[Paper Review] The theory of physical superselection sectors in terms of vertex operator algebra language

Haisheng Li|ArXiv.org|Apr 28, 1995

Algebraic structures and combinatorial models23 references3 citations

TL;DR

This paper formulates the theory of physical superselection sectors using vertex operator algebra (VOA) language, constructing simple currents from primary semisimple elements of weight one. It proves that if a rational VOA admits a simple current satisfying specific conditions, then the direct sum of the VOA and the current naturally acquires a rational vertex (super)algebra structure, enabling extensions and twisted module constructions.

ABSTRACT

We formulate an interpretation of the theory of physical superselection sectors in terms of vertex operator algebra language. Using this formulation we give a construction of simple current from a primary semisimple element of weight one. We then prove that if a rational vertex operator algebra $V$ has a simple current $M$ satisfying certain conditions, then $V\oplus M$ has a natural rational vertex operator (super)algebra structure. Applying our results to a vertex operator algebra associated to an affine Lie algebra, we construct its simple currents and the extension by a simple current. We also present two essentially equivalent constructions for twisted modules for an inner automorphism from the adjoint module or any untwisted module.

Motivation & Objective

• To provide a VOA-based formulation of physical superselection sectors.
• To construct simple currents from primary semisimple elements of weight one in a rational VOA.
• To establish conditions under which a rational VOA extended by a simple current remains rational.
• To apply the framework to affine Lie algebra VOAs to construct explicit simple currents and extensions.
• To present two equivalent constructions for twisted modules under inner automorphisms from adjoint or untwisted modules.

Proposed method

• Uses vertex operator algebra language to reinterpret superselection sector theory in algebraic terms.
• Defines a construction of simple currents from primary semisimple elements of weight one.
• Applies the theory to VOAs associated with affine Lie algebras to generate explicit simple currents.
• Proves that under certain conditions, the direct sum of a rational VOA and a simple current admits a natural rational vertex (super)algebra structure.
• Constructs twisted modules for inner automorphisms using two equivalent methods, starting from the adjoint module or any untwisted module.
• Employs formal calculus and representation theory of VOAs to verify rationality and closure under the extended operations.

Experimental results

Research questions

• RQ1 How can the theory of physical superselection sectors be reformulated within the framework of vertex operator algebras?
• RQ2 What conditions ensure that the extension of a rational vertex operator algebra by a simple current remains rational?
• RQ3 How can simple currents be systematically constructed from primary semisimple elements of weight one?
• RQ4 What is the role of affine Lie algebra VOAs in realizing these constructions explicitly?
• RQ5 How can twisted modules for inner automorphisms be equivalently constructed from untwisted or adjoint modules?

Key findings

• A construction of simple currents is achieved from primary semisimple elements of weight one in a rational vertex operator algebra.
• It is proven that if a rational VOA admits a simple current satisfying specific conditions, then the direct sum V ⊕ M inherits a natural rational vertex (super)algebra structure.
• The framework successfully constructs simple currents and their extensions in VOAs associated with affine Lie algebras.
• Two essentially equivalent constructions for twisted modules under inner automorphisms are presented, one starting from the adjoint module and one from any untwisted module.
• The results demonstrate that the extension by a simple current preserves rationality, providing a systematic method for constructing new rational VOAs.
• The formalism unifies the treatment of superselection sectors, simple currents, and twisted modules within a single VOA-theoretic framework.