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Working within the framework of both the $A_\infty$ and the Berkovits opensuperstring field theory, we derive a necessary and sufficient condition for aNeveu-Schwarz marginal deformation to be exact up to third order in thedeformation parameter. For a specific class of backgrounds, we find that thiscondition localizes on the boundary of the worldsheet moduli space, thusproviding a very simple computational prescription for recovering algebraicconstraints (generalized ADHM equations) which need to be satisfied by themoduli. Applying our results to the $\mathrm{D(-1)/D3}$ system, we confirm upto third order that blowing up the size of the D-instanton inside the D3 braneworldvolume is an exact modulus of the full string theory. We also discussexamples of more complicated backgrounds, such as instantons on unresolved ALEspaces, as well as the spiked instantons.

Generalized ADHM equations from marginal deformations in open superstring field theory