In three dimensions, there are two types of ${\cal N}=2$ anti-de Sitter (AdS) supersymmetry, which are denoted (1,1) and (2,0). They are characterised by different supercurrents and support different families of higher-spin gauge models (massless and massive) which were constructed in arXiv:1807.09098 and arXiv:1809.00802 for the (1,1) and (2,0) cases, respectively, using superspace techniques. It turns out that the precise difference between the (1,1) and (2,0) higher-spin supermultiplets can be pinned down by reducing these gauge theories to (1,0) AdS superspace. The present paper is devoted to the $(1,1) \to (1,0)$ AdS superspace reduction. In conjunction with the outcomes of the $(2,0) \to (1,0)$ AdS superspace reduction carried out in arXiv:1905.05050, we demonstrate that every known higher-spin theory with (1,1) or (2,0) AdS supersymmetry decomposes into a sum of two off-shell (1,0) supermultiplets which belong to four series of inequivalent higher-spin gauge models. The latter are reduced to components.
We propose supertwistor realisations of $(p,q)$ anti-de Sitter (AdS) superspaces in three dimensions and $\cal N$-extended AdS superspaces in four dimensions. For each superspace, we identify a two-point function that is invariant under the corresponding isometry supergroup. This two-point function is a supersymmetric extension (of a function) of the geodesic distance. We also describe a bi-supertwistor formulation for $\cal N$-extended AdS superspace in four dimensions and harmonic/projective extensions of $(p,q)$ AdS superspaces in three dimensions.
Superspace techniques are used to formulate a supersymmetric model on an AdS3 surface embedded in four dimensions. In this model, the supersymmetry transformation is the "square root" of the transformation generated by the isometry generators of AdS3. Since momentum is not an isometry generator, supersymmetry does not result in equal masses for a Bosonic field and its Fermionic partner. We express this model in terms of coordinates that characterize the AdS3 space. In one coordinate system, it is possible to define a subspace with a Minkowski metric. It becomes possible to infer a model in AdS…
In three dimensions, it is known that field theories possessing extended $(p,q)$ anti-de Sitter (AdS) supersymmetry with ${\cal N}=p+q \geq 3$ can be realised in (2,0) AdS superspace. Here we present a formalism to reduce every field theory with (2,0) AdS supersymmetry to ${\cal N}=1$ AdS superspace. As nontrivial examples, we consider supersymmetric nonlinear sigma models formulated in terms of ${\cal N}=2$ chiral and linear supermultiplets. The $(2,0) \to (1,0)$ AdS reduction technique is then applied to the off-shell massless higher-spin supermultiplets in (2,0) AdS superspace constructed i…
We give an introduction to rigid supersymmetry, supergravity and superspace by considering a quantum mechanical model. We analyze the constraints in superspace in this simplified model, and compare the Hamiltonian and Lagrangian BRST formalism.
We investigate a minimal superspace description for 5D superconformal Killing vectors. The vielbein appropriate for AdS symmetry is discussed within the confines of this minimal supergeometry.
Off-shell $(4,q)$ supermultiplets in 2-dimensions are constructed for $q=1,2,4$. These are used to construct sigma models whose target spaces are hyperkähler with torsion. The off-shell supersymmetry implies the three complex structures are simultaneously integrable and allows us to construct actions using extended superspace and projective superspace, giving an explicit construction of the target space geometries.
We use N=1 superspace to construct the supersymmetric matter couplings of vector and hyper multiplets in a five-dimensional anti-de Sitter spacetime background. For hypermultiplets, we find that AdS_5 supersymmetry requires the scalar fields to lie on a hyper-Kahler manifold endowed with a certain type of holomorphic Killing vector.
In this note, we establish the formulation of 6D, N=1 hypermultiplets in terms of 4D chiral-nonminimal (CNM) scalar multiplets. The coupling of these to 6D, N=1 Yang-Mills multiplets is described. A 6D, N=1 projective superspace formulation is given in which the above multiplets naturally emerge. The covariant superspace quantization of these multiplets is studied in details.
We extend the analysis of hep-th/0408069 on a Lorentz invariant interpretation of noncommutative spacetime to field theories on non-anticommutative superspace with half the supersymmetries broken. By defining a Drinfeld-twisted Hopf superalgebra, it is shown that one can restore twisted supersymmetry and therefore obtain a twisted version of the chiral rings along with certain Ward-Takahashi identities. Moreover, we argue that the representation content of theories on the deformed superspace is identical to that of their undeformed cousins and comment on the consequences of our analysis concer…
In this paper, we will analyse three dimensional supersymmetric Yang-Mills theory coupled to matter fields in $SIM(1)$ superspace formalism. The original theory which is invariant under the full Lorentz group has $\mathcal{N} =1$ supersymmetry. However, when we break the Lorentz symmetry down to $SIM(1)$ group, the $SIM(1)$ superspace will break half the supersymmetry of the original theory. Thus, the resultant theory in $SIM(1)$ superspace will have $\mathcal{N} =1/2$ supersymmetry. This is the first time that $\mathcal{N} =1 $ supersymmetry will be broken down to $\mathcal{N} =1/2$ supersymm…
Non commutative superspaces can be introduced as the Moyal-Weyl quantization of a Poisson bracket for classical superfields. Different deformations are studied corresponding to constant background fields in string theory. Supersymmetric and non supersymmetric deformations can be defined, depending on the differential operators used to define the Poisson bracket. Some examples of deformed, 4 dimensional lagrangians are given. For extended superspace (N>1), some new deformations can be defined, with no analogue in the N=1 case.
We review some recent results on the construction in superspace of 3D N=2 AdS supergravities and on the formulation of rigid supersymmetric theories in (1,1) and (2,0) AdS superspaces.
We study supersymmetry conditions for the heterotic pure superstring preserving and N=1 supersymmetry in four dimensions directly from the curved superspace defined by the Berkovits-Howe constraints.
We develop two N=2 superfield formulations of free equations of motion for the joint model of all D=4 massless higher-superspin fields in generating form. The explicit Osp(2|4) supersymmetry is achieved without exploiting the harmonic superspace, and with adding no auxiliary component fields to those of N=1 superfields. The formulations are developed in two different Osp(2|4) homogeneous superspaces which have a structure of a fibre bundle over the standard D=4 AdS superspace, with dimensions (7|4) and (7|8). The N=2 covariant derivatives in these spaces are expressed in terms of N=1 ones whic…
Thermal superspace is characterized by Grassmann variables which are time-dependent and antiperiodic in imaginary time, with a period given by the inverse temperature. The thermal superspace approach allows to define thermal superfields obeying consistent boundary conditions and to formulate a ``super-KMS'' condition for superfield propagators. Upon constructing thermal covariantizations of the superspace derivative operators, we define thermal covariant derivatives and provide a definition of thermal chiral and antichiral superfields. Thermal covariantizations of the generators of the super-P…
We found the pre-potential in the superspace with $AdS_{5}\,\times\,S^{5}$ background. The pre-potential appears as part of the vielbeins, without derivatives. The space-cone gauge destroys the bulk Lorentz covariance, but still preserves boundary Lorentz covariance (projective superspace) $SO\,(\,3,\,1\,)\,\otimes\,SO\,(\,4\,)$, i.e. symmetries of boundary CFT are manifest.
We investigate SUSY of Wess-Zumino models in non(anti-)commutative Euclidean superspaces. Non(anti-)commutative deformations break 1/2 SUSY, then non(anti-)commutative Wess-Zumino models do not have full SUSY in general. However, we can recover full SUSY at specific coupling constants satisfying some relations. We give a general way to construct full SUSY non(anti-)commutative Wess-Zumino models. For a some example, we investigate quantum corrections and $β$-functions behavior.
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 relate the 10D IIB superstring (primarily free supergravity) expanded about AdS_5\timesS^5 to 4D N=4 Yang-Mills directly in superspace. The sphere becomes the CFT internal space of projective superspace, plus a coordinate counting the number of supergluons. The 32 fermionic coordinates of AdS become the 8 of the CFT in a spacecone gauge that manifestly preserves SO(3,1).SO(4). The PSU(2,2|4) algebra of AdS becomes the projective representation (plus the extra coordinate) on the boundary. (A review of projective superspace and the projective lightcone limit is included for self-containment.)
This paper presents a detailed discussion of the issue of supergravity perturbations around the flat five dimensional superspace required for manifest superspace formulations of the supergravity side of the AdS_{5}/CFT_{4} Correspondence.
The coupling of matter to supergravity with $N=1$ supersymmetry in $d=4$ dimensions is described in a geometric manner by Kähler superspace. A straightforward way to implement Kähler superspace is via $\mathrm{U}(1)$ superspace by identifying the $\mathrm{U}(1)$ pre-potential with the Kähler potential, which is a function of the matter (chiral) superfields. In this framework, the components of the supergravity multiplet are contained in the supervielbein and torsion tensor of superspace. Furthermore, interactions with the Yang-Mills (vector) multiplet are formulated by introducing a connection…
We consider a deformed superspace in which the coordinates θdo not anticommute, but satisfy a Clifford algebra. We present results on the properties of N=1/2 supersymmetric theories of chiral superfields in deformed superspace, taking the Wess-Zumino model as the prototype. We prove new (non)renormalization theorems: the F-term is radiatively corrected and becomes indistinguishable from the D-term, while the Fbar-term is not renormalized. Supersymmetric vacua are critical points of the antiholomorphic superpotential. The vacuum energy is zero to all orders in perturbation theory. We illustrate…
In our recent paper arXiv:0704.1185 [hep-th], we presented the differential geometry of five-dimensional N=1 anti-de Sitter superspace AdS^{5|8}=SU(2,2|1)/SO(4,1) x U(1) and developed the harmonic and the projective superspace settings to formulate off-shell supersymmetric theories in AdS^{5|8}. Here we give a brief review of the geometry and the harmonic superspace construction, and also elaborate on vector supermultiplets.
We examine spontaneously broken N=2 supersymmetry in four dimensions and associate a spinor superfield with each Goldstino via a finite supersymmetry transformation with parameters that are the Grassmann coordinates of N=2 superspace. Making use of a special choice of coset parametrization allows us to develop a version of nonlinearly realized N=2 supersymmetry for which the associated Goldstino superfields are defined on harmonic superspace, thereby providing a natural mechanism for construction of a Goldstino action. The corresponding superfield Lagrangian is an O(4) multiplet. This property…
On-shell supergravity vertex operators in an $AdS_5\times S^5$ background are described in the pure spinor formalism by the zero mode cohomology of a BRST operator. After expanding the pure spinor BRST operator in terms of the $AdS_5$ radius variable, this cohomology is computed using N=4 harmonic superspace variables and explicit superfield expressions are obtained for the behavior of supergravity vertex operators near the boundary of $AdS_5$.
We review the superspace technique to determine supersymmetric spacetimes in the framework of off-shell formulations for supergravity in diverse dimensions using the case of 3D N=2 supergravity theories as an illustrative example. This geometric formalism has several advantages over other approaches advocated in the last four years. Firstly, the infinitesimal isometry transformations of a given curved superspace form, by construction, a finite-dimensional Lie superalgebra, with its odd part corresponding to the rigid supersymmetry transformations. Secondly, the generalised Killing spinor equat…
We find the superisometry of the near-horizon superspace, forming the superconformal algebra. We present here the explicit form of the transformation of the bosonic and fermionic coordinates (as well as the compensating Lorentz-type transformation) which keeps the geometry invariant. We comment on i) the BPS condition of the branes in adS background and ii) significant simplification of superisometries of the gauge-fixed Green-Schwarz action in adS(5)*S(5) background.
We present a covariant, supersymmetric and kappa-symmetric action for non-Abelian multiwave system (nAmW) in D=3 AdS superspace. Its flat superspace limit provides a simplest counterpart of the recently proposed action for 11 dimensional system of N nearly coincident M-waves (multiple M0-branes), which is presently known for the case of flat target superspace only.
Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ an…
Three inequivalent real forms of the complex N=4 supersymmetric Toda chain hierarchy (Nucl. Phys. B558 (1999) 545, solv-int/9907004) in the real N=2 superspace with one even and two odd real coordinates are presented. It is demonstrated that the first of them possesses a global N=4 supersymmetry, while the other two admit a twisted N=4 supersymmetry. A new superfield basis in which supersymmetry transformations are local is discussed and a manifest N=4 supersymmetric representation of the N=4 Toda chain in terms of a chiral and an anti-chiral N=4 superfield is constructed. Its relation to the …
We review the spinor moving frame formulations and generalized action principle for super-p-branes, describe in detail the superembedding approach to superstring in general type IIB supergravity background and present the complete superembedding description of type IIB superstring in the AdS(5)xS*5 superspace.
As a step towards formulating projective superspace techniques for supergravity theories with eight supercharges, this work is devoted to field theory in five-dimensional N = 1 anti-de Sitter superspace AdS^{5|8} = SU(2,2|1)/SO(4,1) x U(1) which is a maximally symmetric curved background. We develop the differential geometry of AdS^{5|8} and describe its isometries in terms of Killing supervectors. Various off-shell supermultiplets in AdS^{5|8} x S^2 are defined, and supersymmetric actions are constructed both in harmonic and projective superspace approaches. Several families of supersymmetric…
We study the boundary limit of the bulk isometries of AdS x S. The superconformal symmetry is realized on the coordinates of the AdS boundary, the fermionic superspace coordinates, and the harmonics on the sphere. We show how these may be related to the coordinates of an off-shell harmonic superspace of SCFT living on the boundary. In the special case of d=4, N=2 super Yang-Mills theory, a truncation of the N=4 SYM dual to Type IIB string theory compactified on AdS_5 x S^5, we identify the bosonic space SU(2)/U(1) of the N=2 harmonic superspace (known before as auxiliary) with the S^2 submanif…
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 study four-point functions of spinning operators in the flavor current multiplet in four dimensional $\mathcal{N}=2$ SCFTs, using superspace techniques. In particular we explicitly construct the differential operators relating the different components of the super-correlator. As a byproduct of our analysis, we report the computation of the four-point amplitudes of gluons in bosonic Yang-Mills theories on $\mathrm{AdS}_5$ and we give evidence of an AdS double copy relation between the gluon amplitude and its gravitational counterpart.
In this paper we will discuss non-anticommutative deformations of the harmonic superspace. We will analyse the non-anticommutative deformation of the superspace that break the supersymmetry from $\mathcal{N} =3$ supersymmetry to $\mathcal{N} =2$ supersymmetry. We will then study the ABJ theory in this non-anticommutative superspace. This deformed ABJ theory will be shown to posses $\mathcal{N} =5$ supersymmetry.
We provide a manifestly N=2 supersymmetric formulation of the N=2 U(N_c) gauge model constructed in terms of N=1 superfields in hep-th/0409060. The model is composed of N=2 vector multiplets in harmonic superspace and can be viewed as the N=2 U(N_c) Yang-Mills effective action equipped with the electric and magnetic Fayet-Iliopoulos terms. We generalize this gauge model to an N=2 U(N_c) QCD model by introducing N=2 hypermultiplets in harmonic superspace which include both the fundamental representation of U(N_c) and the adjoint representation of U(N_c). The effect of the magnetic Fayet-Iliopou…
We present an explicit formulation of supersymmetric Yang-Mills theories from $\D=$ 5 to 10 dimensions in the familiar $\N=1,\D=4$ superspace. This provides the rules for globally supersymmetric model building with extra dimensions and in particular allows us to simply write down $\N=1$ SUSY preserving interactions between bulk fields and fields localized on branes. We present a few applications of the formalism by way of illustration, including supersymmetric ``shining'' of bulk fields, orbifolds and localization of chiral fermions, anomaly inflow and super-Chern-Simons theories.
We propose a new formulation which realizes exact twisted supersymmetry for all the supercharges on a lattice by twisted superspace formalism. We show explicit examples of N=2 twisted supersymmetry invariant BF and Wess-Zumino models in two dimensions. We introduce mild lattice noncommutativity to preserve Leibniz rule on the lattice. The formulation is based on the twisted superspace formalism for N=D=2 supersymmetry which was proposed recently. From the consistency condition of the noncommutativity of superspace, we find an unexpected three-dimensional lattice structure which may reduce into…
We present the (0,4) superspace version of Witten's sigma model construction for ADHM instantons. We use the harmonic superspace formalism, which exploits the three complex structures common to both (0,4) supersymmetry and self-dual Yang-Mills theory. A novel feature of the superspace formulation is the manifest interplay between the ADHM construction and its twistor counterpart.
We introduce simple variables for describing the AdS$_5\times S^5$ superspace, i. e. $\frac{PSU(2,2|4)}{SO(4,1)\times SO(5)}$. The idea is to embed the coset superspace into a space described by variables which are in linear (ray) representations of the supergroup $PSU(2,2|4)$ by imposing certain supersymmetric quadratic constraints (up to two overall U(1) factors). The construction can be considered as a supersymmetric generalisation of the elementary realisations of the $AdS_5$ and the $S^5$ spaces by the SO(4,2) and SO(6) invariant quadratic constraints on two six-dimensional flat spaces.
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…
The supersymmetric version of a topological quantum field theory describing flat connections, the super BF-theory, is studied in the superspace formalism. A set of observables related to topological invariants is derived from the curvature of the superspace. Analogously to the non-supersymmetric versions, the theory exhibits a vector-like supersymmetry. The role of the vector supersymmetry and an additional new symmetry of the action in the construction of observables is explained.
There exist two superspace approaches to describe N=2 supersymmetric nonlinear sigma models in four-dimensional anti-de Sitter (AdS_4) space: (i) in terms of N=1 AdS chiral superfields, as developed in arXiv:1105.3111 and arXiv:1108.5290; and (ii) in terms of N=2 polar supermultiplets using the AdS projective-superspace techniques developed in arXiv:0807.3368. The virtue of the approach (i) is that it makes manifest the geometric properties of the N=2 supersymmetric sigma-models in AdS_4. The target space must be a non-compact hyperkahler manifold endowed with a Killing vector field which gene…
A discussion of the AdS/CFT correspondence in IIB is given in a superspace context. The main emphasis is on the properties of SCFT correlators on the boundary which are studied using harmonic superspace techniques. These techniques provide the easiest way of implementing the superconformal Ward identities. The Ward identities, together with analyticity, can be used to give a compelling argument in support of the non-renormalisation theorems for two- and three-point functions, and to establish the triviality of extremal and next-to-extremal correlation functions. The OPE in is also briefly disc…
The N=1 superspace generalization of the 3-brane action in 6 dimensions with partially broken N=2 supersymmetry can be constructed using N=1 chiral, complex linear, or real linear superfields. The physical scalars of these multiplets give equivalent descriptions of the two transverse coordinates. The second supersymmetry is realized nonlinearly in all these actions. We derive the superspace brane actions and their nonlinear supersymmetry for both kinds of linear superfields when we break N=2 supersymmetry spontaneously. This breaking is realized in the free action of hypermultiplets that live …
By using symplectic Majorana spinors as Grassmann coordinates in a superspace associated with the supersymmetric extension of the isometry group on the spherical surface S_2, it proves possible to formulate supersymmetric models on S_2 using superspace techniques.
We study four dimensional supersymmetric gauge theory on the noncommutative superspace, recently proposed by Seiberg. We construct the gauge-invariant action of N=1 super Yang-Mills theory with chiral and antichiral superfields, which has N=1/2 supersymmetry on the noncommutative superspace. We also construct the action of N=2 super Yang-Mills theory. It is shown that this theory has only N=1/2 supersymmetry.
