Dynamical Signature: Complex Manifolds, Gauge Fields and Non-Flat Tangent Space

Theoretical possibilities of models of gravity with dynamical signature are discussed. The different scenarios of the signature change are proposed in the framework of Einstein-Cartan gravity. We consider, subsequently, the dynamical signature in the model of the complex manifold with complex coordinates and complex metrics are introduced, a complexification of the manifold and coordinates through new gauge fields, an additional gauge symmetry for the Einstein-Cartan vierbein fields, and non-flat tangent space for the metric in the Einstein-Cartan gravity. A new small parameter, which characterizes a degree of the deviation of the signature from the background one, is introduced in all models. The zero value of this parameter corresponds to the signature of an initial background metric. In turn, in the models with gauge fields present, this parameter represents a coupling constant of the gauge symmetry group. The mechanism of metric determination through induced gauge fields with defined signatures in the corresponding models is considered. The ways of the signature change through the gauge field dynamics are reviewed, and the consequences and applications of the proposed ideas are discussed as well.