The theory of non-Abelian tensor fields: Gauge transformations and curvature

We take one more step in formulating the theory of non-Abelian two-tensor fields: we find gauge transformation rules and the curvature tensor for them. To define the theory, we use the surface exponential. We derive a differential equation for the exponential and attempt to formulate its definition as a matrix model. We discuss applications of our construction to the Yang-Baxter equation for integrable models and to string field theory.