Supersymmetric Hierarchy Research Articles

Supersymmetric partition function hierarchies and character expansions

We construct the supersymmetric beta and (q, t)-deformed Hurwitz–Kontsevich partition functions through W-representations and present the corresponding character expansions with respect to the Jack and Macdonald superpolynomials, respectively. Based on the constructed beta and (q, t)-deformed superoperators, we further give the supersymmetric beta and (q, t)-deformed partition function hierarchies through W-representations. We also present the generalized super Virasoro constraints, where the constraint operators obey the generalized super Virasoro algebra and null super 3-algebra. Moreover, the superintegrability for these (non-deformed) supersymmetric hierarchies is shown by their character expansions, i.e., <character>sim character.

Time-dependent Darboux transformation and supersymmetric hierarchy of Fokker–Planck equations

A procedure is presented for solving the Fokker–Planck equation with constant diffusion but non-stationary drift. It is based on the correspondence between the Fokker–Planck equation and the non-stationary Schrödinger equation. The formalism of supersymmetric quantum mechanics is extended by applying the Darboux transformation directly to the non-stationary Schrödinger equation. From a solution of a Fokker–Planck equation a solution of the Darboux partner is obtained. For drift coefficients satisfying the condition of shape invariance, a supersymmetric hierarchy of Fokker–Planck equations with solutions related by the Darboux transformation is obtained.

Quasi-exactly solvable extended trigonometric Pöschl-Teller potentials with position-dependent mass

Infinite families of quasi-exactly solvable position-dependent mass Schrodinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one- and two-parameter trigonometric Poschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function W+(x) (and its accompanying function W-(x) . The generalized trigonometric Poschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.

Infinite families of position-dependent mass Schrödinger equations with known ground and first excited states

A construction method of infinite families of quasi-exactly solvable position-dependent mass Schrödinger equations with known ground and first excited states is proposed in a deformed supersymmetric background. Such families correspond to extensions of known potentials endowed with a deformed shape invariance property. Two different approaches are combined. The first one is a generating function method, which enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential, from some generating function W+(x) [and its accompanying function W−(x)]. The second approach is the conditionally deformed shape invariance method, wherein the deformed shape invariance property of the starting potentials is generalized to their extensions by adding some constraints on the parameters and by imposing compatibility conditions between sets of constraints. Detailed results are given for some extensions of the linear and radial harmonic oscillators, as well as the Kepler–Coulomb and Morse potentials.

Exact results for the infinite supersymmetric extensions of the infinite square well

One-dimensional potentials defined by (for integer S) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, , for all V(S)(x) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl–Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions for large and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.

Darboux transformations of new supersymmetric CKP and BKP hierarchies

In this paper, we construct a supersymmetric CKP(SCKP) hierarchy and its Darboux transformations. These Darboux transformations can generate new solutions from seed solutions by using supersymmetric functions. Further, we obtain the supersymmetric Sawada–Kotera equation from the reduction of supersymmetric CKP hierarchy. After that, using a new B type condition, we also construct a new supersymmetric BKP(SBKP) hierarchy which is different from the supersymmetric BKP hierarchy defined by Ramos and Stanciu. Meanwhile, the Darboux transformations of this new SBKP hierarchy are derived.

Two supersymmetric hierarchies related to the super-HS spectral problem

In this paper, we construct Lax pairs and super-bihamiltonian structures of two supersymmetric integrable hierarchies related to a given super-HS spectral problem.

Environmentally selected WIMP dark matter with high-scale supersymmetry breaking

We explore the possibility that both the weak scale and the thermal relic dark matter abundance are environmentally selected in a multiverse. An underlying supersymmetric theory containing the states of the minimal supersymmetric standard model (MSSM) and singlets, with supersymmetry and $R$ symmetry broken at unified scales, has just two realistic low-energy effective theories. One theory ($\mathrm{SM}+\stackrel{\texttildelow{}}{w}$) is the standard model augmented only by the wino, having a mass near 3 TeV, and has a Higgs boson mass in the range of (127--142) GeV. The other theory ($\mathrm{SM}+\stackrel{\texttildelow{}}{h}/\stackrel{\texttildelow{}}{s}$) has Higgsinos and a singlino added to the standard model. The Higgs boson mass depends on the single new Yukawa coupling of the theory, $y$, and is near 141 GeV for small $y$ but grows to be as large as 210 GeV as this new coupling approaches strong coupling at high energies. Much of the parameter space of this theory will be probed by direct detection searches for dark matter that push 2 orders of magnitude below the present bounds; furthermore, the dark matter mass and cross section on nucleons are correlated with the Higgs boson mass. The indirect detection signal of monochromatic photons from the Galactic center is computed, and the range of parameters that may be accessible to LHC searches for trilepton events is explored. Taking a broader view, allowing the possibility of $R$ symmetry protection to the TeV scale or axion dark matter, we find four more theories: ($\mathrm{SM}+\mathrm{\text{axion}}$), two versions of split supersymmetry, and the E-MSSM, where a little supersymmetric hierarchy is predicted. The special Higgs mass value of $(141\ifmmode\pm\else\textpm\fi{}2)\text{ }\text{ }\mathrm{GeV}$ appears in symmetry limits of three of the six theories, ($\mathrm{SM}+\mathrm{\text{axion}}$), ($\mathrm{SM}+\stackrel{\texttildelow{}}{w}$), and ($\mathrm{SM}+\stackrel{\texttildelow{}}{h}/\stackrel{\texttildelow{}}{s}$), motivating a comparison of other signals of these three theories.

The $N=2$ supersymmetric unconstrained matrix GNLS hierarchies

The generalization of the $N=2$ supersymmetric chiral matrix $(k|n,m)$--GNLS hierarchy to the case when matrix entries are bosonic and fermionic unconstrained $N=2$ superfields is proposed. This is done by exhibiting the corresponding matrix Lax--pair representation in terms of $N=2$ unconstrained superfields. It is demonstrated that when matrix entries are chiral and antichiral $N=2$ superfields, it reproduces the $N=2$ chiral matrix $(k|n,m)$-GNLS hierarchy, while in the scalar case, $k=1$, it is equivalent to the $N=2$ supersymmetric multicomponent hierarchy. The simplest example --- the $N=2$ unconstrained $(1|1,0)$--GNLS hierarchy --- and its reduction to the $N=2$ supersymmetric ${\alpha}=1$ KdV hierarchy are discussed in more detail, and its rich symmetry structure is uncovered.

Non-local Conservation Laws and Flow Equations¶ for Supersymmetric Integrable Hierarchies

An infinite series of Grassmann-odd and Grassmann-even flow equations is defined for a class of supersymmetric integrable hierarchies associated with loop superalgebras. All these flows commute with the mutually commuting bosonic ones originally considered to define these hierarchies and, hence, provide extra fermionic and bosonic symmetries that include the built-in N=1 supersymmetry transformation. The corresponding non-local conserved quantities are also constructed. As an example, the particular case of the principal supersymmetric hierarchies associated with the affine superalgebras with a fermionic simple root system is discussed in detail.

LAX PAIRS FOR N=2,3 SUPERSYMMETRIC KdV EQUATIONS AND THEIR EXTENSIONS

We present the Lax operator for the N=3 KdV hierarchy and consider its extensions. We also construct a new infinite family of N=2 supersymmetric hierarchies by exhibiting the corresponding super Lax operators. The new realization of N=4 supersymmetry on the two general N=2 superfields, bosonic spin-1 and fermionic spin-1/2, is discussed.

The N=2 Supersymmetric Matrix GNLS Hierarchies

We construct the matrix generalization of the N=2 supersymmetric GNLS hierarchies. This is done by exhibiting the corresponding matrix super Lax operators in terms of N=2 superfields in two different superfield bases. We present the second Hamiltonian structure and discrete symmetries. We then extend our discussion by conjecturing the Lax operators of different reductions of the N=2 supersymmetric matrix KP hierarchy and discuss the simplest examples.

Fully supersymmetric hierarchies from an energy-dependent super Hill operator

A super Hill operator with energy dependent potentials is proposed and the associated integrable hierarchy is constructed explicitly. It is shown that in the general case, the resulted hierarchy is multi-Hamiltonian system. The Miura type transformations and modified hierarchies are also presented.

N = 4 super NLS-mKdV hierarchies

N=2 extension of affine algebra $\hat{sl(2)\oplus u(1)}$ possesses a hidden global N=4 supersymmetry and provides a second hamiltonian structure for a new N=4 supersymmetric integrable hierarchy defined on N=2 affine supercurrents. This system is an N=4 extension of at once two hierarchies, N=2 NLS and N=2 mKdV ones. It is related to N=4 KdV hierarchy via a generalized Sugawara-Feigin-Fuks construction which relates N=2 $\hat{sl(2)\oplus u(1)}$ algebra to ``small'' N=4 SCA. We also find the underlying affine hierarchy for another integrable system with the N=4 SCA second hamiltonian structure, ``quasi'' N=4 KdV hierarchy. It respects only N=2 supersymmetry. For both new hierarchies we construct scalar Lax formulations. We speculate that any N=2 affine algebra admitting a quaternionic structure possesses N=4 supersymmetry and so can be used to produce N=4 supersymmetric hierarchies. This suggests a way of classifying all such hierarchies.

A COMMENT ON THE ODD FLOWS FOR THE SUPERSYMMETRIC KdV EQUATION

In a recent paper Dargis and Mathieu introduced integrodifferential odd flows for the supersymmetric KdV equation. These flows are obtained from the nonlocal conservation laws associated with the fourth root of its Lax operator. In this note I show that only half of these flows are of the standard Lax form, while the remaining half provide us with Hamiltonians for an SKdV-type reduction of a new supersymmetric hierarchy. This new hierarchy is shown to be closely related to the Jacobian supersymmetric KP-hierarchy of Mulase and Rabin. A detailed study of the algebra of additional symmetries of this new hierarchy reveals that it is isomorphic to the super-W1+∞ algebra, thus making it a candidate for a possible interrelationship between superintegrability and two-dimensional supergravity.

NEW SUPERSYMMETRIZATIONS OF THE GENERALIZED KdV HIERARCHIES

Recently we investigated a new supersymmetrization procedure for the KdV hierarchy inspired in some recent work on supersymmetric matrix models. We extend this procedure here for the generalized KdV hierarchies. The resulting supersymmetric hierarchies are generically nonlocal, except for the case of Boussinesque which we treat in detail. The resulting supersymmetric hierarchy is integrable and bi-Hamiltonian and contains the Boussinesque hierarchy as a subhierarchy. In a particular realization, we extend it by defining supersymmetric odd flows. We end with some comments on a slight modification of this supersymmetrization which yields local equations for any generalized KdV hierarchy.

DOUBLE SCALING LIMIT OF THE SUPER-VIRASORO CONSTRAINTS

We obtain the double scaling limit of a set of superloop equations recently proposed to describe the coupling of two-dimensional supergravity to minimal superconformal matter of type (2,4m). The continuum loop equations are described in terms of a [Formula: see text] theory with a Z2-twisted scalar field and a Weyl–Majorana fermion in the Ramond sector. We have computed correlation functions in genus zero, one and partially in genus two. An integrable supersymmetric hierarchy describing our model has not yet been found. We present a heuristic argument showing that the purely bosonic part of our model is described by the KdV hierarchy.

Gauge-fermion masses in supersymmetric hierarchy models

The gauge-fermion mass is found to be generally nonzero in three scale-supersymmetric hierarchy models. In a large class of models, the mass can be obtained from a simple current-algebra argument for an anomalous $R$ symmetry.