Logarithmic conformal field theory, log-modular tensor categories and modular forms

Abstract

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters on the one hand and its interpretation of modules as objects in a modular tensor category on the other one. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and $C_2$-cofinite vertex operator algebras. We suggest that their modular pillar are trace functions with insertions corresponding to intertwiners of the projective cover of the vacuum, and that the categorical pillar are finite tensor categories $\mathcal{C}$ which are ribbon and whose double is isomorphic to the Deligne product $\mathcal{C}\otimes {\mathcal{C}}^{opp}$. Overarching these pillars is then a logarithmic variant of Verlinde's formula. Numerical data realizing this are the modular $S$-matrix and modified traces of open Hopf links. The representation categories of $C_2$-cofinite and logarithmic conformal field theories that are fairly well understood are those of the ${\mathcal{W}}_p$-triplet algebras and the symplectic fermions. We illustrate the ideas in these examples and especially make the relation between logarithmic Hopf links and modular transformations explicit.

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Cited by in corpus (32)

• Tensor categories for vertex operator superalgebra extensions
• Schur-Weyl Duality for Heisenberg Cosets
• 3d Modularity
• Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• A quasi-Hopf algebra for the triplet vertex operator algebra
• Gluing vertex algebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• SL(2,Z)-action for ribbon quasi-Hopf algebras
• On ribbon categories for singlet vertex algebras
• Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Modularity of logarithmic parafermion vertex algebras
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Logarithmic conformal field theories of type $B_n,\ell =4$ and symplectic fermions
• Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$
• Logarithmic Link Invariants of ${\overline{U}}_q^H({\mathfrak{sl}}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras
• The logarithmic Cardy case: Boundary states and annuli
• The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre
• Modified traces for quasi-Hopf algebras
• On rationality of $\mathbb{C}$-graded vertex algebras and applications to Weyl vertex algebras under conformal flow
• On rationality for $C_2$-cofinite vertex operator algebras
• Tensor Categories arising from the Virasoro Algebra
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Lattice models from CFT on surfaces with holes I: Torus partition function via two lattice cells
• Character Vectors of Strongly Regular Vertex Operator Algebras
• Recent developments of the categorical Verlinde formula
• Fourier expansions of vector-valued automorphic functions with non-unitary twists
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• A duality between vertex superalgebras $L_{-3/2}(\mathfrak{osp}(1|2))$ and ${\mathcal{V}}^{(2)}$ and generalizations to logarithmic vertex algebras