In this chapter we will formulate the free bosonic string and superstrings as a field theories in their critical dimensions in a way that manifestly possesses Lorentz invariance. While it is to be expected that these theories possess the gauge symmetries of the massless modes we will find that such a formulation of string theory possesses an infinite number of local symmetries. The actions have considerable elegance and make a surprising use of the BRST formalism even though they are classical objects.

One of the most remarkable aspects of the evolution of modern physics has been the central role played by symmetry. Themost important are local symmetries which place very strong constraints on the form of the theory that possesses them. This first discovery of a gauge symmetry was by Weyl in the electromagnetic equations of Maxwell and continued in the construction of the Standard Model which possesses the SU(3) ⊗ SU(2) ⊗ U(1) gauge symmetry. Another crucial tool which played an important role in modern physics is the Feynman path integral description of quantum field theory. This formulation possesses the advantage that it is constructed from an action which can be used to manifestly display the symmetries of interest and, as a result, one can more easily derive those properties of the theory which are a consequence of its symmetries.