Three self-similar solutions of Yang-Mills equations in high odd dimensions

TL;DR

This paper constructs and analyzes self-similar solutions to spherically symmetric Yang-Mills equations in high odd dimensions, revealing a pattern in the number of solutions related to polynomial zeros, with some results verified computationally.

Contribution

It establishes the existence and uniqueness of multiple self-similar solutions for high odd-dimensional Yang-Mills equations and provides an explicit method to determine their number.

Findings

Number of solutions N=3 for all tested dimensions from 11 to 31

Explicit polynomial P_m(z) determines the solutions

Computational evidence suggests N=3 for all odd d≥11

Abstract

We consider spherically symmetric Yang-Mills equations with gauge group $SO(d)$ in $d+1$ dimensional Minkowski spacetime. For any given odd $d\geq 11$, we establish existence and uniqueness (modulo reflection symmetry) of exactly $N$ smooth self-similar solutions, where $N$ is the number of zeros of an explicit polynomial $P_m(z)$ of degree $m=(d-5)/2$ in the interval $0<z<1$. The number $N$ can be determined algorithmically by an explicit computation. We find that $N=3$ for all integer $m$ from $3$ to $15$, the upper bound being merely limited by the extent of our computations. A proof that $N=3$ for all odd $d\ge 11$ remains an open problem.