Quantisation of type IIB superstring theory and the matrix model

We discuss the path-integral quantisation of perturbative string theory and show equivalence between the Polyakov-type, Schild-type and Nambu-Goto-type formulations of critical type II superstring theory in the Minkowski and Euclidean signatures. Remarkably, we also find that the Minkowskian path integral realises causality in the sense that a string does not propagate between points at space-like separation, by giving careful consideration to the measure of the world-sheet metric. We also discuss matrix regularisation of the path integral for type IIB perturbative superstring theory. The obtained matrix models are the Euclidean IKKT matrix model and a modified Minkowskian IKKT model, depending on how the matrix regularisation is applied.