Gravity models with matter in two dimensions and their symmetries

Various couplings of massless matter including fermion and/or scalar fields to 2-dim. gravity theory are investigated in a uniform way using a general gauge invariant description. This formalism uses lorentz- and diffeomorphism invariant functions associated with matter fields, e.g. the fermion phases as coordinates. This yields a conformal representation for all cases including minimally coupled matter with full degrees of freedom (both chiralities ... ). In these general coordinates the system of field equations is reduced to two nonlinear third order partial differential equations in the function X associated with curvature (in the minimally coupled cases) or a dilation field. An equivalence of a theory with torsion to a torsionless theory resulting from a redefinition of potential can be shown. The two third order equations can be reduced to a fourth order equation corresponding to all minimally coupled cases as well as to matterless theory in general conformal gauge. Generally solvable models are presented: chiral fermions as well as the corresponding scalar field theory containing only the selfdual or antiselfdual part, furthermore the flat space theories of constant potential being conformally equivalent to CGHS-dilaton theory, and the teleparallelism theory of pure torsion. A special feature of matterless gravity is an absolutely conserved quantity corresponding to a nonlinear global symmetry according to Noether's theorem. By the introduction of matter coupling, this symmetry is broken and the corresponding conservation law is modified by matter coupling contributions. These matter corrections obey a consistency equation. As a consequence the generalization of this conserved quantity to theories with matter involves nonlocal expressions for the matter coupling part. (author)