Harmonic Superspace Research Articles

N=4, 3D supersymmetric quantum mechanics in a non-Abelian monopole background

Using the harmonic superspace approach, we construct the three-dimensional N=4 supersymmetric quantum mechanics of the supermultiplet (3,4,1) coupled to an external SU(2) gauge field. The off-shell N=4 supersymmetry requires the gauge field to be a static form of the 't Hooft ansatz for the 4D self-dual SU(2) gauge fields, that is a particular solution of Bogomolny equations for BPS monopoles. We present the explicit form of the corresponding superfield and component actions, as well as of the quantum Hamiltonian and N=4 supercharges. The latter can be used to describe a more general N=4 mechanics system, with an arbitrary BPS monopole background and on-shell N=4 supersymmetry. The essential feature of our construction is the use of semi-dynamical spin (4,4,0) multiplet with the Wess-Zumino type action.

Three-dimensional N=4 supersymmetry in harmonic N=3 superspace

We consider the map of three-dimensional N=4 superfields to the 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 describing N=4 superconformal couplings of Abelian gauge superfields to hypermultiplets. We investigate the N=4 invariance in the non-Abelian N=3 Yang-Mills theory.

SQM with non-abelian self-dual fields: harmonic superspace description

We present a Lagrangian formulation for N=4 supersymmetric quantum-mechanical systems describing the motion in external non-Abelian self-dual gauge fields. For any such system, one can write a component supersymmetric Lagrangian by introducing extra bosonic variables with topological Chern-Simons type interaction. For a special class of such system when the fields are expressed in the `t Hooft ansatz form, it is possible to give a superfield description using harmonic superspace formalism. As a new explicit example, the N=4 mechanics with Yang monopole is constructed.

Self-duality and supersymmetry

We observe that the Hamiltonian H=D̸2, where D̸ is the flat 4d Dirac operator in a self-dual gauge background, is supersymmetric, admitting 4 different real supercharges. A generalization of this model to the motion on a curved conformally flat 4d manifold exists. For an Abelian self-dual background, the corresponding Lagrangian can be derived from known harmonic superspace expressions.

Three-dimensional % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8WrFz0dbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0dd9qqFj0dXdbba91qpepGe9FjuP0-is % 0dXdbba9pGe9xq-Jbba9suk9fr-xfr-xfrpeWZqaaeaabiGaciaaca % qabeaadaabauaaaOqaamrr1ngBPrwtHrhAXaqehuuDJXwAKbstHrhA % G8KBLbacgaGae8xdX7eaaa!4B62! $$ \mathcal{N} $$ =4 superconformal superfield theories

The mirror map in the D=3, N=4 supersymmetry connects the left and right SU(2) automorphism groups and also the superfield representations of the corresponding N=4 supermultiplets. The mirror N=4 harmonic superspaces use the harmonics of two SU(2) groups and two types of the Grassmann analyticity. The irreducible left and right N=4 supermultiplets are defined in these harmonic superspaces. We analyze the N=4 superconformal interactions of the gauge and matter superfields and the spontaneous breakdown of the superconformal symmetry. The most interesting superconformal action possesses the mirror symmetry and contains two nonlinear terms of the abelian left and right gauge superfields, and also the mixing N=4 BF interaction which yields the topological masses of the gauge fields and the nontrivial interaction of the scalar and pseudoscalar fields. The minimal interactions of the left and right N=4 hypermultiplets can be included to this abelian gauge theory. We consider also the nonlinear N=4 gauge superfield interactions.

Quantum 𝒩 = 3,d= 3 Chern-Simons matter theories in harmonic superspace

We develop the background field method for studying classical and quantum aspects of = 3, d = 3 Chern-Simons and matter theories in = 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 hypermultiplet proves to be the super Yang-Mills action rather than the Chern-Simons one. The hypermultiplet mass is induced by a constant triplet of central charges in the = 3, d = 3 Poincaré superalgebra.

Supersymmetry in superspace: 35 years of the research activity in LTP

This is an overview of the all-years research activity in supersymmetry in the Markov-Ogievetsky group at the Bogoliubov Laboratory of Theoretical Physics. It started shortly after the discovery of supersymmetry and is lasting for more than 30 years. In this survey the main emphasis is made on the superspace geometric approaches and unconstrained superfield formulations. Alongside such milestones as the geometric formulation of supergravity and the harmonic superspace approach to extended supersymmetry, some other developments largely contributed by the Dubna theorists are briefly accounted for.

ABJM models in 𝒩 = 3 harmonic superspace

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 sextic scalar potential of ABJM emerges after the elimination of auxiliary fields. Besides the original U(N)xU(N) ABJM model, we also construct N=3 superfield formulations of some generalizations. For the SU(2)xSU(2) case we give a simple superfield proof of its enhanced N=8 supersymmetry and SO(8) R-symmetry.

The ultra-violet question in maximally supersymmetric field theories

We discuss various approaches to the problem of determining which supersymmetric invariants are permitted as counterterms in maximally supersymmetric super Yang--Mills and supergravity theories in various dimensions. We review the superspace non-renormalisation theorems based on conventional, light-cone, harmonic and certain non-Lorentz covariant superspaces, and we write down explicitly the relevant invariants. While the first two types of superspace admit the possibility of one-half BPS counterterms, of the form $F^4$ and $R^4$ respectively, the last two do not. This suggests that UV divergences begin with one-quarter BPS counterterms, i.e. $d^2 F^4$ and $d^4 R^4$, and this is supported by an entirely different approach based on algebraic renormalisation. The algebraic formalism is discussed for non-renormalisable theories and it is shown how the allowable supersymmetric counterterms can be determined via cohomological methods. These results are in agreement with all the explicit computations that have been carried out to date. In particular, they suggest that maximal supergravity is likely to diverge at four loops in D=5 and at five loops in D=4, unless other infinity suppression mechanisms not involving supersymmetry or gauge invariance are at work.

One-loop effective action in the % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYt % UvgaiyGacqWFneVtaaa!4736! $$ \mathcal{N} $$ =2 supersymmetric massive Yang-Mills field theory

We consider the \( \mathcal{N} \)=2 supersymmetric massive Yang-Mills field theory formulated in the \( \mathcal{N} \)=2 harmonic superspace. We present various gauge-invariant forms of writing the mass term in the action (in particular, using the Stueckelberg superfield), which result in dual formulations of the theory. We develop a gaugeinvariant and explicitly supersymmetric scheme of the loop expansion of the superfield effective action beyond the mass shell. In the framework of this scheme, we calculate gauge-invariant and explicitly \( \mathcal{N} \)=2 supersymmetric one-loop counterterms including new counterterms depending on the Stueckelberg superfield. We analyze the component structure of one of these counterterms.

Gauge theory in deformed % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvga % iyaacqWFneVtaaa!4628! $$ \mathcal{N} $$ = (1, 1) superspace

We review the non-anticommutative Q-deformations of $$ \mathcal{N} $$ = (1, 1) supersymmetric theories in four-dimensional Euclidean harmonic superspace. These deformations preserve chirality and harmonic Grassmann analyticity. The associated field theories arise as a low-energy limit of string theory in specific backgrounds and generalize the Moyal-deformed supersymmetric field theories. A characteristic feature of the Q-deformed theories is the half-breaking of supersymmetry in the chiral sector of the Euclidean superspace. Our main focus is on the chiral singlet Q-deformation, which is distinguished by preserving the SO(4) ∼ Spin(4) “Lorentz” symmetry and the SU(2) R-symmetry. We present the superfield and component structures of the deformed $$ \mathcal{N} $$ = (1, 0) supersymmetric gauge theory as well as of hypermultiplets coupled to a gauge superfield: invariant actions, deformed transformation rules, and so on. We discuss quantum aspects of these models and prove their renormalizability in the Abelian case. For the charged hypermultiplet in an Abelian gauge superfield background we construct the deformed holomorphic effective action.

Formula omitted] superparticle and super-Yang–Mills theory in [formula omitted] harmonic superspace

We study the N = 4 harmonic superparticle model, both with and without central charge and quantize it. Since the central charge breaks the U ( 4 ) R-symmetry group of the N = 4 superalgebra down to USp ( 4 ) , we consider the superparticle dynamics in N = 4 harmonic superspace with USp ( 4 ) / ( U ( 1 ) × U ( 1 ) ) harmonic variables. We show that the quantization of a massive superparticle with central charge leads to a superfield realization of the N = 4 massive vector multiplet in N = 4 harmonic superspace. In the massless case without central charge the superparticle quantization reproduces three different multiplets: the N = 4 SYM multiplet, the N = 4 gravitino multiplet and N = 4 supergravity multiplet. The SYM multiplet is described by six analytic superfield strengths with different types of analyticity. We show that these strengths solve the N = 4 SYM constraints and can be used for the construction of actions in N = 4 harmonic superspace.

Formula omitted] superparticle model

We consider the formulation and quantization of the N = 3 superparticle model, both with and without central charge. Without the central charge the action possesses U ( 3 ) invariance and therefore is naturally quantized in the N = 3 harmonic superspace. The quantization reproduces the N = 3 supergauge strength multiplets, described by analytic N = 3 superfields and a gravitino multiplet as a constrained N = 3 chiral superfield. When the central charge is present, it breaks the U ( 3 ) R-symmetry of N = 3 superalgebra down to SU ( 2 ) × U ( 1 ) , and the corresponding superparticle model is formulated in the N = 2 harmonic superspace extended by a pair of extra Grassmann variables. The quantization of such a model leads to the massive BPS N = 3 vector multiplet. It is shown that upon additional superfield constraints such multiplet reduces to the massive N = 2 vector multiplet.

On black hole effective potential in 6D/7D [formula omitted] supergravity

Using the harmonic superspace method and the duality between real and complex representations of hypermultiplets, we compute the explicit scalar field expression of the quaternionic metric G m n ( φ ) of the moduli space SO ( 4 , k ) × SO ( 1 , 1 ) SO ( 4 ) × SO ( k ) of 6D N = 2 supergravity with generic k Maxwell supermultiplets. The obtained metric includes the particular case k = 20 associated with 10D type IIA superstring on K3. Uplifting to 7D N = 2 supergravity is described and aspects of 6D/7D black attractor effective potentials are studied.

Chern–Simons [formula omitted], [formula omitted] superfield theory

We construct the D=3,N=5 harmonic superspace using the SO(5)/U(1)×U(1) harmonics. Three gauge harmonic superfields satisfy the off-shell constraints of the Grassmann and harmonic analyticities. The corresponding component supermultiplet contains the gauge field Am and an infinite number of bosonic and fermionic fields with the SO(5) vector indices arising from decompositions of gauge superfields in harmonics and Grassmann coordinates. The non-Abelian superfield Chern–Simons action is invariant with respect to the N=6 superconformal supersymmetry realized on the N=5 superfields. The component Lagrangian contains the Chern–Simons interaction of Am and an infinite number of bilinear and trilinear interactions of auxiliary fields. The fermionic and bosonic auxiliary fields from the infinite N=5 multiplet vanish on-shell.

Harmonicity in [formula omitted] supersymmetry and its quantum anomaly

The holomorphicity property of N = 1 superpotentials or of N = 2 F-terms involving vector multiplets is generalized to the case of N = 4 1/2-BPS effective operators defined in harmonic superspace. The resulting harmonicity equations are shown to control the moduli dependence of the couplings of higher-dimensional operators involving powers of the N = 4 Weyl superfield, computed by N = 4 topological amplitudes. These equations can also be derived on the string side, exhibiting an anomaly from world-sheet boundary contributions that leads to recursion relations for the non-analytic part of the couplings.

The common origin of linear and nonlinear chiral multiplets in [formula omitted] mechanics

Elaborating on previous work [F. Delduc, E. Ivanov, Nucl. Phys. B 753 (2006) 211, hep-th/0605211; F. Delduc, E. Ivanov, Nucl. Phys. B 770 (2007) 179, hep-th/0611247], we show how the linear and nonlinear chiral multiplets of N = 4 supersymmetric mechanics with the off-shell content ( 2 , 4 , 2 ) can be obtained by gauging three distinct two-parameter isometries of the “root” ( 4 , 4 , 0 ) multiplet actions. In particular, two different gauge groups, one Abelian and one non-Abelian, lead, albeit in a disguised form in the second case, to the same (unique) nonlinear chiral multiplet. This provides an evidence that no other nonlinear chiral N = 4 multiplets exist. General sigma model type actions are discussed, together with the restricted potential terms coming from the Fayet–Iliopoulos terms associated with Abelian gauge superfields. As in our previous work, we use the manifestly supersymmetric language of N = 4 , d = 1 harmonic superspace. A novel point is the necessity to use in parallel the λ and τ gauge frames, with the “bridges” between these two frames playing a crucial role. It is the N = 4 harmonic analyticity which, though being non-manifest in the τ frame, gives rise to both linear and nonlinear chirality constraints.

Five-dimensional [formula omitted] AdS superspace: Geometry, off-shell multiplets and dynamics

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 ) × 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 × S 2 are defined, and supersymmetric actions are constructed both in harmonic and projective superspace approaches. Several families of supersymmetric theories are presented including nonlinear sigma-models, Chern–Simons theories and vector–tensor dynamical systems. Using a suitable coset representative, we make use of the coset construction to develop an explicit realization for one half of the superspace AdS 5 | 8 as a trivial fiber bundle with fibers isomorphic to four-dimensional Minkowski superspace.

Harmonic superspace, minimal unitary representations and quasiconformal groups

We show that there is a remarkable connection between the harmonic superspace (HSS) formulation of N=2, d=4 supersymmetric quaternionic Kaehler sigma models that couple to N=2 supergravity and the minimal unitary representations of their isometry groups. In particular, for N=2 sigma models with quaternionic symmetric target spaces of the form G/HXSU(2) we establish a one-to-one mapping between the Killing potentials that generate the isometry group G under Poisson brackets in the HSS formulation and the generators of the minimal unitary representation of G obtained by quantization of its geometric realization as a quasiconformal group. Quasiconformal extensions of U-duality groups of four dimensional N=2, d=4 Maxwell-Einstein supergravity theories (MESGT) had been proposed as spectrum generating symmetry groups earlier. We discuss some of the implications of our results, in particular, for the BPS black hole spectra of 4d, N=2 MESGTs.

Gauging [formula omitted] supersymmetric mechanics II: [formula omitted] models from the [formula omitted] ones

Exploiting the gauging procedure developed by us in hep-th/0605211, we study the relationships between the models of N = 4 mechanics based on the off-shell multiplets ( 4 , 4 , 0 ) and ( 1 , 4 , 3 ) . We make use of the off-shell N = 4 , d = 1 harmonic superspace approach as most adequate for treating this circle of problems. We show that the most general sigma-model type superfield action of the multiplet ( 1 , 4 , 3 ) can be obtained in a few non-equivalent ways from the ( 4 , 4 , 0 ) actions invariant under certain three-parameter symmetries, through gauging the latter by the appropriate non-propagating gauge multiplets. We discuss in detail the gauging of both the Pauli–Gürsey SU ( 2 ) symmetry and the Abelian three-generator shift symmetry. We reveal the ( 4 , 4 , 0 ) origin of the known mechanisms of generating potential terms for the multiplet ( 1 , 4 , 3 ) , as well as of its superconformal properties. A new description of this multiplet in terms of unconstrained harmonic analytic gauge superfield is proposed. It suggests, in particular, a novel mechanism of generating the ( 1 , 4 , 3 ) potential terms via coupling to the fermionic off-shell N = 4 multiplet ( 0 , 4 , 4 ) .