The paper by E.S. Fradkin and A.A. Tseytlin explores the effective action for an abelian vector field coupled to virtual open Bose strings. The authors solve the problem exactly in the "tree" and "one-loop" approximations for the case of a constant field strength and 26-dimensional space-time. The resulting tree-level effective Lagrangian is shown to coincide with the Born-Infeld Lagrangian, \(\sqrt{\det(\partial_\mu^\nu + 2\pi d' F_\mu^\nu)}\). The study is based on a new formulation of string theory using an off-shell covariant effective action \(\Gamma\) for an infinite number of fields corresponding to excitation modes of a first-quantized string. This approach allows for a deeper understanding of non-perturbative properties of string theories, particularly in the context of ground-state (compactification) problems and connections to effective field theories. The authors demonstrate how non-perturbative results for the effective action can be derived for the open Bose string theory, focusing on the dependence of \(\Gamma\) on the vector field \(A_\mu\) under the assumption that its strength \(E_{\mu\nu}\) is constant. The "tree" and "one-loop" contributions to \(\Gamma(F)\) are computed exactly, with the "tree" approximation coinciding with the Born-Infeld action. The paper also discusses the general relations and the tree approximation, showing that the Born-Infeld Lagrangian is the exact solution to the constant external vector field problem in open string theory. The authors suggest that there might be a bootstrap mechanism where the effective field theory action corresponding to fundamental string theory should have string-like classical solutions. The paper concludes with a discussion on possible generalizations to non-abelian cases, fermionic strings, and superstrings, as well as the application to heterotic superstrings.The paper by E.S. Fradkin and A.A. Tseytlin explores the effective action for an abelian vector field coupled to virtual open Bose strings. The authors solve the problem exactly in the "tree" and "one-loop" approximations for the case of a constant field strength and 26-dimensional space-time. The resulting tree-level effective Lagrangian is shown to coincide with the Born-Infeld Lagrangian, \(\sqrt{\det(\partial_\mu^\nu + 2\pi d' F_\mu^\nu)}\). The study is based on a new formulation of string theory using an off-shell covariant effective action \(\Gamma\) for an infinite number of fields corresponding to excitation modes of a first-quantized string. This approach allows for a deeper understanding of non-perturbative properties of string theories, particularly in the context of ground-state (compactification) problems and connections to effective field theories. The authors demonstrate how non-perturbative results for the effective action can be derived for the open Bose string theory, focusing on the dependence of \(\Gamma\) on the vector field \(A_\mu\) under the assumption that its strength \(E_{\mu\nu}\) is constant. The "tree" and "one-loop" contributions to \(\Gamma(F)\) are computed exactly, with the "tree" approximation coinciding with the Born-Infeld action. The paper also discusses the general relations and the tree approximation, showing that the Born-Infeld Lagrangian is the exact solution to the constant external vector field problem in open string theory. The authors suggest that there might be a bootstrap mechanism where the effective field theory action corresponding to fundamental string theory should have string-like classical solutions. The paper concludes with a discussion on possible generalizations to non-abelian cases, fermionic strings, and superstrings, as well as the application to heterotic superstrings.