N = 1 / 2 supergravity with matter in four Euclidean dimensions

An N = 1 / 2 supergravity in four Euclidean spacetime dimensions, coupled to both vector- and scalar-multiplet matter, is constructed for the first time. We begin with the standard N = (1, 1) conformally extended supergravity in four Euclidean dimensions, and freeze out the graviphoton field strength to an arbitrary (fixed) self-dual field (the so-called C-deformation). Though a consistency of such procedure with local supersymmetry is not guaranteed, we find a simple consistent set of algebraic constraints that reduce the local supersymmetry by 3/4 and eliminate the corresponding gravitini. The final field theory (after the superconformal gauge-fixing) has the residual local N = (0, frac(1, 2)) or just N = 1 / 2 supersymmetry with only one chiral gravitino as the corresponding gauge field. Our theory is not 'Lorentz'-invariant because of the non-vanishing self-dual graviphoton vacuum expectation value, which is common to the C-deformed N = 1 / 2 rigidly supersymmetric field theories constructed in a non-anticommutative superspace.