Supersymmetry and the Nonlocal Yangian Deformation Symmetry

In the quantized two-dimensional non-linear supersymmetric σ-model, the supercurrent supermultiplet, which contains the energy-momentum tensor, is transformed by the nonlocal symmetry of the model into the isospin current supermultiplet. This effect incorporates supersymmetry into the known infinite-dimensional Yangian deformation symmetry of plain σ-models, leads to precisely the same nontrivial extension of the twodimensional super-Poincaré group as found previously for the Poincaré group, and thus determines the theory’s mass spectrum. A generalization to all higher-order nonlocal charges is conjectured such that their generating function, the so-called “master charge”, has a definite Lorentz spin which depends on the spectral parameter. We contribute this to Prof. Biedenharn’s Festschrift in recognition of his appreciation of symmetries in physics and his timely interest in promising features of Hopf algebras. Allez en avant, et la foi vous viendra. (J. D’ Alembert) One of the more productive proposals for probing the nonperturbative structure of field theory is the utilization of nonlocal disorder variables [1], predicated on the celebrated nonlocal symmetries [2] of solvable two-dimensional models, such as the nonlinear σ-model, the Gross-Neveu model, or their supersymmetric combination. Such nonlocal symmetries have provided determinations of the S-matrices of the respective models [3, 4]. More recently, in the wake of the advent of “Quantum Deformation” Hopf algebra applications in field theory, Bernard [5] noted that the algebraic structure of these nonlocal symmetries, in fact, comprises a “Yangian” algebra [6, 7], i.e. a non-co-commutative coproduct deformation of affine Lie algebras. By dint of its nontrivial coproduct action on composite states, the symmetry evades the Coleman-Mandula theorem, and, due to a remarkable but simple quantum effect, provides a nontrivial extension of the Poincaré group [5, 8, 9]: the lowest order conserved nonlocal charge Q(1) is not invariant under a Lorentz boost. This engenders additional “kinematic” constraints on the physical states of the underlying theory which Belavin [8] subsequently applied to cogently rederive the mass spectrum of the SU(N) nonlinear σ-model. Actually, such a quantal extension of the Poincaré group was to be expected on the basis of known results for local lagrangean field theories. On the one hand, as must be the case in a local field theory, any such extension of Work supported by the NSF grant PHY-92-09978. 2 Work supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38.