RG Equations from Whitham Hierarchy

TL;DR

This work provides a complete, Fay-calc-based derivation of how the second derivatives of the Seiberg-Witten prepotential, with respect to Whitham times, can be written in terms of Riemann theta-functions on Toda-chain spectral curves. The authors derive explicit formulas for first and second derivatives with respect to Whitham times, and for mixed derivatives with respect to both Whitham times and moduli, expressing results in residue form and via the Szegö kernel. A key outcome is a renormalization group type equation linking derivatives with respect to the dynamical scale Lambda to derivatives of theta-functions, along with a consistency framework connecting Lambda- and T1-derivatives of the prepotential. The results extend Kontsevich-model relations to the Seiberg-Witten/Whitham setting, offering a transcendental generalization with potential applications to Donaldson theory and Hitchin-type systems.

Abstract

The second derivatives of prepotential with respect to Whitham time-variables in the Seiberg-Witten theory are expressed in terms of Riemann theta-functions. These formulas give a direct transcendental generalization of algebraic ones for the Kontsevich matrix model. In particular case they provide an explicit derivation of the renormalization group (RG) equation proposed recently in papers on the Donaldson theory.