Integrability in String/Field Theories and Hamiltonian Flows in the Space of Physical Systems

Integrability in string/field theories is known to emerge when considering dynamics in the moduli space of physical theories. This implies that one must study the dynamics with respect to unusual time variables such as coupling constants or other quantities parameterizing the configuration space of physical theories. The dynamics given by variations of coupling constants can be considered as a canonical transformation or, infinitesimally, a Hamiltonian flow in the space of physical systems. We briefly consider an example of integrable mechanical systems. Then any function $T(\vec p,\vec q)$ generates a one-parameter family of integrable systems in the vicinity of a single system. For an integrable system with several coupling constants, the corresponding “Hamiltonians” $T_i(\vec p,\vec q)$ satisfy the Whitham equations and, after quantization (of the original system), become operators satisfying the zero-curvature condition in the coupling-constant space.