Geometrical representation of the multi-dimensional consistency : 1-form case

A new notion of integrability called the multi-dimensional consistency for the integrable systems with the Lagrangian 1-form structure is captured in the geometrical language both classical and quantum levels. A zero-curvature condition, which implies the multi-dimensional consistency, will be a key relation in various contexts, e.g. Hamiltonian vector fields, Lagrangian vector fields, temporal Lax matrices and Hamiltonian operators. Therefore, the existence of the zero-curvature condition directly leads to the path-independent feature in various maps (which will be expressed in terms of the Wilson line), e.g. a family of maps in N-parameter groups on cotangent and tangent bundles, time evolution maps in the Lax pair level and unitary multi-time evolution operators in the Schrödinger picture. The main highlights of this work are the following. The new mathematical objects, called Hamilton vector field and Lagrange vector field defined in the space of independent variables, are introduced to alternatively capture integrability of the systems. To ensure the integrability, these new vector fields must be conservative and irrotational. Another important result is the formulation of the continuous multi-time propagator. With this new type of the propagator, a new perspective on summing all possible paths unavoidably arises as not only all possible paths on the space of dependent variables but also on the space of independent variables must be taken into account. The semi-classical approximation is applied to the multi-time propagator expressing in terms of the classical action and the fluctuation around the classical path and, therefore, the integrability of the quantum system can be captured through the path independent feature on the space of independent variables of the multi-time propapgator. Mathematics Subject Classification: 37J70