Integrability in string/field theories and Hamiltonian Flows in the Space of Physical Systems

TL;DR

This paper explores how integrability in string and field theories can be understood through Hamiltonian flows in the space of physical systems, linking dynamics in moduli space to integrable structures and zero-curvature conditions.

Contribution

It introduces a framework connecting Hamiltonian flows in theory space with integrability, including classical and quantum aspects, and illustrates this with mechanical systems and Whitham equations.

Findings

Hamiltonian flows generate integrable families of systems.

Quantization leads to zero-curvature conditions in coupling constant space.

Example with mechanical systems demonstrates the approach.

Abstract

Integrability in string/field theories is known to emerge when considering dynamics in the moduli space of physical theories. This implies that one has to look at the dynamics with respect to unusual time variables like coupling constants or other quantities parameterizing 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 here an example of mechanical integrable systems. Then, any function $T(\vec p, \vec q)$ generates a one-parametric family of integrable systems in vicinity of a single system. For integrable system with several coupling constants the corresponding "Hamiltonians" $T_i(\vec p, \vec q)$ satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants: $$ [\frac{\partial}{\partial g_a} - \hat T_a(\hat{\vec p},\hat{\vec q}),\ \frac{\partial}{\partial g_b} - \hat T_b(\hat{\vec p},\hat{\vec q})] = 0 $$