On Spinning Particles, their Partition Functions and Picture Changing Operators

TL;DR

The paper computes and interprets the partition function for the $N=1$ spinning particle, including picture-changing and the large Hilbert space, showing that the partition function counts the dimension of BRST cohomology in $D=2$ and $D=4$ target spaces. It builds a covariant target-space action for multiforms via BV techniques and a picture-changing framework, and derives consistent interactions through a derived BV bracket, yielding a non-linear theory whose kinematics reproduce Dirac-type dynamics for multiform fields. A spinorial reformulation maps multiforms to bispinors, establishing a Dirac equation and conserved currents, and clarifies the connection to standard fermionic theories, including chiral anomalies identified as Hodge (star) anomalies within the multiform setup. The work also analyzes the Large Hilbert Space, dimensional reduction from $N=2$ to $N=1$, and the interplay between pictures, PCOs, and dualities, providing a geometric bridge between worldline formalisms and spacetime fermions with potential implications for higher-form interactions and anomaly structures.

Abstract

We compute the partition function for the $N=1$ spinning particle, including pictures and the large Hilbert space, and show that it counts the dimension of the BRST cohomology in two- and four-dimensional target space. We also construct a quadratic action in the target space. Furthermore, we find a consistent interaction as a derived bracket based on the associative product of world line fields, leading to an interacting theory of multiforms in space-time. Finally, we comment on the equivalence of the multiform theory with a Dirac fermion. We also identify the chiral anomaly of the latter with a Hodge anomaly for the multiform theory, which manifests itself as a deformation of the gauge fixing.