Supersymmetrizing Landau models

We review recent progress in constructing and studying superextensions of the Landau problem of a quantum particle on a plane in a uniform magnetic field and also its Haldane S² generalization. We focus on the planar super Landau models that are invariant under the inhomogeneous supergroup ISU (1|1), a contraction of the supergroup SU(2|1), and are minimal superextensions of the original Landau model. Their significant common feature is the presence of a hidden dynamical worldline N = 2 supersymmetry, which exists at both the classical and quantum levels and is revealed most naturally in passing to the new invariant inner products in the space of quantum states in order to make the norms of all states positive. For one of the planar models, the superplane Landau model, we present an off-shell worldline superfield formulation in which the N = 2 supersymmetry becomes explicit. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 409–423, March, 2008.     

Generalized Calabi-Yau manifolds and the chiral de Rham complex

We show that the chiral de Rham complex of a generalized Calabi-Yau manifold carries N=2 supersymmetry. We discuss the corresponding topological twist for this N=2 algebra. We interpret this as an algebroid version of the super-Sugawara or Kac-Todorov construction.     

N=4 Multiplets in N=3 Harmonic Superspace

We show that the N=3 harmonic superfield equations of motion are invariant with respect to the fourth supersymmetry. We also use the SU(3) harmonics to analyze a more flexible form of superfield constraints for the Abelian N=4 vector multiplet and its N=3 decomposition. An unusual alternative representation of the N=4 supersymmetry is realized on infinite multiplets of analytic superfields in the N=3 harmonic superspace. An integer-valued parameter playing the role of a discrete coordinate parameterizes U(1) charges of superfields in these multiplets. Each superfield term of the N=3 Yang–Mills action has an infinite-dimensional N=4 generalization. The gauge group of this model contains an infinite number of superfield parameters.     

On mathematical models and numerical simulation of the fluidization of polydisperse suspensions

The well-known Masliyah–Lockett–Bassoon (MLB) model for sedimentation of small particles is extended to fluidization of polydisperse suspensions. For N particle species that differ in size and density, this model leads to a first-order system of N conservation laws, which in general is of mixed (in the case N = 2, hyperbolic–elliptic) type. By a simple algebraic steady-state analysis, we derive necessary compatibility conditions on the size and density parameters that admit the formation of stationary fluidized beds. We then proceed to determine the composition of polydisperse fluidized beds of given compatible species by varying the fluidization velocity and the initial composition of the suspensions, and prove that, within the framework of the MLB model combined with the Richardson–Zaki formula, the constructed bidisperse beds always cause the equations to be hyperbolic. This means that these states are always predicted to be stable. The transient behaviour of the MLB model applied to fluidization is illustrated by three numerical examples, in which the system of conservation laws is solved for N = 2, N = 3 and N = 5, respectively. These examples illustrate the effects of bed expansion and layer inversion caused by successively increasing the applied fluidization velocity and show that the predicted fluidized states are indeed attained.     

Nonanticommutative deformations of <Emphasis Type="Italic">N</Emphasis>=(1, 1) supersymmetric theories

We discuss chirality-preserving nilpotent deformations of the four-dimensional N=(1, 1) Euclidean harmonic superspace and their implications in N=(1, 1) supersymmetric gauge and hypermultiplet theories. For the SO(4) × SU(2)-invariant deformation, we present nonanticommutative Euclidean analogues of the N=2 gauge multiplet and hypermultiplet off-shell actions.As a new result, we consider a specific nonanticommutative hypermultiplet model with the N=(1, 0) supersymmetry. It involves free scalar fields and interacting right-handed spinor fields.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 235–251, February, 2005.     

Supersymmetric Quantum Field Theory: Indefinite Metric

We study the recently introduced Krein structure (indefinite metric) of the N = 1 supersymmetry and present the way into physical applications outside path integral methods. From the mathematical point of view some perspectives are mentioned at the end of the paper.     

Off-Shell Supersymmetry and Filtered Clifford Supermodules

An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1|N}\), correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1,1|p,q}\), correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.     

Nonlinear supersymmetry and goldstino couplings to the MSSM

We briefly review the nonlinear supersymmetry formalisms in the standard realization and superfield methods. We then evaluate the goldstino couplings to the minimal supersymmetric standard model (MSSM) superfields and discuss their phenomenological consequences. These relate to the tree-level Higgs mass and to invisible Higgs- and Z-boson decays. The Higgs mass is increased from its MSSM tree-level value and brought above the LEP2 mass bound for a low scale of supersymmetry breaking √f ∼ 2 TeV to 7 TeV. The invisible decay rates of the Higgs and Z bosons into goldstino and neutralino are computed and shown to bring stronger constraints on f than their decays into goldstino pairs, which are subleading in 1/f.     

Moufang polygons and irreducible spherical BN-pairs of rank 2, I

Let G be a group with an irreducible spherical BN-pair of rank 2 satisfying the additional condition: (∗) There exists a normal nilpotent subgroup U of B with B=TU, where T=B∩N and |W|≠16 for the Weyl group W=N/B∩N. We show that G corresponds to a Moufang polygon and hence is essentially known.     

Supersymmetry and density of states of the magnetic schrödinger operator with a random potential revisited

Abstract. For large magnetic field and away from the Landau levels, we prove that the density of states of the magnetic Schrodinger operator with a random potential is a C₀ ^∞ function for some appropriate Cq classes of probability distributions. To achieve the above result, we use the method of ohr—Sommerfeld quantization, supersymmetry and an elementary cluster expain—sion. Combining with our previous results [24], we obtain an asymptotic expansion for the density of states as a measure.     

Exact eigensystems for some matrices arising from discretizations

It is known, for example, that the eigenvalues of the N×N matrix A, arising in the discretization of the wave equation, whose only nonzero entries are A_kk+1=A_k+1k=-1,k=1,…,N-1, and A_kk=2,k=1,…,N, are 2{1-cos[pπ/(N+1)]} with corresponding eigenvectors v^(p) given by . We show by considering a simple finite difference approximation to the second derivative and using the summation formulae for sines and cosines that these and other similar formulae arise in a simple and unified way.     

A Liouville type theorem for polyharmonic elliptic systems

In this paper, we consider the polyharmonic system m(−Δ)U=Vq,m(−Δ)V=Up in RN, for m>1, N>2m, with p?1, q?1, but not both equal to 1, where m(−Δ) is the polyharmonic operator. Set α=2m(q+1)/(pq−1), β=2m(p+1)/(pq−1), for α,β∈[(N−2)/2,N−2m), we prove the nonexistence of positive solutions.     

Joint modelling of the total amount and the number of claims by conditionals

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple (S,N), where N is a count variable and S=X₁+?+X_N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S|N and N|S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson-Gamma and the Poisson-Binomial conditionals models. Other conditionals models are proposed, including the Poisson-Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.     

On real moduli spaces of holomorphic bundles over M-curves

Let F be a genus g curve and σ:F→F a real structure with the maximal possible number of fixed circles. We study the real moduli space N^′=Fix(σ#) where σ#:N→N is the induced real structure on the moduli space N of stable holomorphic bundles of rank 2 over F with fixed non-trivial determinant. In particular, we calculate H^?(N^′,Z) in the case of g=2, generalizing Thaddeus' approach to computing H^?(N,Z).     

The Ω Deformed B-model for Rigid N = 2 Theories

We give an interpretation of the Ω deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four-dimensional rigid N = 2 theories explicitly in general Ω-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N = 2 supersymmetric theories. The rigid N = 2 field theories we focus on are the conformal rank one N = 2 Seiberg–Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N = 2 theories arising from compactifications on local Calabi–Yau manifolds, we consider the theory of local ${\mathbb{P}^2}$ . We calculate motivic Donaldson–Thomas invariants for this geometry and make predictions for generalized Gromov–Witten invariants at the orbifold point.     

Landau’s theorem for functions with logharmonic Laplacian

In this paper, we show the existence of Landau constant for functions with logharmonic Laplacian of the form F(z) = ∣z∣²L(z) + K(z), ∣z∣ < 1, where L is logharmonic and K is harmonic. Moreover, the problem of minimizing the area is solved     

Boundary singularities of solutions of N-harmonic equations with absorption

We study the boundary behaviour of solutions u of −ΔNu+|u|^q−1u=0 in a bounded smooth domain Ω⊂RN subject to the boundary condition u=0 except at one point, in the range q>N−1. We prove that if q?2N−1 such an u is identically zero, while, if N−1<q<2N−1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.     

On Chen's theorem (II)

Let N be a sufficiently large even integer and S(N) denote the number of solutions of the equation

N=p+P₂,     

Three-dimensional <Emphasis Type="Italic">N</Emphasis>=4 supersymmetry in harmonic <Emphasis Type="Italic">N</Emphasis>=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.