Spinning particles, their partition functions, and picture changing operators

We compute the partition function for the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>N</a:mi> <a:mo>=</a:mo> <a:mn>1</a:mn> </a:math> spinning particle, including pictures and the large Hilbert space, and show that it counts the dimension of the Becchi-Rouet-Stora-Tyutin 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 worldline 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.