J.W. van Holten

The non-linear supersymmetric σ model on E6/SO(10) × U(1)

Abstract We give the explicit SO (10) × U (1) decomposition of the E 6 algebra. On this basis we find a non-linear realization of the E 6 SO (10) × U (1) coset space. We obtain the Kahler potential of this manifold in closed form and use it to construct the corresponding supersymmetric σ model. It involves a 16 family of quasi-Goldstone fermions whose four-fermion interactions are effectively those of an SO(10) GUT. Possible applications are briefly discussed.

Quantum theory of compact and non-compact σ models

Abstract We discuss quantization of SO( N + 1) σ models and CP N models, and of certain non-compact counterparts, SO( N , 1) and QP N respectively, of these, both in canonical operator formalism and the covariant path integral formulation, showing the equivalence of the two approaches. We discuss also a class of supersymmetric σ models formulated in d ⩽ 3 dimensions and apply the results to the SO( N + 1) and SO( N , 1) cases. This allows us to calculate the Witten index in each case. For SO(2 l + 1,1) we thereby find supersymmetry breaking. However, for SO(2 l , 1), we find supersymmetry is unbroken. Moreover, there is no unique ground state, invariant under SO(2 l , 1), rather an infinite multiple of zero energy states, carrying a unitary irreducible representation of the non-compact SO(2 l , 1) group. We discuss also field theoretic aspects of the models in d ⩾ 2 dimensions, stressing differences of the non-compact to the compact cases. These include infrared instead of ultraviolet asymptotic freedom, lack of an energy gap, failure (in the QP N case) of the auxiliary vector field to become dynamical. A further conclusion that is argued concerns the absence of a consistent particle interpretation for the QP N model in exactly two dimensions. For d > 2 the non-compact symmetry of QP N is broken down to the compact subgroup.

Gauged supersymmetric σ models and E6/SO(10)×U(1)

Abstract We construct gauged supersymmetric σ-models and their coupling to extra fermion families in chiral multiplets. We construct the coset space E 6 /SO(10)×U(1) and describe its Kahler geometry. We apply these results to supersymmetric models of quarks and leptons with non-linear E 6 symmetry. In most of these models supersymmetry is broken spontaneously. The (pseudo)-Goldstone bosons are then found to behave like Higgs scalars inducing a spontaneous breaking of SO(10)×U(1). An analysis of the scalar potential reveals an unstable stationary point with a residual SU(5)×U(1) invariance and a stable minimum with SO(7)×U(1) symmetry.

Matter coupling in supersymmetric σ-models

Abstract The general invariant coupling of matter fields to σ-models on Kahler manifolds is presented. Supersymmetry is seen to impose some restrictions on the allowed representations for the matter fields. The gauging of the global symmetries is discussed.

Composite gauge bosons in sigma models and supergravity

We discuss the non-compact SU(n, 1)SU(n) × U(1) sigma models and show there is no dynamical generation of the composite gauge boson in these models in two and three dimensions. We discuss some necessary conditions for the SU(8) gauge boson in N = 8 supergravity to become dynamical.

Anomalies in supersymmetric σ-models

Abstract We compute the anomaly density for the non-linear symmetries of supersymmetric σ-models on Kahler manifolds formulated in terms of unconstrained fields. We check that our expression for the anomaly satisfies the Wess-Zumino consistency condition.

Symmetry breaking in gauged supersymmetric sigma-models

Abstract We discuss several aspects of gauged supersymmetric sigma-models: off-shell formulation, supersymmetry breaking and breaking of internal symmetry. Some specific examples are given.

Structure of Grassmannian σ-models

The construction of σ-models on compact and non-compact Grassmannian manifolds is discussed. The relation with Kähler geometry is reviewed. General matter couplings are constructed, including the supersymmetric ones. Coupling to vector fields by gauging the global symmetries is discussed as well.

Non-compact sigma models and composite gauge bosons

Abstract It is shown that composite gauge bosons can arise in non-compact sigma models in four-dimensional space-time, provided there is a potential giving masses to the fields. In the case of the SU(1,1)/U(1) model this breaks the non-compact symmetry and reproduces the scalar sector of gauged N =4 supergravity. Non-abelian and supersymmetric generalizations are possible.

Some dynamical aspects of compositeness in quantum field theory

Abstract The dynamics of composite particles, in particular composite vector bosons, is discussed. An example in two dimensions illustrates the relevance of asymptotic freedom. Several dynamical models in four dimensions are investigated. Negative metric ghost fields, such as occur in theories with non-compact symmetries, may cure some of the problems encountered. Possible applications are discussed.