Formula omitted]-Gravities, toda theories and the geometry of the central charge

Abstract

Classical two-dimensional conformal field theories such as Toda models that are invariant under classical Virasoro or W -algebra symmetries with non-zero central charges are discussed. A naive gauging of such symmetries by coupling to gravity or W -gravity is shown not to work if there are central charges, and two successful ways of gauging are presented. The first is Weyl invariant and requires the introduction of spin-zero fields which can be thought of as gauge fields for gauge transformations generated by the central charges. The second involves, for the Virasoro algebra, a “non-geometric” coupling to all components of the metric and violates Weyl invariance. For the formulation with gauge fields corresponding to the central charges, BRST quantization requires ghosts for the transformations generated by the central charge. The resulting BRST charge is nilpotent for any value of the central charge, i.e. Q 2 = 0 for any conformal field theory. However, there are no non-trivial BRST cohomology classes unless the total central charge, including ghost contributions, is zero. Non-critical strings are quantized using this approach. There are no anomalies for any value of the space-time dimension D and if D = 26, the usual string spectrum is obtained. However, if D ≠ 26, all Fock-space physical states are BRST-trivial, so that there are no particles in the spectrum. In this sense, non-critical strings can be thought of as topological field theories. Models in which the central charges arise dynamically are studied. In these models, the role of each central charge is played by the expectation value of a spin-zero field, the field equations of which force it to be constant. The field is the Noether current for a spin-zero symmetry. Examples include Toda theories coupled to certain topological field theories, with the “central charge scalars” arising as currents in the topological sector.