Physical States, Factorization and Nonlinear Structures in Two Dimensional Quantum Gravity

TL;DR

This paper explores the nonlinear structures, physical states, and factorization in two-dimensional quantum gravity coupled with minimal models, using Liouville theory to derive constraints and analyze state decoupling.

Contribution

It provides a detailed analysis of the physical states, Ward identities, and constraints in 2D quantum gravity with minimal models, including explicit derivations of key equations and state decoupling.

Findings

Identification of dressed primary states with gravitational descendants

Derivation of $L_0$, $L_1$, and $W_{-1}^{(3)}$ equations

Explicit demonstration of edge state decoupling

Abstract

The nonlinear structures in 2D quantum gravity coupled to the $(q+1,q)$ minimal model are studied in the Liouville theory to clarify the factorization and the physical states. It is confirmed that the dressed primary states outside the minimal table are identified with the gravitational descendants. Using the discrete states of ghost number zero and one we construct the currents and investigate the Ward identities which are identified with the W and the Virasoro constraints. As nontrivial examples we derive the $L_0$, $L_1$ and $W_{-1}^{(3)}$ equations exactly. $L_n$ and $W^{(k)}_n$ equations are also discussed. We then explicitly show the decoupling of the edge states $O_j ~(j=0~ {\rm mod}~ q) $. We consider the interaction theory perturbed by the cosmological constant $O_1$ and the screening charge $S^+ =O_{2q+1}$. The formalism can be easily generalized to potential models other than the screening charge.