Three-Point Functions in c≤1 Liouville Theory and Conformal Loop Ensembles

Area of Science:

• Theoretical Physics
• Condensed Matter Physics
• String Theory

Background:

• The extension of Liouville conformal field theory (CFT) to central charges c≤1 has been a long-standing problem.
• Previous research focused on c≥25, leaving the lower central charge regime largely unexplored.
• A recent proof established the consistency of c≤1 Liouville CFT, including real critical exponents and analytic continuation of three-point coupling formulas.

Purpose of the Study:

• To interpret the recently established consistent c≤1 Liouville conformal field theory.
• To connect this theory to microscopic loop models.
• To analyze the operator algebra and geometric interpretation of specific operators within this framework.

Main Methods:

• Introduction of a family of geometrical operators.
• Development and application of an efficient lattice algorithm for computing three-point functions.
• Comparison of the computed operator algebra with vertex operators in c≤1 Liouville theory.

Main Results:

• Demonstrated that the consistent c≤1 Liouville CFT can be interpreted via microscopic loop models.
• Showcased that the introduced geometric operators' algebra precisely matches that of vertex operators V_{α[over ^]} in c≤1 Liouville theory.
• Provided a geometric interpretation for the limit α[over ^]→0 of V_{α[over ^]}, explaining its non-identity nature despite zero conformal weight.

Conclusions:

• The study successfully links microscopic loop models to the extended c≤1 Liouville conformal field theory.
• The geometric operators provide a concrete realization of the vertex operator algebra in this regime.
• The findings offer new insights into the structure and interpretation of Liouville CFT at low central charges.