[Paper Review] Logarithmic tensor category theory, III: Intertwining maps and tensor product bifunctors

Yi-Zhi Huang, James Lepowsky+1 more|arXiv (Cornell University)|Dec 19, 2010

Algebraic structures and combinatorial models9 references50 citations

TL;DR

This paper establishes the foundational framework for logarithmic tensor category theory by introducing and rigorously defining $P(z)$- and $Q(z)$-intertwining maps and their associated tensor product bifunctors in the context of strongly $ ilde{A}$-graded generalized modules over a vertex algebra. The key contribution is proving the equivalence of the existence of $P(z)$- and $Q(z)$-tensor products via universal properties and duality, paving the way for associativity isomorphisms and braided tensor category structures in later works.

ABSTRACT

This is the third part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part III), we introduce and study intertwining maps and tensor product bifunctors.

Motivation & Objective

• To formalize the notion of $P(z)$- and $Q(z)$-intertwining maps in the context of strongly graded generalized modules over vertex algebras.
• To define $P(z)$- and $Q(z)$-tensor product bifunctors using universal properties and intertwining maps.
• To establish the equivalence between the existence of $P(z)$- and $Q(z)$-tensor products via duality and contragredient module constructions.
• To lay the categorical groundwork for constructing associativity isomorphisms and braided tensor category structures in subsequent papers.

Proposed method

• Introduce $P(z)$-intertwining maps via a generalized Jacobi identity (equation 4.4) and ${rak{sl}}(2)$-bracket relations (4.5), ensuring grading compatibility.
• Define $P(z)$-tensor product bifunctors as universal objects representing $P(z)$-intertwining maps, using natural universal properties.
• Relate $P(z)$- and $Q(z)$-tensor products via duality, showing that $W_1 oxtimes_{P(z)} W_2$ and $W_1 oxtimes_{Q(z^{-1})} W_2$ are isomorphic as generalized $V$-modules.
• Use contragredient module theory and the action of $e^{zL(1)}$ and $e^{-z^{-1}L(1)}$ to relate matrix coefficients and prove uniqueness of the universal map.
• Apply the universal property of $Q(z^{-1})$-tensor products to show that if $Q(z^{-1})$-tensor product exists, then so does the $P(z)$-tensor product.
• Prove that the $P(z)$-tensor product structure is equivalent to the $Q(z)$-tensor product structure via a canonical isomorphism involving the contragredient map and $e^{zL(1)}$ conjugation.

Experimental results

Research questions

• RQ1 Under what conditions does the $P(z)$-tensor product of two strongly $ ilde{A}$-graded generalized $V$-modules exist?
• RQ2 How are $P(z)$- and $Q(z)$-intertwining maps related, and what is the role of duality in their equivalence?
• RQ3 What is the precise relationship between the $P(z)$-tensor product and the $Q(z^{-1})$-tensor product bifunctors?
• RQ4 How do the universal properties of $P(z)$- and $Q(z)$-tensor products ensure uniqueness and compatibility in the category of generalized $V$-modules?
• RQ5 What structural conditions allow the $P(z)$- and $Q(z)$-tensor product bifunctors to be constructed universally, enabling the development of associativity isomorphisms?

Key findings

• The $P(z)$-tensor product of two strongly $ ilde{A}$-graded generalized $V$-modules exists if and only if the $Q(z)$-tensor product exists, as shown in Corollary 4.52.
• The $P(z)$-tensor product bifunctor is isomorphic to the $Q(z^{-1})$-tensor product bifunctor as generalized $V$-modules, though the intertwining maps differ geometrically.
• The universal property of the $P(z)$-tensor product is characterized by the existence of a unique module map $ar{ heta}^{P(z)}$ such that $I = ar{ heta}^{P(z)} oxtimes_{Q(z^{-1})}$, ensuring uniqueness.
• The construction of the $P(z)$-tensor product relies on the invertibility of $e^{z^{-1}L(1)}$ and $e^{-z^{-1}L(1)}e^{i heta L(0)}$ operators on the modules, which ensures the surjectivity of the associated pairing.
• The proof of uniqueness of the universal map relies on showing that the dual map vanishes on all $w_{(1)} oxtimes_{P(z)} w_{(2)}$, implying the map is zero, thus proving uniqueness.
• The $P(z)$-tensor product of $V$ and $W$ (or $W$ and $V$) is isomorphic to $W$ itself, demonstrating that $V$ acts as a unit in this tensor structure.