Jonathan M. Evans

Supersymmetric Toda field theories

We analyse a class of integrable two-dimensional field theories with N =1 and N = 2 supersymmetry via the Lax, or zero-curvature, construction based on a Lie superalgebra. Some of the theories exhibit extended superconformal symmetries, generalizing the bosonic W-algebras. We also investigate the possible integrable deformations of these models and find that in some cases supersymmetry is broken and in other cases it is preserved.

Integrable theories that are asymptotically CFT

Abstract A series of sigma models with torsion are analysed which generate their mass dynamically but whose ultra-violet fixed points are non-trivial conformal field theories — in fact SU(2) WZW models at level k . In contrast to the more familiar situation of asymptotically free theories in which the fixed points are trivial, the sigma models considered here may be termed “asymptotically CFI”. These theories have previously been conjectured to be quantum integrable; this is confirmed by postulating a factorizable S-matrix to describe their infra-red behaviour and then carrying out a stringent test of this proposal. The test involves coupling the theory to a conserved charge and evaluating the response of the free-energy both in perturbation theory to one loop and directly from the S-matrix via the Thermodynamic Bethe Ansatz with a chemical potential at zero temperature. Comparison of these results provides convincing evidence in favour of the proposed S-matrix; it also yields the universal coefficients of the beta-function and allows for an evaluation of the mass gap (the ratio of the physical mass to the Λ-parameter) to leading order in 1 k .

Complex Toda theories and twisted reality conditions

The Toda equations (based on a finite-dimensional or affine Lie algebra of superalgebra) are discussed as integrable non-linear differential equations for a set of complex scalar fields. We show that such complex Toda fields can either be restricted to take real values the standard way or else they can be subjected to a “twisted” reality condition associated to any Z2 symmetry of the Cartan matrix of Dynkin diagram of the underlying algebra. Different reality conditions give rise to different lagrangian field theories. In the conformal case, however, these theories have the same central charge, while in the affine case they have the same mass spectrum. The construction of N = 2 superconformal theories based on the superalgebras A(n, n−1) is clarified, and a new class of conformal field theories with positive kinetic energy based on the superalgebras C(n) is presented. The ideas developed are also relevant to understanding soliton solutions in affine Toda theories with imaginary coupling constant.

The exact mass-gap of the supersymmetric O(N) sigma model

Abstract A formula for the mass-gap of the supersymmetric O(N) sigma model (N > 4) in two dimensions is derived: m Λ MS =2 2Δ sin (πΔ) (πΔ) , where Δ = 1 (N−2) and m is the mass of the fundamental vector particle in the theory. This result is obtained by comparing two expressions for the free-energy density in the presence of a coupling to a conserved charge; one expression is computed from the exact S-matrix of Shankar and Witten via the thermodynamic Bethe ansatz and the other is computed using conventional perturbation theory. These calculations a provide a stringent test of the S-maxtrix, showing that it correctly reproduces the universal part of the beta-functional and resolving the problem of CDD ambiguities.

Supersymmetry algebras and Lorentz invariance for d=10 super Yang-Mills

Abstract We consider ways in which conventional supersymmetry can be embedded in the set of more general fermionic transformations proposed recently [1] as a framework in which to study d =10 super Yang-Mills. Solutions are exhibited which involve closed algebras of various numbers of supersymmetries together with their invariance groups: nine supersymmetries with G 2 ×SO(1,1) invariance; eight supersymmetries with SO(7)×SO(1,1) invariance; four supersymmetries with SO(3,1)×U(3) invariance. We recover in this manner all previously known ways of adding finite numbers of bosonic auxiliary fields so as to partially close the d =10 superalgebra. A crucial feature of these solutions is that the auxilliary fields transform non-trivially under the residual Lorentz symmetry, even though they are originally introduced as Lorentz scalars.

Integrable N=2 supersymmetric field theories

Abstract A classification is given of Toda-like theories with N = 2 supersymmetry which are integrable by virtue of some underlying Lie superalgebra. In addition to the N = 2 superconformal theories based on sl( m , m −1), which generalize the Liouville model, a family of massive N = 2 theories based on the algebras sl( m , m ) (1) is found, providing natural generalizations of the sine-Gordon theory. A third family of models based on sl( m , m ) which have global supersymmetry, a version of conformal invariance, but no superconformal invariance is also briefly discussed. Unlike their N = 0 and N = 1 cousins, the N = 2 massive theories apparently cannot be directly thought of as integrable deformations of the corresponding N = 2 superconformal theories. It is shown that these massive theories admit supersymmetric soliton solutions and a form for their exact S -matrices is conjectured.

Massive superparticles with Siegel symmetry and their covariant canonical quantisation

Abstract We study massive superparticles with Siegel symmetry as toy models of the Green-Schwarz string, and as mechanics actions furnishing minimal representations of supersymmetry upon quantisation. A general construction yielding all such theories is given, and it is shown that the simplest class can be derived from massless models in higher dimensions. We consider the quantisation of massive superparticles using the canonical methods of Dirac, and show that the first-class and second-class constraints can always be defined so as to respect all global symmetries, giving a completely covariant method of quantisation. (This cannot be done in the massless case.) The quantisation is carried out explicitly for the simplest class of models: Dirac brackets are computed and all necessary consistency checks performed. We prove that no operator-ordering problems arise by finding a set of variables for which the covariant Dirac brackets become free canonical graded commutation relations. By considering features shared by massive superparticles and the Green-Schwarz string (but not present for massless superparticles) we argue that the latter should admit a similar covariant quantisation.

Exact scattering in the SU(n) supersymmetric principal chiral model

Abstract The complete spectrum of states in the supersymmetric principal chiral model based on SU(n) is conjectured, and an exact factorizable S-matrix is proposed to describe scattering amongst these states. The SU(n)L × SU(n)R symmetry of the Lagrangian is manifest in the S-matrix construction. The supersymmetries, on the other hand, are incorporated in the guise of spin - 1 2 charges acting on a set of RSOS kinks associated with su(n) at level n. To test the proposed S-matrix, calculations of the change in the ground-state energy in the presence of a coupling to a background charge are carried out. The results derived from the Lagrangian using perturbation theory and from the S-matrix using the TBA are found to be in complete agreement for a variety of background charges which pick out, in turn, the highest weight states in each of the fundamental representations of SU(n). In particular, these methods rule out the possibility of additional CDD factors in the S-matrix. Comparison of the expressions found for the free-energy also yields an exact result for the mass gap in these models: m Λ MS = ( n π ) sin ( π n ) .

Auxiliary fields for super Yang-Mills from division algebras

Division algebras are used to explain the existence and symmetries of various sets of auxiliary fields for super Yang-Mills in dimensions d = 3, 4, 6,10.

Covariant separation of first-class and second-class constraints for the Green-Schwarz superstring

Abstract We define super-Poincare covariant sets of reducible first-class and second-class constraints for the Green-Schwarz superstring. These constraints can be treated using Diracs methods and allow, in principle, a fully covariant canonical quantisation.