We consider the (1ϩ1)-dimensional Nϭ(8,8) supersymmetric matrix field theory obtained from a dimensional reduction of ten dimensional Nϭ1 super Yang-Mills theory. The gauge groups we consider are U(N) and SU(N), where N is finite but arbitrary. We adopt light-cone coordinates, and choose to work in the light-cone gauge. Quantizing this theory via discretized light-cone quantization ͑DLCQ͒ introduces an integer, K, which restricts the light-cone momentum-fraction of constituent quanta to be integer multiples of 1/K. Solutions to the DLCQ bound state equations are obtained for Kϭ2, 3 and 4 by discretizing the light-cone super charges, which preserves supersymmetry manifestly. We discuss degeneracies in the massive spectrum that appear to be independent of the light-cone compactification, and are therefore expected to be present in the decompactified limit K→ϱ. Our numerical results also support the claim that the SU(N) theory has a mass gap. ͓S0556-2821͑98͒01922-5͔

Spontaneous breakdown of the continuous symmetry is studied in the framework of discretized light-front quantization. We consider linear sigma model in 3+1 dimension and show that the careful treatment of zero modes together with the regularization of the theory by introducing NG boson mass leads to the correct description of Nambu-Goldstone phase on the light-front.Recently the light-front (LF) quantization with a Tamm-Dancoff truncation has attracted much attention as a promising method for solving QCD and other strong coupling theories and indeed it describes the bound state spectra successfully in various field theoretical models in (1+1) dimensions [1]. However, there remain difficulties in applying the present-stage formulation directly to (3+1) dimensional gauge theories which are working tools of modern particle physics. One of them is to understand the phenomenon of spontaneous symmetry breaking (SSB) on the LF and it may provide a cornerstone for the nonperturbative LF QCD. For the discrete symmetry, several authors discussed the possibility of SSB in (1+1) dimensional scalar models and argue that the solution of the zero-mode constraints[2] may realize the broken vacuum [3]. *

The quantum and classical aspects of a deformed c = 1 matrix model proposed by Jevicki and Yoneya are studied. String equations are formulated in the framework of the Toda lattice hierarchy. The Whittaker functions now play the role of generalized Airy functions in c < 1 strings. This matrix model has two distinct parameters. Identification of the string coupling constant is thereby not unique, and leads to several different perturbative interpretations of this model as a string theory. Two such possible interpretations are examined. In both cases, the classical limit of the string equations, which turns out to give a formal solution of Polchinski's scattering equations, shows that the classical scattering amplitudes of massless tachyons are insensitive to deformations of the parameters in the matrix model. †

We study the zero mode and the spontaneous symmetry breaking on the light front (LF). We use the discretized light-cone quantization (DLCQ) of Maskawa-Yamawaki to treat the zero mode in a clean separation from all other modes. It is then shown that the Nambu-Goldstone (NG) phase can be realized on the trivial LF vacuum only when an explicit symmetry-breaking mass of the NG boson m π is introduced. The NG-boson zero mode integrated over the LF must exhibit singular behavior ∼ 1/m 2 π in the symmetric limit m π → 0, which implies that current conservation is violated at zero mode, or equivalently the LF charge is not conserved even in the symmetric limit. We demonstrate this peculiarity in a concrete model, the linear sigma model, where the role of zero-mode constraint is clarified. We further compare our result with the continuum theory. It is shown that in the continuum theory it is difficult to remove the zero mode which is not a single mode with measure zero but the accumulating point causing uncontrollable infrared singularity. A possible way out within the continuum theory is also suggested based on the "ν theory". We finally discuss another problem of the zero mode in the continuum theory, i.e., no-go theorem of Nakanishi-Yamawaki on the non-existence of LF quantum field theory within the framework of Wightman axioms, which remains to be a challenge for DLCQ, "ν theory" or any other framework of LF theory.