Unitarity Relations in c=1 Liouville Theory

TL;DR

This paper analyzes the unitarity constraints on the S-matrix of c=1 Liouville theory, revealing how unitarity fully determines certain amplitudes and explaining the absence of particle creation branch cuts in higher genus results.

Contribution

It demonstrates that unitarity constraints uniquely determine (N,2) amplitudes from (N,1) amplitudes at tree level in c=1 Liouville theory and clarifies the absence of branch cuts in higher genus amplitudes.

Findings

Unitarity fully constrains (N,2) amplitudes in terms of (N,1) amplitudes.

Higher genus amplitudes lack particle creation branch cuts due to unitarity.

Naive field theory limits predict branch cuts, but full string theory does not exhibit them.

Abstract

We consider the S-matrix of c=1 Liouville theory with vanishing cosmological constant. We examine some of the constraints imposed by unitarity. These completely determine (N,2) amplitudes at tree level in terms of the (N,1) amplitudes when the plus tachyon momenta take generic values. A surprising feature of the matrix model results is the lack of particle creation branch cuts in the higher genus amplitudes. In fact, we show the naive field theory limit of Liouville theory would predict such branch cuts. However, unitarity in the full string theory ensures that such cuts do not appear in genus one (N,1) amplitudes. We conclude with some comments about the genus one (N,2) amplitudes.