Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion

Abstract

The rank 1 bosonic ghost vertex algebra, also known as the $\beta \gamma$ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor $C_2$-cofinite. We identify a module category, denoted category $\mathcal{F}$, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category $\mathcal{F}$ has sufficiently many projective and injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusion products. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computed using a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonic ghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationality assumptions such as $C_2$-cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensional conformal weight 0 space. In particular, to the best of the authors' knowledge this is the first example of a proof of rigidity for a logarithmic non-$C_2$-cofinite vertex algebra.

arXiv comment39 pages, 2 figures. Fixed typos and restructured Section 5 for greater clarity of exposition

References in corpus (39)

• Vertex Algebras and Algebraic Curves
• Infinite Chiral Symmetry in Four Dimensions
• A Rational Logarithmic Conformal Field Theory
• Chiral de Rham complex
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• Curiosities at c=-2
• Modular Data and Verlinde Formulae for Fractional Level WZW Models II
• Logarithmic Conformal Field Theories via Logarithmic Deformations
• Coset Constructions of Logarithmic (1,p)-Models
• The tensor structure on the representation category of the ${\mathcal{W}}_p$ triplet algebra
• Logarithmic intertwining operators and W(2,2p-1)-algebras
• From boundary to bulk in logarithmic CFT
• sl^(2)_{-1/2}: A Case Study
• Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators
• Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• W{1+ \infty} algebra, W_3 algebra, and Friedan-Martinec-Shenker bosonization
• 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
• Bosonic Ghosts at $c=2$ as a Logarithmic CFT
• 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
• Consistent systems of correlators in non-semisimple conformal field theory
• On fusion rules and intertwining operators for the Weyl vertex algebra
• On the triplet vertex algebra W(p)
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• On the extended W-algebra of type sl_2 at positive rational level
• Invariant chiral differential operators and the W_3 algebra
• Free Fields and Quasi-Finite Representation of $W_{1+\infty }$ Algebra
• On the applicability of logarithmic tensor category theory
• Restriction and induction of indecomposable modules over the Temperley-Lieb algebras
• From non-semisimple Hopf algebras to correlation functions for logarithmic CFT
• Vertex tensor category structure on a category of Kazhdan--Lusztig
• Logarithmic Link Invariants of ${\overline{U}}_q^H({\mathfrak{sl}}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras
• Admissible level $\mathfrak{osp}(1|2)$ minimal models and their relaxed highest weight modules
• Representation theory of the vertex algebra $W_{1+\infty }$
• Differential equations and logarithmic intertwining operators for strongly graded vertex algebras
• Representations of the vertex algebra $W_{1+\infty }$ with a negative integer central charge

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• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Modularity of Bershadsky-Polyakov minimal models
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• 3d Mirror Symmetry and the $\beta \gamma$ VOA
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions
• On rationality of $\mathbb{C}$-graded vertex algebras and applications to Weyl vertex algebras under conformal flow
• Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson
• Staggered modules of $N=2$ superconformal minimal models
• Non-unitary Wightman CFTs and non-unitary vertex algebras
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
• Bosonic Ghost Correlators: A Case Study