# What is the relationship between the JNR instantons and the BPST instanton?

The JNR instantons are related to the t'Hooft ansatz, and take the form $$A=\sigma_{\mu\nu}\frac{\partial_\nu \rho}{\rho}dx^\mu,$$ where $\rho$ takes the form $$\rho=\sum \frac{\lambda_p}{\lvert x-x_p\rvert^2}.$$ The Belavin et al solution takes the form $$A_{\mu}=\frac{\eta^{a}_{\mu\nu}(x-z)_\nu}{(x-z)^2+c^2},$$ up to a constant, where $\eta^a$ is the t'Hooft symbol (defined on the wikipedia page https://en.wikipedia.org/wiki/BPST_instanton). In principle these should agree for the one instanton solution, but I do not see in what way this is the case. In the singular gauge of the BPST instanton it is still not clear what is the relationship.
I have gotten some of the formulas above from Solitons, Instantons, and Twistors by Dunajski and from wikipedia for the BPST instanton.

• Well, for one thing, the first one is a Lie-algebra (matrix) valued one-form, and the second one a scalar function. So the correspondence is far from straightforward. Are you sure the $\eta_{\mu\nu}$ is not actually an 't Hooft symbol $\eta^a_{\mu\nu}$? – AccidentalFourierTransform Aug 9 '18 at 22:04
• I made a typo on the second one, so it is now the coefficient of a Lie algebra valued one form. You're right, $\eta$ is the t'Hooft symbol, which I also changed. Either way, the forms are not the same, but as far as I know these both describe basic one instantons. – Teddy Baker Aug 10 '18 at 15:16
• 1. What is a "JNR instanton"? You linked a Wikipedia page for the BPST instanton but not for the JNR instanton. 2. What are $\sigma_{\mu\nu}$ and $\lambda_\rho$? 3. Are both of these formulae a) in the same gauge and b) in the same patch (i.e. around the origin as opposed to "around infinity")? 4. Have you tried computing the corresponding field strength and seeing whether it is easier to see that both produce the same field strength than to see that they're gauge-equivalent? – ACuriousMind Aug 11 '18 at 8:09

The BPST instanton is a self dual solution of the Euclidean Yang-Mills equations with instanton number $1$. 't Hooft ansatz generalizes the BPST instanton solution to multi-instanton solutions with instanton number $N$ greater than $1$. 't Hooft's family of solutions has $5N+3$ free parameters which except for $N=1$ does not span the whole $8N$ dimensional instanton moduli space.
The JNR construction is a generalization by Jackiw Nohl and Rebbi to a $5N + 7$ parameter solution space. These solutions support the action of the conformal group. For $N=1,2$ the JNR solution has more parameters than the instanton moduli space and it can be proven that they are redundant. For $N\gt 2$, these solutions do not span the whole instanton moduli space.