[Paper Review] Coadjoint Orbits and Conformal Field Theory

Washington Taylor|ArXiv.org|Oct 7, 1993

Black Holes and Theoretical Physics52 references6 citations

TL;DR

This paper introduces a geometric approach to conformal field theory (CFT) using coadjoint orbits and geometric quantization, constructing Virasoro and affine algebra representations via local gauge choices on line bundles. The key contribution is a new realization of these algebras as dual Verma modules that avoids technical issues in Feigin-Fuchs and Wakimoto constructions, enabling explicit formulae for vertex operators, screening, and intertwining operators in terms of differential operators between sections of line bundles.

ABSTRACT

This thesis describes a new approach to conformal field theory. This approach combines the method of coadjoint orbits with resolutions and chiral vertex operators to give a construction of the correlation functions of conformal field theories in terms of geometrically defined objects. Explicit formulae are given for representations of Virasoro and affine algebras in terms of a local gauge choice on the line bundle associated with geometric quantization of a given coadjoint orbit; these formulae define a new set of explicit bosonic realizations of these algebras. The coadjoint orbit realizations take the form of dual Verma modules, making it possible to avoid the technical difficulties associated with the two-sided resolutions which arise from Feigin-Fuchs and Wakimoto realizations. Formulae are given for screening and intertwining operators on the coadjoint orbit representations. Chiral vertex operators between Virasoro modules are constructed, and related directly to Virasoro algebra generators in certain cases. From the point of view taken in this thesis, vertex operators have a geometric interpretation as differential operators taking sections of one line bundle to sections of another. A suggestion is made that by connecting this description with recent work deriving field theory actions from coadjoint orbits, a deeper understanding of the geometry of conformal field theory might be achieved.

Motivation & Objective

• To develop a new geometric framework for conformal field theory based on coadjoint orbits.
• To resolve technical difficulties in existing realizations of Virasoro and affine algebras, such as those in Feigin-Fuchs and Wakimoto constructions.
• To construct explicit formulae for correlation functions, vertex operators, screening, and intertwining operators using geometric quantization.
• To interpret chiral vertex operators as differential operators between sections of line bundles associated with coadjoint orbits.
• To lay a foundation for deeper geometric understanding of CFT by connecting coadjoint orbit constructions with field theory actions.

Proposed method

• Utilizes geometric quantization of coadjoint orbits to define quantum states as sections of line bundles over symplectic manifolds.
• Applies local gauge choices on the line bundle to construct explicit representations of the Virasoro and affine Lie algebras.
• Represents the algebras in terms of dual Verma modules, avoiding the need for two-sided resolutions common in Feigin-Fuchs and Wakimoto realizations.
• Derives formulae for screening and intertwining operators directly on the coadjoint orbit representations.
• Constructs chiral vertex operators between Virasoro modules as differential operators mapping sections of one line bundle to another.
• Suggests a geometric unification with recent field theory actions derived from coadjoint orbits to deepen insight into CFT geometry.

Experimental results

Research questions

• RQ1 How can coadjoint orbits be used to construct representations of the Virasoro algebra in a geometric and explicit manner?
• RQ2 Can the technical complications of two-sided resolutions in Feigin-Fuchs and Wakimoto realizations be avoided through a dual Verma module formulation on coadjoint orbits?
• RQ3 What is the geometric interpretation of chiral vertex operators in terms of differential operators between line bundle sections?
• RQ4 How can screening and intertwining operators be explicitly realized on coadjoint orbit representations?
• RQ5 What connections exist between coadjoint orbit constructions and the derivation of field theory actions in conformal field theory?

Key findings

• The paper provides explicit bosonic realizations of the Virasoro and affine Lie algebras via local gauge choices on line bundles over coadjoint orbits.
• Representations are constructed as dual Verma modules, circumventing the need for two-sided resolutions and simplifying the algebraic structure.
• Formulae for screening and intertwining operators are derived directly on the coadjoint orbit representations, enabling explicit computation.
• Chiral vertex operators between Virasoro modules are constructed and shown to be related directly to Virasoro algebra generators in specific cases.
• Vertex operators are interpreted geometrically as differential operators mapping sections of one line bundle to another, linking algebraic structures to geometry.
• The framework suggests a potential geometric unification with field theory actions derived from coadjoint orbits, offering a new pathway to understanding CFT geometry.