We present a superspace formulation of the D=3, N=4,5 superconformal Chern-Simons Matter theories, with matter supermultiplets valued in a symplectic 3-algebra. We first construct an N=1 superconformal action, and then generalize a method used by Gaitto and Witten to enhance the supersymmetry from N=1 to N=5. By decomposing the N=5 supermultiplets and the symplectic 3-algebra properly and proposing a new super-potential term, we construct the N=4 superconformal Chern-Simons matter theories in terms of two sets of generators of a (quaternion) symplectic 3-algebra. The N=4 theories can also be derived by requiring that the supersymmetry transformations are closed on-shell. The relationship between the 3-algebras, Lie superalgebras, Lie algebras and embedding tensors (proposed in [E. A. Bergshoeff, O. Hohm, D. Roest, H. Samtleben, and E. Sezgin, J. High Energy Phys. 09 (2008) 101.]) is also clarified. The general N=4,5 superconformal Chern-Simons matter theories in terms of ordinary Lie algebras can be rederived in our 3-algebra approach. All known N=4,5 superconformal Chern-Simons matter theories can be recovered in the present superspace formulation for super-Lie-algebra realization of symplectic 3-algebras.
Within the supertwistor approach, we analyse the superconformal structure of 4D N = 2 compactified harmonic/projective superspace. In the case of 5D superconformal symmetry, we derive the superconformal Killing vectors and related building blocks which emerge in the transformation laws of primary superfields. Various off-shell superconformal multiplets are presented both in 5D harmonic and projective superspaces, including the so-called tropical (vector) multiplet and polar (hyper)multiplet. Families of superconformal actions are described both in the 5D harmonic and projective superspace sett…
A geometrical construction of superconformal transformations in six dimensional (2,0) superspace is proposed. Superconformal Killing vectors are determined. It is shown that the transformation of the tensor multiplet involves a zero curvature non-trivial cochain.
We summarise recent work on superconformal field theories using analytic superspace. All operators of N=4 SYM can be given as unconstrained superfields on analytic superspace. We show how to write down operators as superfields on analytic superspace and how to completely solve the Ward indentities for their correlation functions. We discuss the non-renormalisation of certain operators, and of some of their correlation functions. We discuss the relationship between harmonic and analytic superspace. Finally we discuss applications of these techniques to superconformal field theory in 6 dimension…
Representations of four-dimensional superconformal groups on harmonic superfields are discussed. It is shown how various short representations can be obtained by parabolic induction. It is also shown that such short multiplets may admit several descriptions as superfields on different superspaces. In particular, this is the case for on-shell massless superfields. This allows a description of short representations as explicit products of fundamental fields. Superconformal transformations of analytic fields in real harmonic superspaces are given explicitly.
A quantum deformation of 4-dimensional superconformal algebra realized on quantum superspace is investigated. We study the differential calculus and the action of the quantum generators corresponding to $sl_q(1|4)$ which act on the quantum superspace. We derive deformed $su(1|2,2)$ algebras from the deformed $sl(1|4)$ algebra. Through a contraction procedure we obtain a deformed super-Poincar{é} algebra.
Representations of four dimensional superconformal groups are constructed as fields on many different superspaces, including super Minkowski space, chiral superspace, harmonic superspace and analytic superspace. Any unitary irreducible representation can be given as a field on any one of these spaces if we include fields which transform under supergroups. In particular, on analytic superspaces, the fields are unconstrained. One can obtain all representations of the N=4 complex superconformal group $PSL(4|4)$ with integer dilation weight from copies of the Maxwell multiplet on $(4,2,2)$ analyti…
We study $3d$ SCFTs in the superspace formalism and discuss superfields and on-shell higher spin current multiplets in free $3d$ SCFTs with $\mathcal{N}= 1,2,3,4$ and $6$ superconformal symmetry. For $\mathcal{N}=1$ 3d SCFTs we determine the superconformal invariants in superspace needed for constructing 3-point functions of higher spin operators, find the non-linear relations between the invariants and consequently write down all the independent invariant structures, both parity even and odd, for various 3-point functions of higher spin operators.
We study a deformed $su(m|n)$ algebra on a quantum superspace. Some interesting aspects of the deformed algebra are shown. As an application of the deformed algebra we construct a deformed superconformal algebra. {}From the deformed $su(1|4)$ algebra, we derive deformed Lorentz, translation of Minkowski space, $iso(2,2)$ and its supersymmetric algebras as closed subalgebras with consistent automorphisms.
Representations of four-dimensional superconformal groups on harmonic superfields are discussed. It is argued that any representation can be given as a superfield on many superflag manifolds. Representations on analytic superspaces do not require constraints. We discuss short representations and how to obtain them as explicit products of fundamental fields. We also discuss superfields that transform under supergroups.
The superconformal structure of coset superspaces with AdS_m x S^n geometry of bosonic subspaces is studied. It is shown, in particular, that the conventional superspace extensions of the coset manifolds AdS_2 x S^2, AdS_3 x S^3 and AdS_5 x S^5, which arise as solutions of corresponding D=4,6, 10 supergravities and have been extensively studied in connection with AdS/CFT correspondence, are not superconformally flat, though their bosonic submanifolds are conformally flat. We give a group-theoretical reasoning for this fact. We find that in the AdS_2 x S^2 and AdS_3 x S^3 cases there exist diff…
A study of the superconformal covariantization of superdifferential operators defined on $(1|1)$ superspace is presented. It is shown that a superdifferential operator with a particular type of constraint can be covariantized only when it is of odd order. In such a case, the action of superconformal transformation on the superdifferential operator is nothing but a hamiltonian flow defined by the corresponding supersymmetric second Gelfand-Dickey bracket. The covariant form of a superdifferential operator of odd order is given.
A unique feature of N=6 conformal supergravity in three dimensions is that the super Cotton tensor W^{IJKL} can equivalently be viewed, via the Hodge duality, as the field strength of an Abelian vector multiplet, W^{IJ}. Using this observation and the conformal superspace techniques developed in arXiv:1305.3132 and arXiv:1306.1205, we construct the off-shell action for N=6 conformal supergravity. The complete component action is also worked out.
The superspace formulation of N=1 conformal supergravity in four dimensions is demonstrated to be equivalent to the conventional component field approach based on the superconformal tensor calculus. The detailed correspondence between two approaches is explicitly given for various quantities; superconformal gauge fields, curvatures and curvature constraints, general conformal multiplets and their transformation laws, and so on. In particular, we carefully analyze the curvature constraints leading to the superconformal algebra and also the superconformal gauge fixing leading to Poincare supergr…
We review harmonic superspaces of the D=3, N=3 and 4 supersymmetries and gauge models in these superspaces. Superspaces of the D=3, N=5 supersymmetry use harmonic coordinates of the SO(5) group. The superfield N=5 actions describe the off-shell infinite-dimensional Chern-Simons supermultiplet.
In $D=4$, $\cal{N}=1$ conformal superspace, the Yang-Mills matter coupled supergravity system is constructed where the Yang-Mills gauge interaction is introduced by extending the superconformal group to include the Kähler isometry group of chiral matter fields. There are two gauge-fixing procedures to get to the component Poincaré supergravity: one via the superconformal component formalism and the other via the Poincaré superspace formalism. These two types of superconformal gauge-fixing conditions are analyzed in detail and their correspondence is clarified.
In this paper we apply a supersymmetric generalization of the method of derivative expansion to compute the induced non-Abelian Chern-Simons term in $\mathcal{N}=1$, $d=3$ superspace, for an arbitrary gauge group.
The superspace geometry of Chern-Simons forms is shown to be closely related to that of the 3-form multiplet. This observation allows to simplify considerably the geometric structure of supersymmetric Chern-Simons forms and their coupling to linear multiplets. The analysis is carried through in U_K(1) superspace, relevant at the same time for supergravity-matter couplings and for chirally extended supergravity.
We establish a theorem about non-trivial 11D supergravity fluctuations that are conformally related to flat superspace geometry. Under the assumption that a theory of conformal 11D supergravity exists, similar in form to that of previously constructed theories in lower dimensions, this theorem demands the appearance of non-vanishing dimension 1/2 torsion tensors in order to accommodate a non-trivial 11D conformal compensator and thus M-theory corrections that break super-conformal symmetry. At the complete non-linear level, a presentation of a conventional minimal superspace realization of Wey…
In this paper we will study non-abelian Chern-Simons theory on a deformed superspace. We will deform the superspace in such a way that it includes the noncommutativity between bosonic and fermionic coordinates. We will first analyse the BRST and the anti-BRST symmetries of the Chern-Simons theory on this deformed superspace. Then we will analyse the extended BRST and the extended anti-BRST symmetries of this theory in the Batalin-Vilkovisky (BV) formalism. Finally, we will express these extended BRST and extended anti-BRST symmetries in extended superspace formalism by introducing new Grassman…
Proposals are made to describe the Weyl scaling transformation laws of supercovariant derivatives $\nabla{}_{\underline A}$, the torsion supertensors $T{}_{{\underline A} \, {\underline B}}{}^{\underline C}$, and curvature supertensors $R{}_{{\underline A} \, {\underline B}}{}_{\, \underline c} {}^{\underline d}$ in 10D superspaces. Starting from the proposal that an unconstrained supergravity prepotential for the 11D, $\mathcal{N}$ = 1 theory is described by a scalar superfield, considerations for supergravity prepotentials in the 10D theories are enumerated. We derive infinitesimal 10D super…
We use the manifestly N=2 supersymmetric, off-shell, harmonic (or twistor) superspace approach to solve the constraints implied by four-dimensional N=2 superconformal symmetry on the N=2 non-linear sigma-model target space, known as the special hyper-K"ahler geometry. Our general solution is formulated in terms of a homogeneous (of degree two) function of unconstrained (analytic) Fayet-Sohnius hypermultiplet superfields. We also derive the improved (N=2 superconformal) actions for the off-shell (constrained) N=2 projective hypermultiplets, and relate them (via non-conformal deformations) to th…
We construct an action for the superconformal Chern-Simons theory with non-Abelian gauge groups in three-dimensional N=3 projective superspace. We propose a Lagrangian given by the product of the function of the tropical multiplet, that represents the N=3 vector multiplet, and the O(-1,1) multiplet. We show how the tropical multiplet is embedded into the O(-1,1) multiplet by comparing our Lagrangian with the Chern- Simons Lagrangian in the N=2 superspace. We also discuss N=4 generalization of the action.
We investigate the complex of differential forms in curved, six-dimensional, $N=(1,0)$ superspace. The superconformal group acts on this complex by super-Weyl transformations. An ambi-twistor-like representation of a second conformal group arises on a pure spinor subspace of the cotangent space. The $p$-forms are defined by super-Weyl-covariant tensor fields on this pure spinor subspace. The on-shell dynamics of such fields is superconformal. We construct the superspace de Rham complex by successively obstructing the closure of forms. We then extend the analysis to composite forms obtained by …
Superconformal symmetry in six-dimensions is analyzed in terms of coordinate transformations on superspace. A superconformal Killing equation is derived and its solutions are identified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. The full superconformal symmetry, which is shown to form the group OSp(2,6|N), is possible only if the supersymmetry algebra has N spinorial generators of the same chirality, corresponding to (N,0) supersymmetry. The R-symmetry group is then Sp(N) and the corresponding supers…
We construct N=2 affine current algebras for the superalgebras sl(n|n-1)^{(1)} in terms of N=2 supercurrents subjected to nonlinear constraints and discuss the general procedure of the hamiltonian reduction in N=2 superspace at the classical level. We consider in detail the simplest case of N=2 sl(2|1)^{(1)} and show how N=2 superconformal algebra in N=2 superspace follows via the hamiltonian reduction. Applying the hamiltonian reduction to the case of N=2 sl(3|2)^{(1)}, we find two new extended N=2 superconformal algebras in a manifestly supersymmetric N=2 superfield form. Decoupling of four …
Using the worldline SU(2|1) superfield approach, we construct N=4 superconformally invariant actions for the d=1 multiplets (1, 4, 3) and (2, 4, 2). The SU(2|1) superfield framework automatically implies the trigonometric realization of the superconformal symmetry and the harmonic oscillator term in the corresponding component actions. We deal with the general N=4 superconformal algebra D(2,1;$α$) and its central-extended $α$=0 and $α$=-1 psu(1,1|2)$\oplus$su(2) descendants. We capitalize on the observation that D(2,1;$α$) at $α\neq$0 can be treated as a closure of its two su(2|1) subalgebras,…
The superconformal invariants in analytic superspace are found. Superconformal invariance is shown to imply that the Green's functions of analytic operators are invariant holomorphic sections of a line bundle on a product of certain harmonic superspaces. It is argued that the correlation functions for a class of sufficiently low dimension gauge invariant operators in N=2 and N=4 supersymmetric Yang-Mills theory can be evaluated up to constants.
We present a new, alternative interpretation of the vector-tensor multiplet as a 2-form in central charge superspace. This approach provides a geometric description of the (non-trivial) central charge transformations ab initio and is naturally generalized to include couplings of Chern-Simons forms to the antisymmetric tensor gauge field, giving rise to a N=2 supersymmetric version of the Green-Schwarz anomaly cancellation mechanism.
We discuss the extended BRST and anti-BRST symmetry (including shift symmetry) in the Batalin-Vilkovisky (BV) formulation for the Chern-Simons (CS) theories in $(2+1)$ spacetime dimensions. Further we develop the superspace description of BV formulation for such theories. Interestingly, the extended BRST invariant CS theories can be described in superspace in covariant manner with the help of one more (Grassmann) coordinate. However, a superspace with two Grassmann coordinates are required for a manifestly covariant formulation of the extended BRST and extended anti-BRST invariant actions for …
We investigate the concept of superconformal symmetry in six dimensions, applied to the interacting theory of (2,0) tensor multiplets and self-dual strings. The action of a superconformal transformation on the superspace coordinates is found, both from a six-dimensional perspective and by using a superspace with eight bosonic and four fermionic dimensions. The transformation laws for all fields in the theory are derived, as well as general expressions for the transformation of on-shell superfields. Superconformal invariance is shown for the interaction of a self-dual string with a background c…
We propose an off-shell construction of three-dimensional N = 3 and N = 4 superconformal Abelian Chern-Simons theories in the projective superspace formalism. We also construct coupling terms among the gauge fields and matter hypermultiplets.
Recently the quantum hamiltonian reduction was done in the case of general $s\ell(2)$ embeddings into Lie algebras and superalgebras. In this paper we extend the results to the quantum hamiltonian reduction of $N=1$ affine Lie superalgebras in the superspace formalism. We show that if we choose a gauge for the supersymmetry, and consider only certain equivalence classes of fields, then our quantum hamiltonian reduction reduces to quantum hamiltonian reduction of non-supersymmetric Lie superalgebras. We construct explicitly the super energy-momentum tensor, as well as all generators of spin 1 (…
We discuss the problem of infrared divergences in the N=2 superspace approach to classically marginal three-dimensional Chern-Simons-matter theories. Considering the specific case of ABJM theory, we describe the origin of such divergences and offer a prescription to eliminate them by introducing non-trivial gauge-fixing terms in the action. We also comment on the extension of our procedure to higher loop order and to general three-dimensional Chern-Simons-matter models.
We explore the possibilities for constructing Lagrangian descriptions of three-dimensional superconformal classical gauge theories that contain a Chern-Simons term, but no kinetic term, for the gauge fields. Classes of such theories with N = 1 and N = 2 supersymmetry are found. However, interacting theories of this type with N = 8 supersymmetry do not exist.
We search for superspace Chern-Simons-like higher-derivative terms in the low energy effective actions of supersymmetric theories in four dimensions. Superspace Chern-Simons-like terms are those gauge-invariant terms which cannot be written solely in terms of field strength superfields and covariant derivatives, but in which a gauge potential superfield appears explicitly. We find one class of such four-derivative terms with N=2 supersymmetry which, though locally on the Coulomb branch can be written solely in terms of field strengths, globally cannot be. These terms are classified by certain …
The main focus of this lecture is on extended objects in adS*S bosonic backgrounds with unbroken supersymmetry. The backgrounds are argued to be exact, special consideration are given to the non-maximal supersymmetry case. The near horizon superspace construction is explained. The superconformal symmetry appears in the worldvolume actions as the superisometry of the near horizon superspace, like the superPoincare symmetry of GS superstring and BST supermembrane in the flat superspace. The issues in gauge fixing of local kappa-symmetry are reviewed. We describe the features of the gauge-fixed I…
We gauge the abelian hierarchy of tensor fields in 4D by a Lie algebra. The resulting non-abelian tensor hierarchy can be interpreted via an equivariant chain complex. We lift this structure to N=1 superspace by constructing superfield analogs for the tensor fields, along with covariant superfield strengths. Next we construct Chern-Simons actions, for both the bosonic and N=1 cases, and note that the condition of gauge invariance can be presented cohomologically. Finally, we provide an explicit realization of these structures by dimensional reduction, for example by reducing the three-form of …
We demonstrate that the general D=3, N=5 Chern-Simons matter theory possesses a full OSp(5|4) superconformal symmetry, and construct the corresponding superconformal currents. The closure of the superconformal algebra is verified in detail. We also show that the conserved OSp(6|4) superconformal currents in the general N=6 theory can be obtained as special cases of the OSp(5|4) currents by enhancing the R-symmetry of the N=5 theory from USp(4) to SU(4).
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.
We prove explicitly that the general D=3, N=4 Chern-Simons-matter (CSM) theory has a complete OSp(4|4) superconformal symmetry, and construct the corresponding conserved currents. We re-derive the OSp(5|4) superconformal currents in the general N=5 theory as special cases of the OSp(4|4) currents by enhancing the supersymmetry from N=4 to N=5. The closure of the full OSp(4|4) superconformal algebra is verified explicitly.
We consider the map of three-dimensional N=4 superfields to N=3 harmonic superspace. The left and right representations of the N=4 superconformal group are constructed on N=3 analytic superfields. These representations are convenient for the description of N=4 superconformal couplings of the Abelian gauge superfields with hypermultiplets. We analyze the N=4 invariance in the non-Abelian N=3 Yang-Mills theory.
We derive the superconformal transformation properties of the supercurrent for N=1 supersymmetric QED in four dimensions within the superfield formalism. Superconformal Ward identities for Green functions involving insertions of the supercurrent are conveniently derived by coupling the supercurrent to the appropriate prepotential of a classical curved superspace background, and by combining superdiffeomorphisms and super Weyl transformations. We determine all superconformal anomalies of SQED on curved superspace within an all-order perturbative approach and derive a local Callan-Symanzik equat…
Aharony, Bergman, Jafferis and Maldacena have recently proposed a dual gravitational description for a family of superconformal Chern Simons theories in three spacetime dimensions. In this note we perform the one loop computation that determines the field theory superconformal index of this theory and compare with the index computed over the Fock space of dual supersymmetric gravitons. In the appropriate limit (large $N$ and large $k$) we find a perfect match.
In this paper, we will analyse a three dimensional supersymmetric Chern-Simons theory in SIM(1) superspace formalism. The breaking of the Lorentz symmetry down to the SIM(1) symmetry, breaks half the supersymmetry of the Lorentz invariant theory. So, the supersymmetry of the Lorentz invariant Chern-Simons theory with N=1 supersymmetry will break down to N=1/2 supersymmetry, when the Lorentz symmetry is broken down to the SIM(1) symmetry. First, we will write the Chern-Simons action using SIM(1) projections of N=1 superfields. However, as the SIM(1) transformations of these projections are very…
We consider the superspace of D=3, N=5 supersymmetry using SO(5)/U(2) harmonic coordinates. Three analytic N=5 gauge superfields depend on three vector and six harmonic bosonic coordinates and also on six Grassmann coordinates. Decomposition of these superfields in Grassmann and harmonic coordinates yields infinite-dimensional supermultiplets including a three-dimensional gauge Chern-Simons field and auxiliary bosonic and fermionic fields carrying SO(5) vector indices. The superfield action of this theory is invariant with respect to D=3, N=6 conformal supersymmetry realized on N=5 superfields…
For extended $\mathcal{N}\leq 8$ supersymmetry we classify all possible gauge groups for a scalar multiplet allowed by the algebras of global and local supersymmetry in three dimensions. A detailed discussion of supersymmetry enhancement is included. For the corresponding topologically massive gravity with negative cosmological constant the mass of the graviton is determined algebraically as a function of $\mathcal{N}$ and the possible gauge couplings.
We demonstrate that the four and six-point tree-level amplitudes of N=6 superconformal Chern-Simons theory (ABJM) enjoy OSp(6|4) dual superconformal symmetry if one enlarges the dual superspace to include three additional Grassmann-even coordinates which correspond to an abelian isometry of CP3. The inclusion of these coordinates enables us to match the nontrivial dual superconformal generators with level-one Yangian generators when acting on on-shell amplitudes. We also discuss some implications of dual conformal symmetry for loop-level amplitudes.
Some stochastic evolutions of conformal maps can be described by SLE and may be linked to conformal field theory via stochastic differential equations and singular vectors in highest-weight modules of the Virasoro algebra. Here we discuss how this may be extended to superconformal maps of N=1 superspace with links to superconformal field theory and singular vectors of the N=1 superconformal algebra in the Neveu-Schwarz sector.
By coupling {\cal N}=8 superconformal matter to {\cal N}=8 superconformal Chern-Simons gravity in three dimensions we obtain theories with novel terms in the scalar potential leading to AdS_3 solutions and superconformal symmetry breaking. If we start from the theory derived by Bagger, Lambert and Gustavsson, our coupled theory either inherits the SO(4) gauge group or reduces it to SO(3). If the construction is instead based on a free matter theory we find that the gravitational topological gauging also requires the introduction of a Chern-Simons gauge sector resulting in a consistent theory f…
Constraints of the osp(6|4) symmetry on tree-level scattering amplitudes in N=6 superconformal Chern-Simons theory are derived. Supplemented by Feynman diagram calculations, solutions to these constraints, namely the four- and six-point superamplitudes, are presented and shown to be invariant under Yangian symmetry. This introduces integrability into the amplitude sector of the theory.
We construct in detail an N=1, D=4 superspace with the superconformal algebra as the structure group and discuss its relation to prior component approaches and the existing Poincaré superspaces.
In this work we study some properties of the three dimensional $U(N)$ SUSY Chern-Simons coupled to a scalar field in the fundamental representation in the large $N$ limit. For large $N$ we show that the theory has two phases, one which is conformally invariant, and other where the superconformal symmetry is broken and masses for the matter fields are generated.
We develop the background field method for studying classical and quantum aspects of N=3, d=3 Chern-Simons and matter theories in N=3 harmonic superspace. As one of the immediate consequences, we prove a nonrenormalization theorem implying the ultra-violet finiteness of the corresponding supergraph perturbation theory. We also derive the general hypermultiplet and gauge superfield propagators in a Chern-Simons background. The leading supergraphs with two and four external lines are evaluated. In contrast to the non-supersymmetric theory, the leading quantum correction to the massive charged hy…
We clarify some properties of projective superspace by using a manifestly superconformal notation. In particular, we analyze the N=2 scalar multiplet in detail, including its action, and the propagator and its super-Schwinger parameters. The internal symmetry is taken to be noncompact (after Wick rotation), allowing boundary conditions that preserve it off shell. Generalization to N=4 suggests the coset superspace PSU(2,2|4)/OSp(4|4) for the AdS/CFT superstring.
The multiplets that occur in four dimensional rigidly supersymmetric theories can be described either by chiral superfields in Minkowski superspace or analytic superfields in harmonic superspace. The superconformal Ward identities for Green's functions of gauge invariant operators of these types are derived. It is shown that there are no chiral superconformal invariants. It is further shown that the Green's functions of analytic operators are severely restricted by the superconformal Ward when analyticity is taken into account.
Using N=2 superspace techniques we compute the four-loop spectrum of single trace operators in the SU(2) x SU(2) sector of ABJM and ABJ supersymmetric Chern-Simons theories. Our computation yields a four-loop contribution to the function h^2(λ) (and its ABJ generalization) in the magnon dispersion relation which has fixed maximum transcendentality and coincides with the findings in components given in the revised versions of arXiv:0908.2463 and arXiv:0912.3460. We also discuss possible scenarios for an all-loop function h^2(λ) that interpolates between weak and strong couplings.
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 develop superspace techniques to construct general off-shell N=1,2,3,4 superconformal sigma-models in three space-time dimensions. The most general N=3 and N=4 superconformal sigma-models are constructed in terms of N=2 chiral superfields. Several superspace proofs of the folklore statement that N=3 supersymmetry implies N=4 are presented both in the on-shell and off-shell settings. We also elaborate on (super)twistor realisations for (super)manifolds on which the three-dimensional N-extended superconformal groups act transitively and which include Minkowski space as a subspace.
