What is meant by "smooth" instantons, and is there a constraint on the winding number of $E_8$ instantons?

Question

In the paper, ``Comments on String Dynamics in Six Dimensions" (arXiv:hep-th/9603003) by Seiberg and Witten, there is a sentence on page 8 (section 3) which reads

classically, smooth $E_8$ instantons with $n = 1, 2,$ or $3$ do not exist.

$n$ refers to the winding number (corresponding to the embedding of $SU(2$) in $E_8$).

My question is: why do the winding numbers $n = 1, 2, 3$ not exist? What is meant by smooth instantons?

Also, is the $n \neq \{1,2,3\}$ constraint valid for K3 compactifications or is it more general?

This post imported from StackExchange Physics at 2016-10-15 12:58 (UTC), posted by SE-user leastaction

Comments

Do you have a reason to suspect it means something other than the general notion of an instanton in a gauge theory with gauge group $G=E_8$?

This post imported from StackExchange Physics at 2016-10-15 12:58 (UTC), posted by SE-user ACuriousMind

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No, I do not. But what are smooth instantons and why don't they exist for $n = 1, 2, 3$?

This post imported from StackExchange Physics at 2016-10-15 12:58 (UTC), posted by SE-user leastaction

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@leastaction : smooth instantons got with parameters leading to smooth ( and horizonless ) geometries ?