2D N=(4,4) superspace supergravity and bi-projective superfields

We propose a new superspace formulation for N=(4,4) conformal supergravity in two dimensions. This is based on a geometry where the structure group of the curved superspace is chosen to be SO(1,1) x SU(2)_L x SU(2)_R. The off-shell supergravity multiplet possesses super-Weyl transformations generated by an unconstrained real scalar superfield. The new supergravity formulation turns out to be an extension of the minimal multiplet introduced in 1988 by Gates et. al. and it allows the existence of various off-shell matter supermultiplets. Covariant twisted-II and twisted-I multiplets respectively describe the field strength of an Abelian vector multiplet and its prepotential. Moreover, we introduce covariant bi-projective superfields. These define a large class of matter multiplets coupled to 2D N=(4,4) conformal supergravity. They are the analogue of the covariant projective superfields recently introduced for 4D and 5D matter-coupled supergravity but they differ by the fact that bi-projective superfields are defined with the use of two CP^1 instead of one. We conclude by giving a manifestly locally supersymmetric and super-Weyl invariant action principle in bi-projective superspace.