Equivalence of the self-dual and Nambu-Goto strings

We establish explicitely the relation between the algebraic and Nambu-Goto strings when the target space is a four dimensional flat space. We find that the two theories are exactly equivalent only when the algebraic string is restricted to the self-dual or anti self-dual sectors. In its Hamiltonian formulation, the algebraic string defines a constrained system with first and second class constraints. In the self-dual case, we exhibit the appropriate set of second class constraints such that the resulting physical phase space is formulated in the same way as it is in the standard Nambu-Goto string. We conclude with a discussion on alternative quantisation schemes.