Diverse <Emphasis Type="Italic">N</Emphasis>=(4,4) Twisted Multiplets in the <Emphasis Type="Italic">N</Emphasis>=(2,2) Superspace

We describe four different types of N=(4, 4) twisted supermultiplets in the two-dimensional N=(2, 2) superspace ℝ^(1,1|2,2). All these multiplets are represented by a pair of chiral and twisted chiral superfields and differ in the transformation properties under an extra hidden N=(2, 2) supersymmetry. The sigma-model N=(2, 2) superfield Lagrangians for each type of the N=(4, 4) twisted supermultiplet are real functions subjected to some differential constraints implied by the hidden supersymmetry. We prove that the general sigma-model action including all types of N=(4, 4) twisted multiplets and invariant under the N=(4, 4) supersymmetry reduces to a sum of sigma-model actions for separate types. An interaction between the multiplets of different sorts is possible only through the appropriate mass terms and only for those multiplets that belong to the same “self-dual” pair. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 66–86, October, 2005.     

Three-dimensional $$ \mathcal{N} $$=4 superconformal superfield theories

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.     

Chern-simons theory in the SO(5)/U(2) harmonic superspace

We consider the superspace of D=3, N=5 supersymmetry using SO(5)/U(2) harmonic coordinates. Three analytic N=5 gauge superfields depend on three vector and six harmonic bosonic coordinates and also on six Grassmann coordinates. Decomposing these superfields in Grassmann and harmonic coordinates yields infinite-dimensional supermultiplets including a three-dimensional gauge Chern-Simons field and auxiliary bosonic and fermionic fields carrying SO(5) vector indices. The superfield action of this theory is invariant with respect to the D=3, N=6 conformal supersymmetry realized on N=5 superfields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 217–234, November, 2008.     

Geometry of Solutions of the N=2 SYM Theory in Harmonic Superspace

The classical equations of motion of the D=4, N=2 supersymmetric Yang–Mills (SYM) theory for Minkowski and Euclidean spaces are analyzed in harmonic superspace. We study dual superfield representations of equations and subsidiary conditions corresponding to classical SYM solutions with different symmetries. In particular, alternative superfield constructions of self-dual and static solutions are described in the framework of the harmonic approach.     

The N=4 super Landau models

We briefly describe a new superextended Landau model with a worldline N=4 supersymmetry and an internal target space ISU (2|2) supersymmetry. It shares many features with the previously studied N=2 supersymmetric Landau model, which is also briefly described.     

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.     

Integrability Properties of A Supersymmetric Coupled Dispersionless Integrable System

We study the integrability aspects of an N=1 supersymmetric coupled dispersionless (SUSY-CD) integrable system in detail. We present a superfield Lax representation of the SUSY-CD system by writing its (3×3)-matrix superfield Lax pair and show that the zero-curvature condition corresponds to the SUSY-CD system. From the fermionic superfield Lax representation, we obtain a set of coupled superfield Riccati equations that we further use to obtain an infinite set of superfield conserved currents. We investigate the Darboux transformation of the SUSY-CD system and use it to obtain multisoliton solutions of the system.     

Background harmonic superfields inN=2 supergravity

A modification of the harmonic superfield formalism in D=4, N=2 supergravity is considered using an additional condition of covariance under the background supersymmetry with a central charge (B-covariance). Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 116, No. 2, pp. 288–304, August. 1998.     

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.     

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.     

Superbranes and Super Born–Infeld Theories as Nonlinear Realizations

We use a few instructive examples to outline the characteristic features of the approach to superbranes and super Born–Infeld theories based on the concept of partial spontaneous breaking of global supersymmetry (PBGS). The examples include the N=1, D=4 supermembrane and the space-filling D2- and D3-branes. In addition to a short account of the available results for these systems, we present some new developments. For the supermembrane, we prove that the equation of motion following from the off-shell Goldstone superfield action is equivalent to the one derived directly from the nonlinear-realization formalism. We use a universal procedure inspired by the relation between linear and nonlinear realizations of the PBGS to obtain a new derivation of the off-shell Goldstone superfield actions for these systems.     

The holomorphic effective action inN=2D=4 supergauge theories with various gauge groups

We consider the theory of hypermultiplets in arbitrary representations of arbitrary semisimple gauge groups coupled to gauge superfields. Using the N=2 harmonic superspace formulation of these models, we find the general structure of the holomorphic effective action depending on the gauge superfield with values in the Cartan subalgebra of the gauge algebra. We find explicit expressions for the effective actions in the cases where the hypermultiplets are in the fundamental and adjoint representations of SU(n), SO(n), and Sp(2n). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 444–455, April, 2000.     

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.     

One-loop effective action in the $$ \mathcal{N} $$=2 supersymmetric massive Yang-Mills field theory

We consider the =2 supersymmetric massive Yang-Mills field theory formulated in the =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 =2 supersymmetric one-loop counterterms including new counterterms depending on the Stueckelberg superfield. We analyze the component structure of one of these counterterms. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 22–40, October, 2008.     

One-loop effective action for super-QCD

The path integral method with respect to the scalar (quark) superfields is used to obtain the effective action for fields with the quantum numbers of the mesons and their superpartners inN-1 super-QCD. The background fields include gauge superfields of the groupSU _L (N)xSU _R (N)xU _L(1)xU _R(1) and also Higgs superfields in the adjoint representation of the groupSU _L (N)xSU _R (N).St. Petersburg Wood Technology Academy. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 2, pp. 237–246, February, 1995.     

Points de petite hauteur sur les courbes modulaires X0(N)

Let N be a square free integer, prime to 6. Let φ the imbeding of X ₀(N) in its Jacobian relative to the point ∞. We show that the set is finite and that is infinite. This explicit form of the Bogomolov conjecture is obtained by an estimation of the self-intersection of the dualizing sheaf, in the sense of Arakelov theory, of modular curves. This result is obtained by estimating several quantities attached to the Arakelov metric on X ₀(N), starting with Petersson's trace formula Oblatum 1-I-1997 & 30-IV-1997     

Monotone perturbations of the laplacian inL 1(R N )

The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L ¹(R ^N ),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular, if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense.     

On the singular set of stationary harmonic maps

LetM andN be compact riemannian manifolds, andu a stationary harmonic map fromM toN. We prove thatH ⁿ⁻² (Σ)=0, wheren=dimM and Σ is the singular set ofu. This is a generalization of a result of C. Evans [7], where this is proved in the special caseN is a sphere. We also prove that, ifu is a weakly harmonic map inW ^1,n (M, N), thenu is smooth. This extends results of F. Hélein for the casen=2, or the caseN is a sphere ([9], [10]).     

Trigonometric Approximation of SO(N) Loops

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a H?lder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order O(n^-a+e)\mathcal{O}(n^{-\alpha+\epsilon}) for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.