Dynamical Generation of the VY Superpotential in $N=1$ SYM: A Higher-Form Perspective

TL;DR

This paper offers a semiclassical derivation of the VY superpotential in 4D N=1 SYM using higher-form gauge fields and topological sectors, connecting vortex dynamics and fractional instantons.

Contribution

It introduces a higher-form gauge perspective to derive the VY superpotential semiclassically, linking topological flux sectors to non-perturbative effects.

Findings

Reproduces the VY superpotential from higher-form gauge configurations.

Identifies fractional instantons as key non-perturbative contributions.

Shows the topological sector decomposition aligns with the $ ext{Z}_N$ structure.

Abstract

We present a semiclassical account of the Veneziano-Yankielowicz (VY) superpotential in four-dimensional $N=1$ super Yang-Mills theory. Motivated by two-dimensional gauged linear sigma models, where superpotentials arise from vortex dynamics, we reinterpret domain walls as fundamental objects associated with higher-form gauge fields. In this formulation, the vacuum structure is encoded in a compact three-form gauge field, whose four-form flux labels topological sectors. In the presence of charged matter with total charge $N$, these sectors exhibit a natural $\mathbb{Z}_N$ structure, leading to a decomposition into $N$ semiclassical contributions. These contributions arise from Euclidean point-like configurations in the higher-form sector, analogous to fractional instantons. We show that these configurations provide the relevant non-perturbative contributions to the effective superpotential. Integrating out the associated degrees of freedom reproduces the VY superpotential in the infrared. This gives a semiclassical origin of the VY superpotential in terms of higher-form gauge dynamics.