Soliton-fermion systems and stabilised vortex loops

In several self-coupled quantum field theories when treated in semi-classical limit one obtains solitonic solutions determined by topology of the boundary conditions. Such solutions, e.g. magnetic monopole in unified theories [1] [2] or the skyrme model of hadrons have been proposed as possible non-perturbative bound states which remain stable due to topological quantum numbers. Furthermore when fermions are introduced, under certain conditions one obtains zero-energy solutions [5][6] for the Dirac equations localised on the soliton. An implication of such zero-modes is induced fermion number [7] carried by the soliton. Metastable topological objects were studied in [8]. It was shown in[9] that metastable objects can also carry fermion zero-modes and this can render a metastable object stable if the induced fermion number is fractional. This is true even if the fermion number is violated by a Majorana mass term. This result can be extended to metastable states of the cosmic string, in the form of loops. These can also possess fermion zero modes[10][11] leading to their absolute stability. A summary of these results constitutes this conference contribution. The existence of zero-modes and the possibility of fractional induced charge are well known results for which the reader is referred to [7]. We begin by discussing the induced stability of metastable topological objects. In what follows the bosonic fields constituting the soliton are treated as classical, and the fermion field is quantised in this background. In such a background, the Dirac field is expanded as