Bosonic Ghostbusting: The Bosonic Ghost Vertex Algebra Admits a Logarithmic Module Category with Rigid Fusion

Robert Allen,Simon Wood

doi:10.1007/s00220-021-04305-6

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 mathscr {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 mathscr {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.

Highlights

A vertex algebra is called logarithmic if it admits reducible yet indecomposable modules on which the Virasoro L0 operator acts non-semisimply, giving rise to logarithmic singularities in the correlation functions of the associated conformal field theory
There is a general consensus within the research community that many of the structures familiar from rational vertex algebras such as modular tensor categories [1] and, in particular, the Verlinde formula should generalise in some form to the logarithmic case, at least for sufficiently nice logarithmic vertex algebras
We derive explicit formulae for the decomposition of any fusion product into indecomposable direct summands, and we show that fusion is rigid and matches the Verlinde formula of [16]

Summary

Introduction

A vertex algebra is called logarithmic if it admits reducible yet indecomposable modules on which the Virasoro L0 operator acts non-semisimply, giving rise to logarithmic singularities in the correlation functions of the associated conformal field theory. There is a general consensus within the research community that many of the structures familiar from rational vertex algebras such as modular tensor categories [1] and, in particular, the Verlinde formula should generalise in some form to the logarithmic case, at least for sufficiently nice logarithmic vertex algebras. To this end, considerable work has been. This adjusted argument proves Theorem 5.7, which should prove useful for the generalisations of category F to other vertex algebras such as those constructed from affine Lie algebras at admissible levels

Bosonic Ghost Vertex Algebra

Projective Modules

Classification of Indecomposables

Rigid Tensor Category

Fusion Product Formulae