Quantum Group Structure and Local Fields in the Algebraic Approach to 2D Gravity

Abstract

This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself - and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.

38 pages, LaTex file. Proceedings of the Vth International Conference on Mathematical Physics, Strings and Quantum gravity, Alushta, Ukraine 1994

References in corpus (10)

• W-Geometries
• Gravity-Matter Couplings from Liouville Theory
• Solving the Strongly Coupled 2D Gravity: 2. Fractional-Spin Operators, and Topological Three-Point Functions
• The Braiding of Chiral Vertex Operators with Continuous Spins in 2D Gravity
• Continous Spins in 2D Gravity: Chiral Vertex Operators and Local Fields
• The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap
• q-Discrete Toda Molecule Equation
• The Multivalued Free-Field Maps of Liouville and Toda Gravities
• Canonical Quantization of the Liouville Theory, Quantum Group Structures, and Correlation Functions
• What are we quantizing in integrable field theory?

Cited by in corpus (5)

• Quantum Liouville theory and BTZ black hole entropy
• Poisson Structure and Moyal Quantisation of the Liouville Theory
• Quantum Exchange Algebra and Exact Operator Solution of $A_2$-Toda Field Theory
• Quantum Exchange Algebra and Locality in Liouville Theory
• Some insights in the structure of correlation functions in Liouville and Toda field theories