String Perturbation Theory

String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History	S-Matrix Theory Regge Trajectory
Bosonic String Theory	Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation
Supersymmetric Revolution	Supersymmetry RNS Formalism GS Formalism BPS
Superstring Revolutions
First Superstring Revolution	GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory
Second Superstring Revolution	T-Duality D-Brane S-Duality Horava-Witten String Theory M-Theory Holographic Principle N=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism
After the Revolutions
Phenomenology	String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology

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String Perturbation Theory (or Perturbative String Theory) is a method for calculating the S-Matrix of a String Theory vacuum - the asymptotic scattering behavior of weakly coupled strings moving on that geometric background - as a sum over string histories of different topologies, analogous to the sum over Feynman diagrams employed in perturbative quantum field theory.

Also as in QFT, these scattering amplitudes only describe transitions from a state in the distant past to a state in the distant future. For the finite-time evolution of strings, one needs a more advanced formalism like String Field Theory.

Basics

Schematically, the scattering amplitude to start with strings in a state $|a\rangle$ and to finish with strings in a state $|b\rangle$ is

$\langle b|S|a\rangle = \sum_{\mathrm{topology}} \int \mathcal D X \mbox{ } e^{-S[X]} O_a[X] O_b[X]$

where $DX$ stands for a Measure on the Moduli Space of a particular topological class of Riemann Surfaces, $S[X]$ is the action of a particular history from that class, and $O[X]$ is a set of Vertex Operators encoding the initial state $|a\rangle$ or final state $|b\rangle$.