RG Equations from Whitham Hierarchy

TL;DR

This paper derives RG equations within Seiberg-Witten theory using Whitham hierarchy, expressing second derivatives of the prepotential via Riemann theta-functions, thus generalizing algebraic formulas and connecting to Donaldson theory.

Contribution

It introduces a transcendental approach to RG equations in Seiberg-Witten theory using Riemann theta-functions, extending previous algebraic methods.

Findings

Derived explicit formulas for second derivatives of prepotential

Connected RG equations to Riemann theta-functions

Provided a new derivation of RG equations in Donaldson theory

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.