The Complete Structure of the Cohomology Ring and Associated Symmetries in $D=2$ String Theory

TL;DR

This paper explicitly determines the entire cohomology ring structure in 2D string theory, revealing symmetries and their transformations, including discrete and tachyon states, through algebraic identities and associativity.

Contribution

It provides the first complete explicit calculation of the cohomology ring and associated symmetries in 2D string theory, including new states generated by symmetry actions.

Findings

Explicit structure constants of the cohomology ring are computed.

Symmetry algebra can be derived from the cohomology ring using simple operations.

New states naturally emerge from symmetry transformations.

Abstract

We determine explicitly all structure constants of the whole chiral BRST cohomology ring in $D=2$ string theory including both the discrete states and tachyon states. This is made possible by establishing several identities for Schur polynomials with operator argument and exploring associativity. Furthermore we find that the (chiral) symmetry algebra of the charges obtained by using the descent equations can actually be read off from the cohomology ring structure by simple operation involving the ghost field $b$. We also determine the enlarged symmetry algebra which contains the charges having ghost number $-1$ and $1$. Finally the complete symmetry transformation rules are derived for closed string discrete states by carefully combining the left and right sectors. It turns out that the new states introduced recently by Witten and Zwiebach are naturally created when symmetries act on the old states.