Classical BV cohomology of the $N=1$ spinning particle

TL;DR

This work addresses pathological negative BV cohomology observed for the classical $N=1$ spinning particle and shows it is resolved by a Koszul–Tate expansion of the constrained phase space, restoring the Felder–Kazhdan axioms. By formulating the KT resolution within the BV framework and connecting it to AKSZ, the authors demonstrate the equivalence of KT and BV–BRST formalisms and explain how reducibility ghosts kill the negative ghost-number cohomology. The analysis is carried out in the phase-space setting, with a detailed treatment in 1D where the full KT resolution can be described explicitly; for general $d$, the resolution continues indefinitely with higher Omega-entries. These results clarify the role of saturation reducibility for fermionic constraints and have implications for BV quantisation of related systems such as string theory and supergravity, where similar reducibility phenomena arise.

Abstract

We show that the classical Batalin--Vilkovisky cohomology at negative ghost number of the spinning particle, observed in ref. arXiv:1511.02135, is removed by a Koszul--Tate resolution involving saturation of Grassmann odd variables. The model thus satisfies the axioms of Felder and Kazhdan. The AKSZ formulation of the resolved model is described. We reveal partial information on the resolution of the constrained phase space, which involves resolving the parity-shifted tangent sheaf of the light-cône. Specialising to dimension one, we describe the full resolution.

Figures (1)

• Figure 1: A picture of the ring $S$. Powers of $\theta$ and $p$ are on the horizontal and vertical axis, respectively. Solid dots denote non-vanishing $\mathfrak{so}(d)$ modules. The constraints (generators of the ideal) $T$ and $U$ are drawn as black rings, reducibility ghosts as red rings, and ghosts of the following stage as blue rings.