In this article, we review recent work on constrained Hamiltonian systems with infinite-dimensional constraint algebras. The work is based on an extension of the constraint algebra to include subsidiary (gauge-fixing) constraints and a central element, forming a larger, closed algebra. This extension leads to several immediate results, at both a practical and a conceptual level: 1) A simple method is found for abelizing the original constraints. 2) This leads naturally to a “generalized vielbein” relating Abelian and non-Abelian quantities. 3) The vielbein apparatus makes transparent the full global symmetry of the extended phase space associated with the Batalin, Fradkin and Vilkovisky formalism. An interesting duality also arises between quantities with “Abelian” and “non-Abelian” indices. String theory provides the most important and straightforward application of these general methods. For this special case, one finds that the vielbeins are the Fourier components of the string vertex operator. The vertex operator describes the physical process of photon emission, and determines the physical modes of the string by way of the DDF operators. Due to the relation between the vertex operator and the vielbein, the dual abelized Virasoro operators simplify the usual discussion of the physical Fock space. The correspondence between these two fundamental objects, the vertex and the vielbein, thus provides a geometric understanding of the usual algebraic approach to string theory.
Vertex operator in String Theory on Orbifold: Intertwining Fields between Different Sectors in the Fock Space Representation
