Local conserved charges in principal chiral models

We present a general construction of local, commuting conserved charges, with spins equal to the exponents, modulo the Coxeter number, of the underlying Lie algebra, for principal chiral models based on the classical Lie groups.     

Local conserved charges in principal chiral models

Local conserved charges in principal chiral models in 1+1 dimensions are investigated. There is a classically conserved local charge for each totally symmetric invariant tensor of the underlying group. These local charges are shown to be in involution with the non-local Yangian charges. The Poisson bracket algebra of the local charges is then studied. For each classical algebra, an infinite set of local charges with spins equal to the exponents modulo the Coxeter number is constructed, and it is shown that these commute with one another. Brief comments are made on the evidence for, and implications of, survival of these charges in the quantum theory.     

Anomalous action in gauge invariant, nonlocal, dynamical quark model

Anomalous sector of chiral Lagrangian is calculated in a gauge invariant, nonlocal, dynamical quark model. The Wess–Zumino term is proved coming from two kinds of sources, one is independent of and another dependent on dynamical quark self energy Σ(k²). p⁶ and more higher order anomalous sectors come only from Σ(k²) dependent source. After some cancellation, standard Wess–Zumino action is obtained.     

Ultraviolet properties of noncommutative nonlinear σ-models in two dimensions

We discuss the ultra-violet properties of bosonic and supersymmetric noncommutative nonlinear σ-models in two dimensions, both with and without a Wess–Zumino–Witten term.     

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.     

Intersoliton forces in the Wess–Zumino model

The spectrum of supersymmetric domain wall solitons of the Wess–Zumino model is known to be discontinuous across a curve (of marginal stability) in the moduli space of quartic superpotentials. Here we show how this phenomenon can be understood from the behaviour of the long-range inter-soliton force, which we compute by a method due to Manton.     

Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras

We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p+2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz’s wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz’s original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds. Dedicated to the memory of Tom Branson     

Wess–Zumino actions and Dirichlet boundary conditions for super p-branes with exotic fractions of supersymmetry

The general solutions in the models of closed and open superstring and super p-branes with exotic fractions of the N=1 supersymmetry are considered and the spontaneously broken character of the OSp(1,2M) symmetry of the models is established. It is shown that extending these models by Wess–Zumino terms generates the Dirichlet boundary conditions for superstring and super p-branes. Using the generalized Wess–Zumino terms new OSp(1,2M)-invariant super p-brane and Dp-brane-like actions preserving (M−1)/M fraction of supersymmetry are proposed. For M=32 these models suggest new superbrane vacua of M-theory preserving 31 from 32 global supersymmetries.     

Can supersymmetric theories be bosonized?

We investigate the possibility of transforming supersymmetric theories into pure;y fermionic or bosonic forms. The supersymmetric sine-Gordon lagrangian is rewritten in a purely fermionic form, and the Fermi equivalent of the original supersymmetry transformation is derived. This transformation represents an invariance only at the quantum level, when the effects of the chiral anomaly have been taken into account. When supersymmetric theories are written in purely bosonic forms, a non-local bosonic transformation takes place of the supersymmetry transformation. In both cases the supersymmetry algebra, as realized on the fields, is lost.     

Integrable sigma-models and Drinfeld–Sokolov hierarchies

Local commuting charges in sigma-models with classical Lie groups as target manifolds are shown to be related to the conserved quantities appearing in the Drinfeld–Sokolov (generalized mKdV) hierarchies. Conversely, the Drinfeld–Sokolov construction can be used to deduce the existence of commuting charges in these and in wider classes of sigma-models, including those whose target manifolds are exceptional groups or symmetric spaces. This establishes a direct link between commuting quantities in integrable sigma-models and in affine Toda field theories.     

The algebra formed by the invariant charges of the Nambu-Goto theory: Identification of a maximal Abelian subalgebra

Continuing the analysis of the algebraic structure of the invariant charges of the Nambu-Goto theory, we identify a complete set of commuting observables for the bosonic strings.     

Boson-Fermion equivalence in a two-dimensional anomalous chiral model

A current algebraic approach is taken to realise the two-dimensional chiral model having anomaly analogous to that of Wess and Zumino. The conserved currents are shown to be a combination of bosonic and fermionic ones. The equations of motion of the fermions are realised through its commutation algebra with the Sugawara-like energy-momentum tensor. Quantisation of the anomaly parameter is shown to be a consequence of consistent statistics.     

Integrable chiral theories in 2 + 1 dimensions

Following a recent proposal for integrable theories in higher dimensions based on zero curvature, new Lorentz invariant submodels of the principal chiral model in 2 + 1 dimensions are found. They have infinite local conserved currents, which are explicitly given for the su(2) case. The construction works for any Lie algebra and in any dimension, and it is given explicitly also for su(3). We comment on the application to supersymmetric chiral models.     

New avenues in supersymmetry and supergravity

We analyze the problem of constructing supersymmetric versions of gauge theories of particles and of gravity which have a closed supersymmetric algebra. Inparticular we present the basic no-go theorems that indicate that in four dimensions it is not possible to construct suitably extended supersymmetric versions of the above theories without drastic modification of the supersymmetric algebra. Two ways past the“N=3” barrier are discussed; that of central charges involved highly constrained versions which appearn difficult to quantize effectively, while the use of light-cone variables seems to be the most promising. We give light-cone gauge versions of supersymmetric Yang-Mills theories for all extended cases of interest and briefly consider their ultraviolet divergence properties.     

Systematic approach to conformal systems with extended Virasoro symmetries

The Toda field theories, which exist for every simple Lie group, are shown to give realizations of extended Virasoro algebras that involve generators of spins higher than or equal to two. They are uniquely determined from the canonical lagrangian formalism. The quantization of the Toda field theories gives a systematic treatment of generalized conformal bosonic models. The well-known pattern of conformal field theories with non-extended Virasoro algebra, appears to be repeated for any simple group, leading to a “periodic table”, parallel to the mathematical classification of simple Lie groups.     

Jack Polynomials in Superspace

This work initiates the study of orthogonal symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given explicitly.     

On equal constituent quark masses in different heavy quarkonia potential models

An Azcarraga-Lukierski supersymmetric particle model, which is invariant with respect to theN=2 extended SuperPoincare group, is quantised by the method of Dirac brackets. The central charges, which satisfy the spin reducing conditions, are conjugate to additional bosonic coordinates in the superspace. The resulting superwavefunction doublet is in one-to-one correspondence with the superfield expansion of Sohnius describing the Fayet hypermultiplet.     

Supersymmetry algebras that include topological charges

We show that in supersymmetric theories with solitons, the usual supersymmetry algebra is not valid; the algebra is modified to include the topological quantum numbers as central charges. Using the corrected algebra, we are able to show that in certain four dimensional gauge theories, there are no quantum corrections to the classical mass spectrum. These are theories for which Bogomolny has derived a classical bound; the argument involves showing that Bogomolny's bound is valid quantum mechanically and that it is saturated.     

Supersymmetry and functional integration measures

We complete the proof of a recently proposed new characterization of scalar supersymmetric theories and extend this result to “non-scalar” models such as supersymmetric gauge theories. The new characterization does not make use of anticommuting variables since supersymmetry can now be directly understood as a property of certain purely bosonic functional integration measures where all fermionic variables have been “integrated out”.