Instantons in string theory

Instantons are finite-action, topologically non-trivial solutions of Euclidean Yang-Mills theory and represent non-perturbative contributions to the path integral. This thesis reviews their construction from field theory to string theory, starting from the ADHM description of self-dual Yang-Mills connections and its extension to supersymmetric gauge theories. The same ADHM data are then identified in the D$(-1)$/D$3$ brane system, where the massless open strings reproduce the instanton collective coordinates and the mixed boundary conditions are described by twist fields on the worldsheet. The main goal is to study the blow-up of D$(-1)$-branes into finite-size D$3$-brane instantons within open superstring field theory. The string field is expanded perturbatively in powers of $\rho/\sqrt{\alpha'}$, and the Maurer--Cartan equation of the underlying $A_\infty$ algebra is analysed order by order. This shows how the consistency of the deformation is tied to the ADHM constraints and to the obstruction structure of the superstring field equations. Finally, inspired by the ADHM expression of the instanton connection as a projected pure-gauge-like object, the thesis explores whether the instanton profile can be represented by a singular quasi-gauge string field in the field-theory limit. Two possible solutions are discussed: one based on the physical mixed-sector twist field, and one involving excited twist fields to evade the conformal-weight obstruction. The construction is presented as a candidate mechanism for recovering the BPST profile from the quadratic part of the open string field algebra, leaving a complete non-perturbative treatment for future work.