Virasoro invariance and theory of internal string.

Abstract

Treating the Virasoro invariance of the string as an internal symmetry, we show how to construct gauge theories of the Virasoro group which could be interpreted as a theory of an internal string. To do this we first generalize the Kaplansky-Feigin-Fuks representation of the Virasoro algebra, and construct the invariant tensors which play the crucial role in gauge theories of the Virasoro group. Next we clarify the geometric meaning of the Kaplansky-Feigin-Fuks representation, and show that any five-dimensional field theory defined on the fiber bundle space made of the four-dimensional space-time and the internal string ${S}^{1}$ which is invariant under the space-time-dependent reparametrization of the string can be interpreted as a gauge theory of the Virasoro group from the four-dimensional point of view. We discuss two types of theories: one in which the space-time metric does not depend on the string coordinate and one in which the space-time metric does. In both cases we show that spontaneous symmetry breaking is possible. In the latter case we discuss how spontaneous symmetry breaking can generate an infinite tower of massive spin-two fields, and construct the massive spin-two fields explicitly. We argue that one of the gauge theories could provide a field-theoretic description of a real string, in the point limit of the string.