How "hedgehog" or instanton event changes the configuration of neel vector in 2D antiferromagnet?

The definition of "hedgehog" or instanton event here is "a space-time event where the skyrmion number Q changes by $$\pm1$$ is called a hedgehog" (ref.1 & 2).

A nice figure illustrates such an instanton event is found in ref.3, where the initial configuration has non-zero skyrmion number, and the final configuration after the instanton event has zero skyrmion number. Also, close to this instanton event, a hedgehog figure is usually presented in papers (such as ref.2), where all the neel vectors are putting outward. My questions are

1. Does the instanton event change all the neel vectors? For example, as the first figure, the instanton makes all the neel vector point to up direction. Does the instanton just make a part of neel vector flip their arrow (just the part point to down direction) ? Then maybe I want to know the size of the effect.
2. Does the hedgehog (the second figure posted here)represents all the neel vector of a configuration? (It seems that every slice close to the instanton in a fixed time has skyrmion number 0).
3. Does the instanton event changes the skyrmion number of all possible configuration? For example, if the initial configuration has 0 skyrmion number, does instanton event change it to a configuration with $$\pm1$$ skyrmion number ? (Yes, it is possible for the skyrmion number from 0 to 1 as in ref., this means that instanton event may not changes every neel vector, some of them are "flipped", but remained part is almost unchanged, then they form a skyrmion number 1 configuration. Then as in question 1, may be we can ask how many neel vectors are flipped which is related to the "size" of this hedgehog. ) Or if the initial configuration has 1 skyrmion number, is that possible to have an instanton event and changes the configuration to the one with skyrmion number 2?

At the end, I find Haldane's paper (ref.1) is very difficult to follow, (the main goal is to calculate the Berry phase including the effect from monopole/hedgehog/instanton, Sachdev and Read calculate this term by another method-large N method). I find there is a section about this problem in Sachdev's book (Quantum phase transition), but he says "This is a mathematical step, and the details are given by Haldane" in page 393. Is there any detailed derivation for this paper?

One more question, how to interpret "When we have a monopole event in the spacetime, it must give a branch-cut structure on the image of the solid angle $$\omega_i$$ (i is the lattice index) , because $$\omega_i$$ must change by $$4\pi$$ around the monopole location in the space"(ref.9). I can understand $$\sum_{i\in p}\omega_i=4\pi$$, when p is a plaquette surrounding a hedgehog/instanton/monopole, just like Gauss's law, but how this discontinuity show up when there is a hedgehog/instanton/monopole.

references

1. F.D.M.Haldane, Phys.Rev.Lett 61,1029(1988).
2. F.Alet, et al, Physica A 369 (2006) 122–142.
3. Fa Wang, et al, Nat.Phys. 11,959(2015)
4. C.M.Jian, et al, Phys.Rev.B 93,035114(2016)
5. J.Y.Lee, et al, Phys.Rev.X 9,041037(2019)