Perturbative String Theory and Its Non-Perturbative Probes

This article presents a systematic construction of perturbative string theory, tracing how, exactly, the two-dimensional theory on the string worldsheet fixes the physical spectrum, the structure of scattering amplitudes, and the protected quantities used to probe physics beyond weak coupling. Starting from the Polyakov action
\[
S_P=\frac{1}{4\pi\alpha'}\int_{\Sigma}d^2\sigma\,\sqrt h\,h^{ab}\partial_aX^\mu\partial_bX_\mu ,
\]
the Faddeev--Popov gauge fixing of worldsheet diffeomorphisms and Weyl rescalings is carried out in detail, giving the \(bc\)-ghost system and a nilpotent BRST charge \(Q_B\) whose cohomology \(H(Q_B)\) is identified with the physical spectrum. Imposing \(Q_B^2=0\), equivalently the cancellation of the total conformal anomaly, fixes the critical dimension to \(D=26\) for the bosonic string and \(D=10\) for the superstring. This article aims to show, next, how string scattering amplitudes are built as integrals over the moduli space of punctured Riemann surfaces,
\[
{\cal A}_{h,n}\sim g_s^{2h-2+n}\int_{{\cal M}_{h,n}}d\mu\,\left\langle \prod_{i=1}^{n}V_i\right\rangle_{\Sigma_h},
\]
where physical vertex operators enforce the spacetime mass-shell condition while encoding gauge equivalence through BRST cohomology. At genus one it is shown that modular invariance of the torus amplitude restricts the spectrum, selects the allowed spin structures, and forces the GSO projection in the superstring; for open and unoriented strings the same factorization is shown to expose massless closed-string tadpoles, and the condition under which their cancellation fixes the type-I Chan--Paton gauge group to \(SO(32)\) is derived. Finally, we show how equating two descriptions of the cylinder amplitude, an open-string loop and a closed-string exchange between boundary states, fixes the D-brane tension and extracts the Ramond--Ramond charge carried by the brane. These protected quantities are then used to organize the \((p,q)\)-string spectrum of type-IIB theory, to identify D0-branes with Kaluza--Klein momentum modes of strongly coupled type-IIA theory, and to supply the microscopic states used in black-hole entropy counting.