Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models

TL;DR

This review explores the development of the effective prepotential in supersymmetric quantum field theories, highlighting its relation to matrix models and deformation theory over two decades of research.

Contribution

It provides a comprehensive overview of the evolution of the theory of effective prepotentials, emphasizing the connection to matrix models and deformation theory in supersymmetric QFT.

Findings

Effective prepotential corresponds to the free energy of matrix models.

Deformation theory on Riemann surfaces plays a key role in understanding prepotentials.

Historical development highlights the link between Seiberg-Witten theory and matrix models.

Abstract

This is a semi-pedagogical review of a medium size on the exact determination of and the role played by the low energy effective prepotential ${\cal F}$ in QFT with (broken) extended supersymmetry, which began with the work of Seiberg and Witten in 1994. While paying an attention to an overall view of this subject lasting long over the two decades, we probe several corners marked in the three major stages of the developments, emphasizing uses of the deformation theory on the attendant Riemann surface as well as its close relation to matrix models. Examples picked here in different contexts tell us that the effective prepotential is to be identified as the suitably defined free energy $F$ of a matrix model: ${\cal F} = F$. To be submitted to PTEP as an invited review article and based in part on the talk delivered by one of the authors (H.I.) in the workshop held at Shizuoka University, Shizuoka, Japan, on December 5, 2014.