String representation for a field theory with internal symmetry

Abstract

It is possible to represent certain quantum field theories as theories of interacting strings when both are defined on a suitable null-plane lattice. This representation is discussed for scalar field theories with internal degrees of freedom and quartic self-couplings. The internal-symmetry structure of the lattice string is that of an appropriate two-dimensional statistical-mechanical vertex model. The internal degrees of freedom are frozen out in the continuum limit (confinement) unless the vertex model is critical. The simplest internal symmetry, U(1), corresponds to Pauling's model of two-dimensional ice, which is critical. We compute the excitation energies of the internal degrees of freedom of the $F$ model, a generalization of the ice model. The $F$ model is characterized by a parameter $\Delta <1\mathrm{}$, and ice corresponds to $\Delta =\frac{1}{2}$. It is critical for $-1<\Delta <1\mathrm{}$. For $\Delta <-1\mathrm{}$ charged states have infinite energy in the continuum limit. For $-1<\Delta <1\mathrm{}$ the spectrum of long-wavelength excitations is that of a free one-dimensional semiperiodic boson field defined on a finite space. When the space is a ring there is a conserved topological charge as well as ordinary charge. The effect of these excitations on the resulting dual model is to contribute one degree of freedom reducing the critical dimension of space-time by 1.