Fixed Points of the Yang-Mills Gradient Flow in the Vortex Sector
Abstract
We study the Yang-Mills gradient flow of center vortices in SU(2) gauge theory. For axially symmetric vortices the flow reduces exactly to a linear diffusion equation for the radial profile: a Gaussian core keeps its shape while its width grows as √(R2+4t), the flux is exactly conserved, and the thin vortex is the unique bounded fixed point, though not an attractor. Off-diagonal perturbations are governed by a pair of radial Schrödinger operators, and a variational computation shows that, frozen at any instant, the Abelian core has a tachyonic mode for every winding n≥ 1. Since the core spreads while the mode grows, we pass to self-similar variables, where the unstable modes are known in closed form, with growth exponents μn,m=n−1+m. Multiply wound vortices are therefore unstable, while the fundamental vortex is exactly marginal and stays Abelian. A scaling argument rules out any static vortex in the transverse plane, and the instability acquires a simple geometric meaning that shortens the path traced in SU(2) by the holonomy of a growing loop, lowering the winding by two. The flow thus drives every vortex toward the Abelian core of winding n bmod 2, with non-Abelian structure appearing only as the transient through which multiply wound cores unwind.
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BibTeX
@misc{silva2026fixed,
title = {{Fixed Points of the Yang-Mills Gradient Flow in the Vortex Sector}},
author = {A. M. Silva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33874},
eprint = {2610.33874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33874},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fixed Points of the Yang-Mills Gradient Flow in the Vortex Sector
AU - A. M. Silva
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33874
UR - https://arxiv.org/abs/2610.33874
AB - We study the Yang-Mills gradient flow of center vortices in SU(2) gauge theory. For axially symmetric vortices the flow reduces exactly to a linear diffusion equation for the radial profile: a Gaussian core keeps its shape while its width grows as √(R^2+4t), the flux is exactly conserved, and the thin vortex is the unique bounded fixed point, though not an attractor. Off-diagonal perturbations are governed by a pair of radial Schrödinger operators, and a variational computation shows that, frozen at any instant, the Abelian core has a tachyonic mode for every winding n≥ 1. Since the core spreads while the mode grows, we pass to self-similar variables, where the unstable modes are known in closed form, with growth exponents μ_n,m=n−1+m. Multiply wound vortices are therefore unstable, while the fundamental vortex is exactly marginal and stays Abelian. A scaling argument rules out any static vortex in the transverse plane, and the instability acquires a simple geometric meaning that shortens the path traced in SU(2) by the holonomy of a growing loop, lowering the winding by two. The flow thus drives every vortex toward the Abelian core of winding n bmod 2, with non-Abelian structure appearing only as the transient through which multiply wound cores unwind.
ER -