Path integral for the closed superstring and the matrix model

TL;DR

This paper reviews the Minkowskian path-integral formulation of superstring theory, derives a related matrix model, and discusses how stringy causality is realized in this framework.

Contribution

It introduces a Minkowskian version of the IKKT matrix model derived from the path integral of perturbative superstring theory, addressing previous ambiguities.

Findings

Derived Minkowskian path integral equivalent to Euclidean formulation.

Showed stringy causality is realized in the path-integral formulation.

Obtained a Minkowskian matrix model with properties similar to stringy causality.

Abstract

The IKKT matrix model, which is proposed as a non-perturbative formulation of superstring theory, has an issue typical of zero-dimensional theory -- ambiguity in the definition of its path integral. To tackle this issue, we revisit the path-integral formulation of perturbative string theory. In this article, we review recent progress in the string world-sheet path-integral formulation, especially in the Minkowski signature. We first derive the Minkowskian path integral of the Nambu-Goto type equivalent to Polyakov's Euclidean path integral for critical closed string theory, showing equivalences among the Nambu-Goto-, Schild- and Polyakov-type formulations both in the Minkowskian and Euclidean signatures. We also show that ``stringy causality'' is realised in the path-integral formulation at the level of string perturbation theory. We then obtain the matrix model with a property like the stringy causality, which turns out to be a Minkowskian version of the NBI-type IKKT matrix model, by matrix regularisation of the path integral for perturbative type IIB string theory.