Solving N=2 SYM by Reflection Symmetry of Quantum Vacua

TL;DR

This paper demonstrates how reflection symmetries in quantum vacua of N=2 SU(2) SYM theory determine the moduli space structure and derive key functions matching Seiberg-Witten solutions from fundamental principles.

Contribution

It establishes the exact reflection symmetries and derives the functions a(u) and a_D(u) directly from first principles, confirming their equivalence with Seiberg-Witten solutions.

Findings

Reflection symmetry $ar{u}( au)=u(-ar{ au})$ and $u( au+1)=-u( au)$ hold exactly.

Functions a(u) and a_D(u) derived from first principles match Seiberg-Witten solutions.

The inverse relation between $ au$ and the uniformizing coordinate u clarifies the moduli space structure.

Abstract

The recently rigorously proved nonperturbative relation between u and the prepotential, underlying N=2 SYM with gauge group SU(2), implies both the reflection symmetry $\overline{u(\tau)}=u(-\bar\tau)$ and $u(\tau+1)=-u(\tau)$ which hold exactly. The relation also implies that $\tau$ is the inverse of the uniformizing coordinate u of the moduli space of quantum vacua. In this context, the above quantum symmetries are the key points to determine the structure of the moduli space. It turns out that the functions a(u) and a_D(u), which we derive from first principles, actually coincide with the solution proposed by Seiberg and Witten. We also consider some relevant generalizations.