Deducing the existence of a disconnected solution

[ShayanJ]

I'm trying to read this paper. At some point, it tries to find the minimums of an action. At first it deduces the existence of two independent solutions and says that one of them is a minimum and the other is a saddle point. Then goes on to mention that this means that actually there is another minimum and the saddle point is actually a maximum. But I don't understand the argument presented. Here's what it says:

We see that for small l the equation of motion (13) has two independent solutions, one with
large $U^∗$ and the other with $U^∗ ≃ U_0$ . The former is a local minimum of the action (15) while the latter is a saddle point. We can interpolate between them with a sequence of curves which differ in the minimal value of U, such that the solution with large U ∗ is a local minimum along this sequence, while the one with $U^∗ ≃ U_0$ is a local maximum.This implies that there must be another local minimum of the effective action, with $U_∗$ smaller than that of the saddle point. This solution cannot correspond to a smooth $U(x)$, since then it would be captured by the above analysis. Therefore, it must correspond to a disconnected solution, which formally has $U^∗ = U_0$ , but is better described as two disconnected surfaces that are extended in all spatial directions except for x, and are located at $x = ± \frac l 2$ .

$U^*$ is the maximum of the curve but because of the particular direction of the coordinate U, the maximum of the curve happens at a smaller U than the rest of the curve and so the paper calls it the minimum of the curve.
Can anyone explain this argument to me?
Thanks

 

[Paul C]

The half-integer is by itself characteristic of a link, and I assume you've made reference to the abelian U. On the whole, what you've stated is quite reasonable insofar as to describe it as "two disconnected surfaces". It's a bit over my head, so sorry if I can't break it down further.