[Paper Review] Twisted sectors for tensor product VOAs associated to permutation groups

Katrina Barron, Chongying Dong+1 more|arXiv (Cornell University)|Mar 24, 1998

Advanced Algebra and Logic7 citations

TL;DR

This paper establishes that the categories of weak, admissible, and ordinary g-twisted modules for the tensor product vertex operator algebra (VOA) $ V^{ imes k} $ are isomorphic to the corresponding categories of $ V $-modules when $ g $ is a $ k $-cycle automorphism. For arbitrary permutation automorphisms $ g $, it proves that the category of admissible $ g $-twisted $ V^{ imes k} $-modules is semisimple, with simple objects explicitly determined if $ V $ is rational, using a construction of weak $ g $-twisted modules from weak $ V $-modules.

ABSTRACT

Let V be a vertex operator algebra. It is shown that the categories of weak, admissible and ordinary g-twisted modules for the tensor product VOA V ⊗k are isomorphic to the categories of weak, admissible and ordinary V-modules respectively where g is a k cycle automorphism of V ⊗k. For arbitrary permutation automorphism g of V ⊗k the category of admissible g-twisted modules for V ⊗k is semi simple and the simple objects are determined if V is rational. The key result is a construction of the weak g-twisted V ⊗k-modules from weak V-modules. 1

Motivation & Objective

• To understand the structure of twisted modules for tensor product VOAs under permutation automorphisms.
• To establish isomorphisms between twisted module categories of $ V^{ imes k} $ and $ V $ when $ g $ is a $ k $-cycle.
• To classify simple objects in the category of admissible $ g $-twisted $ V^{ imes k} $-modules for arbitrary permutation automorphisms $ g $.
• To provide a systematic construction of weak $ g $-twisted $ V^{ imes k} $-modules from weak $ V $-modules.

Proposed method

• Utilize the action of a $ k $-cycle automorphism $ g $ on the tensor product VOA $ V^{ imes k} $ to define $ g $-twisted modules.
• Construct weak $ g $-twisted $ V^{ imes k} $-modules from weak $ V $-modules via a lifting procedure.
• Prove that the category of admissible $ g $-twisted $ V^{ imes k} $-modules is semisimple for arbitrary permutation automorphisms $ g $.
• Characterize the simple objects in the category of admissible $ g $-twisted modules when $ V $ is rational.
• Establish category isomorphisms between $ g $-twisted modules of $ V^{ imes k} $ and $ V $-modules for $ k $-cycle $ g $.
• Apply representation theory of permutation groups to analyze the structure of twisted modules in the tensor product setting.

Experimental results

Research questions

• RQ1 How do the categories of weak, admissible, and ordinary $ g $-twisted modules for $ V^{ imes k} $ relate to those of $ V $ when $ g $ is a $ k $-cycle automorphism?
• RQ2 What is the structure of the category of admissible $ g $-twisted $ V^{ imes k} $-modules when $ g $ is an arbitrary permutation automorphism?
• RQ3 Can weak $ g $-twisted $ V^{ imes k} $-modules be systematically constructed from weak $ V $-modules?
• RQ4 Under what conditions is the category of admissible $ g $-twisted $ V^{ imes k} $-modules semisimple?
• RQ5 What are the simple objects in the category of admissible $ g $-twisted $ V^{ imes k} $-modules when $ V $ is rational?

Key findings

• The categories of weak, admissible, and ordinary $ g $-twisted $ V^{ imes k} $-modules are isomorphic to the corresponding categories of $ V $-modules when $ g $ is a $ k $-cycle automorphism.
• For arbitrary permutation automorphisms $ g $, the category of admissible $ g $-twisted $ V^{ imes k} $-modules is semisimple.
• The simple objects in the category of admissible $ g $-twisted $ V^{ imes k} $-modules are completely determined when $ V $ is rational.
• A construction of weak $ g $-twisted $ V^{ imes k} $-modules from weak $ V $-modules is provided, establishing a direct link between the module categories.
• The isomorphism between module categories for $ k $-cycle $ g $ implies that the representation theory of $ V^{ imes k} $ under such symmetries reduces to that of $ V $.
• The semisimplicity result holds regardless of the specific permutation, provided $ g $ is an arbitrary permutation automorphism of $ V^{ imes k} $.