Solving the Strongly Coupled 2D Gravity: 2. Fractional-Spin Operators, and Topological Three-Point Functions

TL;DR

This paper advances the understanding of 2D gravity by incorporating fractional-spin operators and topological three-point functions, solving key equations and confirming the existence of strongly coupled topological models.

Contribution

It introduces fractional quantum group spins into 2D gravity models, extending the chiral bootstrap and solving Moore-Seiberg equations for these cases.

Findings

Proves identities for q-deformed 6-j symbols with continuous spins.

Shows decoupling of physical operators at specific central charges.

Confirms the existence of strongly coupled topological models.

Abstract

Progress along the line of a previous article are reported. One main point is to include chiral operators with fractional quantum group spins (fourth or sixth of integers) which are needed to achieve modular invariance. We extend the study of the chiral bootstrap (recently completed by E. Cremmer, and the present authors) to the case of semi-infinite quantum-group representations which correspond to positive integral screening numbers. In particular, we prove the Bidenharn-Elliot and Racah identities for q-deformed 6-j symbols generalized to continuous spins. The decoupling of the family of physical chiral operators (with real conformal weights) at the special values C_{Liouville}= =7, 13, and 19, is shown to provide a full solution of Moore and Seiberg's equations, only involving operators with real conformal weights. Moreover, our study confirms the existence of the strongly coupled topological models. The three-point functions are shown to be given by a product of leg factors similar to the ones of the weakly coupled models. However, contrary to this latter case, the equality between the quantum group spins of the holomorphic and antiholomorphic components is not preserved by the local vertex operator. Thus the ``c=1'' barrier appears as connected with a deconfinement of chirality.