Abstract
We present the N=4 superspace constraints for the two-dimensional (2d) off-shell (4,4) supergravity with the superfield strengths expressed in terms of a (4,4) twisted (scalar) multiplet TM-I in a form suitable for applications. The constraints are shown to be invariant under the N=4 super-Weyl transformations, whose N=4 superfield parameters form another twisted (scalar) multiplet TM-II. The full (4,4) superconformally invariant (improved) TM-II matter action is found. The latter can be extended to the fully covariant and (4,4) locally supersymmetric Liouville action which is suitable for describing (4,4) supersymmetric non-critical strings.
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