the sympelctic 3-algebra and the fields are decomposed in a proper fashion. By specifying the 3-brackets, some presently known N=5, 6 superconformal theories are described in terms o…
Manin triple construction of N=4 superconformal field theories is considered. The correspondence between quasi Frobenius finite-dimensional Lie algebras and N=4 superconformal field theories is established.
We show that an infinite family of N=6 d=3 superconformal Chern-Simons-matter theories has hidden N=8 superconformal symmetry and hidden parity on the quantum level. This family of theories is different from the one found by Aharony, Bergman, Jafferis and Maldacena, as well as from the theories constructed by Bagger and Lambert, and Gustavsson. We also test several conjectural dualities between BLG theories and ABJ theories by comparing superconformal indices of these theories.
In this paper we construct and classify novel Drukker-Trancanelli (DT) type BPS Wilson loops along infinite straight lines and circles in $\mathcal N=2,3$ quiver superconformal Chern-Simons-matter theories, Aharony-Bergman-Jafferis-Maldacena (ABJM) theory, and $\mathcal N=4$ orbifold ABJM theory. Generally we have four classes of Wilson loops, and all of them preserve the same supersymmetries as the BPS Gaiotto-Yin (GY) type Wilson loops. There are several free complex parameters in the DT type BPS Wilson loops, and for two classes of Wilson loops in ABJM theory and $\mathcal N=4$ orbifold ABJ…
