Operator Product Expansion and Topological States in $c = 1$ Matter Coupled to 2-D Gravity

TL;DR

This paper analyzes the factorization of $N$-tachyon amplitudes in $c=1$ 2D quantum gravity, revealing the role of operator product expansion and topological states, with implications for understanding singularities and state contributions.

Contribution

It provides a detailed analysis of the operator product expansion and the role of topological states in $c=1$ matter coupled to 2D gravity, extending previous short communications.

Findings

Short-distance singularities are fully accounted for by tachyon interactions.

Other potential singularities are absent due to vanishing residues.

Infinitely many topological states contribute to intermediate states.

Abstract

Factorization of the $N$-tachyon amplitudes in two-dimensional $c=1$ quantum gravity is studied by means of the operator product expansion of vertex operators after the Liouville zero mode integration. Short-distance singularities between two tachyons with opposite chiralities account for all singularities in the $N$-tachyon amplitudes. Although the factorization is valid, other possible short-distance singularities corresponding to other combinations of vertex operators are absent since the residue vanishes. Apart from the tachyon states, there are infinitely many topological states contributing to the intermediate states. This is a more detailed account of our short communication on the factorization.