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 is a simple example of conformal field theory which neither rational, nor $$C_2$$ C 2 -cofinite. We identify module category, denoted category $$\mathscr {F}$$ F , satisfies three necessary conditions coming from considerations: closure under restricted duals, fusion and action modular group on characters. prove second these conditions, with other two already being known. Further, we show that has sufficiently many projective injective modules, give classification all indecomposable rigid compute products. product formulae turn out to perfectly match previously proposed Verlinde formula, was computed using conjectured generalisation usual rational called standard formalism. ghosts therefore exhibit essentially rich structure theories despite satisfying none rationality assumptions such -cofiniteness, algebra isomorphic its dual having one-dimensional weight 0 space. In particular, best authors’ knowledge this first proof rigidity for logarithmic non- -cofinite algebra.