International Journal of Modern Physics A

We review the recent work on the light-cone quantization of (1+1)-dimensional large N QCD coupled to adjoint matter. The spectrum of these models consists of glueball-like bound states which, in the large N limit, can be pictured as closed strings that cannot split or join. The pair creation (annihilation) inside the string is not suppressed even in the large N limit, giving rise to complex dynamics similar to that of genuine higher dimensional gauge theories. The mass spectra of such models exhibit an exponential growth of the density of states (Hagedorn behavior) and the absence of massless glueballs even in the case of massless constituents.

The tricritical Ising model perturbed by the subleading energy operator was known to be an integrable scattering theory of massive kinks,¹⁴ and in fact it preserves supersymmetry. We consider here the model defined on the half-plane with a boundary and compute the associated factorizable boundary S matrix. The conformal boundary conditions of this model are identified and the corresponding S matrices are found. We also show how some of these S matrices can be perturbed and generate “flows” between different boundary conditions.

A bicovariant calculus on the twisted inhomogeneous multiparametric q groups of the B_n, C_n, D_n types, and on the corresponding quantum planes, is found by means of a projection from the bicovariant calculus on B_n+1, C_n+1, D_n+1. In particular we obtain the bicovariant calculus on a dilatation-free q Poincaré group ISO_q(3, 1), and on the corresponding quantum Minkowski space.

The classical limit of the B_n, C_n, D_n bicovariant calculus is discussed in detail.

Using Dirac’s analysis of second class constraints, a Hamiltonian formulation of higher order Lagrangians is reached. The new Hamiltonian description recovers the conventional description of such systems due to Ostrogradsky, but unlike the latter, it provides a simpler structure for the phase space and consequently the underlying symplectic structure.

Curvature–induced phase transition is thoroughly investigated in a four-fermion theory with N components of fermions for arbitrary space–time dimensions (2≤D<4). We adopt the 1/N expansion method and calculate the effective potential for a composite operator . The resulting effective potential is expanded asymptotically in terms of the space–time curvature R by using the Riemann normal coordinate. We assume that the space–time curves slowly, and we keep only terms independent of R and terms linear in R. In evaluating the effective potential it is found that first order phase transition is caused and the broken chiral symmetry is restored for a large positive curvature. In the space–time with a negative curvature the chiral symmetry is broken down even if the coupling constant of the four-fermion interaction is sufficiently small. We present the behavior of the dynamically generated fermion mass. The critical curvature, R_cr, which divides the symmetric and asymmetric phases, is obtained analytically as a function of the space–time dimension D. At the four-dimensional limit our result R_cr agrees with the exact results known in de Sitter space and the Einstein universe.

We investigate the Chung–Fukuma–Shapere theory, or Kuperberg theory, of three-dimensional lattice topological field theory. We construct a functor which satisfies Atiyah’s axioms of topological quantum field theory by reformulating the theory as a Turaev–Viro type state sum theory on a triangulated manifold. This corresponds to giving the Hilbert space structure to the original theory. The theory can be extended to give a topological invariant of manifolds with boundary.

A general construction is found for “topological” singular vectors of the twisted N=2 superconformal algebra. It demonstrates many parallels with the known construction for affine sℓ(2) singular vectors due to Malikov–Feigin–Fuchs, but is formulated independently of the latter. The two constructions taken together provide an isomorphism between the topological and affine sℓ(2) singular vectors. The general formula for topological singular vectors can be reformulated as a chain of direct recursion relations that allow one to derive a given singular vector | S(r, s)〉 from the lower ones | S(r, s′<s)〉. We also introduce generalized Verma modules over the twisted N=2 algebra and show that they provide a natural setup for the new construction for topological singular vectors.