RG-Whitham dynamics and complex Hamiltonian systems

TL;DR

This paper explores the complex Hamiltonian systems inspired by Seiberg-Witten solutions, revealing how the Argyres-Douglas point acts as a fixed point with reduced degrees of freedom and connecting quantum mechanics relations to Whitham dynamics.

Contribution

It introduces a novel link between complex Hamiltonian dynamics, RG-like behavior, and the Whitham equations, especially at the Argyres-Douglas point, with implications for quantum mechanics.

Findings

At AD point, degrees of freedom reduce in Hamiltonian systems.

Anomalous dimensions at AD point match Berry indexes.

Dunne-"Unsal relation coincides with Whitham equations in Omega deformation.

Abstract

Inspired by the Seiberg-Witten exact solution, we consider some aspects of the Hamiltonian dynamics with the complexified phase space focusing at the renormalization group(RG)-like Whitham behavior. We show that at the Argyres-Douglas(AD) point the number of degrees of freedom in Hamiltonian system effectively reduces and argue that anomalous dimensions at AD point coincide with the Berry indexes in classical mechanics. In the framework of Whitham dynamics AD point turns out to be a fixed point. We demonstrate that recently discovered Dunne-\"Unsal relation in quantum mechanics relevant for the exact quantization condition exactly coincides with the Whitham equation of motion in the Omega - deformed theory.