(4,4) Superfield supergravity

We present the N = 4 superspace constraints for the two-dimensional off-shell (4,4) supergravity with the superfield strengths expressed in terms of a (4, 4) twisted (scalar) multiplet TM-I, as well as the corresponding component results, 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. To solve the constraints, we propose an ansatz which makes the N = 4 superconformal flatness of the N = 4 supergravity curved superspace manifest. Locally (4, 4) supersyrnmetric TM-I matter couplings are constructed, with potential terms resulting from spontaneous supersymmetry breaking. We also find the full (4, 4) superconformally invariant (improved) TM-II matter action. The latter can be extended to the (4, 4) locally supersyrnmetric Liouville action which is suitable for describing (4, 4) supersyrnmetric non-critical strings.