Note on Massless Bosonic States in Two-Dimensional Field Theories

In a wide class of GL × GR invariant two-dimensional super-renormalizable field theories, the parity-odd part of the two-point function of global currents is completely determined by a fermion one-loop diagram. For any non-trivial fermion content, the two-point function possesses a massless pole which corresponds to massless bosonic physical states. As an application, we show that two-dimensional supersymmetric gauge theory without a superpotential possesses U(1) L × U(1) R symmetry and contains one massless bosonic state per fixed spatial momentum. The supersymmetric pure Yang-Mills theory possesses SU(2) L × SU(2) R symmetry, and there exist at least three massless bosonic states.