  In a wide class of $G_L\times G_R$ 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 $\mathcal{N}=(2,2)$ supersymmetric
gauge theory without a superpotential possesses $U(1)_L\times U(1)_R$ symmetry
and contains one massless bosonic state per fixed spatial momentum. The
$\mathcal{N}=(4,4)$ supersymmetric pure Yang-Mills theory possesses
$SU(2)_L\times SU(2)_R$ symmetry, and there exist at least three massless
bosonic states.
