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Affleck-Dine inflation is a recently proposed model in which a single complex scalar field, nonminimally coupled to gravity, drives inflation and simultaneously generates the baryon asymmetry of universe via Affleck-Dine mechanism. In this paper we investigate the supersymmetric implementation of Affleck-Dine inflation in the use of two chiral superfields with appropriate superpotential and K\"ahler potential. The scalar potential has a similar form to the potential of original Affleck-Dine inflation, and it gives successful inflation and baryogenesis. We also consider the isocurvature perturbation evolving after crossing the horizon, and find that it is ignorable and hence consistent with the observations., Comment: 14 pages, 5 figures

We revisit the novel symmetries in $\mathcal{N}$ = 2 supersymmetric (SUSY) quantum mechanical (QM) models by considering specific examples of coupled systems. Further, we extend our analysis to a general case and list out all the novel symmetries. In each case, we show the existence of two sets of discrete symmetries that correspond to the Hodge duality operator of differential geometry. Thus, we are able to provide a proof of the conjecture which points out the existence of more than one set of discrete symmetry transformations corresponding to the Hodge duality operator. Moreover, we derive on-shell nilpotent symmetries for a generalized superpotential within the framework of supervariable approach., 20 pages, typos fixed, text modified, references added, version to appear in CTP

We present the supersymmetric completion of the auxiliary vector modified polynomial $f(R)$ theories in their dual scalar-tensor theory formulation that interpolate between the auxiliary vector modified polynomial $f(R)$ theories and chaotic inflation with the power-law potential $V(\phi) \propto \phi^p$. The supersymmetrization is achieved in two steps: First, we introduce a superconformal theory for three chiral multiplets by choosing a conformal K\"ahler potential and a conformal superpotential. In the second step, we use one of the chiral multiplets to compensate for the superconformal symmetries and achieve the K\"ahler potential and the superpotential while the other two are used to realize inflation with a stable inflationary trajectory. The stability of the inflationary trajectory requires certain deformations to the K\"ahler potential which we discuss their compatibility against the inflationary observables from the latest Planck data., Comment: v3., title changed, various clarifications added, typos fixed, 8 pages, 2 figures, version appeared in JCAP

We establish the existence of a quasi-Hopf algebraic structure underlying the Leigh-Strassler N=1 superconformal marginal deformations of the N=4 Super-Yang-Mills theory. The scalar-sector R-matrix of these theories, which is related to their one-loop spin chain Hamiltonian, does not generically satisfy the Quantum Yang-Baxter Equation. By constructing a Drinfeld twist which relates this R-matrix to that of the N=4 SYM theory, but also produces a non-trivial co-associator, we show that the generic Leigh-Strassler R-matrix satisfies the quasi-Hopf version of the QYBE. We also use the twist to define a suitable star product which directly relates the N=4 SYM superpotential to that of the marginally deformed gauge theories. We expect our results to be relevant to studies of integrability (and its breaking) in these theories, as well as to provide useful input for supergravity solution-generating techniques., Comment: 38 pages, 2 figures, Mathematica notebook submitted. v2: Typos fixed, references added

We consider primordial black hole (PBH) production in inflationary $\alpha$-attractors. We discuss two classes of models, namely models with a minimal polynomial superpotential as well as modulated chaotic ones that admit PBHs. We find that a significant amplification of the curvature power spectrum ${\cal P_R}$ can be realized in this class of models with a moderate tuning of the potential parameters. We consistently examine the PBH formation during radiation and additionally during reheating eras where the background pressure is negligible. It is shown that basic features of the curvature power spectrum are explicitly related with the postinflationary cosmic evolution and that the PBH mass and abundance expressions are accordingly modified. PBHs in the mass range $10^{-16}-10^{-14} \, M_{\odot}$ can form with a cosmologically relevant abundance for a power spectrum peak ${\cal P_R} \sim 10^{-2}$ and large reheating temperature and, furthermore, for a moderate peak ${\cal P_R} \sim 10^{-5}$ and reheating temperature $T_\text{rh}\sim 10^7$ GeV, characteristic of the position of the power spectrum peak. Regarding the CMB observables, the $\alpha$-attractor models utilized here to generate PBH in the low-mass region predict in general a smaller $n_s$ and larger $r$ and $\alpha_s$ parameter values compared to the conventional inflationary $\alpha$-attractor models., Comment: 34 pages, 12 figures with 33 plots. Minor corrections. References added. In this version there are updated parameters for our inflationary models in accordance with the Planck 2018 results. Journal version

We investigate a new realization of extended quantum-mechanical supersymmetry. We first show that an $\mathcal{N}=2$ quantum-mechanical supersymmetry is hidden in the four-dimensional (4D) spectrum of the Kaluza-Klein decomposition for the higher dimensional Dirac field, that is, Kaluza-Klein mode functions of 4D right-handed spinors and 4D left-handed ones form $\mathcal{N}=2$ supermultiplets. In addition to $\mathcal{N}=2$ supersymmetry, we discover that an $\mathcal{N}$-extended supersymmetry ($\mathcal{N} = d+2\ (d+1)$ for $d=$ even (odd) extra dimensions) is further hidden in the 4D spectrum. The extended symmetry can explain additional degeneracy of the spectrum. Furthermore, we show that a superpotential can be introduced into the $\mathcal{N}$-extended supercharges and clarify the condition to preserve the supersymmetry. The partial breaking of the supersymmetry is also demonstrated., Comment: 11 pages, 3 figures

The factorisation method commonly used in linear supersymmetric quantum mechanics is extended, such that it can be applied to nonlinear quantum mechanical systems. The new method is distinguishable from the linear formalism, as the superpotential is forced to become eigenfunction-dependent. An example solution is given for the nonlinear Schroedinger equation and its supersymmetric partner equation. This method allows new nonlinear evolution equations to be constructed from the solutions of known nonlinear equations, and has the potential to be a useful tool for mathematicians and physicists working in the field of nonlinear systems, allowing the discovery of previously unknown `dualities' amongst soliton solutions and their respective equations., Comment: Some minor imprecisions were corrected, new figures, improved discussion of Lax pairs for the scale-free equation

We find that in a supersymmetric quantum mechanics (SUSY QM) system, in addition to supersymmetric algebra, an associated SU(2) algebra can be obtained by using semiunitary (SUT) operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z-axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SUτ (2) ⊗ SUσ (2) symmetry in the context of the relativistic Pauli Hamiltonian squared. We show that there exists also an SU(2) symmetry associated with the supersymmetry of the Dirac particle.

We find the most general tetrads which give a regular charged spacetime in tetrad theory of gravitation. The metric is a static one and it includes the Schwarzschild and Reissner Nordstrom black holes. The energy content contained in a sphere of radius R is calculated using the superpotential given by Moller in the context of Weitzenbock spacetime.

We describe the intersecting brane configuration of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua in four dimensional N=1 supersymmetric SU(N_c) gauge theory with an antisymmetric flavor, a conjugate symmetric flavor, eight fundamental flavors, m_f fundamental flavors and m_f antifundamental flavors. This is done by analyzing the N=1 supersymmetric SU(2m_f-N_c+4) magnetic gauge theory with dual matters and the corresponding dual superpotential., Comment: 20 pp, 3 figures; Pages 11,12, and 14 improved; to appear in CQG

Minimal supersymmetric hybrid inflation based on a minimal Kahler potential predicts a spectral index n s ≥ 0.98. On the other hand, WMAP three-year data prefer a central value n s ≈ 0.95. We propose a class of supersymmetric hybrid inflation models based on the same minimal superpotential but with a non-minimal Kahler potential. Including radiative corrections using the one-loop effective potential, we show that the prediction for the spectral index is sensitive to the small non-minimal corrections, and can lead to a significantly red-tilted spectrum, in agreement with WMAP.

We comment on several incorrect results given in a recent paper by Jia and co-workers. In particular, it is pointed out that their discussion with the help of the shape invariance approach and the supersymmetry WKB approximation is wrong, since the superpotential of the four-parameter diatomic molecule potential employed in their calculation is not suitable in the case where the deformation parameter λ is λ < 0 or 0 < λ < 1. The correct results for the energy levels and wavefunctions can be obtained in standard quantum mechanics through resolution of Schrodinger's equation by taking into account the different ranges of the shape parameter λ of the potential.

The asymptotic behaviour of the superpotential of general SUSY transformations for a coupled-channel Hamiltonian with different thresholds is analyzed. It is shown that asymptotically the superpotential can tend to a diagonal matrix with an arbitrary number of positive and negative entries depending on the choice of the factorization solution. The transformation of the Jost matrix is generalized to "non-conservative" SUSY transformations introduced in Sparenberg et al (2006 J. Phys. A: Math. Gen. 39 L639). Applied to the zero initial potential the method permits to construct superpartners with a nontrivially coupled Jost-matrix. Illustrations are given for two- and three-channel cases., Comment: 17 pages, 3 explicit examples and figures added

We present the configurations of intersecting branes in type IIA string theory corresponding to the meta-stable supersymmetry breaking vacua(hep-th/0608063) in the four-dimensional N=1 supersymmetric Yang-Mills theory coupled massive flavors with adjoint matter where the superpotential has three deformed terms., Comment: 15pp, 3 figures; The last two sentences in page 7 corrected; The last figure added;improved pages 10 and 11;figures corrected;the references to colors improved in the text and to appear in CQG

This paper suggests a systematic method based on supersymmetric quantum mechanics for generating conditionally exactly soluble potentials, and uses the variational supersymmetric WKB method to obtain the approximate values of the energy spectrum of the whole class.

The super-algebraic structure of a generalized version of the Jaynes–Cummings model is investigated. We find that a Z2 graded extension of the so(2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the four-dimensional super-algebra u(1/1) with two odd and two even elements. Differential matrix operators are taken as realization of the elements of the superalgebra to which the model Hamiltonian belongs. Several examples with various choices of superpotentials are presented. The energy spectrum and corresponding wavefunctions are obtained analytically.

In the context of the gauge-string correspondence, we discuss the spontaneous partial breaking of supersymmetry. Starting from the orbifold of S5, supersymmetry breaking leads us to consider the (resolved) conifold background and some of the gauge dynamics encoded in that geometry. Using this gravity dual, we compute the low energy effective superpotential for such = 1 theories. We are naturally led to extend the Veneziano-Yankielowicz one: glueball fields appear.

Following the lectures given at CERN Winter School 2006, we present a pedagogical overview of flux compactifications and generalized geometries, concentrating on closed string fluxes in type II theories. We start by reviewing the supersymmetric flux configurations with maximally symmetric four-dimensional spaces. We then discuss the no-go theorems (and their evasion) for compactifications with fluxes. We analyse the resulting four-dimensional effective theories for Calabi-Yau and Calabi-Yau orientifold compactifications, concentrating on the flux-induced superpotentials. We discuss the generic mechanism of moduli stabilization and illustrate with two examples: the conifold in IIB and a T 6 /(Z 3 x Z 3 ) torus in IIA. We finish by studying the effective action and flux vacua for generalized geometries in the context of generalized complex geometry.

Supersymmetric quantum mechanics (SUSY QM) is used to determine the class of potentials that are solvable as well as to find their isospectral partners. To do that SUSY QM assumes the knowledge of the Witten superpotential as a particular solution of the Riccati equation involved in order to obtain the corresponding generalizations. Consequently, due to the fact that in statistical mechanics the internal energy is indeed known as a single temperature derivative of the Helmholtz free energy, a simple transformation has been proposed here in order to get, in an heuristic way, a Schrodinger-like thermodynamic (SLT) equation. In this proposal, the quantum Witten superpotential and the wavefunction correspond, in statistical physics, to the internal energy and to the canonical partition function respectively. With that, it is shown how all the results from SUSY QM can be incorporated in the statistical case, to find the generalized internal energy and partition functions as well as to generalize all the other related thermodynamical properties. As an example of the usefulness of the proposed SLT equation, the case of Boltzmann statistics has been considered explicitly and it was found that the corresponding generalized internal energy contain, as a particular case, the Bose–Einstein or the Fermi–Dirac probability distributions.

We consider supersymmetric SL(3,R) deformations of various type IIB supergravity backgrounds which exhibit flows away from an asymptotically locally AdS_5 x S^5 fixed point. This includes the gravity dual of the Coulomb branch of N=1 super Yang Mills theory, for which the deformed superpotential is known. We also consider the gravity duals of field theories which live on various curved backgrounds, such as Minkowski_2 x H^2, AdS_3 x S^1 and R x S^3. Some of the deformed theories flow from a four-dimensional N=1 superconformal UV fixed point to a two-dimensional (2,2) superconformal IR fixed point. We study nonsupersymmetric generalizations of the deformations of the above Coulomb branch flows., Comment: 29 pages, additional references and comments

The fundamental matrix factorisations of the D-model superpotential are found and identified with the boundary states of the corresponding conformal field theory. The analysis is performed for both GSO-projections. We also comment on the relation of this analysis to the theory of surface singularities and their orbifold description., Comment: 23 pages, LaTeX

We show how to compute the mass of a Kerr-anti-de Sitter spacetime with respect to the anti-de Sitter background in any dimension, using a superpotential which has been derived from standard Noether identities. The calculation takes no account of the source of the curvature and confirms results obtained for black holes via the first law of thermodynamics., minor changes; accepted by CQG

The relation of supersymmetric quantum mechanics (SUSYQM) is discussed with two other symmetry-based approaches: symmetry and a differential realization of the su(1, 1) and su(2) algebra. It is demonstrated that symmetry imposes conditions on the even and odd parts of the real and imaginary components of the superpotential W(x), and these are expressed in terms of a system of first-order linear differential equations, which is homogeneous when the factorization energy is real and inhomogeneous when it is complex. The formal solution of this system is presented for various special cases as well as for the general case. It is shown that a trivial solution of this system corresponds to the unbroken symmetry for the Scarf II potential. The formalism of SUSYQM is also linked with that of the potential algebra approach, and it is demonstrated that the J+ and J? ladder operators of some su(1, 1) or su(2) algebras act on series of degenerate levels of different potentials essentially in the same way as the A and A? shift operators of SUSYQM. Examples are presented for su(1, 1) and su(2) potential algebras, as well as for spectrum generating algebras of the same type. Possible generalizations of this construction are also pointed out.

A detailed analysis of matrix Darboux transformations under the condition that the derivative of the superpotential be self-adjoint is given. As a onsequence, a class of the symmetries associated to Schr\"odinger matrix Hamiltonians is characterized. The applications are oriented towards the Jaynes-Cummings eigenvalue problem, so that exactly solvable $2\times 2$ matrix Hamiltonians of the Jaynes-Cummings type are obtained. It is also established that the Jaynes-Cummings Hamiltonian is a quadratic function of a Dirac-type Hamiltonian., Comment: 15 pages

Using the associated hypergeometric differential equation, we analytically solve the bound states corresponding to a hierarchy of the radial potential ?v0 e??r/(1 ? e??r) + c e??r/(1 ? e??r)2 as a generalization of the Hulth?n potential. Then, an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l(l + 1) is also derived. Meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics, it is shown that these bound states can be calculated by two different algebraic methods. Based on these two approaches, it is noted that the bound states realize an extended supersymmetry structure.

We review in a pedagogical way the computation of gravitational corrections to the effective superpotential of N = 1 supersymmetric gauge theories coupled to supergravity, by using the method of the generalized Konishi anomaly. We focus on the first non-trivial gravitational corrections to U (N) gauge theories with a single adjoint chiral multiplet, and briefly comment on some generalizations.

We review the supergravity derivation of some non-perturbatively generated effective superpotentials for N=1 gauge theories. Specifically, we derive the Veneziano-Yankielowicz superpotential for pure N=1 Super Yang-Mills theory from the warped deformed conifold solution, and the Affleck-Dine-Seiberg superpotential for N=1 SQCD from a solution describing fractional D3-branes on a C^3 / Z_2 x Z_2 orbifold., LaTeX, iopart class, 8 pages, 3 figures. Contribution to the proceedings of the workshop of the RTN Network "The quantum structure of space-time and the geometric nature of fundamental interactions", Copenhagen, September 2003; v2: published version with minor clarifications

In this paper we review the Dijfgraaf-Vafa conjecture for U(N c ) super Yang-Mills theories with matter in the adjoint and fundamental representation. Secondly, we discuss a proposal to derive the effective superpotential in terms of the mesonic superfields and show how the Affleck-Dine-Seiberg non-perturbative superpotential arises directly from the matrix model. Finally, we review how the random matrix model can be used to obtain factorization data of Seiberg-Witten curves with matter.

We begin with a brief introduction on N=1 gauge theories, focusing on the importance of the effective superpotential in light of the new techniques to compute it systematically. We then proceed to consider theories for which the Konishi anomaly proves to be enough to solve exactly for the effective superpotential. As an example we study a chiral SO(10) gauge theory, where we also discuss the occurrence of dynamical supersymmetry breaking.

Using Noether's identities, we define a superpotential with respect to a background for the Einstein Gauss-Bonnet theory of gravity. As an example, we show that its associated conserved charge yields the mass-energy of a D-dimensional Gauss-Bonnet black hole in an anti-de Sitter spacetime., Comment: 17 pages, LaTeX, references added, typos corrected, version to appear in Class. Quant. Grav

In the SO(N_c) gauge theory with N_f quarks for N_f=N_c-2, its instanton effects indicate the signal of dynamical flavor symmetry breakdown of SU(N_f) to SO(N_f), which is not described by the conventional "magnetic" degree's of freedom. It is argued that this breaking is well described by our effective superpotential consisting of "electric" quarks and gluons instead of monopoles of SO(N_c). The low-energy particles include the Nambu-Goldstone superfields associated with this breakdown. The proposed superpotential is found to exhibit the holomorphic decoupling property and the anomaly-matching property on a residual chiral U(1) symmetry.

By examining the generic form of the superpotential of translational shape invariant potentials (TSIPs), we explicitly show the exactness of the lowest order supersymmetric WKB (SWKB) formula for TSIPs. Remarkably, our method applies to both unbroken and broken supersymmetric systems. We also demonstrate the equivalence of one-parameter and multi-parameter TSIPs, thus establishing the exactness of the SWKB formula for all TSIPs.

Given a SUSY potential �(x) in nonrelativistic quantum mechanics, a pair of potentials V±(x) can be constructed to describe the bosonic and fermionic sectors of the theory. We apply a form of electric–magnetic duality to such a pair of potentials and derive conditions that must be satisfied in order for the images of the original potentials under duality to still form a supersymmetric pair in one or two dimensions. These conditions serve as compatibility conditions between the supersymmetry and duality transformations.

We discuss the stability properties of the solutions of the general nonlinear \Schrodinger\ equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time ($\PT$) symmetric superpotential $W(x)$ that we considered earlier \cite{PhysRevE.92.042901}. In particular we consider the nonlinear partial differential equation $ \{ i \, \partial_t + \partial_x^2 - V(x) + g | \psi(x,t) |^{2\kappa} \} \, \psi(x,t) = 0 \>, $ for arbitrary nonlinearity parameter $\kappa$, where $g= \pm1$ and $V$ is the well known P{\"o}schl-Teller potential which we allow to be repulsive as well as attractive. Using energy landscape methods, linear stability analysis as well as a time dependent variational approximation, we derive consistent analytic results for the domains of instability of these new exact solutions as a function of the strength of the external potential and $\kappa$. For the repulsive potential (and $g=+1$) we show that there is a translational instability which can be understood in terms of the energy landscape as a function of a stretching parameter and a translation parameter being a saddle near the exact solution. In this case, numerical simulations show that if we start with the exact solution, the initial wave function breaks into two pieces traveling in opposite directions. If we explore the slightly perturbed solution situations, a 1\% change in initial conditions can change significantly the details of how the wave function breaks into two separate pieces. For the attractive potential (and $g=+1$), changing the initial conditions by 1 \% modifies the domain of stability only slightly. For the case of the attractive potential and negative $g$ perturbed solutions merely oscillate with the oscillation frequencies predicted by the variational approximation., Comment: 16 pages, 10 figures

A novel method, based on superpotentials is proposed for obtaining the quasi-normal modes of anti-de Sitter black holes. This is inspired by the case of the three-dimensional BTZ black hole, where the quasi-normal modes can be obtained exactly and are proportional to the surface gravity. Using this approach, the quasi-normal modes of the five dimensional Schwarzschild anti-deSitter black hole are computed numerically. The modes again seem to be proportional to the surface gravity for very small and very large black holes. They reflect the well-known instability of small black holes in anti-deSitter space., LaTeX, 17 pages, 5 eps figures, 1 eepic figure, minor typos corrected

The conserved charges associated to gauge symmetries are defined at a boundary component of space-time because the corresponding Noether current can be rewritten on-shell as the divergence of a superpotential. However, the latter is afflicted by ambiguities. Regge and Teitelboim found a procedure to lift the arbitrariness in the Hamiltonian framework. An alternative covariant formula was proposed by one of us for an arbitrary variation of the superpotential, it depends only on the equations of motion and on the gauge symmetry under consideration. Here we emphasize that in order to compute the charges, it is enough to stay at a boundary of spacetime, without requiring any hypothesis about the bulk or about other boundary components, so one may speak of holographic charges. It is well known that the asymptotic symmetries that lead to conserved charges are really defined at infinity, but the choice of boundary conditions and surface terms in the action and in the charges is usually determined through integration by parts whereas each component of the boundary should be considered separately. We treat the example of gravity (for any space-time dimension, with or without cosmological constant), formulated as an Affine theory which is a natural generalization of the Palatini and Cartan-Weyl (vielbein) first order formulations. We then show that the superpotential associated to a Dirichlet boundary condition on the metric (the one needed to treat asymptotically flat or AdS spacetimes) is the one proposed by Katz, Bi\u{c}{\'a}k and Lynden-Bell and not that of Komar. We finally discuss the KBL superpotential at null infinity., Comment: 16 pages, minor corrections and references added. Final version to appear in CQG

We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q-deformed operators, we also give the q-deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q-supercoherent states associated with the q-deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q-supercoherent states are discussed as well.

An extended minimum supersymmetric standard model, characterized by bilinear R-parity violations through the third generation lepton number (τ) in superpotential and the soft breaking SUSY tefms, is constructed. The masses of CP-odd Higgs bosons and those of charged Higgs bosons are estimated with proper formulation and possible parametrization. Several interestingfeatures, such as the mixing among τ lepton and charginos etc., are discussed precisely. Taking an example, we compute the 'mixing production' cross sections in e+e- colliders, where denote the physical τ lepton and the lightest charginos respectively.

Usual prescriptions for the supersymmetric Wentzel-Kramers-Brillouin (SWKB) formulation are applicable either in the case of broken or unbroken supersymmetry. However, they fail for one-dimensional potentials where there is a mixture of broken and unbroken supersymmetry, which corresponds to an oscillating superpotential. Such a situation arises when two or more identical wells interact by tunnelling through a potential barrier, exhibiting quasi-degeneracy. The simplest examples of twofold quasi-degeneracy are the double oscillator well and the double finite square well which are also very important from the point of view of application in real physics problems. The quasi-degenerate states are associated with six supersymmetric turning points in contrast with four classical turning points. This disagreement in the number of classical and supersymmetric turning points is responsible for the failure of the usual SWKB formulation and makes the case more interesting. Here we propose a modified version of the SWKB formulation using the partner potential to calculate the quasi-degenerate states. We apply our reformulated SWKB prescription to the double finite square well, which gives an excellent result.

Two different forms of d-dimensional supersymmetric quantum mechanics are constructed from 1-dimensional one by means of two complex Clifford algebras C(d, 0) and C(0, d) respectively. The two new d-dimensional realizations are based upon the more general supercharges, which include not only a vector superpotentiaI but also a complex scalar superpotential, and bring out the richer structures.

We apply supersymmetric quantum mechanics (SUSY QM) to multidimensional Schrodinger equations involving nonseparable potentials, which result in a system of coupled differential equations, where instead of the conventional definition of a scalar superpotential we introduce a superpotential matrix and succeed in applying SUSY QM to the coupled system. Finally, we discuss the shape-invariance condition for the potential matrix of such a system.

Using an adaptation of the Gelfand-Dickey analysis of two-dimensional integrable Hamiltonian systems to the harmonic superspace (HS), we derive a class of completely solvable four-dimensional N = 2 hypermultiplet self-couplings. The known harmonic non-localities are shown to factor out in the HS integrable models. As an illustration, we rederive the Taub-Nut result and obtain a new hyper-Kahler metric associated with a HS superpotential having a pure imaginary coupling constant. A comparison with classes of metrics in the literature is given.

On the basis of local n = 2 conformal supersymmetry we have constructed the supersymmetric action for a set of complex scalar supermultiplets in the FRW model. This action corresponds to dilaton - axion and chiral components of supergravity theory, and it is shown that the supersymmetric action of this model has the form of a local version of the supersymmetric quantum mechanics with spontaneous breaking of supersymmetry when the vacuum energy is zero.

The Schr\"odinger equation with attractive delta potential has been previously studied in the supersymmetric quantum mechanical approach by a number of authors, but they all used only the particular superpotential solution. Here, we introduce a one-parameter family of strictly isospectral attractive delta function potentials, which is based on the general superpotential (general Riccati) solution, we study the problem in some detail and suggest possible applications, Comment: 7 pages in LaTex, 4 eps figures, published version v4

In this paper, we carefully analyze the unitarity of the simplest Kazama–Suzuki model based on the noncompact group. The chiral ring structure in this N = 2 theory is clarified. In general, it is infinitely dimensional and innumerable. The primary chiral states belonging to the ring can be constructed by means of the direct products of the highest weight states of the coset SU(1,1)/U(1) and the scalar representation of U(1) for the fermions. The primary chird and anti-chiral states live individually in the positive discrete series D+ and negative discrete series D- of SU(1,1), respectively. With appropriate restrictions, this theory can be regarded as one at the fixed point of a Landau–Ginzburg theory, in which the superpotential is given by S12 in the fourteen exceptional singularities.

Supersymmetric quantum mechanics is illustrated by taking the example of Landau levels. These highly degenerate energy levels occur for the planar motion of a charged particle in a uniform magnetic field. The degeneracy of the system allows the levels to be grouped into different towers, connected according to supersymmetric quantum mechanics with appropriate operators.

Supersymmetric quantum mechanics is considered in one-, two- and three-dimensional spaces characterized by a non-vanishing constant curvature. In particular, a basic superalgebra is pointed out. Interactions such as those of oscillator-like or Calogero type are analysed in this context with respect to their spectra and associated accidental degeneracies.

We generalize the formalism and the techniques of the supersymmetric (susy) quantum mechanics to the cases where the superpotential is generated/defined by higher excited eigenstates. The generalization is technically almost straightforward but physically quite nontrivial since it yields an infinity of new classes of susy-partner potentials, whose spectra are exactly identical except for the lowest m+1 states, if the superpotential is defined in terms of the (m+1)-st eigenfunction, with m=0 reserved for the ground state. It is shown that in case of the infinite 1-dim potential well nothing new emerges (the partner potential is still of P\"oschl-Teller type I, for all m), whilst in case of the 1-dim harmonic oscillator we get a new class of infinitely many partner potentials: for each m the partner potential is expressed as the sum of the quadratic harmonic potential plus rational function, defined as the derivative of the ratio of two consecutive Hermite polynomials. These partner potentials of course have m singularities exactly at the locations of the nodes of the generating (m+1)-st wavefunction. The susy formalism applies everywhere between the singularities. A systematic application of the formalism to other potentials with known spectra would yield an infinitely rich class of "solvable" potentials, in terms of their partner potentials. If the potentials are shape invariant they can be solved at least partially and new types of analytically obtainable spectra are expected. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq, Comment: 15 pages LaTeX file, no figures, submitted to J. Phys. A: accepted for publication

The superfield action for Bianchi type models is formulated on the basis of local n = 2 supersymmetry. It is shown that the supersymmetric action for these models has the form of the localized version of n = 2 supersymmetric quantum mechanics. This local symmetry procedure provides a well defined scheme for including matter.