Isomonodromic Properties of the Seiberg-Witten Solution

TL;DR

This paper explores the Seiberg-Witten solution of N=2 supersymmetric SU(2) gauge theories as an isomonodromy problem, revealing connections to conformal field theory, braid and fusion operations, and implications for BPS spectra and superconformal points.

Contribution

It demonstrates that Seiberg-Witten sections can be deformed via isomonodromy, linking them to conformal blocks and elucidating the monodromy structure and BPS spectrum.

Findings

Monodromies with affine terms are explicitly derived.

Braiding and fusing operations obey Yang-Baxter and Pentagonal identities.

Connections established between supersymmetric sections and logarithmic minimal models.

Abstract

The Seiberg-Witten solution of N=2 supersymmetric SU(2) gauge theories with matter is analysed as an isomonodromy problem. We show that the holomorphic section describing the effective action can be deformed by moving its singularities on the moduli space while keeping their monodromies invariant. Well-known examples of isomonodromic sections are given by the correlators of two-dimensional rational conformal field theories -- the conformal blocks. The Seiberg-Witten section similarly admits the operations of braiding and fusing of its singularities, which obey the Yang-Baxter and Pentagonal identities, respectively. Using them, we easily find the complete expressions of the monodromies with affine term, and the full quantum numbers of the BPS spectrum. While the braiding describes the quark-monopole transmutation, the fusing implies the superconformal points in the moduli space. In the simplest case of three singularities, the supersymmetric sections are directly related to the conformal blocks of the logarithmic minimal models.