Search

In the study of integrable non-linear $\sigma$-models which are assemblies and/or deformations of principal chiral models and/or WZW models, a rational function called the twist function plays a central role. For a large class of such models, we show that they are one-loop renormalizable, and that the renormalization group flow equations can be written directly in terms of the twist function in a remarkably simple way. The resulting equation appears to have a universal character when the integrable model is characterized by a twist function., Comment: v1: 1+47 pages, Latex, v2: JHEP version

In the approach recently proposed by K. Costello and M. Yamazaki, which is based on a four-dimensional variant of Chern-Simons theory, we derive a simple and unifying two-dimensional form for the action of many integrable $\sigma$-models which are known to admit descriptions as affine Gaudin models. This includes both the Yang-Baxter deformation and the $\lambda$-deformation of the principal chiral model. We also give an interpretation of Poisson-Lie $T$-duality in this setting and derive the action of the $\mathsf{E}$-model., Comment: 37 pages

We explain how to obtain new classical integrable field theories by assembling two affine Gaudin models into a single one. We show that the resulting affine Gaudin model depends on a parameter $\gamma$ in such a way that the limit $\gamma \to 0$ corresponds to the decoupling limit. Simple conditions ensuring Lorentz invariance are also presented. A first application of this method for $\sigma$-models leads to the action announced in [Phys. Rev. Lett. 122 (2019) 041601] and which couples an arbitrary number $N$ of principal chiral model fields on the same Lie group, each with a Wess-Zumino term. The affine Gaudin model descriptions of various integrable $\sigma$-models that can be used as elementary building blocks in the assembling construction are then given. This is in particular used in a second application of the method which consists in assembling $N-1$ copies of the principal chiral model each with a Wess-Zumino term and one homogeneous Yang-Baxter deformation of the principal chiral model., Comment: 72 pages

A systematic procedure for constructing classical integrable field theories with arbitrarily many free parameters is outlined. It is based on the recent interpretation of integrable field theories as realizations of affine Gaudin models. In this language, one can associate integrable field theories with affine Gaudin models having arbitrarily many sites. We present the result of applying this general procedure to couple together an arbitrary number of principal chiral model fields on the same Lie group, each with a Wess-Zumino term.

A three-parameter integrable deformation of ℤ4 permutation supercosets is constructed. These supercosets are of the form Fˆ/F0 where F0 is the bosonic diagonal subgroup of the product supergroup Fˆ=Gˆ×Gˆ. They include the AdS3 × S3 and AdS3 × S3 × S3 supercosets. This deformation encompasses both the bi-Yang-Baxter deformation of the semi-symmetric space σ-model on ℤ4 permutation supercosets and the mixed flux model. Truncating the action at the bosonic level, we show that one recovers the bi-Yang-Baxter deformation of the principal chiral model plus Wess-Zumino term., Journal of High Energy Physics, 2019 (1), ISSN:1126-6708, ISSN:1029-8479

We define a two-parameter family of integrable deformations of the principal chiral model on an arbitrary compact group. The Yang-Baxter sigma-model and the principal chiral model with a Wess-Zumino term both correspond to limits in which one of the two parameters vanishes., 12 pages

A multi-parameter integrable deformation of the principal chiral model is presented. The Yang-Baxter and bi-Yang-Baxter σ-models, the principal chiral model plus a Wess-Zumino term and the TsT transformation of the principal chiral model are all recovered when the appropriate deformation parameters vanish. When the Lie group is SU(2), we show that this four-parameter integrable deformation of the SU(2) principal chiral model corresponds to the Lukyanov model., Journal of High Energy Physics, 2017 (10), ISSN:1126-6708, ISSN:1029-8479

The Yang-Baxter $\sigma$-model is an integrable deformation of the principal chiral model on a Lie group $G$. The deformation breaks the $G \times G$ symmetry to $U(1)^{\textrm{rank}(G)} \times G$. It is known that there exist non-local conserved charges which, together with the unbroken $U(1)^{\textrm{rank}(G)}$ local charges, form a Poisson algebra $\mathscr U_q(\mathfrak{g})$, which is the semiclassical limit of the quantum group $U_q(\mathfrak{g})$, with $\mathfrak{g}$ the Lie algebra of $G$. For a general Lie group $G$ with rank$(G)>1$, we extend the previous result by constructing local and non-local conserved charges satisfying all the defining relations of the infinite-dimensional Poisson algebra $\mathscr U_q(L \mathfrak{g})$, the classical analogue of the quantum loop algebra $U_q(L \mathfrak{g})$, where $L \mathfrak{g}$ is the loop algebra of $\mathfrak{g}$. Quite unexpectedly, these defining relations are proved without encountering any ambiguity related to the non-ultralocality of this integrable $\sigma$-model., Comment: 21 pages, references added

We construct manifestly N = 4 supersymmetric off-shell superfield actions for the HKT d = 1 sigma models on the group manifolds U(2) and SU(3), using the harmonic d = 1 approach. The underlying ( 4 , 4 , 0 ) and ( 4 , 4 , 0 ) ⊕ ( 4 , 4 , 0 ) multiplets are described, respectively, by one and two harmonic analytic superfields q + satisfying the appropriate nonlinear harmonic constraints. The invariant actions in both cases are bilinear in the superfields. We present the corresponding superfield realizations of the U(2) and SU(3) isometries and show that in fact they are enlarged to the products U(2)×SU(2) and SU(3)×U(2). We prove the corresponding invariances at both the superfield and component levels and present the bosonic d = 1 sigma model actions, as integral over t in the U(2) case and over t and S U ( 2 ) harmonics in the SU(3) case. In the U(2) case we also give a detailed comparison with the general harmonic approach to HKT models and establish a correspondence with a particular action of the off-shell nonlinear multiplet ( 3 , 4 , 1 ) . A possible way of generalizing U(2) model to the matrix U(2n) case is suggested.

We consider σ-models on para-complex Z T -cosets, which are analogues of those on complex homogeneous target spaces considered recently by D. Bykov. For these models, we show the existence of a gauge-invariant Lax connection whose Poisson brackets are ultralocal. Furthermore, its light-cone components commute with one another in the sense of Poisson brackets. This extends a result of O. Brodbeck and M. Zagermann obtained twenty years ago for hermitian symmetric spaces.

Yang-Baxter type models are integrable deformations of integrable field theories, such as the principal chiral model on a Lie group $G$ or $\sigma$-models on (semi-)symmetric spaces $G/F$. The deformation has the effect of breaking the global $G$-symmetry of the original model, replacing the associated set of conserved charges by ones whose Poisson brackets are those of the $q$-deformed Poisson-Hopf algebra $\mathscr U_q(\mathfrak g)$. Working at the Hamiltonian level, we show how this $q$-deformed Poisson algebra originates from a Poisson-Lie $G$-symmetry. The theory of Poisson-Lie groups and their actions on Poisson manifolds, in particular the formalism of the non-abelian moment map, is reviewed. For a coboundary Poisson-Lie group $G$, this non-abelian moment map must obey the Semenov-Tian-Shansky bracket on the dual group $G^*$, up to terms involving central quantities. When the latter vanish, we develop a general procedure linking this Poisson bracket to the defining relations of the Poisson-Hopf algebra $\mathscr U_q(\mathfrak g)$, including the $q$-Poisson-Serre relations. We consider reality conditions leading to $q$ being either real or a phase. We determine the non-abelian moment map for Yang-Baxter type models. This enables to compute the corresponding action of $G$ on the fields parametrising the phase space of these models., Comment: 45 pages

The bi-Yang-Baxter sigma-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimcik who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset sigma-model on G x G / G_diag, we show that it also admits a Lax matrix whose Poisson bracket is of the standard r/s-form characterised by a twist function which we determine. A number of results immediately follow from this, including the identification of certain complex Poisson commuting Kac-Moody currents as well as an explicit description of the q-deformed symmetries of the model. Moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations of sigma-models. Finally, we show that although the Poisson bracket of the Lax matrix still takes the r/s-form after fixing the G_diag gauge symmetry, it is no longer characterised by a twist function., Comment: 26 pages, 1 figure

We construct the manifestly N = 4 supersymmetric off-shell superfield “master” action for any number n of the N = 4 supermultiplets ( 4 , 4 , 0 ) described by harmonic analytic superfields q + a ( ζ , u ) , a = 1 , … , 2 n , subjected to the most general harmonic constraints. The action consists of the sigma-model and Wess–Zumino parts. We present the general expressions for the target space metric, torsion and background gauge fields. The generic target space geometry is shown to be weak HKT (hyper-Kahler with torsion), with the strong HKT and HK ones as particular cases. The background gauge fields obey the self-duality condition. Our formulation suggests that the weak HKT geometry is fully specified by the two primary potentials: an unconstrained scalar potential L ( q + , q − , u ) | θ = 0 which is the θ = 0 projection of the superfield sigma-model Lagrangian, and a charge 3 harmonic analytic potential L + 3 a ( q + , u ) | θ = 0 coming from the harmonic constraint on q + a . The reductions to the strong HKT and HK geometries amount to simple restrictions on the underlying potentials. We also show, using the N = 2 superfield approach, that the most general bosonic target geometry of the N = 4 , d = 1 sigma models, of which the weak HKT geometry is a particular case, naturally comes out after adding the mirror ( 4 , 4 , 0 ) multiplets with different transformation laws under N = 4 supersymmetry and SO ( 4 ) R symmetry. Thus the minimal dimension of the target spaces exhibiting such a “weakest” geometry is 8, which corresponds to a pair of the mutually mirror ( 4 , 4 , 0 ) multiplets.

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.

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–Gursey 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).

We argue that off-shell dualities between d = 1 supermultiplets with different sets of physical bosonic components and the same number of fermionic ones are related to gauging some symmetries in the actions of the supermultiplets with maximal sets of physical bosons. Our gauging procedure uses off-shell superfields and so is manifestly supersymmetric. We focus on N = 4 supersymmetric mechanics and show that various actions of the multiplet ( 3 , 4 , 1 ) amount to some gauge choices in the gauged superfield actions of the linear or nonlinear ( 4 , 4 , 0 ) multiplets. In particular, the conformally invariant ( 3 , 4 , 1 ) superpotential is generated by the Fayet–Iliopoulos term of the gauge superfield. We find a new nonlinear variant of the multiplet ( 4 , 4 , 0 ) , such that its simplest superfield action produces the most general 4-dim hyper-Kahler metric with one triholomorphic isometry as the bosonic target metric. We also elaborate on some other instructive examples of N = 4 superfield gaugings, including a non-Abelian gauging which relates the free linear ( 4 , 4 , 0 ) multiplet to a self-interacting ( 1 , 4 , 3 ) multiplet.

The Poisson algebra of the Lax matrix associated with the Pohlmeyer reduction of the AdS 5 × S 5 superstring is computed from first principles. The resulting non-ultralocality is mild, which enables to write down a corresponding lattice quadratic Poisson algebra, a key step towards the integrable quantization of this theory from first principles.

We recently proposed an integrable q-deformation of the AdS_5 x S^5 superstring action. Here we give details on the hamiltonian origin and construction of this deformation. The procedure is a generalization of the one previously developed for deforming principal chiral and symmetric space sigma-models. We also show that the original psu(2,2|4) symmetry is replaced in the deformed theory by a classical analog of the quantum group U_q(psu(2,2|4)) with q real. The relation between q and the deformation parameter entering the action is given. The framework used to derive the deformation also enables to prove that at the hamiltonian level, the "maximal deformation" limit corresponds to an undeformed semi-symmetric space sigma-model with bosonic part dS_5 x H^5. Finally, we discuss the various freedoms in the construction., 42 pages, 1 figure; v2: minor typos corrected and references added

An integrable deformation of the type IIB ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ superstring action is presented. The deformed field equations, Lax connection, and $\ensuremath{\kappa}$-symmetry transformations are given. The original $\mathfrak{psu}(2,2|4)$ symmetry is expected to become $q$ deformed.

We construct the worldline superfield massive superparticle actions which preserve 1/4 portion of the underlying higher-dimensional supersymmetry. The rest of supersymmetry is spontaneously broken and realized by nonlinear transformations. We consider the cases of N=4 to N=1 and N=8 to N=2 partial breaking. In the first case we present the corresponding Green-Schwarz type target superspace action with one $\kappa$-supersymmetry. It is related to the superfield action via a field redefinition. In the second case we find out two possible models, one of which is a direct generalization of the N=4 to N=1 case, while another is essentially different. For the first model we formulate Green-Schwarz type action with two kappa supersymmetries. We elaborate on the bosonic part of the superfield action for the second model and find that only in two special limits it takes the standard Nambu-Goto form. In the general case it is determined by a fourth-order algebraic equation. The characteristic common feature of these new superparticle models is that the algebras of their spontaneously broken supersymmetries are non-trivial truncations of the general extensions of N=1 and N=2 Poincare D=4 superalgebras by tensorial central charges., Comment: 22 pages

Subject to some conditions, the input data for the Drinfeld-Sokolov construction of KdV type hierarchies is a quadruplet $(\A,\Lambda, d_1, d_0)$, where the $d_i$ are $\Z$-gradations of a loop algebra $\A$ and $\Lambda\in \A$ is a semisimple element of nonzero $d_1$-grade. A new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented. The proposal relies on splitting the $d_1$-grade zero part of $\A$ into a vector space direct sum of two subalgebras. This permits one to interpret certain Gelfand-Dickey type systems associated with a nonstandard splitting of the algebra of pseudo-differential operators in the Drinfeld-Sokolov framework., Comment: 19 pages, LaTeX file

We study the Poisson bracket algebra of the $N=2$ supersymmetric chiral WZNW model in superspace. It involves two classical r-matrices, one of which comes from the geometrical constraints implied by $N=2$ supersymmetry. The phase space itself consists of superfields satisfying constraints involving this r-matrix. An attempt is made to relax these constraints. The symmetries of the model are investigated., 21 pages

The Faddeev-Reshetikhin procedure corresponds to a removal of the non-ultralocality of the classical SU(2) principal chiral model. It is realized by defining another field theory, which has the same Lax pair and equations of motion but a different Poisson structure and Hamiltonian. Following earlier work of M. Semenov-Tian-Shansky and A. Sevostyanov, we show how it is possible to alleviate in a similar way the non-ultralocality of symmetric space sigma models. The equivalence of the equations of motion holds only at the level of the Pohlmeyer reduction of these models, which corresponds to symmetric space sine-Gordon models. This work therefore shows indirectly that symmetric space sine-Gordon models, defined by a gauged Wess-Zumino-Witten action with an integrable potential, have a mild non-ultralocality. The first step needed to construct an integrable discretization of these models is performed by determining the discrete analogue of the Poisson algebra of their Lax matrices., Comment: 31 pages; v2: minor changes

We perform an Hamiltonian reduction on a classical \cw(\cg, \ch) algebra, and prove that we get another \cw(\cg, \ch$'$) algebra, with $\ch\subset\ch'$. In the case $\cg=S\ell(n)$, the existence of a suitable gauge, called Generalized Horizontal Gauge, allows to relate in this way two \cw-algebras as soon as their corresponding \ch-algebras are related by inclusion.

We propose a new formulation of the heterotic $D=10$ Green-Schwarz superstring whose worldsheet is a superspace with two even and eight odd coordinates. The action is manifestly invariant under both target-space supersymmetry and a worldsheet reparametrisation supergroup. It contains only a finite set of auxiliary fields. The key ingredient are the commuting spinor (twistor) variables, which naturally arise as worldsheet superpartners of the target space Grassmann coordinates. These spinors parametrise the sphere $S^8$ regarded as a coset space of the $D=10$ Lorentz group. The sphere is associated with the lightlike vector of one of the string Virasoro constraints. The origin of the on-shell $D=10$ fermionic kappa symmetry of the standard Green-Schwarz formulation is explained. An essential and unusual feature is the appearance of the string tension only on shell as an integration constant., Comment: 29 pages, LATEX file, BONN-HE-92-19, JHU-TIPAC-920018, ENSLAPP-L-392-92

We establish a correspondence between (integral and half-integral) gradings of a simple Lie algebra G specifying Toda models and SL(2)×U(1) subalgebras of G . The U(1)factor is submitted to special conditions which insure in each case that the theory is not degenerate. The conformal spin content of the W algebra underlying each Toda model can easily be computed in this approach.

The authors consider the superparticle action of Sorokin, Tkach and Volkov (1989), which has both spacetime and extended worldline supersymmetry. They give a full off-shell description of the D=4,6 (N=2,4 worldline) cases in the presence of supergravity and gauge background fields. The actions have the form of the geometric relations defining the chiral D=4 and harmonic-analytic D=6 bases in curved superspace, multiplied by Lagrange multipliers. Thus the form of the action is determined by the prepotentials of conformal supergravity and super-Yang-Mills theory.

We apply the recently developed concept of a compact Lorentz-harmonic space, which is isomorphic to the sphere S D −2 , to the problem of covariant quantization of the superparticle in D = 3, 4, 6, 10. We study the structure of the representation of the Lorentz group realized on harmonic functions on S D −2 . The crucial difference between compact and non-compact harmonic analysis is explained. The massless harmonic fields depend on one space-time coordinate x ++ only, and on D −2 harmonic coordinates. It is shown how ordinary massless fields can be obtained from the harmonic ones by means of covariant integration on S D −2 . We construct a Lorentz-harmonic superspace, which involves only one quarter of the usual number of Grassmann coordinates. It closely resembles the light-cone superspace, however it is Lorentz covariant. This framework is used to formulate a superparticle action, in which all the constraints are first class, Lorentz-covariant and allow a straightforward canonical quantization.

We exploit the supersymmetric structure of the Chern-Simons theory in the Landau gauge in order to prove its full scale invariance to all orders of perturbation theory. Not only the β-function, but also the anomalous dimensions vanish. Therefore the theory is completely finite. This result also implies the absence of any radiative correction to the self-energy. Our main tool is the supermultiplet of the currents and a supertrace identity.

We discuss various descriptions of a quantum particle on noncommutative space in a (possibly non-constant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the literature., Dedicated to the memory of Julius Wess. Work presented by F. Gieres at the conference `Non-commutative Geometry and Physics' (Orsay, April 2007)

Using an N=4, d=1 superfield approach, we construct an N=8 supersymmetric action of the self-interacting off-shell N=8 multiplet {\bf (1, 8, 7)}. This action is found to be invariant under the exceptional N=8, d=1 superconformal group F(4) with the R-symmetry subgroup SO(7). The general N=8 supersymmetric {\bf (1, 8, 7)} action is a sum of the superconformal action and the previously known free bilinear action. We show that the general action is also superconformal, but with respect to redefined superfield transformation laws. The scalar potential can be generated by two Fayet-Iliopoulos N=4 superfield terms which preserve N=8 supersymmetry but break the superconformal and SO(7) symmetries., Comment: 10 pages, LaTeX

We develop a pseudo-differential approach to the N=2 supersymmetric unconstrained matrix (k|n,m)-Generalized Nonlinear Schroedinger hierarchies and prove consistency of the corresponding Lax-pair representation (nlin.SI/0201026). Furthermore, we establish their equivalence to the integrable hierarchies derived in the super-algebraic approach of the homogeneously-graded loop superalgebra sl(2k+n|2k+m)\otimes C[{lambda},{lambda}^{-1}] (nlin.SI/0206037). We introduce an unconventional definition of N=2 supersymmetric strictly pseudo-differential operators so as to close their algebra among themselves., Comment: 12 pages, LaTex

We consider a system of two qubits at the ends of a finite length 1D cavity. This problem is mapped onto the double-Kondo model which is also shown to describe the low energy physics of a finite length quantum wire with resonant levels at its ends. At the Toulouse point the ground state energy and the average populations and correlations of the qubits or resonant levels at zero temperature are computed. These results show that the effective interactions between the qubits or resonant levels can be used to probe their associated Kondo length scale., New version (accepted in Europhysics Letters)

On the example of nonabelian Toda type theory associated with the Lie superalgebra $osp(2|4)$ we show that this integrable dynamical system is relevant to a black hole background metric in the corresponding target space. In the even sector the model under consideration reduces to the exactly solvable conformal theory (nonabelian $B_2$ Toda system) in the presence of a black hole recently proposed in the article "Black holes from non-abelian Toda theories" by the last two authors (hep-th 9203039)., Comment: 4 pages, Latex

We show that the equations derived by LeBrun to describe scalar-flat Kahler surfaces with a conformal S 1 -action are the continuous long wave approximation for the Toda-type system associated with the semi-direct sum of the algebra A r and the commutative multiplet which transform according to the adjoint representation of A r . Moreover, we display the corresponding continuum Lie algebra, on which the Maurer-Cartan 1-form (written explicitly) takes values for the equations under consideration, the improved energy-momentum tensor, etc.

We construct the worldline superfield massive superparticle actions which preserve 1/4 portion of the underlying higher-dimensional supersymmetry. We consider the cases of N=4/N=1 and N=8/N=2 partial breaking. In the first case we present the corresponding Green-Schwarz type target superspace action with one kappa supersymmetry. In the second case we find out two possibilities, one of which is a direct generalization of the N=4/N=1 case, while another is essentially different., 15 pages, LaTeX, Talk given by S.K. at the XIV-th Max Born Symposium, Karpacz, Poland, September 21-25, 1999

A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its reductions which are characterized by a finite number of fields are described. This class includes the one-dimensional reduction of the two-dimensional N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry -- the N=4 Toda chain hierarchy -- which may be relevant in the construction of supersymmetric matrix models. The Lax pair representations of the bosonic and fermionic flows, corresponding local and nonlocal Hamiltonians, finite and infinite discrete symmetries, the first two Hamiltonian structures and the recursion operator connecting all evolution equations and the Hamiltonian structures of the N=4 Toda chain hierarchy are constructed in explicit form. Its secondary reduction to the N=2 supersymmetric alpha=-2 KdV hierarchy is discussed., 26 pages, LaTeX, a few misprints corrected

The standard format of matrices belonging to Lie superalgebras consists of partitioning the matrices into even and odd blocks. In this paper, we present a systematic study of other possible matrix formats and in particular of the so-called diagonal format which naturally occurs in various physical applications, e.g. for the supersymmetric versions of conformal field theory, integrable models. W algebras and quantum groups.

We propose a formulation of the massive spinning particle in terms of physical bosonic and fermionic fields only. We make use neither of auxiliary objects of the type of $\gamma_5$ nor of gauge fields. The model is to be used as a suitable starting point for a forthcoming study of the rigid superparticle., Comment: The proof of equivalence of the one-parameter family of actions simplified; references added

We present a new integrable extension of the a=-2, N=2 SKdV hierarchy, with the "small" N=4 superconformal algebra (SCA) as the second hamiltonian structure. As distinct from the previously known N=4 supersymmetric KdV hierarchy associated with the same N=4 SCA, the new system respects only N=2 rigid supersymmetry. We give for it both matrix and scalar Lax formulations and consider its various integrable reductions which complete the list of known SKdV systems with the N=2 SCA as the second hamiltonian structure. We construct a generalized Miura transformation which relates our system to the $\alpha = -2$, N=2 super Boussinesq hierarchy and, respectively, the ``small'' N=4 SCA to the N=2 W_3 superalgebra., Comment: 13 pages, LaTeX file, minor mistakes and misprints corrected

We give the formulation in extended superspace of an $N=2$ supersymmetric KP hierarchy using chirality preserving pseudo-differential operators. We obtain two quadratic hamiltonian structures, which lead to different reductions of the KP hierarchy. In particular we find two different hierarchies with the $N=2$ classical super-${\cal W}_n$ algebra as a hamiltonian structure. The relation with the formulation in $N=1$ superspace is carried out., 18 pages, LaTeX file, important reference added

The cohomology of the BRS operator corresponding to a group of rigid symmetries is studied in a space of local field functionals subjected to a condition of gauge invariance. We propose a procedure based on a filtration operator counting the degree in the infinitesimal parameters of the rigid symmetry transformations. An application to Witten's topological Yang-Mills theory is given., Comment: appendix and refs added

Using a superspace approach, it is proved that aN=1 Super-Toda theory can be seen as a constrained WZW model based on a supergroup. The gauge transformations which survive the constraints are then useda la Drinfeld Sokolov to determine explicity the super-W algebra underlying this theory. The conformal spin content of any such super-Walgebra is provided in the general case.

Factoring out the spin $1$ subalgebra of a $ W $ algebra leads to a new $ W $ structure which can be seen either as a rational finitely generated $ W $ algebra or as a polynomial non-linear $ W_\infty$ realization., Comment: 11 pages, LATEX, preprint ENSLAPP-AL-429/93 and NORDITA-93/47-P

We construct $N=4$ supersymmetric KdV equation as a hamiltonian flow on the $N=4\;SU(2)$ super Virasoro algebra. The $N=4$ KdV superfield, the hamiltonian and the related Poisson structure are concisely formulated in $1D \;N=4$ harmonic superspace. The most general hamiltonian is shown to necessarily involve $SU(2)$ breaking parameters which are combined in a traceless rank 2 $SU(2)$ tensor. First nontrivial conserved charges of $N=4$ super KdV (of dimensions 2 and 4) are found to exist if and only if the $SU(2)$ breaking tensor is a bilinear of some $SU(2)$ vector with a fixed length proportional to the inverse of the central charge of $N=4\;SU(2)$ algebra. After the reduction to $N=2$ this restricted version of $N=4$ super KdV goes over to the $a=4$ integrable case of $N=2$ super KdV and so is expected to be integrable. We show that it is bi-hamiltonian like its $N=2$ prototype., Comment: 11 pages, preprint ENSLAPP-L-415-92

We study a non linear sigma model $O(N)\otimes O(2)/O(N-2)\otimes O(2)$ describing the phase transition of N-components helimagnets up to two loop order in $D=2+\epsilon$ dimensions. It is shown that a stable fixed point exists as soon as $N$ is greater than 3 (or equal). In the N=3 case, the symmetry of the system is dynamically enlarged at the fixed point to O(4) We show that the order parameter for Heisenberg helimagnets involves a tensor representation of $O(4)$. We show that for large $N$ and in the neighborhood of two dimensions this nonlinear sigma model describes the same critical theory as the Landau-Ginzburg linear theory. We deduce that there exists a dividing line $N_c (D)$ in the plane $(N, D)$ separating a first-order region containing the Heisenberg point at $D=4$ and a second-order region containing the whole $D=2$ axis. We conclude that the phase transition of Heisenberg helimagnets in dimension 3 is either first order or second order with $O(4)$ exponents involving a tensor representation or tricritical., Comment: 25 pages and 1 figure not included, LateX, Saclay preprint SPhT/93-044

We derive a chirally split expression for the superdiffeomorphism anomaly in the two-dimensional superplane. The effective action from which this anomaly was obtained is related to the WZ-action associated with the superdiffeomorphism group. Furthermore, we show that the anomalous Ward identities generated by supercoordinate transformations encompass the wellknown OPE’s for the stress-energy supertensor. In conclusion, the extension to generic super Riemann surfaces is discussed.