Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms

Abstract

This is the sixth 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 VI), we construct the appropriate natural associativity isomorphisms between triple tensor product functors. In fact, we establish a "logarithmic operator product expansion" theorem for logarithmic intertwining operators. In this part, a great deal of analytic reasoning is needed; the statements of the main theorems themselves involve convergence assertions.

Part VI of a series of 8 papers generalizing the results in and collectively replacing arXiv:0710:2687, with new titles. 108 pages. Enhanced version. Minor changes

References in corpus (6)

• Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators
• Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra
• Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition
• Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps
• Logarithmic tensor category theory, III: Intertwining maps and tensor product bifunctors
• Logarithmic tensor category theory, IV: Constructions of tensor product bifunctors and the compatibility conditions

Cited by in corpus (41)

• Logarithmic Conformal Field Theory: a Lattice Approach
• Schur-Weyl Duality for Heisenberg Cosets
• Lattice fusion rules and logarithmic operator product expansions
• Simple current extensions beyond semi-simplicity
• Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra
• Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition
• Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps
• Logarithmic tensor category theory, III: Intertwining maps and tensor product bifunctors
• Logarithmic tensor category theory, IV: Constructions of tensor product bifunctors and the compatibility conditions
• On the tensor structure of modules for compact orbifold vertex operator algebras
• The Nichols algebra of screenings
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Twisted modules and $G$-equivariantization in logarithmic conformal field theory
• Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion
• Modularity of logarithmic parafermion vertex algebras
• Logarithmic ^sl(2) CFT models from Nichols algebras. 1
• Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications
• On the applicability of logarithmic tensor category theory
• Logarithmic Intertwining Operators and Genus-One Correlation Functions
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
• Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p>1$
• Tensor products of strongly graded vertex algebras and their modules
• Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
• C_2-cofinite W-algebras and their logarithmic representations
• Vertex algebraic intertwining operators among generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$
• Intertwining operators among modules for affine Lie algebra and lattice vertex operator algebras which respect integral forms
• A general mirror equivalence theorem for coset vertex operator algebras
• Representation theory of $L_k\left(\mathfrak{osp}(1|2)\right)$ from vertex tensor categories and Jacobi forms
• On rationality for $C_2$-cofinite vertex operator algebras
• Tensor Categories arising from the Virasoro Algebra
• Differential equations and logarithmic intertwining operators for strongly graded vertex algebras
• Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson
• False Theta Functions and the Verlinde formula
• The quantum group dual of the first-row subcategory for the generic Virasoro VOA
• Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms
• Algebras of Non-Local Screenings and Diagonal Nichols Algebras
• Representation theory of vertex operator algebras and orbifold conformal field theory